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Universal cycle classes. (English) Zbl 0538.14009
The objective of the paper is to prove the following theorem: For each positive integer \(p\geq 1\), there exists a smooth simplicial scheme \(BL^ p_.\), with a smooth, closed subsimplicial scheme \(Z^ p_.\) of codimension p in each degree, having the property that if X is any noetherian scheme and \(Y\subset X\) any codimension p subscheme locally a complete intersection in X, then there exists an open cover \(\{U_{\alpha}\}\) of X and a map of simplicial schemes \(\chi_ Y: N_.\{U_{\alpha}\}\to BL^ p_.\) such that \(\chi_ Y^{-1}(Z^ p_.)=N_.\{U_{\alpha}\cap Y\}\subset N_.\{U_{\alpha}\}.\) Furthermore the subscheme \(Z^ p_.\) has cycle classes in three cohomology theories: The K-theoretic version of the Chow ring, étale cohomology and crystalline cohomology, which one may regard as universal cycle classes for local complete intersections. - The primary motivation for proving these results is to improve the understanding of intersection theory on singular varieties and schemes.
Reviewer: A.Conte

MSC:
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
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