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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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