Etale cohomological dimension and the topology of algebraic varieties

The purpose of this paper is twofold: to develop a theory of etale cohomological dimension in the context of schemes of finite type over a separably closed field that would be analogous to the well-known theory of quasicoherent cohomological dimension, and to apply our theory to prove new results about the topology of algebraic varieties of small codimension in n-space. The cohomological dimension of a scheme X relative to a closed subscheme Y, denoted by cd(X, Y), is the largest integer r such that the local cohomology group H' (X, F) $& 0 for some abelian torsion sheaf F on Xet, where F consists only of torsion prime to all the residual characteristics of X. The gist of our theory is a technique for proving various upper bounds on cd(X, Y), especially in the case where Y has small codimension in X. Local bounds are the most important as well as the easiest to state. Accordingly, for the purposes of this introduction, let X = Spec A, where A is a strictly Henselian local ring of a finite-type scheme over a separably closed field. Just what kind of bounds on cd(X, Y) one should expect is indicated by the theory of quasicoherent cohomological dimension, which has been developed by several authors; see, for example, [Fa], [Grl], [Hal], [Ha2], [HaSp], [HuLy], [01] and [PesSz]. The quasicoherent cohomological dimension of X relative to Y, denoted by qccd(X, Y), is the largest integer r such that there exists a quasicoherent sheaf F on Xzar with H'(X, F) 54 0. In the above-stated local case, the main results of the theory of quasicoherent cohomological dimension are the following (where n = dim X): (i) qccd(X, Y) < n (see [Gri], 1.12). (ii) qccd(X, Y) < t if Y is set-theoretically defined by t equations (see [Gri], 2.3).