Construction of 𝑡-structures and equivalences of derived categories

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

doi:10.1090/S0002-9947-03-03261-6

Construction of $t$-structures and equivalences of derived categories
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by Leovigildo Alonso Tarrío, Ana Jeremías López and María José Souto Salorio PDF

Trans. Amer. Math. Soc. 355 (2003), 2523-2543 Request permission

Abstract:

We associate a $t$-structure to a family of objects in $\boldsymbol {\mathsf {D}}(\mathcal {A})$, the derived category of a Grothendieck category $\mathcal {A}$. Using general results on $t$-structures, we give a new proof of Rickard’s theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Beĭlinson’s equivalences.

References

Leovigildo Alonso Tarrío, Ana Jeremías López, and Joseph Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 1, 1–39. MR 1422312, DOI 10.1016/S0012-9593(97)89914-4
Leovigildo Alonso Tarrío, Ana Jeremías López, and María José Souto Salorio, Localization in categories of complexes and unbounded resolutions, Canad. J. Math. 52 (2000), no. 2, 225–247. MR 1755776, DOI 10.4153/CJM-2000-010-4
Beĭlinson, A. A.: Coherent sheaves on $\mathbf {P}^{n}$ and problems in linear algebra. Functional Anal. Appl. 12 (1978), no. 3, 214–216 (1979).
A. A. Beĭlinson, The derived category of coherent sheaves on $\textbf {P}^n$, Selecta Math. Soviet. 3 (1983/84), no. 3, 233–237. Selected translations. MR 863137
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Marcel Bökstedt and Amnon Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), no. 2, 209–234. MR 1214458
Nicolae Popesco and Pierre Gabriel, Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes, C. R. Acad. Sci. Paris 258 (1964), 4188–4190 (French). MR 166241
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Grothendieck, A.; Dieudonné, J. A.: Eléments de Géométrie Algébrique I, Grundlehren der math. Wiss. 166, Springer–Verlag, Heidelberg, 1971.
A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
Bernhard Keller and Dieter Vossieck, Sous les catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 6, 225–228 (French, with English summary). MR 907948
B. Keller and D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Sér. A 40 (1988), no. 2, 239–253. Deuxième Contact Franco-Belge en Algèbre (Faulx-les-Tombes, 1987). MR 976638
Illusie, L.: Existence de résolutions globales, in Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), Lecture Notes in Math., no. 225, Springer-Verlag, New York, 1971, pp.160-222
Azriel Lévy, Basic set theory, Springer-Verlag, Berlin-New York, 1979. MR 533962, DOI 10.1007/978-3-662-02308-2
Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 0202787
Amnon Neeman, 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), no. 5, 547–566. MR 1191736, DOI 10.24033/asens.1659
Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953, DOI 10.1007/978-3-642-66066-5
R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI 10.1007/978-0-8176-4576-2_{1}0
Verdier, J.-L.: Categories dérivées. Quelques résultats (Etat 0). Semin. Geom. algebr. Bois-Marie, SGA 4$\frac {1}{2}$, Lect. Notes Math. 569, 262-311 (1977).