Publications of Michel Van den Bergh


Generated Mon Nov 17 12:00:08 2014 using btparse, SWIG and Python.

Monographs

  1. L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Graded orders, Birkhauser Boston Inc., Boston, MA, pp. vi+208, 1988.
  2. I. Reiten and M. Van den Bergh, Two-dimensional tame and maximal orders of finite representation type, Mem. Amer. Math. Soc. 80 (1989), viii+72.
  3. I. M. Musson and M. Van den Bergh, Invariants under tori of rings of differential operators and related topics, Mem. Amer. Math. Soc. 136 (1998), viii+85.

    Abstract: If $G$ is a reductive algebraic group acting rationally on a smooth affine variety $X$ then it is generally believed that $D(X)^G$ has properties very similar to those of enveloping algebras of semisimple Lie algebras. In this paper we show that this is indeed the case when $G$ is a torus and $X=k^r\times (k^*)^s$. We give a precise description of the primitive ideals in $D(X)^G$ and we study in detail the ring theoretical and homological properties of the minimal primitive quotients of $D(X)^G$. The latter are of the form $D(X)^G/(\g-\chi(\g))$ where $\g=\Lie(G)$, $\chi\in \g^\ast$ and $\g-\chi(\g)$ is the set of all $v-\chi(v)$ with $v\in \g$. They occur as rings of twisted differential operators on toric varieties. As a side result we prove that if $G$ is a torus acting rationally on a smooth affine variety then $D(X\quot G)$ is a simple ring.

  4. M. Van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc. 154 (2001), x+140.

    Abstract: In this paper we will think of certain abelian categories with favorable properties as non-commutative surfaces. We show that under certain conditions a point on a non-commutative surface can be blown up. This yields a new non-commutative surface which is in a certain sense birational to the original one. This construction is analogous to blowing up a Poisson surface in a point of the zero-divisor of the Poisson bracket. By blowing up $\le 8$ points in the elliptic quantum plane one obtains global non-commutative deformations of Del-Pezzo surfaces. For example blowing up six points yields a non-commutative cubic surface. Under a number of extra hypotheses we obtain a formula for the number of non-trivial simple objects on such non-commutative surfaces.

Top

Articles

  1. M. Van den Bergh, A duality theorem for Hopf algebras, Methods in ring theory (Antwerp, 1983), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 129, Reidel, Dordrecht, pp. 517-522, 1984.
  2. S. Caenepeel, M. Van den Bergh, and F. Van Oystaeyen, Generalized crossed products applied to maximal orders, Brauer groups and related exact sequences, J. Pure Appl. Algebra 33 (1984), 123-149.
  3. M. Van den Bergh and J. Van Geel, A duality theorem for orders in central simple algebras over function fields, J. Pure Appl. Algebra 31 (1984), 227-239.
  4. M. Van den Bergh and J. Van Geel, Algebraic elements in division algebras over function fields of curves, Israel J. Math. 52 (1985), 33-45.
  5. M. Van den Bergh, On a theorem of Cohen and Montgomery, Proc. Amer. Math. Soc. 94 (1985), 562-564.
  6. M. Van den Bergh, Graded Dedekind rings, J. Pure Appl. Algebra 35 (1985), 105-115.
  7. M. Van den Bergh, A note on graded $K$-theory, Comm. Algebra 14 (1986), 1561-1564.
  8. L. Le Bruyn and M. Van den Bergh, An explicit description of $T_{3,2}$, Ring theory (Antwerp, 1985), Lecture Notes in Math., vol. 1197, Springer, Berlin, pp. 109-113, 1986.
  9. L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Proj of generic matrices and trace rings, Comm. Algebra 14 (1986), 1687-1706.
  10. M. Van den Bergh, The algebraic index of a division algebra, Ring theory (Antwerp, 1985), Lecture Notes in Math., vol. 1197, Springer, Berlin, pp. 190-206, 1986.
  11. M. Van den Bergh, Regular rings of dimension three, S\'eminaire d'alg\`ebre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986), Lecture Notes in Math., vol. 1296, Springer, Berlin, pp. 228-234, 1987.
  12. M. Van den Bergh, A note on graded Brauer groups, Bull. Soc. Math. Belg. S\'er. B 39 (1987), 177-179.
  13. M. Van den Bergh, Linearisations of binary and ternary forms, J. Algebra 109 (1987), 172-183.
  14. L. Le Bruyn and M. Van den Bergh, The ramification divisor of regular tame orders. I, Comm. Algebra 15 (1987), 1815-1840.
  15. M. Van den Bergh, Division algebras over function fields of varieties, Academiae Analecta 49 (1987), 127-135.
  16. M. Van den Bergh, The Brauer-Severi scheme of the trace ring of generic matrices, Perspectives in ring theory (Antwerp, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, pp. 333-338, 1988.
  17. L. Le Bruyn and M. Van den Bergh, Regularity of trace rings of generic matrices, J. Algebra 117 (1988), 19-29.
  18. M. Awami, M. Van den Bergh, and F. Van Oystaeyen, Note on derivations of graded rings and classification of differential polynomial rings, Bull. Soc. Math. Belg. S\'er. A 40 (1988), 175-183.
  19. M. Van den Bergh, Group rings over Dedekind rings, Israel J. Math. 61 (1988), 295-300.
  20. M. Van den Bergh, Algebraic splitting fields of division algebras, Ring theory 1989 (Ramat Gan and Jerusalem, 1988/1989), Israel Math. Conf. Proc., vol. 1, Weizmann, Jerusalem, pp. 381-388, 1989.
  21. M. Van den Bergh, The center of the generic division algebra, J. Algebra 127 (1989), 106-126.
  22. C. N{\u{a}}st{\u{a}}sescu, M. Van den Bergh, and F. Van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), 397-413.
  23. M. Van den Bergh, Trace rings of generic matrices are Cohen-Macaulay, J. Amer. Math. Soc. 2 (1989), 775-799.

    Abstract: In this paper we prove that trace rings of generic matrices are Cohen-Macaulay. This is done by relating this problem to a conjecture of Stanley about modules of invariants under a reductive group. We prove a slightly weakened version of this conjecture in special cases. In particular we obtain that the conjecture is true for $SL_2$.

  24. M. J. Asensio, M. Van den Bergh, and F. Van Oystaeyen, A new algebraic approach to microlocalization of filtered rings, Trans. Amer. Math. Soc. 316 (1989), 537-553.
  25. M. Van den Bergh and F. Van Oystaeyen, Lifting maximal orders, Comm. Algebra 17 (1989), 341-349.
  26. M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol.\ I, Progr. Math., vol. 86, Birkh\"auser Boston, Boston, MA, pp. 33-85, 1990.
  27. M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), 249-271.
  28. H. S. Li, M. Van den Bergh, and F. Van Oystaeyen, Note on the $K\sb 0$ of rings with Zariskian filtration, $K$-Theory 3 (1990), 603-606.
  29. H. S. Li, M. Van den Bergh, and F. Van Oystaeyen, Global dimension and regularity of Rees rings for non-Zariskian filtrations, Comm. Algebra 18 (1990), 3195-3208.
  30. M. Van den Bergh, Differential operators on semi-invariants for tori and weighted projective spaces, Topics in invariant theory (Paris, 1989/1990), Lecture Notes in Math., vol. 1478, Springer, Berlin, pp. 255-272, 1991.

    Abstract: In this paper we study rings of di erential operators for modules of covariants for one-dimensional tori. In particular we analyze when they are Morita equivalent, when they are simple, and when they have nite global dimension. As a side result we obtain an extension of the Bernstein-Beilinson equivalence to weighted projective spaces.

  31. M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math. 106 (1991), 335-388.
  32. M. Van den Bergh, Cohen-Macaulayness of modules of covariants, Invent. Math. 106 (1991), 389-409.

    Abstract: We prove a very slightly weakened version of a conjecture by Richard Stanley

  33. L. Le Bruyn and M. Van den Bergh, Algebraic properties of linear cellular automata, Linear Algebra Appl. 157 (1991), 217-234.
  34. M. Van den Bergh, Cohen-Macaulayness of modules of invariants for $\rm {S}{L}\sb 2$, J. Algebra 142 (1991), 273-284.

    Abstract: Let $U$, $W$ be finite dimensional representations of $G = \mathrm{SL}_2$. We give conditions under which $(U\otimes k[W])^G$ is a Cohen- Macaulay $k[W]^G$-module. In particular we obtain an invariant theoretic proof of the fact that the trace ring of generic $2\times 2$ matrices is Cohen-Macaulay

  35. M. Van den Bergh, Explicit rational forms for the Poincar\'e series of the trace rings of generic matrices, Israel J. Math. 73 (1991), 17-31.

    Abstract: In this paper we derive explicit rational forms for the Poincare series of the commutative and the non-commutative trace ring (5.3 and 5.5). To this end we use the Molien Weyl formula to reduce the question to a problem about ows in a particular graph.

  36. C. Apostolopoulos, M. Van den Bergh, and F. Van Oystaeyen, On Schur rings of group rings of finite groups, Comm. Algebra 20 (1992), 2139-2152.
  37. A. Schofield and M. Van den Bergh, The index of a Brauer class on a Brauer-Severi variety, Trans. Amer. Math. Soc. 333 (1992), 729-739.

    Abstract: Let D and E be central division algebras over k; let K be the generic splitting field of E; we show that the index of $D \otimes_k K$ is the minimum of the indices of $D \otimes_k E^{\otimes i}$. We use this to calculate the index of D under related central extensions and to construct division algebras with special properties.

  38. A. Jensen, S. J{\o}ndrup, and M. Van den Bergh, Artinian quotient rings of filtered rings, J. Algebra 161 (1993), 230-236.
  39. L. Le Bruyn and M. Van den Bergh, On quantum spaces of Lie algebras, Proc. Amer. Math. Soc. 119 (1993), 407-414.
  40. M. Van den Bergh, Cohen-Macaulayness of semi-invariants for tori, Trans. Amer. Math. Soc. 336 (1993), 557-580.

    Abstract: In this paper we give a new method, in terms of one-parameter sub- groups, to study semi-invariants for algebraic tori. In some cases we obtain extensions to results in [7]. In other cases we obtain different proofs.

  41. M. Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), vol. 8, pp. 213-230, 1994.

    Abstract: We compute the Hochschild and cyclic homology of certain three dimensional quantum spaces (type A algebras), introduced by Artin and Schelter. We show that the Hochschild homology is determined by the quasi-classical limit

  42. M. Van den Bergh, A converse to Stanley's conjecture for $\rm {S}l\sb 2$, Proc. Amer. Math. Soc. 121 (1994), 47-51.

    Abstract: In this note we prove, in the case of $\Sl_2$, a converse to Stanley's conjecture about Cohen-Macaulayness of invariant modules for reductive algebraic groups.

  43. A. Schofield and M. Van den Bergh, Division algebra coproducts of index $n$, Trans. Amer. Math. Soc. 341 (1994), 505-517.

    Abstract: Given a family of separable finite dimensional extensions $(L_i)_i$ of a field $k$, we construct a division algebra of dimension $n^2$ over its center which is freely generated over $k$ by the fields $(L_i)_i$.

  44. M. Van den Bergh, Modules of covariants, Proceedings of the International Congress of Mathematicians, Vol.\ 1, 2 (Z\"urich, 1994), Birkh\"auser, Basel, pp. 352-362, 1995.

    Abstract: Let G be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let W, U two finite dimensional representations of G. In this paper we give a survey on the computation of the local cohomology of (U tensor SW)^G.

  45. J. Tate and M. Van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), 619-647.

    Abstract: To a pair consisting of an elliptic curve and a point on it, Odeskii and Feigin associate certain quadratic algebras (``Sklyanin algebras''), having the Hilbert series of a polynomial algebra. In this paper we show that Sklyanin algebras have good homological properties and we obtain some information about their so-called linear modules. We also show how the construction by Odeskii and Feigin may be generalized so as to yield other ``Sklyanin-type'' algebras.

  46. L. Le Bruyn, S. P. Smith, and M. Van den Bergh, Central extensions of three-dimensional Artin-Schelter regular algebras, Math. Z. 222 (1996), 171-212.

    Abstract: For a 3-dimensional Artin-Schelter-regular algebra A with Hilbert series 1/(1-t)^3 we study central extensions; that is, graded algebras D with a regular central element z in degree 1, such that D/(z)=A. We classify such D and we also classify certain D-modules (point modules and line modules) which proved to be important in the study of 3-dimensional Artin-Schelter-regular algebras.

  47. M. Van den Bergh, A translation principle for the four-dimensional Sklyanin algebras, J. Algebra 184 (1996), 435-490.

    Abstract: In this paper we prove a translation principle for the central quotients of four-dimensional Sklyanin algebras, which is analogous to the translation principle for semi-simple Lie algebras. In the course of the proof we construct an ``elliptic'' analog of the sheaf of differential operators on P^1. The translation principle may be used to construct the fat points of a four-dimensional, non-PI, Sklyanin algebra.

  48. M. Van den Bergh, Some rings of differential operators for $\rm {S}l\sb 2$-invariants are simple, J. Pure Appl. Algebra 107 (1996), 309-335.

    Abstract: It has been conjectured that the ring of differential operators of the algebraic quotient of a connected smooth affine variety under a reductive group action is simple. This is known in the case that the group in question is the extension of a finite group with a torus and in the case of classical representation of classical groups. In these notes we present some tools relevant to this conjecture. In particular we show that it is true for some representations of Sl2.

  49. J. Alev, A. Ooms, and M. Van den Bergh, A class of counterexamples to the Gelfand-Kirillov conjecture, Trans. Amer. Math. Soc. 348 (1996), 1709-1716.

    Abstract: Let G be a connected non-special semisimple algebraic group and let W be a finite dimensional G-representation such that W has trivial generic stabilizer. Let g=Lie(G). Then the semi-direct product g+W is a counter example to the Gel'fand-Kirillov conjecture.

  50. M. Van den Bergh, Division algebras on ${\bf P}\sp 2$ of odd index, ramified along a smooth elliptic curve are cyclic, Alg\`ebre non commutative, groupes quantiques et invariants (Reims, 1995), S\'emin. Congr., vol. 2, Soc. Math. France, Paris, pp. 43-53, 1997.

    Abstract: The simplest non-trivial division algebras that can be constructed over a rational function field in two variables are those that ramify along a smooth cubic curve. In this note we show that these division algebras are cyclic if they have odd index.

  51. M. Van den Bergh and M. Van Gastel, Graded modules of Gelfand-Kirillov dimension one over three-dimensional Artin-Schelter regular algebras, J. Algebra 196 (1997), 251-282.

    Abstract: Let $A$ be a three dimensional Artin-Schelter regular algebra. We give a description of the category of finitely generated $A$-modules of Gelfand-Kirillov dimension one (modulo those of finite dimension over the ground field). The proof is based upon a result by Gabriel which says that locally finite categories can be described by module categories over topological rings.

  52. M. Van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), 662-679.

    Abstract: In this note we prove existence theorems for dualizing complexes over graded and filtered rings, thereby generalizing some results by Zhang, Yekutieli and J{\o}rgensen.

  53. K. E. Smith and M. Van den Bergh, Simplicity of rings of differential operators in prime characteristic, Proc. London Math. Soc. (3) 75 (1997), 32-62.

    Abstract: Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p>0 and if the inclusion R-> S splits, then D_k(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic zero.

  54. M. Van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), 1345-1348.

    Abstract: Let ``$HH$'' stand for Hochschild (co)homology. In this note we show that for many rings $A$ there exists $d\in\NN$ such that for an arbitrary $A$-bimodule $N$ we have $HH^i(N)=HH_{d-i}(N) $. Such a result may be viewed as an analog of Poincare duality. Combining this equality with a computation of Soergel allows one to compute the Hochschildt homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo. (Also see erratum below.)

  55. T. Gateva-Ivanova and M. Van den Bergh, Semigroups of $I$-type, J. Algebra 206 (1998), 97-112.

    Abstract: Assume that S is a semigroup generated by {x_1,...,x_n}, and let U be the multiplicative free commutative semigroup generated by {u_1,...,u_n}. We say that S is of I-type if there is a bijection v:U->S such that for all a in U, {v(u_1a),....,v(u_na)}={x_1v(a),....,x_nv(a)}. This condition appeared naturally in the work on Sklyanin algebras by John Tate and the second author. In this paper we show that the condition for a semigroup to be of I-type is related to various other mathematical notions found in the literature. In particular we show that semigroups of I-type appear in the study of the settheoretic solutions of the Yang-Baxter equation, in the theory of Bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author.

  56. K. Bauwens and M. Van den Bergh, Normalizing extensions of the two-Veronese of a three-dimensional Artin-Schelter regular algebra on two generators, J. Algebra 205 (1998), 368-390.

    Abstract: We classify regular algebras of global dimension four that map surjectively onto the two-Veronese of a regular algebra of global dimension three on two generators. We also study the point modules.

  57. B. Sevenhant and M. Van den Bergh, On the number of absolutely indecomposable representations of a quiver, J. Algebra 221 (1999), 29-49.

    Abstract: A conjecture of Kac states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we give a combinatorial reformulation of Kac's conjecture in terms of a property of $q$-multinomial coefficients. As a side result we give a formula for certain inverse Kostka-Foulkes polynomials.

  58. B. Sevenhant and M. Van den Bergh, On the double of the Hall algebra of a quiver, J. Algebra 221 (1999), 135-160.

    Abstract: We show that it is possible to define reflection isomorphisms on the double of the (twisted) Hall algebra of a quiver. Combining these reflections with Fourier transform yields an alternative construction of Lusztig's braid group action on a quantum enveloping algebra.

  59. M. Van den Bergh, Local cohomology of modules of covariants, Adv. Math. 144 (1999), 161-220.

    Abstract: Let G be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let W, U two finite dimensional representations of G. In this paper we compute the local cohomology of (U tensor SW)^G provided a certain relatively weak technical condition is true.

  60. K. Ajitabh and M. Van den Bergh, Presentation of critical modules of GK-dimension $2$ over elliptic algebras, Proc. Amer. Math. Soc. 127 (1999), 1633-1639.

    Abstract: We show that critical modules of Gelfand-Kirillov dimension 2 and multiplicity $d$ over an elliptic algebra have (up to modules of lower GK-dimension and shifting) a presentation by $d\times d$-matrices of linear forms. In the language of non-commutative algebraic geometry this amounts to a generic description of ``curves'' of degree $d$ in a projective quantum plane.

  61. J. Alev, A. I. Ooms, and M. Van den Bergh, The Gelfand-Kirillov conjecture for Lie algebras of dimension at most eight, J. Algebra 227 (2000), 549-581.

    Abstract: We show that the Gelfand-Kirillov holds for Lie algebras of dimension at most eigth. Recall that in dimension nine the authors have constructed a counter example.

  62. M. Van den Bergh, Abstract blowing down, Proc. Amer. Math. Soc. 128 (2000), 375-381.

    Abstract: Assume that $X$ is a surface over an algebraically closed field $k$. Let $\tilde{X}$ be obtained from $X$ by blowing up a smooth point and let $L$ be the exceptional curve. Let $\coh(X)$ be the category of coherent sheaves on $X$. In this note we show how to recover $\coh({X})$ from $\coh(\tilde{X})$, if we know the object $\Oscr_L(L)$.

  63. A. Schofield and M. Van den Bergh, Semi-invariants of quivers for arbitrary dimension vectors, Indag. Math. (N.S.) 12 (2001), 125-138.

    Abstract: The representations of dimension vector $\alpha$ of the quiver $Q$ can be parametrised by a vector space $R(Q,\alpha)$ on which an algebraic group $\Gl(\alpha)$ acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi--invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate to a suitable map between projective representations a semi--invariant polynomial function that describes when this map is inverted on the representation and we show that these semi--invariant polynomial functions form a spanning set of all semi--invariant polynomial functions in characteristic $0$. If the quiver has no oriented cycles, we may replace consideration of inverting maps between projective representations by consideration of representations that are left perpendicular to some representation of dimension vector $\alpha$. These left perpendicular representations are just the cokernels of the maps between projective representations that we consider.

  64. B. Sevenhant and M. Van den Bergh, A relation between a conjecture of Kac and the structure of the Hall algebra, J. Pure Appl. Algebra 160 (2001), 319-332.

    Abstract: In this paper we show that the Hall algebra of a quiver, as defined by Ringel, is the positive part of the quantived enveloping algebra of a generalized Kac-Moody Lie algebra. We give a potential application of this result to a conjecture of Kac which states that the constant coefficient of the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra.

  65. I. Reiten and M. Van den Bergh, Grothendieck groups and tilting objects, Algebr. Represent. Theory 4 (2001), 1-23.

    Abstract: Let $\Cscr$ be a connected noetherian hereditary abelian category with Serre functor over an algebraically closed field $k$, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from \cite{RV2}, we prove that if $\Cscr$ has some object of infinite length, then the Grothendieck group of $\Cscr$ is finitely generated if and only if $\Cscr$ has a tilting object.

  66. J. T. Stafford and M. Van den Bergh, Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 171-216 (electronic).

    Abstract: In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.

  67. I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295-366 (electronic).

    Abstract: In this paper we classify $\Ext$-finite noetherian hereditary abelian categories over an algebraically closed field $k$ satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary abelian categories. As a side result we show that when our hereditary abelian categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost split sequences.

  68. M. Van den Bergh, Erratum to: ``A relation between Hochschild homology and cohomology for Gorenstein rings'' [Proc. Amer. Math. Soc. \bf 126 (1998), no. 5, 1345--1348; MR 99m:16013], Proc. Amer. Math. Soc. 130 (2002), 2809-2810 (electronic).

    Abstract: The paper "A relation between Hochschild homology and cohomology for Gorenstein rings" contains an error in the sense that Theorem 1 (the "duality theorem") is false in the generality stated. As a result the same is true for its corollaries: Proposition 3 and Corollary 6. The main conclusion, which is an affirmative answer to a question by Patrick Polo, remains valid however

  69. M. Van den Bergh and M. Van Gastel, On the structure of non-commutative regular local rings of dimension two, Comm. Algebra 30 (2002), 4575-4588.

    Abstract: In this paper we conjecture that the center of a non-commutative complete regular local ring of global dimension two is a formal power series ring in two variables. We prove this conjecture in the special case of Ore extensions.

  70. M. Van den Bergh, Non-commutative crepant resolutions, The Legacy of Niels Hendrik Abel, Springer, pp. 749-770, 2002.

    Abstract: We introduce the notion of a ``non-commutative crepant'' resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.

  71. A. Bondal and M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Moscow Mathematical Journal 3 (2003), 1-36.

    Abstract: We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. In contrast the similar category for a smooth compact analytic surface with no curves is not saturated.

  72. W. Crawley-Boevey and M. Van den Bergh, Absolutely indecomposable representations and Kac-Moody Lie algebras, Invent. Math. 155 (2004), 537-559.

    Abstract: A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors.

  73. M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), 423-455.

    Abstract: {For $Y,Y^+$ three-dimensional smooth varieties related by a flop, Bondal and Orlov conjectured that the derived categories $D^b(\coh(Y))$ and $D^b(\coh(Y^+))$ are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using non-commutative rings. The new proof also covers some mild singular and higher dimensional situations (including the one in the recent paper by Chen: ``Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities''). }

  74. K. de Naeghel and M. van den Bergh, Ideal classes of three-dimensional Sklyanin algebras, J. Algebra 276 (2004), 515-551.

    Abstract: In this paper we classify graded reflexive ideals, up to isomorphism and shift, in certain three dimensional Artin-Schelter regular algebras. This classification is similar to the classification of right ideals in the first Weyl algebra, a problem that was completely settled recently. The situation we consider is substantially more complicated however.

  75. M. Van den Bergh, A remark on a theorem by Deligne, Proc. Amer. Math. Soc. 132 (2004), 2857-2858 (electronic).

    Abstract: We give a proof avoiding spectral sequences of Deligne's decomposition theorem for objects in a triangulated category admitting a Lefschetz homomorphism.

  76. K. De Naeghel and M. Van den Bergh, Ideal classes of three dimensional Artin-Schelter regular algebras, J. Algebra 283 (2005), 399-429.

    Abstract: We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional Artin-Schelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of torsion free rank one modules on a non-commutative P^2 is connected. A different proof of this fact, based on deformation theoretic methods and the known commutative case has recently been given by Nevins and Stafford. For the Weyl algebra it was proved by Wilson

  77. M. Van den Bergh, On the ${\mathbb Z}D\sb \infty$ category, Proceedings of the 37th Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent Theory Organ. Comm., Osaka, pp. 103-112, 2005.

    Abstract: In this paper we give a direct proof of the properties of the $\ZZ D_\infty$ category which was introduced in the classification of noetherian, hereditary categories with Serre duality by Idun Reiten and the author.

  78. W. Lowen and M. Van den Bergh, Hochschild cohomology of abelian categories and ringed spaces, Adv. Math. 198 (2005), 172-221.

    Abstract: This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild cohomology coincides with the one defined by Gerstenhaber, Schack and Swan in the case of module categories over diagrams and schemes and also with the Hochschild cohomology for exact categories introduced recently by Keller. In addition we show in complete generality that Hochschild cohomology satisfies a Mayer-Vietoris property and that for constantly ringed spaces it coincides with the cohomology of the structure sheaf.

  79. W. Lowen and M. Van den Bergh, Deformation theory of abelian categories, Trans. Amer. Math. Soc. 358 (2006), 5441-5483.

    Abstract: In this paper we develop the basic infinitesimal deformation theory of \emph{abelian categories}. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations and we construct several equivalences between deformation problems.

  80. L. Hille and M. Van den Bergh, Fourier-Mukai transforms, Handbook of tiltingtheory, London Mathematical Society Lecture Note Series, vol. 332, Cambridge University Press, pp. 147-173, 2007.

    Abstract: In this paper we discuss some of the recent developments on derived equivalences in algebraic geometry.

  81. K. De Naeghel and M. Van den Bergh, On incidence between strata of the Hilbert scheme of points on $\mathbb P\sp 2$, Math. Z. 255 (2007), 897-922.

    Abstract: The Hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand studied this problem for strata whose Hilbert series are as close as possible. Preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata. In this paper we present a new approach, based on deformation theory, to Guerimand's result. This allows us to show that the technical condition is not necessary.

  82. M. Van den Bergh, On global deformation quantization in the algebraic case, Journal of Algebra 315 (2007), 326-395.

    Abstract: We give a proof of Yekutieli's global quantization result which does not rely on the choice of local sections of the bundle of affine coordinate systems. Instead we use an argument inspired by algebraic De Rham cohomology.

  83. M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. 360 (2008), 5711-5769.

    Abstract: In this paper we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by Crawley-Boevey and Shaw carry a natural Poisson structure. This follows the fact that appropriately localized path algebras of double quivers carry a certain kind of non-commutative quasi-Hamiltonian structure.

  84. J. T. Stafford and M. Van den Bergh, Noncommutative resolutions and rational singularities, Michigan Math. J. 57 (2008), 659-674.

    Abstract: Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.

  85. M. Van den Bergh, Non-commutative quasi-Hamiltonian spaces, Poisson geometry in mathematics and physics, Contemp. Math., vol. 450, Amer. Math. Soc., Providence, RI, pp. 273-299, 2008.

    Abstract: In this paper we introduce non-commutative analogues for the quasi-Hamiltonian $G$-spaces introduced by Alekseev, Malkin and Meinrenken. We outline the connection with the non-commutative analogues of quasi-Poisson algebras which the author had introduced earlier.

  86. M. Van den Bergh, The Kontsevich weight of a wheel with spokes pointing outward, Algebr. Represent. Theory 12 (2009), 443-479.

    Abstract: This is a companion note to ``Hochschild cohomology and Atiyah classes'' by Damien Calaque and the author. We compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernouilli numbers. Our computation uses Stokes theorem together with some basic properties proved by Kontsevich.

  87. D. Calaque and M. Van den Bergh, Global formality at the $G_\infty$-level, Mosc. Math. J. 10 (2010), 31-64, 271.

    Abstract: In this paper we prove that the sheaf of $\Lscr$-differential operators for a locally free Lie algebroid $\Lscr$ is formal when viewed as a sheaf of $G_\infty$-algebras via Tamarkin's morphism of DG-operads $B_\infty\r G_\infty$.

  88. D. Calaque and M. Van den Bergh, Hochschild cohomology and Atiyah classes, Adv. Math. 224 (2010), 1839-1889.

    Abstract: In this paper we provide a proof of a result announced by Kontsevich that the HKR-morphism, twisted by the square root of the Todd genus, gives an isomorphism between the Hochschild cohomology of a smooth algebraic variety and the cohomology of poly-vector fields, both considered as Gerstenhaber algebras. Our proof is set in the framework of Lie algebroids and so applies in more general settings as well.

  89. R. Buchweitz, G. J. Leuschke, and M. Van den Bergh, Non-commutative desingularization of determinantal varieties I, Invent. Math. 182 (2010), 47-115.

    Abstract: We show that determinantal varieties defined by maximal minors of a generic matrix have a kind of non-commutative desingularization. More precisely we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.

  90. D. Calaque, C. A. Rossi, and M. van den Bergh, Hochschild (co)homology for Lie algebroids, Int. Math. Res. Not. IMRN (2010), 4098-4136.

    Abstract: We define the Hochschild (co)homology of a ringed space relative to a locally free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an algebraic variety. We show that our (co)homology groups can be computed using suitable standard complexes. Our formulae depend on certain natural structures on jetbundles over Lie algebroids. In an appendix we explain this by showing that such jetbundles are formal groupoids which serve as the formal exponentiation of the Lie algebroid.

  91. A. I. Ooms and M. Van den Bergh, A degree inequality for Lie algebras with a regular Poisson semi-center, J. Algebra 323 (2010), 305-322.

    Abstract: For Lie algebras whose Poisson semi-center is a polynomial ring we give a bound for the sum of the degrees of the generating semi-invariants. This bound was previously known in many special cases.

  92. W. Lowen and M. van den Bergh, A Hochschild cohomology comparison theorem for prestacks, Trans. Amer. Math. Soc. 363 (2011), 969-986.

    Abstract: We generalize and clarify Gerstenhaber and Schack's "Special Cohomology Comparison Theorem". More specifically we obtain a fully faithful functor between the derived categories of bimodules over a prestack over a small category U and the derived category of bimodules over its corresponding fibered category. In contrast to Gerstenhaber and Schack we do not have to assume that U is a poset.

  93. M. Van den Bergh, Noncommutative quadrics, Int. Math. Res. Not. IMRN (2011), 3983-4026.

    Abstract: In this paper we describe non-commutative versions of $\PP^1x\PP^1$. These contain the class of non-commutative deformations of $\PP^1x\PP^1$.

  94. B. Keller, D. Murfet, and M. Van den Bergh, On two examples by Iyama and Yoshino, Compos. Math. 147 (2011), 591-612.

    Abstract: In the recent paper "Mutation in triangulated categories and rigid Cohen-Macaulay modules" Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal Cohen-Macaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second approach we use Orlov's result on the graded singularity category. We obtain some new results on the singularity category of isolated singularities which may be interesting in their own right.

  95. M. Van den Bergh, Non-commutative $\Bbb{P}^1$-bundles over commutative schemes, Trans. Amer. Math. Soc. 364 (2012), 6279-6313.

    Abstract: In this paper we give a new definition for non-commutative $\PP^1$-bundles over commutative schemes. The main motivation for this is that every non-commutative deformation of a Hirzebruch surface is given by a non-commutative $\PP^1$-bundle over $\PP^1$.

  96. D. Calaque, C. A. Rossi, and M. Van den Bergh, C\u ald\u araru's conjecture and Tsygan's formality, Ann. of Math. (2) 176 (2012), 865-923.

    Abstract: In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet (the existence of the latter was conjectured by Tsygan). We also rely on the fact - recently proved by the first two authors - that Shoikhet's quasi-isomorphism is compatible with cap product after twisting with a Maurer-Cartan element.

  97. M. Van den Bergh, Notes on formal deformations of abelian categories, Derived categories in algebraic geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Z\"urich, pp. 319-344, 2012.

    Abstract: In these notes we provide the foundation for the deformation theoretic parts of ``Non-commutative quadrics'' and ``Non-commutative $\PP^1$-bundles over commutative schemes''.

  98. S. P. Smith and M. Van den Bergh, Non-commutative quadric surfaces, Journal of Noncommutative Geometry 7 (2013), 817-856.

    Abstract: The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of non-commutative quadric surfaces, We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by non-commutative lines. The Picard groups of the smooth quadrics are free abelian of rank two. The alternating sum of dimensions of Ext groups allows us to define an intersection pairing on the Picard group of the smooth noncommutative quadrics. A surprise is that a smooth noncommutative quadric can sometimes contain a "curve" having self-intersection number -2. Many of the methods used in our paper are noncommutative versions of methods developed by Buchweitz, Eisenbud and Herzog: in particular, the correspondence between the geometry of a quadric hypersurface and maximal Cohen-Macaulay modules over its homogeneous coordinate ring plays a key role. An important aspect of our work is to introduce definitions of non-commutative analogues of the familiar commutative terms used in this abstract. We expect the ideas we develop here for 2-dimensional non-commutative quadric hypersurfaces will apply to higher dimensional non-commutative quadric hypersurfaces and we develop them in sufficient generality to make such applications possible.

  99. L. de Thanhoffer de V\"olcsey and M. Van den Bergh, Some new examples of non-degenerate quiver potentials, International Mathematics Research Notices 20 (2013), 4672-4686.

    Abstract: We prove a technical result which allows us to establish the non-degeneracy of potentials on quivers in some previously unknown cases. Our result applies to McKay quivers and also to potentials derived from geometric helices on Del Pezzo surfaces. On the other hand we also give an example of a skew group ring with a degenerate potential. This shows that for 3-CY orders Iyama-Reiten mutations cannot always be iterated indefinitely.

  100. W. Lowen and M. Van den Bergh, On compact generation of deformed schemes, Advances in Mathematics 244 (2013), 441-464.

    Abstract: We obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's result on compact generation of quasi-compact separated schemes. We prove an application of our theorem to non-commutative deformations of such schemes, based upon a change from Koszul complexes to Chevalley-Eilenberg complexes.

Top

Accepted for publication

  1. B. Kriegk and M. Van den Bergh, Representations of non-commutative quantum groups, to appear in Proceedings of the London Mathematical Society.

    Abstract: We discuss the representation theory of the bialgebra end(A) introduced by Manin. As a side result we give a new proof that Koszul algebras are distributive and furthermore we show that some well-known N-Koszul algebras are also distributive.

  2. L. de Thanhoffer de V\"olcsey and M. Van den Bergh, Explicit models for some stable categories of maximal Cohen-Macaulay modules, to appear in Mathematical Research Letters.

    Abstract: We give concrete DG-descriptions of certain stable categories of maximal Cohen-Macaulay modules.

  3. M. Van den Bergh, Calabi-Yau algebras and superpotentials, to appear in Selecta Mathematica.

    Abstract: We prove that complete d-Calabi-Yau algebras in the sense of Ginzburg are derived from superpotentials.

  4. R. Buchweitz, G. J. Leuschke, and M. Van den Bergh, On the derived category of Grassmannians in arbitrary characteristic, to appear in Compositio Mathematica.

    Abstract: In this paper we consider Grassmannians in arbitrary characteristic. Generalizing Kapranov's well-known characteristic-zero results we construct dual exceptional collections on them (which are however not strong) as well as a tilting bundle. We show that this tilting bundle has a quasi-hereditary endomorphism ring and we identify the standard, costandard, projective and simple modules of the latter.

  5. L. de Thanhoffer de V\"olcsey and M. Van den Bergh, Calabi-Yau deformations and negative cyclic homology, to appear in Journal of Noncommutative Geometry.

    Abstract: In this paper we relate the deformation theory of Ginzburg Calabi-Yau algebras to negative cyclic homology. We do this by exhibiting a DG-Lie algebra that controls this deformation theory and whose homology is negative cyclic homology. We show that the bracket induced on negative cyclic homology coincides with Menichi's string topology bracket. We show in addition that the obstructions against deforming Calabi-Yau algebras are annihilated by the map to periodic cyclic homology. In the commutative we show that our DG-Lie algebra is homotopy equivalent to $(T^poly[[u]],-u div)$.

  6. M. Van den Bergh, On involutivity of p-support, to appear in Int. Math. Res. Not. IMRN.

    Abstract: The p-support of a holonomic D-module was introduced by Kontsevich. Thomas Bitoun in his PhD thesis proved several properties of p-support conjectured by Kontsevich. In this note we give an alternative proof for involutivity by reducing it to a slight extension of Gabber's theorem on the integrability of the characteristic variety. For the benefit of the reader we review how this extension follows from Kaledin's proof of Gabber's theorem.

  7. G. Tabuada and M. Van den Bergh, Noncommutative motives of Azumaya algebras, to appear in Journal de l'Institut Math\'ematiques de Jussieu.

    Abstract: Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients. Assume that 1/r belongs to R. Under this assumption, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Making use of these isomorphisms, we then computer the R-linear additive invariants of differential operators in positive characteristic, of cubic fourfolds containing a plane, of Severi-Brauer varieties, of Clifford algebras, of quadrics, and of finite dimensional k-algebras of finite global dimension. Along the way we establish two results of independent interest. The first one asserts that every element of the Grothendieck group of X which has rank r becomes invertible in the R-linearized Grothendieck group, and the second one that every additive invariant of finite dimensional algebras of finite global dimension is unaffected under nilpotent extensions.

  8. A. Rizzardo and M. Van den Bergh, Scalar extensions of derived categories and non-Fourier-Mukai functors, to appear in Advances in Mathematics.

    Abstract: The famous representability theorem by Orlov asserts that any fully faithful functor between the derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. This result has been extended by Lunts and Orlov to include functors from perfect complexes to quasicoherent complexes. In this paper we show that the latter extension is false without the full faithfulness hypothesis. Our results are based on the properties of scalar extensions of derived categories whose investigation was started by Pawel Sosna and the first author.

Top

Submitted

  1. R. Buchweitz, G. J. Leuschke, and M. Van den Bergh, Non-commutative desingularization of determinantal varieties, II, submitted.

    Abstract: In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the \emph{existence} of non-commutative resolutions in the general case in a manner which is still characteristic free. The explicit description of the resolution by generators and relations is deferred to a later paper. As an application of our results we prove that there is a fully faithful embedding between the bounded derived categories of the two canonical (commutative) resolutions of a determinantal variety, confirming a well-known conjecture of Bondal and Orlov in this special case.

  2. A. Rizzardo and M. Van den Bergh, An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves, submitted.

    Abstract: Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.

  3. D. Presotto and M. Van den Bergh, Noncommutative versions of some classical birational transformations, submitted.

    Abstract: In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct and analogue of the Cremona transform for non-commutative projective planes.

Top

Notes

  1. M. Van den Bergh, Some generalities on $G$-equivariant quasi-coherent $O_X$ and $D_X$-modules, notes.

    Abstract: These are some notes I wrote many years ago. I occasionally get requests for them so I decided to tidy them up and put them here. The main motivation is that there are two possible definitions for the notion of a G-equivariant D-module. The first one is the standard category theoretic definition in terms of the schemes GxX and GxGxX. The second, more convenient one, is in terms of the Lie algebra of G. In these notes we show that these two definitions are equivalent for a connected group. This is certainly well-known but I haven't been able to locate an elementary proof.

  2. M. Van den Bergh, Notes on de Jong's period$=$index theorem for central simple algebras over fields of transcendence degree two, notes.

    Abstract: These are notes on de Jong's proof of the period$=$index theorem over fields of transcendence degree two. They are actually about the ``simplified'' proof sketched by de Jong in the last section of his paper. These notes were meant as support for my lectures at the summer school ``Central Simple Algebras over Function Fields of Surfaces'' at the Universit\"at Konstanz between August, 26 and September, 1 2007 but I did not finish them in time.

Top

Appendices

  1. Y. Berest and G. Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry (with an appendix by Michel Van den Bergh), Int. Math. Res. Not. 26 (2002), 1347-1396.
  2. B. Keller and I. Reiten, Acyclic Calabi-Yau categories, Compos. Math. 144 (2008), 1332-1348.
  3. R. Bocklandt, Graded Calabi Yau algebras of dimension 3, J. Pure Appl. Algebra 212 (2008), 14-32.
  4. B. Keller, Deformed Calabi-Yau completions, J. Reine Angew. Math. 654 (2011), 125-180.
Top