Alexander Schmidt, Kirill Zainoulline: Generic Injectivity for Étale Cohomology and Pretheories

Alexander Schmidt, Kirill Zainoulline:

Generic Injectivity for Étale Cohomology and Pretheories

       Authors: Alexander Schmidt, Kirill Zainoulline
       Title: Generic Injectivity for Étale Cohomology and Pretheories
       Year: 2002
       In: J. of Algebra 263, 215-227 (2003)

dvi file gen.dvi (needs xy-pic), zipped ps-file gen.ps.zip, pdf-file gen.pdf

	Abstract: 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. This file differs in two points from the published version. First of all, it doesn't contain the typing error in theorem 1 (full-modules). Secondly, we added the forgotten condition AB5 in the definition of `universal monomorphism' on page 1. (This inaccuracy is already contained in [CHK].) I am grateful to C. Scheiderer who pointed out this mistake to me.

Abstract: 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.

This file differs in two points from the published version. First of all, it doesn't contain the typing error in theorem 1 (full-modules). Secondly, we added the forgotten condition AB5 in the definition of `universal monomorphism' on page 1. (This inaccuracy is already contained in [CHK].) I am grateful to C. Scheiderer who pointed out this mistake to me.

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