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The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. (English) Zbl 0868.19001

Let \({\mathcal R}\to{\mathcal S}\) be an inclusion of triangulated categories and let \({\mathcal T}\) be the quotient \({\mathcal S}/{\mathcal R}\). For each category \({\mathcal A}\), let \({\mathcal A}^c\) be the subcategory of compact objects \(A\in{\mathcal A}\) such that \(\operatorname{Hom}(A,- )\) respects coproducts. The author shows under certain conditions that there are induced maps \({\mathcal R}^c\to{\mathcal S}^c\to{\mathcal T}^c\) that the map \({\mathcal S}^c/{\mathcal R}^c\to{\mathcal T}^c\) is fully faithful, and that the épaisse closure of the image of the latter map is all of \({\mathcal T}^c\).
Let \(U\to X\) be an open inclusion of schemes satisfying appropriate (quite mild) hypotheses, \({\mathcal S}\) the derived category of quasicoherent sheaves on \(X\), \({\mathcal T}\) the derived category of quasicoherent sheaves on \(U\) and \({\mathcal R}\subset{\mathcal S}\) the full subcategory of complexes with cohomology supported on \(X-U\). Then the theorem quoted above specializes to a theorem of R. W. Thomason and 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 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 \({\mathcal R}\), \({\mathcal S}\) and \({\mathcal T}\), but also the structure of the abelian categories from which \({\mathcal R}\), \({\mathcal S}\) and \({\mathcal T}\), are constructed. By contrast, the paper under review works with the triangulated categories directly.
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.

MSC:

19E08 \(K\)-theory of schemes
18E30 Derived categories, triangulated categories (MSC2010)
19D10 Algebraic \(K\)-theory of spaces
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
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References:

[1] M. BÖKSTEDT and A. NEEMAN , Homotopy Limits in Triangulated Categories , Preprint. · Zbl 0802.18008
[2] A. K. BOUSFIELD , The Localization of Spaces with Respect to Homology , (Topology, vol. 14, 1975 , pp. 133-150). MR 52 #1676 | Zbl 0309.55013 · Zbl 0309.55013
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[8] F. WALDHAUSEN , Algebraic K-Theory of Spaces (Lect. Notes in Math., vol. 1126, 1985 , pp. 318-419). MR 86m:18011 | Zbl 0579.18006 · Zbl 0579.18006
[9] F. WALDHAUSEN , Algebraic K-Theory of Spaces, Localization, and the Chromatic Filtration of Stable Homotopy , in Proceedings of Alg. Top. Conf. in Aarhus, 1982 , SLN 1051, 1984 , pp. 173-195. MR 86c:57016 | Zbl 0562.55002 · Zbl 0562.55002
[10] D. YAO , Higher Algebraic K-Theory of Admissible Abelian Categories and Localization Theorems , Preprint. · Zbl 0746.19006
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