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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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