e 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.