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Universal exactness in algebraic K-theory. (English) Zbl 0558.18007
An exact sequence of abelian groups is defined to be ’universally exact’ if the application to it of any additive functor, commuting with filtering direct limits, again yields an exact sequence. Refining some arguments of D. Quillen [Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)], the author proves a result which shows that many exact sequences occurring in algebraic K-theory are universally exact.
This is true of the Gersten resolution so that one obtains, for example, Theorem: \(A^ q(X)\otimes {\mathbb{Z}}/m\cong H^ q_{Zar}(X;{\mathcal K}_ q\otimes {\mathbb{Z}}/m)\) when X is a nonsingular algebraic variety and \(A^ q\) denotes codimension q cycles modulo linear equivalence.
Reviewer: V.P.Snaith

MSC:
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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