Thm B for hens sch in pos char

This post is a follow-up on this post. There we gave an example of a henselian affine scheme which does not satisfy theorem B.

In this post p will be a prime number and we will show that for an affine henselian scheme in characteristic p we do have theorem B. In fact, it turns out this is almost an immediate consequence of Gabber’s affine analogue of proper base change. I’m a little embarrassed that I didn’t see it earlier. The trick is to use the following trivial lemma which will be added to the Stacks project soonish.

Lemma. Let X be an affine scheme. Let F be an abelian sheaf on the small etale site X_{et} of X. If H^i_{et}(U, F) = 0 for all i > 0 and for every affine object U of X_{et}, then H^i_{Zar}(X, F) = 0 for all i > 0.

Proof. Namely, let U = U_1 ∪ … ∪ U_n be an affine open covering of an affine open U of X. Then all the finite intersections of the U_i are affine too. Hence the Cech to cohomology spectral sequence for F in the etale topology degenerates and we see that the Cech complex is exact in degrees > 0. But by a well know criterion this implies vanishing of higher cohomology groups of F on X_{Zar}. See Tag 01EW. □

OK, now suppose that X = Spec(A) and that A is one half of a henselian pair (A, I) with p = 0 in A. Let Z = Spec(A/I) and denote i : Z → X the inclusion morphism. The corresponding henselian scheme is gotten by taking the underlying topological space of Z and endowing this with a structure sheaf O^h obtained by a process of “henselization” on affine opens. We prefer to do this as described below (it gives the same thing).

For any quasi-coherent module F on X, viewed as a sheaf of O_X-modules on the small etale site of X, we set F^h = (i_{et}^{-1}F)|_{Z_{Zar}}. This is a sheaf of modules over the structure sheaf O^h = (O_X)^h of our henselian affine scheme, in other words on the underlying topological space of Z.

Theorem B. Let (A, I) be a henselian pair with p = 0 in A for some prime number p. Let (Z, O^h) be the henselian affine scheme associated with the pair. Then H^i(Z, F^h) = 0 for i > 0 and any O^h module F^h coming from a quasi-coherent module on Spec(A) as in the construction above.

Proof. We will show that the lemma applies to i_{et}^{-1}F on Z_{et} which will prove that H^i(Z, F^h) = 0 for i > 0 and this will finish the proof of Theorem B. For any affine object V in the site Z_{et} we can find an affine object U in X_{et} such that V = Z xX U. Denote U’ the henselization of U along the inverse image of Z. Denote F’ the pullback of F to U’. Then we see that the restriction of i_{et}^{-1}F to V_{et} is just the pullback of F’ to V by the closed immersion V → U’ (pullback in the etale topology as before). Hence by Gabber’s result (Tag 09ZI) we see that H^i_{et}(V, i_{et}^{-1}F) = H^i_{et}(U’, F’) = 0 because U’ is affine and F’ is quasi-coherent. We may use Gabber’s theorem exactly because p = 0 in A and hence F’ is a torsion sheaf! □


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