Localization in Categories of Complexes and Unbounded Resolutions

Tarrío, Leovigildo Alonso ; López, Ana Jeremías ; Salorio, María José Souto

Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 225-247

Résumé

In this paper we show that for a Grothendieck category $\mathcal{A}$ and a complex $E$ in $\mathbf{C}(\mathcal{A})$ there is an associated localization endofunctor $\ell$ in $\mathbf{D}(\mathcal{A})$ . This means that $\ell$ is idempotent (in a natural way) and that the objects that go to 0 by $\ell$ are those of the smallest localizing (= triangulated and stable for coproducts) subcategory of $\mathbf{D}(\mathcal{A})$ that contains $E$ . As applications, we construct $\text{K}$ -injective resolutions for complexes of objects of $\mathcal{A}$ and derive Brown representability for $\mathbf{D}(\mathcal{A})$ from the known result for $\mathbf{D}(R-\mathbf{mod})$ , where $R$ is a ring with unit.

Détail

DOI : 10.4153/CJM-2000-010-4

Mots-clés : 18E30, 18E15, 18E35

Bibliographie

[AJL] [AJL] Tarrío, Leovigildo Alonso, López, Ana Jeremías and Lipman, Joseph, Local homology and cohomology on schemes. Ann. Sci. École Norm. Sup. (4) 30(1997), 1–39. Google Scholar

[BBD] [BBD] Beĭlinson, A. A., Bernstein, J. and Deligne, P., Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171; Astérisque 100, Soc. Math. France, Paris, 1982. Google Scholar

[BN] [BN] Bökstedt, Marcel and Neeman, Amnon, Homotopy limits in triangulated categories. Compositio Math. (2) 86(1993), 209–234. Google Scholar

[Bo1] [Bo1] Bousfield, A. K., The localization of spectra with respect to homology. Topology (4) 18(1979), 257–281. Google Scholar

[Bo2] [Bo2] Bousfield, A. K., The Boolean algebra of spectra. Comment.Math. Helv. (3) 54(1979), 368–377. Google Scholar

[Bo2a] [Bo2a] Bousfield, A. K., Correction to: “The Boolean algebra of spectra”. Comment.Math. Helv. (4) 58(1983), 599–600. Google Scholar

[BK] [BK] Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations. LectureNotes in Math. 304, Springer-Verlag, Berlin-New York, 1972. Google Scholar

[CtHK] [CtHK] Colliot-Thélène, Jean-Louis, Hoobler, Raymond T. and Kahn, Bruno, The Bloch-Ogus-Gabber theorem. In: Algebraic K-theory (Toronto, ON, 1996), Amer. Math. Soc., Providence, RI, 1998, 31–94. Google Scholar

[F] [F] Franke, J., On the Brown representability theorem for triangulated categories. Preprint. Google Scholar

[GP] [GP] Popesco, Nicolae and Gabriel, Pierre, Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes. C. R. Acad. Sci. Paris 258(1964), 4188–4190. Google Scholar

[I] [I] Iversen, Birger, Cohomology of sheaves. Universitext. Springer-Verlag, Berlin-New York, 1986. Google Scholar

[K] [K] Keller, Bernhard, Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1994), 63–102. Google Scholar

[L] [L] Lévy, Azriel, Basic set theory. Springer-Verlag, Berlin-New York, 1979. Google Scholar

[N1] [N1] Neeman, Amnon, The connection between the K-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. (4) 25(1992), 547–566. Google Scholar

[N2] [N2] Neeman, Amnon, The chromatic tower for D(R). Topology (3) 31(1992), 519–532. Google Scholar

[N3] [N3] Neeman, Amnon, The Grothendieck duality theoremvia Bousfield's techniques and Brown representability. J. Amer. Math. Soc. (1) 9(1996), 205–236. Google Scholar

[N4] [N4] Neeman, Amnon, Triangulated categories. Book in preparation. Google Scholar

[P] [P] Puppe, Dieter, On the formal structure of stable homotopy theory. Colloq. algebr. Topology, Aarhus, 1962, 65–71. Google Scholar

[Ri] [Ri] Rickard, Jeremy, Derived categories and stable equivalence. J. Pure Appl. Algebra (3) 61(1989), 303–317. Google Scholar

[Sp] [Sp] Spaltenstein, N., Resolutions of unbounded complexes. Compositio Math. (2) 65(1988), 121–154. Google Scholar

[St] [St] Stenström, Bo, Rings of quotients. An introduction to methods of ring theory. GrundlehrenMath. Wiss. 217. Springer-Verlag, Berlin-Heidelberg-New York, 1975. Google Scholar

[Ve] [Ve] Verdier, Jean-Louis, Catégories dérivées. Quelques résultats (Etat 0). Sémin. Géom. algébr. Bois-Marie, SGA 4½, LectureNotes in Math. 569(1977), 262–311. Google Scholar

[W] [W] Weibel, Charles A., An introduction to homological algebra. Cambridge Stud. Adv. Math. 38. Cambridge University Press, Cambridge, 1994. Google Scholar