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

This article is cited in 36 scientific papers (total in 36 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
\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
\jour Russian Acad. Sci. Izv. Math.
\yr 1994
\vol 42
\issue 2
\pages 219--260

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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  crossref  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
    4. Koen de Naeghel, Michel van den Bergh, “Ideal classes of three-dimensional Sklyanin algebras”, Journal of Algebra, 276:2 (2004), 515  crossref
    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
    9. Wendy Lowen, “Obstruction Theory for Objects in Abelian and Derived Categories”, Communications in Algebra, 33:9 (2005), 3195  crossref
    10. Daniel Chan, “Noncommutative cyclic covers and maximal orders on surfaces”, Advances in Mathematics, 198:2 (2005), 654  crossref
    11. Koen De Naeghel, Michel Van den Bergh, “Ideal classes of three dimensional Artin–Schelter regular algebras”, Journal of Algebra, 283:1 (2005), 399  crossref
    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
    13. Aaron Bergman, “Undoing orbifold quivers”, J High Energy Phys, 2007:3 (2007), 112  crossref  mathscinet
    14. L. COSTA, R. M. MIRÓ–ROIG, “Geometric collections and Castelnuovo–Mumford regularity”, Math Proc Camb Phil Soc, 143:3 (2007)  crossref  zmath  isi
    15. Costa L., Miro-Roig R.M., “m-Blocks collections and Castelnuovo-Mumford regularity in multiprojective spaces”, Nagoya Mathematical Journal, 186 (2007), 119–155  isi
    16. Michel Dubois-Violette, “Multilinear forms and graded algebras”, Journal of Algebra, 317:1 (2007), 198  crossref
    17. Aaron Bergman, Nicholas Proudfoot, “Moduli spaces for Bondal quivers”, Pacific J Math, 237:2 (2008), 201  crossref  mathscinet  zmath  isi
    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
    22. Bernhard Keller, Daniel Murfet, Michel Van den Bergh, “On two examples by Iyama and Yoshino”, Compositio Math, 2011, 1  crossref
    23. Lutz Hille, Markus Perling, “Exceptional sequences of invertible sheaves on rational surfaces”, Compositio Math, 2011, 1  crossref
    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
    31. Vial Ch., “Exceptional Collections, and the Neron-Severi Lattice For Surfaces”, Adv. Math., 305 (2017), 895–934  crossref  isi
    32. Belmans P., Presotto D., “Construction of Non-Commutative Surfaces With Exceptional Collections of Length 4”, J. Lond. Math. Soc.-Second Ser., 98:1 (2018), 85–103  crossref  isi
    33. A. G. Kuznetsov, “Embedding derived categories of Enriques surfaces in derived categories of Fano varieties”, Izv. Math., 83:3 (2019), 534–539  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    34. Liu W. Yang S. Yu X., “Classification of Full Exceptional Collections of Line Bundles on Three Blow-Ups of P-3”, J. Korean. Math. Soc., 56:2 (2019), 387–419  crossref  mathscinet  isi  scopus
    35. Harder A. Thompson A., “Pseudolattices, Del Pezzo Surfaces, and Lefschetz Fibrations”, Trans. Am. Math. Soc., 373:3 (2020), 2071–2104  crossref  isi
    36. Eric M. Rains, “Elliptic Double Affine Hecke Algebras”, SIGMA, 16 (2020), 111, 133 pp.  mathnet  crossref
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