Etale cohomological dimension and the topology of algebraic varieties (Q1803522)

The purpose of this paper is twofold: to develop a theory of étale cohomological dimension that would be analogous to the well-known quasicoherent theory and to apply it to prove new results on the topology of algebraic varieties of small codimension in \(n\)-space. The gist of the theory is a technique for proving various upper bounds for étale cohomological dimension. It is proven, for example, that if \(A\) is an \(n\)-dimensional strictly Henselian local ring of a non-singular variety and \(Y\) is an irreducible Zariski closed subset of \(X= \text{Spec } A\) of codimension \(\leq c\), then \(\text{écd} (\text{Spec } A-Y)\leq 2n-2- [(n-1)/c]\) and if, in addition, the punctured spectrum of \(Y\) is analytically irreducible, then \(\text{écd} (\text{Spec } A-Y)\leq 2n-1- [n/(c+ 1)]- [(n-1)/ (c+1)]\). -- Upper bounds on étale cohomological dimension yield new results for the topology of varieties of small codimension in \(n\)-space. For example, the above two estimates imply that if \(V\) is an irreducible subvariety of codimension \(\leq c\) of complex projective \(n\)-space \(\mathbb{P}^ n_ \mathbb{C}\), then the higher homotopy groups \(\pi_ i (\mathbb{P}^ n_ \mathbb{C}, V)=0\) for \(1\leq [(n-1)/c]\) and this estimate is sharp. If, in addition, \(V\) is analytically irreducible, then \(\pi_ i( \mathbb{P}^ n_ \mathbb{C}, V)=0\) for \(1\leq [(n+1)/ (c+1)]+ [n/ (c+1)]-1\). In addition, there are consequences for the Picard group of \(V\subset \mathbb{P}_ k^ n\), where \(k\) is any algebraically closed field. For example, if \(V\) is normal of codimension \(\leq n/2-1\), then \(\text{Pic } V\) is generated by \({\mathcal O}_ V(1)\) (up to \(p\)-torsion, if the characteristic of the ground field is \(p>0\)).