Generic Injectivity for Etale Cohomology and Pretheories, by Alexander Schmidt and Kirill Zainoulline

This is a comprehensively revised version of the second named author's preprint "On rational injectivity for pretheories. Relative case." (no. 501 on this server). We prove a generic injectivity result for etale cohomology which can be formulated as follows.

Theorem: Let W be a connected smooth semi-local scheme over a field k, X --> W a smooth and proper morphism, n an integer prime to char(k) and K a complex of etale sheaves of Z/nZ-modules on X such that the cohomology sheaves are locally constant constructible and bounded below. If w denotes the generic point of W, then for all integers q the natural restriction map of etale hypercohomology groups

H^q_et(X, K) --> H^q_et (X_w, K)

is a (universal) monomorphism.

The result applies to any "extensible pretheory". The proof extends the techniques of section 4 of V. Voevodsky's paper "Cohomological theory of presheaves with transfers" to the relative case.

Alexander Schmidt <>
Kirill Zainoulline <>