The Étale Cohomology of p-Torsion Sheaves, II

A formula due to Grothendieck, Ogg, and Shafarevich gives the EulerPoincaré characteristic of a constructible sheaf of Fl-modules on a smooth, proper curve over an algebraically closed field k of characteristic p > 0, as a sum of a global term and local terms, where l 6= p. A previously known result removes the restriction on l in the case of p-torsion sheaves trivialized by p-extensions. The author conjectured a p-torsion analogue of the formula in an earlier paper and proved it there for the case of a constructible sheaf of Fp-modules when the generic stalk has rank p. A general proof of the conjecture is given in the presence of the Étale Core Hypothesis (ECH). The étale core of a suitable finite, flat, height 1, commutative, p-torsion group scheme is the subsheaf fixed by the pth power endomorphism of the tangent space at the identity and it is an étale group scheme. Étale cohomology with coefficients in an étale core is relatively easy to compute and can be used to compute the cohomology with coefficients in a p-torsion constructible sheaf. Results on surfaces and the Fq-vector schemes of Raynaud are included, where q = p, r ≥ 1. In the appendix, we establish conditions under which an étale core can be found via a finite étale extension of the finite étale group scheme corresponding to the generic stalk of a p-torsion sheaf. Introduction The étale cohomology of a scheme of characteristic p > 0 with coefficients in a p-torsion constructible sheaf is poorly understood in contrast ∗ Research partially supported by NSF Grant No. RII-8501567