Previous Up Next Article Citations From References: 8 From Reviews: 0
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@MISC{Walker_previousup,
author = {Daniel R. (-il) Walker and Mark E. (-ne},
title = {Previous Up Next Article Citations From References: 8 From Reviews: 0},
year = {}
}
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Abstract
Geometric models for algebraic K-theory. (English summary) Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday, Part IV. K-Theory 20 (2000), no. 4, 311–330. In the paper the authors construct the K-theory space K d (X, Y) using the full subcategory P d (X, Y) of the category of coherent sheaves on X × Y, which consists of those coherent sheaves F that are flat over X and such that the support of F maps properly to X with fibers of dimension at most d. They further consider the space K d (X × ∆ · , Y) which is the geometric realization of the simplicial space n ← K d (X × ∆ n, Y). The space K d (X × ∆., Y) is shown to be homotopy equivalent to the homotopy-theoretic group completion of the Γ-space [see G. Segal, Topology 13 (1974), 293–312; MR0353298 (50 #5782)] n ← Hom(X × ∆ n, KY,d), where KY,d is a Γ-object in the category of ind-schemes. The construction is given in Section 2. The main theorem of the paper is the following. Theorem 2.3: For any quasi-projective varieties X and Y, there is a natural weak equivalence of spaces Ω|BHom(X × ∆., KY,d) | ← K d (X × ∆., Y). Here Ω|BHom(X × ∆., KY,d) | stands for the “homotopy-theoretic ” group completion of the H-space Hom(X × ∆., KY,d) [see op. cit.]. In Theorem 3.3 the authors show that this equivalence is in fact a weak equivalence of infinite loop spaces. Reviewed by Piotr Krasoń
Keyphrases
previous next article citation coherent sheaf homotopy-theoretic group completion geometric realization full subcategory k-theory space infinite loop space piotr kraso space bhom geometric model h-space hom main theorem quasi-projective variety weak equivalence english summary algebraic k-theory daniel quillen sixtieth birthday natural weak equivalence special issue part iv simplicial space
