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					     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      Rings of Integer-Valued Polynomials
Integer-valued
polynomial rings

Krull domains
and PVMDs                                   Jesse Elliott
Universal                 California State University, Channel Islands
properties of IVP
rings

Summmary of
open problems

                                 Number Theory Seminar
                                         UCLA
                                   February 25, 2008




  UCLA NTS 02/25/08                   [1]                            jesse.elliott@csuci.edu
                    Outline

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction          1   Introduction
Prüfer domains

Integer-valued
polynomial rings
                      2   Prüfer domains
Krull domains
and PVMDs             3   Integer-valued polynomial rings
Universal
properties of IVP
rings                 4   Krull domains and PVMD’s
Summmary of
open problems         5   Universal properties of IVP rings
                      6   Summary of open problems




  UCLA NTS 02/25/08                           [2]             jesse.elliott@csuci.edu
                    Integer-valued polynomials (IVP’s)

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A polynomial f (X) ∈ Q[X] is said to be integer-valued if
Introduction
                      f (Z) ⊆ Z.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [3]                        jesse.elliott@csuci.edu
                    Integer-valued polynomials (IVP’s)

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A polynomial f (X) ∈ Q[X] is said to be integer-valued if
Introduction
                      f (Z) ⊆ Z.
Prüfer domains

Integer-valued
polynomial rings      Example: The binomial coefficient polynomial
Krull domains
and PVMDs
                                    X        X(X − 1)(X − 2) · · · (X − n + 1)
Universal                                =
properties of IVP
rings
                                    n                     n!
Summmary of
open problems         is integer-valued for every positive integer n.




  UCLA NTS 02/25/08                             [3]                         jesse.elliott@csuci.edu
                    Integer-valued polynomials (IVP’s)

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A polynomial f (X) ∈ Q[X] is said to be integer-valued if
Introduction
                      f (Z) ⊆ Z.
Prüfer domains

Integer-valued
polynomial rings      Example: The binomial coefficient polynomial
Krull domains
and PVMDs
                                    X        X(X − 1)(X − 2) · · · (X − n + 1)
Universal                                =
properties of IVP
rings
                                    n                     n!
Summmary of
open problems         is integer-valued for every positive integer n.
                                                           p
                                                         −X
                      Example: The Fermat binomial X p is integer-valued for any
                      prime number p (⇐⇒ Fermat’s little theorem)




  UCLA NTS 02/25/08                             [3]                         jesse.elliott@csuci.edu
                    Describing all IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
  Jesse Elliott
                      Q[X].
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
                    Describing all IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
  Jesse Elliott
                      Q[X].
Introduction

Prüfer domains        Int(Z) is a subring of Q[X] containing Z[X] that is closed under
Integer-valued
polynomial rings
                      composition.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
                    Describing all IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
  Jesse Elliott
                      Q[X].
Introduction

Prüfer domains        Int(Z) is a subring of Q[X] containing Z[X] that is closed under
Integer-valued
polynomial rings
                      composition.
Krull domains
and PVMDs
                      Int(Z) is generated freely as a Z-module by the binomial
Universal
properties of IVP     coefficient polynomials:
rings

Summmary of                                             ∞
open problems                                                 X
                                             Int(Z) =           Z.
                                                        n=0
                                                              n




  UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
                    Describing all IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
  Jesse Elliott
                      Q[X].
Introduction

Prüfer domains        Int(Z) is a subring of Q[X] containing Z[X] that is closed under
Integer-valued
polynomial rings
                      composition.
Krull domains
and PVMDs
                      Int(Z) is generated freely as a Z-module by the binomial
Universal
properties of IVP     coefficient polynomials:
rings

Summmary of                                             ∞
open problems                                                 X
                                             Int(Z) =           Z.
                                                        n=0
                                                              n


                      Int(Z) is generated, as a subring of Q[X] closed under
                                                                     p
                                                                       −X
                      composition, by X and the Fermat binomials X p for p prime.


  UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
                    IVP’s and prime numbers

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                 [5]       jesse.elliott@csuci.edu
                    IVP’s and prime numbers

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Theorem (Tao and Ziegler, 2006)
Prüfer domains

Integer-valued
                      Given any integer-valued polynomials P1 , P2 , . . . , Pk with vanish-
polynomial rings
                      ing constant terms, there are infinitely many integers x and m
Krull domains
and PVMDs             such that the integers x+P1 (m), . . . , x+Pk (m) are simultaneously
Universal             prime.
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [5]                          jesse.elliott@csuci.edu
                    IVP’s and prime numbers

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Theorem (Tao and Ziegler, 2006)
Prüfer domains

Integer-valued
                      Given any integer-valued polynomials P1 , P2 , . . . , Pk with vanish-
polynomial rings
                      ing constant terms, there are infinitely many integers x and m
Krull domains
and PVMDs             such that the integers x+P1 (m), . . . , x+Pk (m) are simultaneously
Universal             prime.
properties of IVP
rings

Summmary of
open problems         The special case when the polynomials are m, 2m, . . . , km implies
                      that there are arithmetic progressions of primes of length k for
                      any k.




  UCLA NTS 02/25/08                            [5]                          jesse.elliott@csuci.edu
                    Int(Z)

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      Like Z[X], the ring Int(Z) is integrally closed of Krull dimension 2.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [6]                          jesse.elliott@csuci.edu
                    Int(Z)

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      Like Z[X], the ring Int(Z) is integrally closed of Krull dimension 2.
Integer-valued
polynomial rings
                      However, Int(Z) is not Noetherian. The ideal XQ[X] ∩ Int(Z) of
Krull domains
and PVMDs             polynomials with vanishing constant term is not finitely generated.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [6]                          jesse.elliott@csuci.edu
                    Int(Z)

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      Like Z[X], the ring Int(Z) is integrally closed of Krull dimension 2.
Integer-valued
polynomial rings
                      However, Int(Z) is not Noetherian. The ideal XQ[X] ∩ Int(Z) of
Krull domains
and PVMDs             polynomials with vanishing constant term is not finitely generated.
Universal
properties of IVP
rings                 Int(Z) is like a “non-Noetherian Dedekind domain” in that the
Summmary of
open problems
                      nonzero finitely generated ideals of Int(Z) are invertible and
                      2-generated.




  UCLA NTS 02/25/08                            [6]                          jesse.elliott@csuci.edu
                    Spec(Int(Z))

     Rings of
 Integer-Valued       One has
   Polynomials

  Jesse Elliott
                        Spec(Q[X]) ←→   {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
                         q(X)Q[X] −→    pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                     [7]                        jesse.elliott@csuci.edu
                    Spec(Int(Z))

     Rings of
 Integer-Valued       One has
   Polynomials

  Jesse Elliott
                        Spec(Q[X]) ←→     {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
                         q(X)Q[X] −→      pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
                            Zp   ←→   {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
                             α   −→   mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems




  UCLA NTS 02/25/08                        [7]                       jesse.elliott@csuci.edu
                    Spec(Int(Z))

     Rings of
 Integer-Valued       One has
   Polynomials

  Jesse Elliott
                         Spec(Q[X]) ←→         {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
                          q(X)Q[X] −→          pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
                              Zp   ←→     {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
                               α   −→     mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems         mp,α ⊇ pq(X) iff q(α) = 0 in Qp .




  UCLA NTS 02/25/08                             [7]                       jesse.elliott@csuci.edu
                    Spec(Int(Z))

     Rings of
 Integer-Valued       One has
   Polynomials

  Jesse Elliott
                         Spec(Q[X]) ←→         {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
                          q(X)Q[X] −→          pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
                              Zp   ←→     {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
                               α   −→     mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems         mp,α ⊇ pq(X) iff q(α) = 0 in Qp .

                                          max-Spec(Int(Z)) ←→        Zp
                                                                 p




  UCLA NTS 02/25/08                             [7]                       jesse.elliott@csuci.edu
                    Spec(Int(Z))

     Rings of
 Integer-Valued       One has
   Polynomials

  Jesse Elliott
                         Spec(Q[X]) ←→         {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
                          q(X)Q[X] −→          pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
                              Zp   ←→     {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
                               α   −→     mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems         mp,α ⊇ pq(X) iff q(α) = 0 in Qp .

                                          max-Spec(Int(Z)) ←→        Zp
                                                                 p

                      mp,α has height 2 iff α ∈ Zp is algebraic over Q.

                      mp,α has height 1 iff α ∈ Zp is transcendental over Q.
  UCLA NTS 02/25/08                             [7]                       jesse.elliott@csuci.edu
                    Stone-Weierstrass

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains        Any continuous function f : Zp −→ Zp can be expanded uniquely
Integer-valued
polynomial rings
                      as
                                                    ∞
Krull domains
                                                          X
and PVMDs                                     f=       an
                                                   n=0
                                                          n
Universal
properties of IVP
rings                 with an ∈ Zp for all n and limn an = 0.
Summmary of
open problems




  UCLA NTS 02/25/08                            [8]                    jesse.elliott@csuci.edu
                    Stone-Weierstrass

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains        Any continuous function f : Zp −→ Zp can be expanded uniquely
Integer-valued
polynomial rings
                      as
                                                    ∞
Krull domains
                                                          X
and PVMDs                                     f=       an
                                                   n=0
                                                          n
Universal
properties of IVP
rings                 with an ∈ Zp for all n and limn an = 0.
Summmary of
open problems
                      Int(Z) is dense in C(Zp , Zp ).




  UCLA NTS 02/25/08                             [8]                   jesse.elliott@csuci.edu
                    Dedekind domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Recall that a domain D is a Dedekind domain if any of the
                      following equivalent conditions holds.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [9]                       jesse.elliott@csuci.edu
                    Dedekind domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Recall that a domain D is a Dedekind domain if any of the
                      following equivalent conditions holds.
Introduction

Prüfer domains
                       1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [9]                          jesse.elliott@csuci.edu
                    Dedekind domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Recall that a domain D is a Dedekind domain if any of the
                      following equivalent conditions holds.
Introduction

Prüfer domains
                       1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
                       2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [9]                        jesse.elliott@csuci.edu
                    Dedekind domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Recall that a domain D is a Dedekind domain if any of the
                      following equivalent conditions holds.
Introduction

Prüfer domains
                       1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
                       2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
                       3   D is Noetherian, and Dp is a DVR (or PID) for every nonzero
properties of IVP
rings
                           prime ideal p of D.
Summmary of
open problems




  UCLA NTS 02/25/08                           [9]                       jesse.elliott@csuci.edu
                    Dedekind domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Recall that a domain D is a Dedekind domain if any of the
                      following equivalent conditions holds.
Introduction

Prüfer domains
                       1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
                       2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
                       3   D is Noetherian, and Dp is a DVR (or PID) for every nonzero
properties of IVP
rings
                           prime ideal p of D.
Summmary of
open problems
                       4   D is Noetherian integrally closed of Krull dimension at most
                           1.




  UCLA NTS 02/25/08                           [9]                        jesse.elliott@csuci.edu
                    Dedekind domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Recall that a domain D is a Dedekind domain if any of the
                      following equivalent conditions holds.
Introduction

Prüfer domains
                       1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
                       2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
                       3   D is Noetherian, and Dp is a DVR (or PID) for every nonzero
properties of IVP
rings
                           prime ideal p of D.
Summmary of
open problems
                       4   D is Noetherian integrally closed of Krull dimension at most
                           1.

                      The ideal class group of D Dedekind is the group of nonzero
                      fractional ideals of D modulo the subgroup of nonzero principal
                      fractional ideals of D.


  UCLA NTS 02/25/08                           [9]                        jesse.elliott@csuci.edu
                    Picard group

     Rings of
 Integer-Valued
   Polynomials
                      The Picard group Pic(D) of a domain D is the group
  Jesse Elliott
                      I(D)/P(D), where
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [10]                      jesse.elliott@csuci.edu
                    Picard group

     Rings of
 Integer-Valued
   Polynomials
                      The Picard group Pic(D) of a domain D is the group
  Jesse Elliott
                      I(D)/P(D), where
Introduction

Prüfer domains
                          I(D) is the group of nonzero invertible (or projective)
Integer-valued
                          fractional ideals of D
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [10]                        jesse.elliott@csuci.edu
                    Picard group

     Rings of
 Integer-Valued
   Polynomials
                      The Picard group Pic(D) of a domain D is the group
  Jesse Elliott
                      I(D)/P(D), where
Introduction

Prüfer domains
                          I(D) is the group of nonzero invertible (or projective)
Integer-valued
                          fractional ideals of D
polynomial rings

Krull domains
                          P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [10]                        jesse.elliott@csuci.edu
                    Picard group

     Rings of
 Integer-Valued
   Polynomials
                      The Picard group Pic(D) of a domain D is the group
  Jesse Elliott
                      I(D)/P(D), where
Introduction

Prüfer domains
                          I(D) is the group of nonzero invertible (or projective)
Integer-valued
                          fractional ideals of D
polynomial rings

Krull domains
                          P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
                      D Dedekind domain =⇒ Pic(D) = the ideal class group of D.
open problems




  UCLA NTS 02/25/08                           [10]                        jesse.elliott@csuci.edu
                    Picard group

     Rings of
 Integer-Valued
   Polynomials
                      The Picard group Pic(D) of a domain D is the group
  Jesse Elliott
                      I(D)/P(D), where
Introduction

Prüfer domains
                          I(D) is the group of nonzero invertible (or projective)
Integer-valued
                          fractional ideals of D
polynomial rings

Krull domains
                          P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
                      D Dedekind domain =⇒ Pic(D) = the ideal class group of D.
open problems

                      Invertible ideals are necessarily finitely generated. If the converse
                      is true in a domain D, we say that D is a Prüfer domain (Prüfer
                      1932, Krull 1936).




  UCLA NTS 02/25/08                           [10]                         jesse.elliott@csuci.edu
                    Picard group

     Rings of
 Integer-Valued
   Polynomials
                      The Picard group Pic(D) of a domain D is the group
  Jesse Elliott
                      I(D)/P(D), where
Introduction

Prüfer domains
                          I(D) is the group of nonzero invertible (or projective)
Integer-valued
                          fractional ideals of D
polynomial rings

Krull domains
                          P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
                      D Dedekind domain =⇒ Pic(D) = the ideal class group of D.
open problems

                      Invertible ideals are necessarily finitely generated. If the converse
                      is true in a domain D, we say that D is a Prüfer domain (Prüfer
                      1932, Krull 1936).

                      D Dedekind ⇐⇒ D Noetherian and Prüfer.


  UCLA NTS 02/25/08                           [10]                         jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A domain D with quotient field K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [11]                       jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A domain D with quotient field K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
                       1   Every nonzero finitely generated ideal of D is invertible.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [11]                        jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A domain D with quotient field K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
                       1   Every nonzero finitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero finitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [11]                        jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A domain D with quotient field K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
                       1   Every nonzero finitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero finitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
                       3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [11]                        jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A domain D with quotient field K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
                       1   Every nonzero finitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero finitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
                       3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
                       4   D is integrally closed, and every element of K is a root of a
open problems              polynomial in D[X] at least one of whose coefficients is a unit
                           of D.




  UCLA NTS 02/25/08                           [11]                        jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A domain D with quotient field K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
                       1   Every nonzero finitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero finitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
                       3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
                       4   D is integrally closed, and every element of K is a root of a
open problems              polynomial in D[X] at least one of whose coefficients is a unit
                           of D.
                       5   Every submodule of a projective D-module is projective.




  UCLA NTS 02/25/08                           [11]                        jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A domain D with quotient field K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
                       1   Every nonzero finitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero finitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
                       3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
                       4   D is integrally closed, and every element of K is a root of a
open problems              polynomial in D[X] at least one of whose coefficients is a unit
                           of D.
                       5   Every submodule of a projective D-module is projective.
                       6   Every D-torsionfree D-module is D-flat.



  UCLA NTS 02/25/08                           [11]                        jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [12]    jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [12]    jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
                         Int(Z)
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [12]    jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
                         Int(Z)
polynomial rings

Krull domains
                         Any Bézout domain, i.e. a domain every finitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [12]                     jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
                         Int(Z)
polynomial rings

Krull domains
                         Any Bézout domain, i.e. a domain every finitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers.
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [12]                     jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
                         Int(Z)
polynomial rings

Krull domains
                         Any Bézout domain, i.e. a domain every finitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [12]                     jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
                         Int(Z)
polynomial rings

Krull domains
                         Any Bézout domain, i.e. a domain every finitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of              The integral closure of any Dedekind or Prüfer domain in any
open problems
                         algebraic extension of its quotient field.




  UCLA NTS 02/25/08                          [12]                      jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
                         Int(Z)
polynomial rings

Krull domains
                         Any Bézout domain, i.e. a domain every finitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of              The integral closure of any Dedekind or Prüfer domain in any
open problems
                         algebraic extension of its quotient field.
                         D[X] Prüfer ⇐⇒ D field.




  UCLA NTS 02/25/08                          [12]                      jesse.elliott@csuci.edu
                    Prüfer domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Prüfer domains:
Introduction
                         Any Dedekind domain.
Prüfer domains

Integer-valued
                         Int(Z)
polynomial rings

Krull domains
                         Any Bézout domain, i.e. a domain every finitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of              The integral closure of any Dedekind or Prüfer domain in any
open problems
                         algebraic extension of its quotient field.
                         D[X] Prüfer ⇐⇒ D field.

                      D is a Bézout domain ⇐⇒ D is a Prüfer domain and Pic(D) = 0.



  UCLA NTS 02/25/08                          [12]                      jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      A non-Noetherian Prüfer of Krull dimension 1, every finitely
Integer-valued
                      generated ideal principal.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      A non-Noetherian Prüfer of Krull dimension 1, every finitely
Integer-valued
                      generated ideal principal.
polynomial rings

Krull domains
and PVMDs             Int(Z) non-Noetherian Prüfer of Krull dimension 2, every finitely
Universal             generated ideal 2-generated.
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      A non-Noetherian Prüfer of Krull dimension 1, every finitely
Integer-valued
                      generated ideal principal.
polynomial rings

Krull domains
and PVMDs             Int(Z) non-Noetherian Prüfer of Krull dimension 2, every finitely
Universal             generated ideal 2-generated.
properties of IVP
rings

Summmary of
open problems
                      D Prüfer of Krull dimension n =⇒ every finitely generated ideal of
                      D can be generated by n + 1 elements.




  UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
                    Prüfer domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      A non-Noetherian Prüfer of Krull dimension 1, every finitely
Integer-valued
                      generated ideal principal.
polynomial rings

Krull domains
and PVMDs             Int(Z) non-Noetherian Prüfer of Krull dimension 2, every finitely
Universal             generated ideal 2-generated.
properties of IVP
rings

Summmary of
open problems
                      D Prüfer of Krull dimension n =⇒ every finitely generated ideal of
                      D can be generated by n + 1 elements. This is the best result
                      possible for arbitrary Prüfer domains.




  UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
                    Generalized IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let D an integral domain with quotient field K. The ring of
  Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
                      of K[X].
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [14]                       jesse.elliott@csuci.edu
                    Generalized IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let D an integral domain with quotient field K. The ring of
  Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
                      of K[X].
Krull domains
and PVMDs

Universal             Int(D) was first studied, for number fields D, by Pólya and
properties of IVP
rings                 Ostrowski circa 1917. They sought a D-module basis for Int(D).
Summmary of
open problems




  UCLA NTS 02/25/08                          [14]                      jesse.elliott@csuci.edu
                    Generalized IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let D an integral domain with quotient field K. The ring of
  Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
                      of K[X].
Krull domains
and PVMDs

Universal             Int(D) was first studied, for number fields D, by Pólya and
properties of IVP
rings                 Ostrowski circa 1917. They sought a D-module basis for Int(D).
Summmary of
open problems
                      D Dedekind =⇒ Int(D) is free as a D-module . . . but a basis may
                      be hard to compute.




  UCLA NTS 02/25/08                          [14]                       jesse.elliott@csuci.edu
                    Generalized IVP’s

     Rings of
 Integer-Valued
   Polynomials        Let D an integral domain with quotient field K. The ring of
  Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
                      of K[X].
Krull domains
and PVMDs

Universal             Int(D) was first studied, for number fields D, by Pólya and
properties of IVP
rings                 Ostrowski circa 1917. They sought a D-module basis for Int(D).
Summmary of
open problems
                      D Dedekind =⇒ Int(D) is free as a D-module . . . but a basis may
                      be hard to compute.

                      D Dedekind domain with finite residue fields =⇒ Int(D) is a
                      non-Noetherian Prüfer domain of Krull dimension 2 with every
                      finitely generated ideal 2-generated.

  UCLA NTS 02/25/08                          [14]                       jesse.elliott@csuci.edu
                    IVP’s over DVR’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      D DVR with finite residue field D/m =⇒ Int(D) is dense in
Integer-valued        C(Dm , Dm ) for the uniform convergence topology.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [15]                      jesse.elliott@csuci.edu
                    IVP’s over DVR’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      D DVR with finite residue field D/m =⇒ Int(D) is dense in
Integer-valued        C(Dm , Dm ) for the uniform convergence topology.
polynomial rings

Krull domains
and PVMDs             In fact, Int(D) (= D[X]) has a D-basis 1, X, f2 (X), f3 (X), . . .
Universal
properties of IVP     such that every continuous function ϕ : Dm −→ Dm can be written
rings
                      uniquely in the form
Summmary of                                               ∞
open problems
                                                     ϕ=         ai fi ,
                                                          i=0

                      where ai ∈ Dm and limi ai = 0.




  UCLA NTS 02/25/08                           [15]                        jesse.elliott@csuci.edu
                    An example: Int(Z[i])

     Rings of
 Integer-Valued
   Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
                                  2
  Jesse Elliott       Int(Z) is not a subset of Int(Z[i]).
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [16]                        jesse.elliott@csuci.edu
                    An example: Int(Z[i])

     Rings of
 Integer-Valued
   Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
                                  2
  Jesse Elliott       Int(Z) is not a subset of Int(Z[i]). Int(D) is not functorial in D.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [16]                          jesse.elliott@csuci.edu
                    An example: Int(Z[i])

     Rings of
 Integer-Valued
   Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
                                  2
  Jesse Elliott       Int(Z) is not a subset of Int(Z[i]). Int(D) is not functorial in D.
Introduction

Prüfer domains
                      A result of Pólya and Ostrowski yields a Z[i]-basis
Integer-valued
                                               4
                              2        3   2       −2X 3 −iX 2 +(1+i)X (5+(1+i)4 )X 5 +···
                      1, X, X1+i , X 1+i , X
                              −X      −X
polynomial rings
                                                        (1+i)3        ,    5(1+i)3         ,...
Krull domains
and PVMDs

Universal
properties of IVP     for Int(Z[i]).
rings

Summmary of
open problems




  UCLA NTS 02/25/08                                 [16]                             jesse.elliott@csuci.edu
                    An example: Int(Z[i])

     Rings of
 Integer-Valued
   Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
                                  2
  Jesse Elliott       Int(Z) is not a subset of Int(Z[i]). Int(D) is not functorial in D.
Introduction

Prüfer domains
                      A result of Pólya and Ostrowski yields a Z[i]-basis
Integer-valued
                                                  4
                              2         3     2       −2X 3 −iX 2 +(1+i)X (5+(1+i)4 )X 5 +···
                      1, X, X1+i , X 1+i , X
                              −X      −X
polynomial rings
                                                           (1+i)3        ,    5(1+i)3         ,...
Krull domains
and PVMDs

Universal
properties of IVP     for Int(Z[i]).
rings

Summmary of
open problems         Int(Z[i]) is generated under the D-algebra operations and
                      composition by the so-called “Fermat binomials”
                                  X 2 −X
                           F2 =    1+i
                                  X p −X
                           Fp =      p      for p ≡ 1 (mod 4)
                                       p2
                                   X     −X
                           Fp2 =        p     for p ≡ 3 (mod 4).

  UCLA NTS 02/25/08                                    [16]                             jesse.elliott@csuci.edu
                    Bases

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      (Pólya and Ostrowski) If D is a number ring (or any Dedekind
Introduction          domain) such that
Prüfer domains
                                                Πq =       m
Integer-valued
polynomial rings                                      N (m)=q
Krull domains
and PVMDs             is principal for all prime powers q, then Int(D) is generated under
Universal             the D-algebra operations and composition by X and the
properties of IVP
rings                 binomials
Summmary of                                         Xq − X
open problems                                  Fq =          for all q,
                                                      πq
                      where Πq = πq D for all q.




  UCLA NTS 02/25/08                           [17]                        jesse.elliott@csuci.edu
                    Bases

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      (Pólya and Ostrowski) If D is a number ring (or any Dedekind
Introduction          domain) such that
Prüfer domains
                                                Πq =       m
Integer-valued
polynomial rings                                      N (m)=q
Krull domains
and PVMDs             is principal for all prime powers q, then Int(D) is generated under
Universal             the D-algebra operations and composition by X and the
properties of IVP
rings                 binomials
Summmary of                                         Xq − X
open problems                                  Fq =          for all q,
                                                      πq
                      where Πq = πq D for all q.

                      In that case they also constructed a D-basis for Int(D).



  UCLA NTS 02/25/08                           [17]                        jesse.elliott@csuci.edu
                    Int(D) as a D-module

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Problem
Introduction

Prüfer domains
                      Find an example, if one exists, of a domain D such that Int(D) is
Integer-valued
                      not free. For which domains D is Int(D) free as a D-module?
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [18]                       jesse.elliott@csuci.edu
                    Int(D) as a D-module

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Problem
Introduction

Prüfer domains
                      Find an example, if one exists, of a domain D such that Int(D) is
Integer-valued
                      not free. For which domains D is Int(D) free as a D-module?
polynomial rings

Krull domains
and PVMDs
                      Problem
Universal
properties of IVP
rings                 For which domains D is Int(D) flat as a D-module?
Summmary of
open problems




  UCLA NTS 02/25/08                          [18]                       jesse.elliott@csuci.edu
                    Int(D) as a D-module

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Problem
Introduction

Prüfer domains
                      Find an example, if one exists, of a domain D such that Int(D) is
Integer-valued
                      not free. For which domains D is Int(D) free as a D-module?
polynomial rings

Krull domains
and PVMDs
                      Problem
Universal
properties of IVP
rings                 For which domains D is Int(D) flat as a D-module?
Summmary of
open problems

                      Proposition
                      Int(D) is locally free (hence faithfully flat) as a D-module for any
                      “Krull domain” D.




  UCLA NTS 02/25/08                           [18]                        jesse.elliott@csuci.edu
                    More examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      If D is a field, then Int(D) = D[X].
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [19]          jesse.elliott@csuci.edu
                    More examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      If D is a field, then Int(D) = D[X].
Integer-valued
polynomial rings

Krull domains
and PVMDs
                      Int(D) might equal D[X] even if D is not a field. This happens, for
Universal
                      example, if D = D [T ] and D is not a finite field, or if D/m is
properties of IVP
rings
                      infinite for every maximal ideal m of D.
Summmary of
open problems




  UCLA NTS 02/25/08                           [19]                       jesse.elliott@csuci.edu
                    More examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      If D is a field, then Int(D) = D[X].
Integer-valued
polynomial rings

Krull domains
and PVMDs
                      Int(D) might equal D[X] even if D is not a field. This happens, for
Universal
                      example, if D = D [T ] and D is not a finite field, or if D/m is
properties of IVP
rings
                      infinite for every maximal ideal m of D.
Summmary of
open problems
                      What about Int(Int(D))?




  UCLA NTS 02/25/08                           [19]                       jesse.elliott@csuci.edu
                    Multivariable integer-valued polynomials

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Generally, for any set X, we let
Introduction

Prüfer domains
                                  Int(DX ) = {f (X) ∈ F [X] : f (DX ) ⊆ D}.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [20]                       jesse.elliott@csuci.edu
                    Multivariable integer-valued polynomials

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Generally, for any set X, we let
Introduction

Prüfer domains
                                   Int(DX ) = {f (X) ∈ F [X] : f (DX ) ⊆ D}.
Integer-valued
polynomial rings

Krull domains
and PVMDs
                      Just as
Universal
properties of IVP                            (R[X])[Y ] = R[X       Y]
rings

Summmary of
open problems
                      for any commutative ring R, one has

                                         Int(Int(DX )Y ) = Int(DX        Y
                                                                             ),

                      as long as D is infinite (not a finite field).




  UCLA NTS 02/25/08                            [20]                               jesse.elliott@csuci.edu
                    Multivariable integer-valued polynomials

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      For any set X, one has a canonical D-algebra homomorphism
Introduction

Prüfer domains

Integer-valued
                                       θX :           Int(D) −→ Int(DX ),
polynomial rings                              T ∈X
Krull domains
and PVMDs
                      where the tensor product is over D.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [21]                         jesse.elliott@csuci.edu
                    Multivariable integer-valued polynomials

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      For any set X, one has a canonical D-algebra homomorphism
Introduction

Prüfer domains

Integer-valued
                                        θX :           Int(D) −→ Int(DX ),
polynomial rings                               T ∈X
Krull domains
and PVMDs
                      where the tensor product is over D.
Universal
properties of IVP
rings

Summmary of
                      If Int(D) = D[X] (e.g. if D is a field), then θX is an isomorphism
open problems         for every set X.




  UCLA NTS 02/25/08                             [21]                         jesse.elliott@csuci.edu
                    Multivariable integer-valued polynomials

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      For any set X, one has a canonical D-algebra homomorphism
Introduction

Prüfer domains

Integer-valued
                                        θX :           Int(D) −→ Int(DX ),
polynomial rings                               T ∈X
Krull domains
and PVMDs
                      where the tensor product is over D.
Universal
properties of IVP
rings

Summmary of
                      If Int(D) = D[X] (e.g. if D is a field), then θX is an isomorphism
open problems         for every set X.

                      If Int(D) is flat as a D-module (e.g if D is a Prüfer domain), then
                      θX is injective for every set X.




  UCLA NTS 02/25/08                             [21]                         jesse.elliott@csuci.edu
                    Multivariable integer-valued polynomials

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Proposition
Introduction

Prüfer domains        If D is a “Krull domain,” or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
                      momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs
                      for every set X.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [22]                      jesse.elliott@csuci.edu
                    Multivariable integer-valued polynomials

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Proposition
Introduction

Prüfer domains        If D is a “Krull domain,” or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
                      momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs
                      for every set X.
Universal
properties of IVP
rings

Summmary of           Problem
open problems
                      Find an example, if one exists, of a domain D and set X for which
                      θX is not surjective/injective/an isomorphism. For which domains
                      D is θX an isomorphism for all sets X?




  UCLA NTS 02/25/08                          [22]                       jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
                      “Krull.”
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [23]                       jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
                      “Krull.”
Integer-valued
polynomial rings

Krull domains         A domain D is a Krull domain if there exists a family {vi } of
and PVMDs
                      discrete valuations on the fraction field K of D such that
Universal
properties of IVP
rings                      For all x ∈ K ∗ , vi (x) = 0 for almost all i.
Summmary of
open problems
                           For all x ∈ K ∗ , x ∈ D iff vi (x) ≥ 0 for all i.




  UCLA NTS 02/25/08                           [23]                        jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
                      “Krull.”
Integer-valued
polynomial rings

Krull domains         A domain D is a Krull domain if there exists a family {vi } of
and PVMDs
                      discrete valuations on the fraction field K of D such that
Universal
properties of IVP
rings                      For all x ∈ K ∗ , vi (x) = 0 for almost all i.
Summmary of
open problems
                           For all x ∈ K ∗ , x ∈ D iff vi (x) ≥ 0 for all i.

                      Every Krull domain is integrally closed.




  UCLA NTS 02/25/08                           [23]                        jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
                      “Krull.”
Integer-valued
polynomial rings

Krull domains         A domain D is a Krull domain if there exists a family {vi } of
and PVMDs
                      discrete valuations on the fraction field K of D such that
Universal
properties of IVP
rings                      For all x ∈ K ∗ , vi (x) = 0 for almost all i.
Summmary of
open problems
                           For all x ∈ K ∗ , x ∈ D iff vi (x) ≥ 0 for all i.

                      Every Krull domain is integrally closed.

                      D Dedekind ⇐⇒ D Krull of Krull dimension 1.



  UCLA NTS 02/25/08                           [23]                        jesse.elliott@csuci.edu
                    Krull domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Krull domains:
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [24]   jesse.elliott@csuci.edu
                    Krull domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Krull domains:
Introduction

Prüfer domains
                         Any UFD or integrally closed Noetherian domain.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [24]                      jesse.elliott@csuci.edu
                    Krull domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Krull domains:
Introduction

Prüfer domains
                         Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [24]                      jesse.elliott@csuci.edu
                    Krull domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Krull domains:
Introduction

Prüfer domains
                         Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains            The integral closure of any Noetherian or Krull domain in any
and PVMDs
                         finite extension of its quotient field.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [24]                       jesse.elliott@csuci.edu
                    Krull domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Krull domains:
Introduction

Prüfer domains
                         Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains            The integral closure of any Noetherian or Krull domain in any
and PVMDs
                         finite extension of its quotient field.
Universal
properties of IVP
rings
                         D[X] Krull ⇐⇒ D Krull.
Summmary of
open problems




  UCLA NTS 02/25/08                          [24]                       jesse.elliott@csuci.edu
                    Krull domains: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of Krull domains:
Introduction

Prüfer domains
                         Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains            The integral closure of any Noetherian or Krull domain in any
and PVMDs
                         finite extension of its quotient field.
Universal
properties of IVP
rings
                         D[X] Krull ⇐⇒ D Krull.
Summmary of
open problems

                      Bouvier’s Conjecture (1985)
                      There exists a UFD or Krull domain D such that dim D[X] > 1 +
                      dim D.




  UCLA NTS 02/25/08                          [24]                       jesse.elliott@csuci.edu
                    v-class group

     Rings of
 Integer-Valued
   Polynomials
                      A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
  Jesse Elliott
                      be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [25]                         jesse.elliott@csuci.edu
                    v-class group

     Rings of
 Integer-Valued
   Polynomials
                      A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
  Jesse Elliott
                      be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [25]                         jesse.elliott@csuci.edu
                    v-class group

     Rings of
 Integer-Valued
   Polynomials
                      A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
  Jesse Elliott
                      be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP     A fractional ideal I of D is said to be v-invertible if (II −1 )v = D,
rings

Summmary of
                      or, equivalently, if Iv is a unit in Iv (D).
open problems




  UCLA NTS 02/25/08                             [25]                         jesse.elliott@csuci.edu
                    v-class group

     Rings of
 Integer-Valued
   Polynomials
                      A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
  Jesse Elliott
                      be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP     A fractional ideal I of D is said to be v-invertible if (II −1 )v = D,
rings

Summmary of
                      or, equivalently, if Iv is a unit in Iv (D).
open problems

                      The v-class group Clv (D) = the group of v-invertible divisorial
                      fractional ideals under v-multiplication modulo the subgroup of
                      nonzero principal fractional ideals.




  UCLA NTS 02/25/08                             [25]                         jesse.elliott@csuci.edu
                    v-class group

     Rings of
 Integer-Valued
   Polynomials
                      A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
  Jesse Elliott
                      be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP     A fractional ideal I of D is said to be v-invertible if (II −1 )v = D,
rings

Summmary of
                      or, equivalently, if Iv is a unit in Iv (D).
open problems

                      The v-class group Clv (D) = the group of v-invertible divisorial
                      fractional ideals under v-multiplication modulo the subgroup of
                      nonzero principal fractional ideals.

                      D is a UFD ⇐⇒ D is a Krull domain such that Clv (D) = 0.


  UCLA NTS 02/25/08                             [25]                         jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [26]                         jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
                       1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [26]                         jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
                       1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings
                       2   The set of all nonzero divisorial fractional ideals of D is a
Krull domains
and PVMDs                  group under v-multiplication.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [26]                          jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
                       1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings
                       2   The set of all nonzero divisorial fractional ideals of D is a
Krull domains
and PVMDs                  group under v-multiplication.
Universal
properties of IVP
                       3   D = p∈X 1 (D) Dp , where X 1 (D) is the set of height one
rings

Summmary of
                           prime ideals of D, where Dp is a DVR for all p ∈ X 1 (D), and
open problems              where every element of D lies in only finitely many
                           p ∈ X 1 (D).




  UCLA NTS 02/25/08                           [26]                         jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
                       1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings
                       2   The set of all nonzero divisorial fractional ideals of D is a
Krull domains
and PVMDs                  group under v-multiplication.
Universal
properties of IVP
                       3   D = p∈X 1 (D) Dp , where X 1 (D) is the set of height one
rings

Summmary of
                           prime ideals of D, where Dp is a DVR for all p ∈ X 1 (D), and
open problems              where every element of D lies in only finitely many
                           p ∈ X 1 (D).
                       4   D is a Mori domain, that is, D satisfies the ACC on divisorial
                           ideals, and D is “completely integrally closed.”




  UCLA NTS 02/25/08                           [26]                         jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction          D Krull domain ⇐⇒ {nonzero divisorial fractional ideals of D} is
Prüfer domains
                      an abelian group under v-multiplication, in which case it is free on
Integer-valued
polynomial rings      the set X 1 (D) of prime ideals of height one.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [27]                        jesse.elliott@csuci.edu
                    Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction          D Krull domain ⇐⇒ {nonzero divisorial fractional ideals of D} is
Prüfer domains
                      an abelian group under v-multiplication, in which case it is free on
Integer-valued
polynomial rings      the set X 1 (D) of prime ideals of height one.
Krull domains
and PVMDs

Universal
                      Every nonzero divisorial fractional ideal I in a Krull domain D has
properties of IVP
rings
                      a unique primary decomposition
Summmary of
open problems
                                               I=                 p(np ) .
                                                      p∈X 1 (D)




  UCLA NTS 02/25/08                            [27]                          jesse.elliott@csuci.edu
                    t-class group

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A fractional ideal I of a domain D is said to be v-finite if Iv = Jv
Introduction          for some finitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-finite.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [28]                         jesse.elliott@csuci.edu
                    t-class group

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A fractional ideal I of a domain D is said to be v-finite if Iv = Jv
Introduction          for some finitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-finite.
Integer-valued
polynomial rings

Krull domains
                      The v-finite divisorial fractional ideals of D form a submonoid
and PVMDs
                      Ivf (D) of Iv (D).
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [28]                         jesse.elliott@csuci.edu
                    t-class group

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A fractional ideal I of a domain D is said to be v-finite if Iv = Jv
Introduction          for some finitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-finite.
Integer-valued
polynomial rings

Krull domains
                      The v-finite divisorial fractional ideals of D form a submonoid
and PVMDs
                      Ivf (D) of Iv (D).
Universal
properties of IVP
rings
                      I is said to be t-invertible t-ideal if I is a unit in the monoid
Summmary of
open problems         Ivf (D) (that is, I is a v-invertible divisorial fractional ideal and
                      both I and I −1 are v-finite).




  UCLA NTS 02/25/08                             [28]                           jesse.elliott@csuci.edu
                    t-class group

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       A fractional ideal I of a domain D is said to be v-finite if Iv = Jv
Introduction          for some finitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-finite.
Integer-valued
polynomial rings

Krull domains
                      The v-finite divisorial fractional ideals of D form a submonoid
and PVMDs
                      Ivf (D) of Iv (D).
Universal
properties of IVP
rings
                      I is said to be t-invertible t-ideal if I is a unit in the monoid
Summmary of
open problems         Ivf (D) (that is, I is a v-invertible divisorial fractional ideal and
                      both I and I −1 are v-finite).

                      Clt (D) = group of t-invertible t-ideals mod principal fractional
                      ideals (= Clv (D) if D is Mori).



  UCLA NTS 02/25/08                             [28]                           jesse.elliott@csuci.edu
                    PVMD’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
                      PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [29]                      jesse.elliott@csuci.edu
                    PVMD’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
                      PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
                       1   Every nonzero v-finite divisorial ideal of D is v-invertible.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [29]                          jesse.elliott@csuci.edu
                    PVMD’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
                      PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
                       1   Every nonzero v-finite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-finite divisorial fractional ideals of D
and PVMDs

Universal
                           is a group under v-multiplication.
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [29]                          jesse.elliott@csuci.edu
                    PVMD’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
                      PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
                       1   Every nonzero v-finite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-finite divisorial fractional ideals of D
and PVMDs

Universal
                           is a group under v-multiplication.
properties of IVP
rings
                       3   Dp is a valuation domain for every “t-maximal ideal” p of D.
Summmary of
open problems




  UCLA NTS 02/25/08                            [29]                          jesse.elliott@csuci.edu
                    PVMD’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
                      PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
                       1   Every nonzero v-finite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-finite divisorial fractional ideals of D
and PVMDs

Universal
                           is a group under v-multiplication.
properties of IVP
rings
                       3   Dp is a valuation domain for every “t-maximal ideal” p of D.
Summmary of
open problems
                       4   D is integrally closed and “(I ∩ J)t = It ∩ Jt ” for all ideals I
                           and J of D.




  UCLA NTS 02/25/08                             [29]                         jesse.elliott@csuci.edu
                    PVMD’s

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
                      PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
                       1   Every nonzero v-finite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-finite divisorial fractional ideals of D
and PVMDs

Universal
                           is a group under v-multiplication.
properties of IVP
rings
                       3   Dp is a valuation domain for every “t-maximal ideal” p of D.
Summmary of
open problems
                       4   D is integrally closed and “(I ∩ J)t = It ∩ Jt ” for all ideals I
                           and J of D.

                      D Krull ⇐⇒ D Mori PVMD.




  UCLA NTS 02/25/08                             [29]                         jesse.elliott@csuci.edu
                    PVMD’s: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of PVMD’s:
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [30]   jesse.elliott@csuci.edu
                    PVMD’s: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [30]              jesse.elliott@csuci.edu
                    PVMD’s: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every finite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [30]                     jesse.elliott@csuci.edu
                    PVMD’s: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every finite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
                         The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
                         its quotient field.
Summmary of
open problems




  UCLA NTS 02/25/08                         [30]                       jesse.elliott@csuci.edu
                    PVMD’s: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every finite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
                         The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
                         its quotient field.
Summmary of
open problems
                         D[X] PVMD ⇐⇒ D PVMD.




  UCLA NTS 02/25/08                         [30]                       jesse.elliott@csuci.edu
                    PVMD’s: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every finite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
                         The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
                         its quotient field.
Summmary of
open problems
                         D[X] PVMD ⇐⇒ D PVMD.
                         D Krull =⇒ Int(D) PVMD!




  UCLA NTS 02/25/08                         [30]                       jesse.elliott@csuci.edu
                    PVMD’s: examples

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Examples of PVMD’s:
Introduction

Prüfer domains            Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings          Any GCD domain, a domain in which every finite set of
Krull domains             elements has a GCD.
and PVMDs

Universal
                          The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
                          its quotient field.
Summmary of
open problems
                          D[X] PVMD ⇐⇒ D PVMD.
                          D Krull =⇒ Int(D) PVMD!

                      D is a GCD domain ⇐⇒ D is a PVMD such that Clt (D) = 0.




  UCLA NTS 02/25/08                          [30]                       jesse.elliott@csuci.edu
                    Some classes of integrally closed domains

     Rings of
 Integer-Valued
   Polynomials
                               Z NNN           DVR            k[X]
  Jesse Elliott
                                    NNN                     q
                                       NNN               qqq
Introduction
                                          NNN         qqq
Prüfer domains                               N'  xqqq
Integer-valued                              Euclidean
polynomial rings

Krull domains
and PVMDs
                                                  
Universal
                                                   PID MM
                                               oo         MMM
properties of IVP

                                           ooo
rings

                                        oo                   MMM
                                       o                        MMM
                                   wooo
Summmary of
open problems
                                                                  &
                           DedekindOOO             UFD M          Bézout
                                       OOOooooo          MMM qqq
                                                            MMM
                                        ooOOO               qqq
                                wo  ooo       OO'     xqqqq MMM& 
                             Krull O             Prüfer              GCD
                                     OOO
                                        OOO                     qqq
                                            OOO             qqq
                                                O'  xqqqq
  UCLA NTS 02/25/08                       [31]
                                                 PVMD                      jesse.elliott@csuci.edu
                    Some classes of integrally closed domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                                                  PID K
Prüfer domains
                                               q        KKK
                                            qqq            KKK
                                         qqq
Integer-valued
                                                              KKK
                                       qq
polynomial rings

Krull domains                        xq                         %
and PVMDs
                             DedekindMMM          UFD K         Bézout
                                         MMMqqqq        KKK ss
Universal
                                                           KKKs
                                                            ss
properties of IVP
                                        qqqqMMM          sss KKK% 
rings
                                  xqq         M&     yss
Summmary of
open problems                  Krull M          Prüfer           GCD
                                       MMM                      s
                                          MMM                sss
                                             MMM          sss
                                                 &  ysss
                                                PVMD




  UCLA NTS 02/25/08                       [32]                           jesse.elliott@csuci.edu
                    In a larger context

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                                                                              PID K
                                                                         qq             KKK
                                                                      qqq
Introduction
                                                                                           KKK
Prüfer domains
                                                                   qqq                        KKK
Integer-valued                                                xqqq                              %
polynomial rings
                                              Dedekind                        UFD K             Bézout
                                              n  MMMM qqq                              KKK ss
                                           nnn                      MMMq                   KKKs
Krull domains

                                        nnn                         qqMM                   ss
                                                                                         sss KKK% 
and PVMDs

                                    nnnn                        qqqq MM                s
Universal
properties of IVP                 wn                        xq            &          ys
rings
                      Noetherian
                            PPP               n Krull MMMM                Prüfer               GCD
                                      PPP nnnn                            q                     s
Summmary of
                                        P
                                         n                          Mqqqqq
                                                                     M                     ssss
                                      nnn PPPPP                    qq MMMMM  ssss
open problems
                                                             xqqq
                                 wnnn           '                            &  ys
                        Mori PP               Coherent
                                                                          PVMD
                                     PPP                                 qq
                                        PPP
                                           PPP
                                                     
                                                                    qqqqq
                                              P'  xqqqq
                                             v-coherent


  UCLA NTS 02/25/08                                 [33]                                   jesse.elliott@csuci.edu
                    Krull dimension

     Rings of
 Integer-Valued
   Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
  Jesse Elliott       hard to compute for non-Noetherian domains D.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [34]                       jesse.elliott@csuci.edu
                    Krull dimension

     Rings of
 Integer-Valued
   Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
  Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                   1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [34]                       jesse.elliott@csuci.edu
                    Krull dimension

     Rings of
 Integer-Valued
   Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
  Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                     1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal             It is not hard to show for all D that
properties of IVP
rings

Summmary of                              dim(Int(D)) ≥ dim(D[X]) − 1.
open problems




  UCLA NTS 02/25/08                             [34]                    jesse.elliott@csuci.edu
                    Krull dimension

     Rings of
 Integer-Valued
   Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
  Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                     1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal             It is not hard to show for all D that
properties of IVP
rings

Summmary of                              dim(Int(D)) ≥ dim(D[X]) − 1.
open problems


                      “dim Int” Conjecture
                      dim(Int(D)) ≤ dim(D[X]) for every domain D.




  UCLA NTS 02/25/08                             [34]                    jesse.elliott@csuci.edu
                    Krull dimension

     Rings of
 Integer-Valued
   Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
  Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                     1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal             It is not hard to show for all D that
properties of IVP
rings

Summmary of                              dim(Int(D)) ≥ dim(D[X]) − 1.
open problems


                      “dim Int” Conjecture
                      dim(Int(D)) ≤ dim(D[X]) for every domain D.


                      Equality holds for Noetherian domains and PVMD’s.

  UCLA NTS 02/25/08                             [34]                    jesse.elliott@csuci.edu
                    A conjecture

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Proposition
Introduction

Prüfer domains        If D is a Krull domain, or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
                      momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs             for every set X.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [35]                      jesse.elliott@csuci.edu
                    A conjecture

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Proposition
Introduction

Prüfer domains        If D is a Krull domain, or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
                      momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs             for every set X.
Universal
properties of IVP
rings

Summmary of           Conjecture
open problems
                      If D is a PVMD such that Int(Dm ) = Int(D)m for every maximal
                      t-ideal m of D (which holds if D is a domain “of Krull-type”), then
                      the homomorphism θX is an isomorphism for every set X.




  UCLA NTS 02/25/08                           [35]                        jesse.elliott@csuci.edu
                    A universal property of IVP rings

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Int(D) is not functorial in D. Nevertheless, it is possible to
                      characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
                    A universal property of IVP rings

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Int(D) is not functorial in D. Nevertheless, it is possible to
                      characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
                      A domain A containing D, say, with fraction field L, is a
polynomial rings
                      polynomially complete (PC) extension of D if any polynomial
Krull domains
and PVMDs             f (X) ∈ L[X] that maps D into A actually maps all of A into A.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
                    A universal property of IVP rings

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Int(D) is not functorial in D. Nevertheless, it is possible to
                      characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
                      A domain A containing D, say, with fraction field L, is a
polynomial rings
                      polynomially complete (PC) extension of D if any polynomial
Krull domains
and PVMDs             f (X) ∈ L[X] that maps D into A actually maps all of A into A.
Universal
properties of IVP
rings                 Proposition
Summmary of
open problems
                      For any infinite domain D and any set X, the domain Int(DX )
                      is the free PC extension of D generated by X.




  UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
                    A universal property of IVP rings

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Int(D) is not functorial in D. Nevertheless, it is possible to
                      characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
                      A domain A containing D, say, with fraction field L, is a
polynomial rings
                      polynomially complete (PC) extension of D if any polynomial
Krull domains
and PVMDs             f (X) ∈ L[X] that maps D into A actually maps all of A into A.
Universal
properties of IVP
rings                 Proposition
Summmary of
open problems
                      For any infinite domain D and any set X, the domain Int(DX )
                      is the free PC extension of D generated by X. In other words,
                      Int(DX ) is a PC extension of D, and for any PC extension A of D
                      and any map ϕ : X −→ A, there is a unique D-algebra homomor-
                      phism Int(DX ) −→ A sending T to ϕ(T ) for all T ∈ X.



  UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
                    Example

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings      Example. A domain A containing Z is a PC extension of Z iff
                       a
Krull domains
and PVMDs
                       n ∈ A ⊗Z Q lies in A for every a ∈ A and every positive integer
                      n. Such a domain A is said to be a binomial domain.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [37]                       jesse.elliott@csuci.edu
                    Example

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings      Example. A domain A containing Z is a PC extension of Z iff
                       a
Krull domains
and PVMDs
                       n ∈ A ⊗Z Q lies in A for every a ∈ A and every positive integer
                      n. Such a domain A is said to be a binomial domain.
Universal
properties of IVP
rings

Summmary of
                      Int(ZX ) is the free binomial domain generated by X.
open problems




  UCLA NTS 02/25/08                           [37]                       jesse.elliott@csuci.edu
                    Over Dedekind domains

     Rings of
 Integer-Valued
   Polynomials
                      Theorem (Gerboud, 1993)
  Jesse Elliott

Introduction
                      For any extension A of an infinite Dedekind domain D, the follow-
Prüfer domains
                      ing conditions are equivalent.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
                    Over Dedekind domains

     Rings of
 Integer-Valued
   Polynomials
                      Theorem (Gerboud, 1993)
  Jesse Elliott

Introduction
                      For any extension A of an infinite Dedekind domain D, the follow-
Prüfer domains
                      ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
                    Over Dedekind domains

     Rings of
 Integer-Valued
   Polynomials
                      Theorem (Gerboud, 1993)
  Jesse Elliott

Introduction
                      For any extension A of an infinite Dedekind domain D, the follow-
Prüfer domains
                      ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
                        2  Int(D) ⊆ Int(A).
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
                    Over Dedekind domains

     Rings of
 Integer-Valued
   Polynomials
                      Theorem (Gerboud, 1993)
  Jesse Elliott

Introduction
                      For any extension A of an infinite Dedekind domain D, the follow-
Prüfer domains
                      ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
                        2  Int(D) ⊆ Int(A).
and PVMDs

Universal
                        3  Int(A) is the A-module generated by Int(D).
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
                    Over Dedekind domains

     Rings of
 Integer-Valued
   Polynomials
                      Theorem (Gerboud, 1993)
  Jesse Elliott

Introduction
                      For any extension A of an infinite Dedekind domain D, the follow-
Prüfer domains
                      ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
                        2  Int(D) ⊆ Int(A).
and PVMDs

Universal
                        3  Int(A) is the A-module generated by Int(D).
properties of IVP
rings                   4  For every ideal m of D with finite residue field, and for every
Summmary of
open problems
                           prime ideal P of A lying over m, one has mAP = PAP and
                           A/P = D/m.




  UCLA NTS 02/25/08                           [38]                       jesse.elliott@csuci.edu
                    Over Dedekind domains

     Rings of
 Integer-Valued
   Polynomials
                      Theorem (Gerboud, 1993)
  Jesse Elliott

Introduction
                      For any extension A of an infinite Dedekind domain D, the follow-
Prüfer domains
                      ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
                        2  Int(D) ⊆ Int(A).
and PVMDs

Universal
                        3  Int(A) is the A-module generated by Int(D).
properties of IVP
rings                   4  For every ideal m of D with finite residue field, and for every
Summmary of
open problems
                           prime ideal P of A lying over m, one has mAP = PAP and
                           A/P = D/m.


                      Condition (4) can be rephrased as saying that A is unramified,
                      and has trivial residue field extensions, at every maximal ideal
                      m of D with finite residue field.

  UCLA NTS 02/25/08                           [38]                       jesse.elliott@csuci.edu
                    Example

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      For example, Int(Z) is a polynomially complete extension of Z.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [39]                        jesse.elliott@csuci.edu
                    Example

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains
                      For example, Int(Z) is a polynomially complete extension of Z.
Integer-valued
polynomial rings      Therefore for every prime ideal P = mp,α of Int(Z) lying over a
Krull domains
and PVMDs             prime p in Z one has
Universal
properties of IVP
rings
                                            PInt(Z)P = pInt(Z)P
Summmary of
open problems         and
                                              Int(Z)/P = Z/pZ.




  UCLA NTS 02/25/08                           [39]                        jesse.elliott@csuci.edu
                    Over Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Theorem
Prüfer domains

Integer-valued        For any flat extension A of an infinite Krull domain D, the following
polynomial rings
                      conditions are equivalent.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
                    Over Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Theorem
Prüfer domains

Integer-valued        For any flat extension A of an infinite Krull domain D, the following
polynomial rings
                      conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
                    Over Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Theorem
Prüfer domains

Integer-valued        For any flat extension A of an infinite Krull domain D, the following
polynomial rings
                      conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP       2 Int(D) ⊆ Int(A).
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
                    Over Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Theorem
Prüfer domains

Integer-valued        For any flat extension A of an infinite Krull domain D, the following
polynomial rings
                      conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP       2 Int(D) ⊆ Int(A).
rings

Summmary of
                        3 Int(A) is the A-module generated by Int(D).
open problems




  UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
                    Over Krull domains

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Theorem
Prüfer domains

Integer-valued        For any flat extension A of an infinite Krull domain D, the following
polynomial rings
                      conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP       2 Int(D) ⊆ Int(A).
rings

Summmary of
                        3 Int(A) is the A-module generated by Int(D).
open problems
                        4 A is unramified, and has trivial residue field extensions, at
                          every height one prime ideal of D with finite residue field.




  UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomiallly complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                            [41]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomiallly complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
                      If (APC) holds, we will say that A is an almost polynomially
Summmary of
open problems         complete extension of D.




  UCLA NTS 02/25/08                            [41]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomiallly complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
                      If (APC) holds, we will say that A is an almost polynomially
Summmary of
open problems         complete extension of D.

                      If (WPC) holds, we will say that A is a weakly polynomially
                      complete extension of D.




  UCLA NTS 02/25/08                            [41]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
                      In general one has (PC) ⇒ (APC) ⇒ (WPC).
Summmary of
open problems




  UCLA NTS 02/25/08                            [42]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
                      In general one has (PC) ⇒ (APC) ⇒ (WPC).
Summmary of
open problems

                      (PC) ⇔ (WPC) if D is an infinite Krull domain and A is flat as a
                      D-module.




  UCLA NTS 02/25/08                            [42]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
                      In general one has (PC) ⇒ (APC) ⇒ (WPC).
Summmary of
open problems

                      (PC) ⇔ (WPC) if D is an infinite Krull domain and A is flat as a
                      D-module.

                      (APC) ⇔ (WPC) if for all X the homomorphism
                      θX : T ∈X Int(D) −→ Int(DX ) is surjective.


  UCLA NTS 02/25/08                            [42]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
                      The extension Z[T /2] of Z[T ] is APC but not PC.
Summmary of
open problems




  UCLA NTS 02/25/08                            [43]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
                      The extension Z[T /2] of Z[T ] is APC but not PC.
Summmary of
open problems

                      Indeed, Int(Z[T ]) = Z[T ][X] and every extension of Z[T ] is APC.
                                                    2
                                                      −X
                      However, the polynomial X 2 maps Z[T ] into Z[T /2] but does
                      not map all of Z[T /2] into itself, and therefore the extension
                      Z[T /2] of Z[T ] is not PC.



  UCLA NTS 02/25/08                            [43]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
                      conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
                       3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings

Summmary of
                      Problem
open problems
                      Does there exist an extension A of a some domain D that is WPC
                      but not APC?




  UCLA NTS 02/25/08                            [44]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Proposition
Prüfer domains

Integer-valued        For any domain D and any set X, the domain Int(DX ) is the free
polynomial rings
                      APC extension of D generated by X.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [45]                      jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction
                      Proposition
Prüfer domains

Integer-valued        For any domain D and any set X, the domain Int(DX ) is the free
polynomial rings
                      APC extension of D generated by X.
Krull domains
and PVMDs

Universal
properties of IVP
rings
                      Problem
Summmary of
open problems
                      For which domains D is Int(DX ) a free WPC extension of D gen-
                      erated by X for every set X? Find an example, if one exists, of a
                      domain D for which this does not hold.




  UCLA NTS 02/25/08                          [45]                       jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Proposition
Introduction
                      Let D be a domain. For any set X, let Intw (DX ) denote the inter-
Prüfer domains

Integer-valued
                      section of every subring of Int(DX ) containing D[X] that is closed
polynomial rings      under pre-composition by every element of Int(D). The domain
Krull domains
and PVMDs
                      Intw (DX ) is the free WPC extension of D generated by X.
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                           [46]                        jesse.elliott@csuci.edu
                    Other universal properties

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      Proposition
Introduction
                      Let D be a domain. For any set X, let Intw (DX ) denote the inter-
Prüfer domains

Integer-valued
                      section of every subring of Int(DX ) containing D[X] that is closed
polynomial rings      under pre-composition by every element of Int(D). The domain
Krull domains
and PVMDs
                      Intw (DX ) is the free WPC extension of D generated by X.
Universal
properties of IVP
rings

Summmary of
                      Proposition
open problems
                      Let D be a domain. Every WPC extension of D is APC iff
                      Int(DX ) = Intw (DX ) for every set X. Both of these conditions
                      hold if the homomorphism θX : T ∈X Int(D) −→ Int(DX ) is sur-
                      jective for every set X.




  UCLA NTS 02/25/08                           [46]                        jesse.elliott@csuci.edu
                    Summary of open problems

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                [47]        jesse.elliott@csuci.edu
                    Summary of open problems

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
                          any domain D and any set X?
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [47]                       jesse.elliott@csuci.edu
                    Summary of open problems

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
                          any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
                          extension of a domain D APC?
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                         [47]                       jesse.elliott@csuci.edu
                    Summary of open problems

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
                          any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
                          extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not flat as a D-module?
Universal
properties of IVP
rings

Summmary of
open problems




  UCLA NTS 02/25/08                          [47]                       jesse.elliott@csuci.edu
                    Summary of open problems

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
                          any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
                          extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not flat as a D-module?
Universal
properties of IVP
                      3   Does there exist a domain D and a set X such that the
rings
                          natural homomorphism T ∈X Int(D) −→ Int(DX ) is not
Summmary of
open problems             injective? is not surjective? is neither injective nor surjective?




  UCLA NTS 02/25/08                            [47]                         jesse.elliott@csuci.edu
                    Summary of open problems

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
                          any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
                          extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not flat as a D-module?
Universal
properties of IVP
                      3   Does there exist a domain D and a set X such that the
rings
                          natural homomorphism T ∈X Int(D) −→ Int(DX ) is not
Summmary of
open problems             injective? is not surjective? is neither injective nor surjective?
                      4   Is dim(Int(D)) ≤ dim(D[X]) for every domain D?




  UCLA NTS 02/25/08                            [47]                         jesse.elliott@csuci.edu
                    Summary of open problems

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott
                      1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
                          any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
                          extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not flat as a D-module?
Universal
properties of IVP
                      3   Does there exist a domain D and a set X such that the
rings
                          natural homomorphism T ∈X Int(D) −→ Int(DX ) is not
Summmary of
open problems             injective? is not surjective? is neither injective nor surjective?
                      4   Is dim(Int(D)) ≤ dim(D[X]) for every domain D?
                      5   Does there exist a UFD or Krull domain D such that
                          dim D[X] > 1 + dim D?




  UCLA NTS 02/25/08                            [47]                         jesse.elliott@csuci.edu
                    References

     Rings of
 Integer-Valued
   Polynomials

  Jesse Elliott

Introduction

Prüfer domains

Integer-valued        P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials,
polynomial rings
                      Mathematical Surveys and Monographs, V. 48, American
Krull domains
and PVMDs             Mathematical Society, 1997.
Universal
properties of IVP
rings                 J. Elliott, Universal properties of integer-valued polynomial rings,
Summmary of
open problems         Journal of Algebra 318 (2007) 68–92.




  UCLA NTS 02/25/08                            [48]                         jesse.elliott@csuci.edu

				
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