Reconstruction algebras of type 𝐴

Wemyss, Michael

doi:10.1090/S0002-9947-2011-05130-5

Reconstruction algebras of type $A$
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Trans. Amer. Math. Soc. 363 (2011), 3101-3132 Request permission

Abstract:

We introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities $\mathbb {C}^2/G$ where $G=\frac {1}{r}(1,a)\leq GL(2,\mathbb {C})$.

References

Ken A. Brown and Ken R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492, DOI 10.1007/978-3-0348-8205-7
K. A. Brown and C. R. Hajarnavis, Homologically homogeneous rings, Trans. Amer. Math. Soc. 281 (1984), no. 1, 197–208. MR 719665, DOI 10.1090/S0002-9947-1984-0719665-5
Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI 10.1090/S0894-0347-01-00368-X
Tom Bridgeland, Stability conditions on a non-compact Calabi-Yau threefold, Comm. Math. Phys. 266 (2006), no. 3, 715–733. MR 2238896, DOI 10.1007/s00220-006-0048-7
A. Craw, The special McKay correspondence as a derived equivalence, arXiv:0704.3627 (2007).
Heiko Cassens and Peter Slodowy, On Kleinian singularities and quivers, Singularities (Oberwolfach, 1996) Progr. Math., vol. 162, Birkhäuser, Basel, 1998, pp. 263–288. MR 1652478, DOI 10.1007/978-3-0348-8770-0_{1}4
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Hélène Esnault, Reflexive modules on quotient surface singularities, J. Reine Angew. Math. 362 (1985), 63–71. MR 809966, DOI 10.1515/crll.1985.362.63
K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, 2nd ed., London Mathematical Society Student Texts, vol. 61, Cambridge University Press, Cambridge, 2004. MR 2080008, DOI 10.1017/CBO9780511841699
Lutz Hille and Michel Van den Bergh, Fourier-Mukai transforms, Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332, Cambridge Univ. Press, Cambridge, 2007, pp. 147–177. MR 2384610, DOI 10.1017/CBO9780511735134.007
Akira Ishii, On the McKay correspondence for a finite small subgroup of $\textrm {GL}(2,\Bbb C)$, J. Reine Angew. Math. 549 (2002), 221–233. MR 1916656, DOI 10.1515/crll.2002.064
Rie Kidoh, Hilbert schemes and cyclic quotient surface singularities, Hokkaido Math. J. 30 (2001), no. 1, 91–103. MR 1815001, DOI 10.14492/hokmj/1350911925
A. D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 515–530. MR 1315461, DOI 10.1093/qmath/45.4.515
—, Tilting bundles on some rational surfaces, preprint available on the Internet at the page http://www.maths.bath.ac.uk/$\sim$masadk/papers/ (1997).
M. Macleod Generalizing the Cohen-Macaulay condition and other homological properties, Ph.D. thesis, (2010).
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
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
C. Quarles and S. Paul Smith, On an example of Leuschke, private communication (2006).
John Rainwater, Global dimension of fully bounded Noetherian rings, Comm. Algebra 15 (1987), no. 10, 2143–2156. MR 909958, DOI 10.1080/00927878708823527
M. Reid, Surface cyclic quotient singularities and Hirzebruch-Jung resolutions, available on the Internet at the page http://www.maths.warwick.ac.uk/$\sim$miles/surf/ (1997).
Oswald Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann. 209 (1974), 211–248 (German). MR 367276, DOI 10.1007/BF01351850
Oswald Riemenschneider, Die Invarianten der endlichen Untergruppen von $\textrm {GL}(2,\textbf {C})$, Math. Z. 153 (1977), no. 1, 37–50. MR 447230, DOI 10.1007/BF01214732
Oswald Riemenschneider, Special representations and the two-dimensional McKay correspondence, Hokkaido Math. J. 32 (2003), no. 2, 317–333. MR 1996281, DOI 10.14492/hokmj/1350657526
J. T. Stafford and M. Van den Bergh, Noncommutative resolutions and rational singularities, Michigan Math. J. 57 (2008), 659–674. Special volume in honor of Melvin Hochster. MR 2492474, DOI 10.1307/mmj/1220879430
Michel van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749–770. MR 2077594
Michel Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. MR 2057015, DOI 10.1215/S0012-7094-04-12231-6
M. Wemyss, The $\mathrm {GL}(2,\mathbb {C})$ McKay Correspondence, arXiv:0809.1973 (2008).
J. Wunram, Reflexive modules on cyclic quotient surface singularities, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 221–231. MR 915177, DOI 10.1007/BFb0078846
Jürgen Wunram, Reflexive modules on quotient surface singularities, Math. Ann. 279 (1988), no. 4, 583–598. MR 926422, DOI 10.1007/BF01458530
Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, LMS Lecture Notes 146 (1990), 177pp.