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Derived categories of sheaves on singular schemes with an application to reconstruction. (English) Zbl 1213.14031

If \(X\) is a (quasi-)projective variety, there are several triangulated categories naturally associated to it: Probably the most common ones are the bounded derived category of coherent sheaves \({\text D}^{\text b}(X)\) and its full triangulated subcategory of perfect complexes \({\text{Perf}}(X)\), which is roughly the smallest triangulated subcategory of \({\text D}^{\text b}(X)\) containing all finite-rank locally free sheaves. These categories coincide if and only if \(X\) is smooth.
In recent years it has become commonplace to investigate the geometry of a smooth projective variety through its bounded derived category of coherent sheaves, so most of the results are formulated, and the proofs usually only work, under the smoothness assumption. In the article under review the author extends two such results to the singular case.
Firstly, A.Bondal and M.van den Bergh proved [Mosc.Math.J.3, No.1, 1–36 (2003; Zbl 1135.18302)] that for a smooth and proper variety \(X\) any covariant or contravariant locally-finite (a certain boundedness condition) cohomological functor from \({\text D}^{\text b}(X)\) to the category of vector spaces is representable. Furthermore, \({\text D}^{\text b}(X)\) is equivalent to either of these categories of functors. The first main result of this paper states that dropping the smoothness one has that \({\text D}^{\text b}(X)\) is equivalent to the category of locally-finite, cohomological functors on \({\text{Perf}}(X)\). The proof uses the machinery of compactly-generated triangulated categories.
Secondly, a result of A.Bondal and D.Orlov [Compos.Math.125, No.3, 327–344 (2001; Zbl 0994.18007)] says that if a smooth and projective variety \(X\) has ample or anti-ample canonical bundle, then any variety \(Y\) satisfying \({\text D}^{\text b}(X)\cong{\text D}^{\text b}(Y)\) is isomorphic to \(X\). The author extends this result to the case where \(X\) is projective Gorenstein using, in particular, a relativization of the notion of a Serre functor.
Besides the above mentioned results the author also proves that two projective schemes have equivalent bounded derived categories if and only if the respective categories of perfect complexes are equivalent. Furthermore, for a projective scheme \(X\) the groups of autoequivalences of \({\text D}^{\text b}(X)\) and of \({\text{Perf}}(X)\) coincide. This result is proved using so-called pseudo-adjoint functors.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
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[2] Bökstedt, M.; Neeman, A., Homotopy limits in triangulated categories, Compos. Math., 86, 2, 209-234 (1993) · Zbl 0802.18008
[3] Bondal, A.; Kapranov, M., Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat., 53, 6, 1183-1205 (1989), 1337
[4] Bondal, A.; Orlov, D., Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math., 125, 3, 327-344 (2001) · Zbl 0994.18007
[5] Bondal, A.; van den Bergh, M., Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J., 3, 1, 1-36 (2003), 258 · Zbl 1135.18302
[7] Christensen, J. D.; Keller, B.; Neeman, A., Failure of Brown representability in derived categories, Topology, 40, 6, 1339-1361 (2001) · Zbl 0997.18007
[8] Hartshorne, R., Residues and Duality, Lecture Notes in Math., vol. 20 (1966), Springer-Verlag: Springer-Verlag Berlin, New York, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne
[9] Huybrechts, D., Fourier-Mukai Transforms in Algebraic Geometry, Oxford Math. Monogr. (2006), The Clarendon Press/Oxford University Press: The Clarendon Press/Oxford University Press Oxford · Zbl 1095.14002
[10] Illusie, L., Existence de résolutions globals, (Théorie des intersections et théorème de Riemann-Roch (1971), Springer-Verlag: Springer-Verlag Berlin), Séminaire de Géométrie Algébrique du Bois-Marie 1966-1967 (SGA 6), pp. xii+700
[12] Krause, H., Smashing subcategories and the telescope conjecture—an algebraic approach, Invent. Math., 139, 1, 99-133 (2000) · Zbl 0937.18013
[13] Lunts, V.; Orlov, D., Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc., 23, 853-908 (2010) · Zbl 1197.14014
[14] Neeman, A., 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, 5, 547-566 (1992) · Zbl 0868.19001
[15] Neeman, A., The Grothendieck duality theorem via Bousfieldʼs techniques and Brown representability, J. Amer. Math. Soc., 9, 1, 205-236 (1996) · Zbl 0864.14008
[16] Orlov, D., Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Algebr. Geom. Metody, Svyazi i Prilozh.. Algebr. Geom. Metody, Svyazi i Prilozh., Tr. Mat. Inst. Steklova, 246, 240-262 (2004) · Zbl 1101.81093
[17] Orlov, D., Triangulated categories of singularities, and equivalences between Landau-Ginzburg models, Mat. Sb., 12, 197, 117-132 (2006) · Zbl 1161.14301
[18] Roberts, P., Homological Invariants of Modules over Commutative Rings, Séminaire de Mathématiques Supérieures, vol. 72 (1980), Presses de lʼUniversité de Montréal: Presses de lʼUniversité de Montréal Montreal, Que. · Zbl 0467.13007
[19] Rouquier, R., Catégories dérivées et géométrie algébrique. Exposés à la semaine « Géométrie algébrique complexe » (2003), Luminy
[20] Rouquier, R., Dimensions of triangulated categories, J. K-Theory, 1, 2, 193-256 (2008) · Zbl 1165.18008
[21] Thomason, R. W.; Trobaugh, T., Higher algebraic \(K\)-theory of schemes and of derived categories, (The Grothendieck Festschrift, vol. III. The Grothendieck Festschrift, vol. III, Progr. Math., vol. 88 (1990), Birkhäuser Boston: Birkhäuser Boston Boston, MA) · Zbl 0731.14001
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