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Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 6, Pages 1183–1205 (Mi izv1153)  

This article is cited in 75 scientific papers (total in 75 papers)

Representable functors, Serre functors, and mutations

A. I. Bondal, M. M. Kapranov

Abstract: This paper studies the categorical version of the concept of mutations of an exceptional set, as used in the theory of vector bundles. The basic object of study is a triangulated category with a family of subcategories satisfying the so-called admissibility condition. A natural notion arising here is that of a Serre functor, effecting a certain duality in the triangulated category.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 35:3, 519–541

Bibliographic databases:

UDC: 513.6
MSC: Primary 14F05, 18E30; Secondary 18F20
Received: 21.06.1988

Citation: A. I. Bondal, M. M. Kapranov, “Representable functors, Serre functors, and mutations”, Izv. Akad. Nauk SSSR Ser. Mat., 53:6 (1989), 1183–1205; Math. USSR-Izv., 35:3 (1990), 519–541

Citation in format AMSBIB
\by A.~I.~Bondal, M.~M.~Kapranov
\paper Representable functors, Serre functors, and mutations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1989
\vol 53
\issue 6
\pages 1183--1205
\jour Math. USSR-Izv.
\yr 1990
\vol 35
\issue 3
\pages 519--541

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    1. M. M. Kapranov, “Mutations and serre functors on constructive bundles”, Funct. Anal. Appl., 24:2 (1990), 155–156  mathnet  crossref  mathscinet  zmath  isi
    2. A. I. Bondal, M. M. Kapranov, “Enhanced triangulated categories”, Math. USSR-Sb., 70:1 (1991), 93–107  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. D. O. Orlov, “Projective bundles, monoidal transformations, and derived categories of coherent sheaves”, Russian Acad. Sci. Izv. Math., 41:1 (1993), 133–141  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. A. I. Bondal, A. E. Polishchuk, “Homological properties of associative algebras: the method of helices”, Russian Acad. Sci. Izv. Math., 42:2 (1994), 219–260  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. S. A. Kuleshov, D. O. Orlov, “Exceptional sheaves on del Pezzo surfaces”, Russian Acad. Sci. Izv. Math., 44:3 (1995), 479–513  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. Eric Zaslow, “Solitons and helices: The search for a math-physics bridge”, Comm Math Phys, 175:2 (1996), 337  crossref  mathscinet  zmath
    7. Valery A Lunts, “Coherent Sheaves on Configuration Schemes”, Journal of Algebra, 244:2 (2001), 379  crossref
    8. D. O. Orlov, “Derived categories of coherent sheaves and equivalences between them”, Russian Math. Surveys, 58:3 (2003), 511–591  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. A. I. Bondal, M. Van den Bergh, “Generators and representability of functors in commutative and noncommutative geometry”, Mosc. Math. J., 3:1 (2003), 1–36  mathnet  crossref  mathscinet  zmath  elib
    10. Sophie Térouanne, “Sur la catégorie pour G réductif fini”, Comptes Rendus Mathematique, 336:6 (2003), 483  crossref
    11. R. V. Bezrukavnikov, D. B. Kaledin, “McKay Equivalence for Symplectic Resolutions of Quotient Singularities”, Proc. Steklov Inst. Math., 246 (2004), 13–33  mathnet  mathscinet  zmath
    12. A. I. Bondal, “A symplectic groupoid of triangular bilinear forms and the braid group”, Izv. Math., 68:4 (2004), 659–708  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. A. L. Gorodentsev, S. A. Kuleshov, “Helix theory”, Mosc. Math. J., 4:2 (2004), 377–440  mathnet  crossref  mathscinet  zmath
    14. A. A. Beilinson, R. V. Bezrukavnikov, I. Mirković, “Tilting exercises”, Mosc. Math. J., 4:3 (2004), 547–557  mathnet  crossref  mathscinet  zmath
    15. Vladimir Drinfeld, “DG quotients of DG categories”, Journal of Algebra, 272:2 (2004), 643  crossref
    16. Meltzer H., “Exceptional Vector Bundles, Tilting Sheaves and Tilting Complexes for Weighted Projective Lines”, Mem. Am. Math. Soc., 171:808 (2004), III+  mathscinet  isi
    17. A. G. Kuznetsov, “Derived Categories of Cubic and $V_{14}$”, Proc. Steklov Inst. Math., 246 (2004), 171–194  mathnet  mathscinet  zmath
    18. Neeman A., “A Survey of Well Generated Triangulated Categories”, Representations of Algebras and Related Topics, Fields Institute Communications, 45, eds. Buchweitz R., Lenzing H., Amer Mathematical Soc, 2005, 307–329  mathscinet  zmath  isi
    19. A. G. Kuznetsov, “Hyperplane sections and derived categories”, Izv. Math., 70:3 (2006), 447–547  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    20. D. O. Orlov, “Triangulated categories of singularities and equivalences between Landau–Ginzburg models”, Sb. Math., 197:12 (2006), 1827–1840  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    21. Rouquier R., “Derived categories and birational geometry [according to Bondal, Orlov, Bridgeland, Kawamata...]”, Asterisque, 2006, no. 307, 283–307  isi  elib
    22. Kawamata Yu., “Derived categories of toric varieties”, Michigan Mathematical Journal, 54:3 (2006), 517–535  crossref  isi  elib
    23. Okada S., “On stability manifolds of Calabi-Yau surfaces”, International Mathematics Research Notices, 2006, 58743  isi  elib
    24. Georg Hein, “Raynaud’s vector bundles and base points of the generalized Theta divisor”, Math Z, 257:3 (2007), 597  crossref  mathscinet  zmath  isi
    25. Calin I Lazaroiu, “Graded D-branes and skew categories”, J High Energy Phys, 2007:8 (2007), 088  crossref  mathscinet
    26. Macri E., “Stability conditions on curves”, Mathematical Research Letters, 14:4 (2007), 657–672  isi
    27. Kuznetsov A., “Homological Projective Duality”, Publications Mathematiques de l'IHES, No 105, Publications Mathematiques de l IHES, 105, Springer-Verlag Berlin, 2007, 157–220  crossref  isi
    28. Osamu Iyama, Yuji Yoshino, “Mutation in triangulated categories and rigid Cohen–Macaulay modules”, Invent math, 172:1 (2008), 117  crossref  mathscinet  zmath  adsnasa  isi
    29. Kuznetsov A., “Derived categories of quadric fibrations and intersections of quadrics”, Advances in Mathematics, 218:5 (2008), 1340–1369  crossref  isi  elib
    30. Kuznetsov A., “Lefschetz decompositions and categorical resolutions of singularities”, Selecta Mathematica-New Series, 13:4 (2008), 661–696  crossref  isi  elib
    31. Donagi R., Pantev T., “Torus fibrations, gerbes, and duality”, Memoirs of the American Mathematical Society, 193:901 (2008), 1–+  isi
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    33. Marcello Bernardara, “A semiorthogonal decomposition for Brauer-Severi schemes”, Math Nachr, 282:10 (2009), 1406  crossref  isi
    34. Kuznetsov, A, “Symplectic structures on moduli spaces of sheaves via the Atiyah class”, Journal of Geometry and Physics, 59:7 (2009), 843  crossref  isi  elib
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    41. D. Kaledin, “Derived Mackey functors”, Mosc. Math. J., 11:4 (2011), 723–803  mathnet  crossref  mathscinet
    42. Kuznetsov A., “Base change for semiorthogonal decompositions”, Compos Math, 147:3 (2011), 852–876  crossref  isi
    43. V. A. Lunts, “Categorical Resolutions, Poset Schemes, and Du Bois Singularities”, International Mathematics Research Notices, 2011  crossref
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    65. Kuznetsov A., “Derived Categories View on Rationality Problems”, Rationality Problems in Algebraic Geometry, Lect. Notes Math., Lecture Notes in Mathematics, 2172, eds. Pardini R., Pirola G., Springer International Publishing Ag, 2016, 67–104  crossref  isi
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    75. Stanley D. van Roosmalen A.-Ch., “T-Structures on Hereditary Categories”, Math. Z., 293:1-2 (2019), 731–766  crossref  isi
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