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Izv. RAN. Ser. Mat., 1993, Volume 57, Issue 2, Pages 3–50 (Mi izv877)  

This article is cited in 34 scientific papers (total in 34 papers)

Homological properties of associative algebras: the method of helices

A. I. Bondal, A. E. Polishchuk


Abstract: Homological properties of associative algebras arising in the theory of helices are studied. A class of noncommutative algebras is introduced in which it is natural (from the viewpoint of the theory of helices) to deform projective spaces and also certain Fano varieties. It is shown that in the case of deformations of the projective plane this approach leads to algebras associated with automorphisms of two-dimensional cubic curves.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1994, 42:2, 219–260

Bibliographic databases:

UDC: 513.6
MSC: 14H45, 14F05, 18F20, 18E30
Received: 24.03.1992

Citation: A. I. Bondal, A. E. Polishchuk, “Homological properties of associative algebras: the method of helices”, Izv. RAN. Ser. Mat., 57:2 (1993), 3–50; Russian Acad. Sci. Izv. Math., 42:2 (1994), 219–260

Citation in format AMSBIB
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\by A.~I.~Bondal, A.~E.~Polishchuk
\paper Homological properties of associative algebras: the method of helices
\jour Izv. RAN. Ser. Mat.
\yr 1993
\vol 57
\issue 2
\pages 3--50
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\transl
\jour Russian Acad. Sci. Izv. Math.
\yr 1994
\vol 42
\issue 2
\pages 219--260
\crossref{https://doi.org/10.1070/IM1994v042n02ABEH001536}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. E. Polishchuk, “I. N. Bernstein–I. M. Gel'fand–S. I. Gel'fand equivalence for triangulated categories generated by helixes”, Russian Acad. Sci. Izv. Math., 43:1 (1994), 127–140  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. L. Gorodentsev, S. A. Kuleshov, “Helix theory”, Mosc. Math. J., 4:2 (2004), 377–440  mathnet  mathscinet  zmath
    3. Polishchuk A., “Noncommutative two-tori with real multiplication as noncommutative projective varieties”, Journal of Geometry and Physics, 50:1–4 (2004), 162–187  crossref  isi
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    5. Meltzer H., “Exceptional Vector Bundles, Tilting Sheaves and Tilting Complexes for Weighted Projective Lines”, Mem. Am. Math. Soc., 171:808 (2004), III+  mathscinet  isi
    6. Costa L., Miro-Roig R.M., “Cohomological characterization of vector bundles on multiprojective spaces”, Journal of Algebra, 294:1 (2005), 73–96  crossref  isi
    7. Bridgeland T., “T-structures on some local Calabi-Yau varieties”, Journal of Algebra, 289:2 (2005), 453–483  crossref  isi
    8. I GORDON, J STAFFORD, “Rational Cherednik algebras and Hilbert schemes”, Advances in Mathematics, 198:1 (2005), 222  crossref
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    12. Auroux D., Katzarkov L., Orlov D., “Mirror Symmetry for Del Pezzo Surfaces: Vanishing Cycles and Coherent Sheaves”, Invent. Math., 166:3 (2006), 537–582  crossref  mathscinet  zmath  adsnasa  isi
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    14. L. COSTA, R. M. MIRÓ–ROIG, “Geometric collections and Castelnuovo–Mumford regularity”, Math Proc Camb Phil Soc, 143:3 (2007)  crossref  zmath  isi
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    18. Herzog Ch.P., Karp R.L., “On the Geometry of Quiver Gauge Theories (Stacking Exceptional Collections)”, Adv. Theor. Math. Phys., 13:3 (2009), 599–636  mathscinet  zmath  isi
    19. M. Van den Bergh, “Noncommutative Quadrics”, Internat Math Res Notices, 2010  crossref
    20. Bridgeland T., Stern D., “Helices on del Pezzo surfaces and tilting Calabi-Yau algebras”, Advances in Mathematics, 224:4 (2010), 1672–1716  crossref  isi
    21. Buchweitz R.-O., Leuschke G.J., Van den Bergh M., “Non-Commutative Desingularization of Determinantal Varieties I”, Invent. Math., 182:1 (2010), 47–115  crossref  mathscinet  zmath  adsnasa  isi
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    24. Efimov A.I., Lunts V.A., Orlov D.O., “Deformation theory of objects in homotopy and derived categories III: Abelian categories”, Adv Math, 226:5 (2011), 3857–3911  crossref  isi
    25. Chan D., Ingalls C., “Conic Bundles and Clifford Algebras”, New Trends in Noncommutative Algebra, Contemporary Mathematics, 562, eds. Ara P., Brown K., Lenagan T., Letzter E., Stafford J., Zhang J., Amer Mathematical Soc, 2012, 53–75  crossref  mathscinet  zmath  isi
    26. L. de Thanhoffer de Volcsey, M. Van den Bergh, “Some New Examples of Nondegenerate Quiver Potentials”, International Mathematics Research Notices, 2012  crossref
    27. Galkin S., Shinder E., “Exceptional Collections of Line Bundles on the Beauville Surface”, Adv. Math., 244 (2013), 1033–1050  crossref  isi
    28. D. O. Orlov, “Geometric realizations of quiver algebras”, Proc. Steklov Inst. Math., 290:1 (2015), 70–83  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    29. Galkin S., Katzarkov L., Mellit A., Shinder E., “Derived Categories of Keum'S Fake Projective Planes”, Adv. Math., 278 (2015), 238–253  crossref  isi
    30. Galkin S., Golyshev V., Iritani H., “Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures”, Duke Math. J., 165:11 (2016), 2005–2077  crossref  mathscinet  zmath  isi  elib  scopus
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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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