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Zbl 0868.19001
Neeman, Amnon
The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 25, No. 5, 547-566 (1992). ISSN 0012-9593

Let ${\cal R}\to{\cal S}$ be an inclusion of triangulated categories and let ${\cal T}$ be the quotient ${\cal S}/{\cal R}$. For each category ${\cal A}$, let ${\cal A}^c$ be the subcategory of compact objects $A\in{\cal A}$ such that $\Hom(A,- )$ respects coproducts. The author shows under certain conditions that there are induced maps ${\cal R}^c\to{\cal S}^c\to{\cal T}^c$ that the map ${\cal S}^c/{\cal R}^c\to{\cal T}^c$ is fully faithful, and that the épaisse closure of the image of the latter map is all of ${\cal T}^c$. \par Let $U\to X$ be an open inclusion of schemes satisfying appropriate (quite mild) hypotheses, ${\cal S}$ the derived category of quasicoherent sheaves on $X$, ${\cal T}$ the derived category of quasicoherent sheaves on $U$ and ${\cal R}\subset{\cal S}$ the full subcategory of complexes with cohomology supported on $X-U$. Then the theorem quoted above specializes to a theorem of {\it R. W. Thomason} and {\it T. F. Trobaugh} [C. R. Acad. Sci., Paris, Sér. I 307, No. 16, 829-831 (1988; Zbl 0697.18004); cf. also Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)], which was earlier generalized by {\it D. Yao} [J. Pure Appl. Algebra 77, No. 3, 263-339 (1992; Zbl 0746.19006)]. The Thomason-Trobaugh proof uses not just the triangulated structure of the categories ${\cal R}$, ${\cal S}$ and ${\cal T}$, but also the structure of the abelian categories from which ${\cal R}$, ${\cal S}$ and ${\cal T}$, are constructed. By contrast, the paper under review works with the triangulated categories directly. \par The body of the paper is quite readable as it stands. The appendix, giving an application to $K$-theory, requires familiarity with notation developed in the author's earlier papers.
[S.E.Landsburg (MR 93k:18015)]

MSC 2000:: *19E08 K-theory of schemes
18E30 Derived categories, etc.
19D10 Algebraic K-theory of spaces
14C35 Appl. of methods of algebraic K-theory

Keywords: localization theorem for $K$-theory; triangulated categories; épaisse closure; derived category of quasicoherent sheaves; abelian categories

Citations: Zbl 0697.18004; Zbl 0731.14001; Zbl 0746.19006

Cited in: Zbl 1189.19003 Zbl 1181.18008 Zbl 1120.19003

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