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A personal note on smooth base change theorem in tale cohomology

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  A personal note on smooth base change theorem in ´tale cohomology
                                          by Jinhyun Park
                                          January 31, 2004

                       1. Proper-smooth base change theorem
   In this section, all varieties are over an algebraically closed field k. Consider the following
diagram with F a sheaf on X:

                                      f
                                    −−
                                  X −−→ X
                                       
                              π
                                       π           X = X ×S T.
                                      f
                               −−
                           T −−→ S
  We then have a natural morphism
                                                         ∗
                                   f ∗ Rr π∗ F → Rr π∗ (f F).
Recall that the proper base change theorem says that it is an isomorphism if π is proper and
F is a torsion sheaf. Also if F is constructible and π is proper, then, Rr π∗ F is constructible
as well. We have similar results:
Theorem 1.1 (smooth base change theorem). In f is smooth (π does not have to be
proper), then above base change morphism is an isomorphism for all constructible sheaf F
whose torsion is prime to the characteristic of k.
   Recall that being locally costant was bad under (higher) direct images, and that was the
reason why we extended the class a little bit, so that we worked with constructible sheaves
to prove the proper base change theorem. But, if we add some more conditions, then, being
locally constant is also preserved under (higher) direct image:
Theorem 1.2 (proper-smooth base change theorem). If π is proper and smooth and F is
locally constant with finite stalks, then, Rr π∗ F is locally constant with finite stalks, provided
that the torsion of F is prime to the characteristic of k.
   Let x0 , x1 ∈ X with x0 being a specialization of x1 . Choose geometric points x0 and x1 .
                                                                                   ¯      ¯
        e                            ¯                                   e
For an ´tale neighborhood (U, u) of x0 , we can give the structure of an ´tale neighborhood
of x1 because the image of U → X contains x1 , being a generalization of x0 . Hence for each
   ¯
e
´tale neighborhood of x0 , we have one for x1 . (But, the converse is not always possible.)
Hence by the definition of stalks, we have a cospecialization map
                                            Fx0 → Fx1
                                             ¯     ¯

for F ∈ Sh(Xet ).
Example 1.3. If F = Ga , then, we have a map on the strictly local rings OX,x0 → OX,x1 .
                                                                            ¯       ¯

  Obviously the following one is true.
Proposition 1.4 (criterion for local constancy). A constructible sheaf F on X is locally
constant iff all cospecialization maps Fx0 → Fx1 are isomorphisms.
                                       ¯     ¯

Example 1.5 (topological case). Let F be a locally constant sheaf on the punctured disk
U , with j : U → X. F corresponds to a module M with an action of π1 (X, u), u ∈ U . The
stalk of j∗ F at u is M , but the stalk at 0 ∈ X is M π1 (U,u) . Hence they are isomorphic to
each other iff π1 acts trivially on M iff j∗ F is constant.
                                                 1
2

  Using the language of cospecialization map, we can reformulate the proper-smooth base
change theorem in the following fashion:
Theorem 1.6 (2nd version of proper-smooth base change theorem). Let π : X → S be a
proper and smooth morphism, F be locally constant with finite stalks with torsion prime to
                                                           ¯ ¯
the characteristic of k. Then for any geometric points s0 , s1 with s0 a specialization of s1 ,
the cospecialization map
                                 H r (Xs¯ , F) → H r (Xs¯ , F)
                                        0               1

is an isomorphism where Xs = X ×S s is the geometric fibre of π over s.
                            ¯          ¯

                                         2. Application
Definition 2.1. Let X be a complete nonsingular variety over an algebraically closed field
k of characteristic p = 0. We say that X can be lifted to characteristic 0 if
    (1) there is a DVR (R, m) with K = FracR of characteristic 0 and R/m = k.
    (2) there is a scheme π : X → S = SpecR which is proper and smooth, whose special
        fibre is X → Speck.
Example 2.2. A variety X ⊂ Pn can be lifted to characteristic 0 if there are homogeneous
                                       k
polynomials fi (T0 , · · · , Tn ) ∈ R[T0 , · · · , Tn ] such that
        ¯
    (1) fi ∈ k[T0 , · · · , Tn ] generate I(X) in k[T0 , · · · , Tn ]
    (2) fi ∈ R[T0 , · · · , Tn ] ⊂ K[T0 , · · · , Tn ] define a variety X1 ⊂ Pn with dim X1 = dim X.
                                                                             K

Example 2.3.        (1) Any nonsingular hypersurface in Pn can be lifted to characteristic
                                                           k
      0: Just lift the single definining polynomial of X in k[T0 , · · · , Tn ] to R[T0 , · · · , Tn ].
  (2) Similarly, any smooth complete intersection in Pn can be lifted to characteristic 0.
                                                        k
  (3) Curves and abelian varieties can be lifted to characteristic 0.
Remark. Common obstruction to lifting is the following situation: Let X be of dimension
r in Pn , but I(X) requires s > r generators. Then, when the generators of I(X) are lifted
      k
to R[T0 , · · · , Tn ], then, the variety X1 manytimes has dimension less than that of X, i.e.
                                n − s ≤ dim X1 ≤ dim X = n − r.
    We have the following consequence of the proper-smooth base change theorem:
Theorem 2.4. Suppose a variety X0 over an algebraically closed field of characteristic
p = 0 can be lifted to a variety X1 over K with char 0. Then, for any finite abelian group
Λ,
                                 H r (X0 , Λ) H r (X1 K , Λ).
                                                      ¯
In particular, the betti numbers of X0 are equal to the betti numbers of X1 .
Proof. A simple application of the proper-smooth base change theorem, applied to the
closed point and the generic point of a DVR R.

				
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