Generic injectivity for étale cohomology and pretheories. (Q1812162)

The main result of this paper is the following: Theorem 1. Let \(W\) be a connected smooth semi-local scheme over a field \(k\) and let \(\eta\) be its generic point. Let \(X\to W\) be a proper smooth morphism, \(n\) an integer prime to \(\text{char}(k)\) and \({\mathcal K}^ \bullet\) a complex of sheaves of \({\mathbb Z}/n{\mathbb Z}\) full-modules on \(X_{\text{ét}}\) whose cohomology sheaves are locally constant constructible and bounded below. Then for every \(q\in \mathbb Z\) the canonical map \(H^q_{\text{ét}}(X,{\mathcal K}^ \bullet )\to H^q_{\text{ét}}(X_\eta,{\mathcal K}^ \bullet )\) is a universal monomorphism. Some special cases of this theorem are known in the literature (cf. the paper under review).