Fix a field k. Let X be a variety. Let \u03b5 : X_* —> X be a proper hypercovering (see earlier post). Let \u039b be a finite ring. Then we have H^*(X, \u039b) = H^*(X_*, \u039b) for H^* = etale cohomology. This follows from the proper base change theorem combined with the case where X is a (geometric) point, in which case it follows from the fact that there is a section X —> X_*. (This is a very rough explanation.)<\/p>\n
Now let k be a perfect field of characteristic p > 0. For a smooth proper variety X over k denote H^*(X\/W) the crystalline cohomology of X with W coefficients. So for example H^0(Spec(k)\/W) = W. Similarly H^*(X\/k) is crystalline cohomology with coefficients in k, so that H^*(X\/k) = H^*(X, \u03a9_{X\/k}).<\/p>\n
It turns out that crystalline cohomology does not satisfy descent for proper hypercoverings. What I mean is this: If X_* —> X is a proper hypercovering and all the varieties X, X_n are smooth and projective, then it is not the case that H^*(X\/W) = H^*(X_*\/W). (Note: It takes some work to even define H^*(X_*\/W)……….) Descent fails for two reasons.<\/p>\n
Nobuo Tsuzuki has shown that descent for proper hypercoverings does<\/strong> hold for rigid cohomology, and hence for H^*(X\/W)[1\/p] in the situation above. The point of this post is to consider coefficients where p is not invertible.<\/p>\n The first is purely inseparable morphisms. Namely, if X_0 —> X is a finite universal homeomorphism, then the constant simplicial scheme X_* with X_n = X_0 is a proper hypercovering of X. In particular, if k is a perfect field of characteristic p,\u00a0 X_* is the constant simplicial scheme with value P<\/strong>^1_k, X = P<\/strong>^1_k and the augmentation is given by \u03c0 : X_0 —> X which raises the coordinate on P<\/strong>^1 to the pth power. Hence, if H^* is a cohomology theory which satisfies descent for proper hypercoverings, then \u03c0^* : H^*(P<\/strong>^1) —> H^*(P<\/strong>^1) should be an isomorphism, which isn’t the case for crystalline cohomology because \u03c0^* : H^2(P<\/strong>^1) —> H^2(P<\/strong>^1) is multiplication by p which is not an automorphism of W.<\/p>\n Let X_* —> X be a proper hypercovering (of smooth proper varieties over k) with the following additional property: For every separably algebraically closed field K\/k the map of simplicial sets X_*(K) —> X(K) is an equivalence (not sure what the correct language is here and I am too lazy to look it up; what I mean is that it is a hypercovering in the category of sets with the canonical topology). This does not hold for our example above because the generic point of X does not lift to X_0 even after any separable extension. It turns out that crystalline cohomology does not satisfy descent for such proper hypercoverings either.<\/p>\n To construct an example, note that if the desent is true with W coefficients, then it is true with k-coefficients because R\u0393(X\/k) = R\u0393(X\/W) \\otimes_W k (again details on general theory have to be filled in here, but this is just a blog…….). OK, now go back to the hypercovering I described in this post<\/a>. Namely, assume char(k) = 2, let X = P<\/strong>^1, let X_0 —> X be an Artin-Schreier covering ramified only above infinity, let X_1 the disjoint union of 2 copies of X_0, let X_2 be the disjoint union of 4 copies of X_0 plus 4 extra points over infinity, and so on. Then, I claim, you get one of these proper hypercoverings described above. I claim that H^1(X_*\/k) is not zero which will prove that the descent for crystalline cohomology fails. To see this note that there is a spectral sequence H^p(X_q\/k) => H^{p + q}(X_*\/k). Look at the term H^1(X_0\/k). This has dimension 2g where g is the genus of X_0. The kernel of the differential to H^1(X_0\/k) —> H^1(X_1\/k) is the subspace of invariants under the involution on X_0. Hence it has dimension at least g (because of the structure of actions of groups of order 2 on vector spaces in characteristic 2). The next differential maps into a subquotient of H^0(X_2\/k). But since I needed only to add 4 points to construct my proper hypercovering, it follows that dim H^0(X_2\/k)\u00a0\u2264 8. Hence we see that dim H^1(X_*\/k) is at least g – 8. As we can make the genus of a Artin-Schreyer covering arbitrarily large, we find that this is nonzero in general.<\/p>\n This is exactly the kind of negative result nobody would ever put in an article. I think in stead of a journal publishing papers that were rejected by journals, it might be fun to have a place where we collect arguments that do not work, or even just things that aren’t true. What do you think?<\/p>\n","protected":false},"excerpt":{"rendered":" Fix a field k. Let X be a variety. Let \u03b5 : X_* —> X be a proper hypercovering (see earlier post). Let \u039b be a finite ring. Then we have H^*(X, \u039b) = H^*(X_*, \u039b) for H^* = etale … Continue reading