Ragnar-Olaf Buchweitz

Ragnar-Olaf Buchweitz
University of Toronto | U of T · Department of Computer and Mathematical Sciences at Scarborough

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71
Publications
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2,551
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September 1987 - present
University of Toronto Scarborough, Toronto, Canada
Position
  • University of Toronto

Publications

Publications (71)
Article
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We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups G⊆...
Preprint
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We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider c...
Preprint
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We show how Coxeter's work implies a bijection between complex reflection groups of rank two and real reflection groups in $O(3)$. We also consider this magic square of reflections and rotations in the framework of Clifford algebras: we give an interpretation using (s)pin groups and explore these groups in small dimensions.
Article
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In representation theory, commutative algebra and algebraic geometry, it is an important problem to understand when the triangulated category $\mathsf{D}_{\operatorname{sg}}^{\mathbb{Z}}(R)=\text{}\underline{\mathsf{CM}}_{0}^{\mathbb{Z}}R$ admits a tilting (respectively, silting) object for a $\mathbb{Z}$ -graded commutative Gorenstein ring $R=\big...
Chapter
We give an introduction to the McKay correspondence and its connection to quotients of Cⁿ by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants of these reflection groups. This paper is an extended version of E. F.’s talk with the same title delivered at the ICRA.
Article
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We construct a noncommutative desingularization of the discriminant of a finite reflection group $G$ as a quotient of the skew group ring $A=S*G$. If $G$ is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement $\mathcal{A}(G)$ viewed as a module over the coordinate ring $S^G/(\De...
Article
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We give an introduction to the McKay correspondence and its connection to quotients of $\mathbb{C}^n$ by finite reflection groups. This yields a natural construction of noncommutative resolutions of the discriminants of these reflection groups. This paper is an extended version of E.F.'s talk with the same title delivered at the ICRA.
Article
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We give normal forms of determinantal representations of a smooth projective plane cubic in terms of Moore matrices. Building on this, we exhibit matrix factorizations for all indecomposable vector bundles of rank 2 and degree 0 without nonzero sections, also called Ulrich bundles, on such curves.
Article
We determine the product structure on Hochschild cohomology of commutative algebras in low degrees, obtaining the answer in all degrees for complete intersection algebras. As applications, we consider cyclic extension algebras as well as Hochschild and ordinary cohomology of finite abelian groups.
Article
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Platonic solids, Felix Klein, H.S.M. Coxeter and a flap of a swallowtail: The five Platonic solids tetrahedron, cube, octahedron, icosahedron and dodecahedron have always attracted much curiosity from mathematicians, not only for their sheer beauty but also because of their many symmetry properties. In this snapshot we will start from these symmetr...
Article
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Let $\varphi :X\to S$ be a morphism between smooth complex analytic spaces and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi $ is a Cohen–Macaulay $\mathcal {O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi $, then $f\circ \varphi =0$ defines...
Article
The strong global dimension of a ring is the supremum of the length of perfect complexes that are indecomposable in the derived category. In this note we characterize the noetherian commutative rings that have finite strong global dimension. We also give a similar characterization for arbitrary noetherian schemes.
Chapter
We introduce the fundamental group of a morphism in a triangulated category and show that the groupoid of distinguished triangles containing a given extension of objects from an abelian category is equivalent to the Quillen groupoid of the corresponding extension category as studied by Retakh (Uspekhi Mat. Nauk 41:179–180, 1986), Neeman–Retakh (Com...
Article
Let W be a polynomial or power series in several variables, or, more generally, a nonzero element in some regular commutative ring. A matrix factorization of W consists of a pair of square matrices X and Y of the same size, with entries in the given ring, such that the matrix product XY is W multiplied by the identity matrix. For example, if X is a...
Article
We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through the chain rule or by extending them into the tangent bundle....
Article
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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 characteri...
Article
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A topological interpretation of Hochster's Theta pairing of two modules on a hypersurface ring is given in terms of linking numbers. This generalizes results of M. Hochster and proves a conjecture of J. Steenbrink. As a corollary we get that the Theta pairing vanishes for isolated hypersurface singularities in an odd number of variables, as was con...
Article
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We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that 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-commutati...
Article
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We construct a characteristic free tilting bundle on Grassmannians. Comment: Due to some confusion between the authors the posted version was not the latest one. In particular it did not contain an acknowledgement. This is hereby corrected. Mathematically there are some minor clarifications in the proofs
Article
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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, cost...
Article
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Given a k k –scheme X X that admits a tilting object T T , we prove that the Hochschild (co-)homology of X X is isomorphic to that of A = End X ⁡ ( T ) A=\operatorname {End}_{X}(T) . We treat more generally the relative case when X X is flat over an affine scheme Y = Spec ⁡ R Y=\operatorname {Spec} R , and the tilting object satisfies an appropriat...
Article
We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X→Y, the Hochschild complex HX/Y, as introduced in [R.-O. Buchweitz, H. Flenner, Global Hochschild...
Article
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We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C^5. It is shown that such free divis...
Article
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We show that a formal power series ring A[[X]] over a noetherian ring A is not a projective module unless A is artinian. However, if (A, ) is any local ring, then A[[X]] behaves like a projective module in the sense that Extp A (A[[X]], M)=0 for all -adically complete A-modules. The latter result is shown more generally for any flat A-module B inst...
Chapter
This paper analyzes the graded maximal Cohen-Macaulay modules over rings of the form R=k[x1,...,xr]/Q, when Q is a quadratic form defining a regular projective hypersurface, and k is an arbitrary field (the case when k is algebraically closed of characteristic ≠2 is a special case of the theory developed by Knörrer [1986]). For any nonzero quadrati...
Article
It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications in...
Article
We introduce Hochschild (co-)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so-called (derived) Hochschild complex of a morphism; the Hochschild cohomology and homology groups are then the Ext and Tor groups of that complex. We prove that these objects...
Article
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A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a substitute for the length of a free complex--and on the rank of a differential module in terms of invariants of its...
Article
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Thick subcategories of triangulated categories arise in various mathematical areas, for instance in algebraic geometry, in representation theory of groups and algebras, or in stable homotopy theory. The aim of this workshop has been to bring together experts from these fields and to stimulate interaction and exchange of ideas.
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Linear free divisors are free divisors, in the sense of K.Saito, with linear presentation matrix (example: normal crossing divisors). Using techniques of deformation theory on representations of quivers, we exhibit families of linear free divisors as discriminants in representation spaces for real Schur roots of a finite quiver. We review some basi...
Article
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Let Λ = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution 𝔽 of the graded simple modules over Λ is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultip...
Article
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A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix has even size. Establishing a correspondence between such factorizations and extensions of maximal Cohen--Macau...
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We show that the adjoint matrix of a generic square matrix of even size can be factored nontrivially, answering a question of G. Bergman. This note is a preliminary report on work in progress.
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Dieter Happel asked the following question: If the $n$-th Hochschild cohomology group of a finite dimensional algebra $\Gamma$ over a field vanishes for all sufficiently large $n$, is the global dimension of $\Gamma$ finite? We give a negative answer to this question.
Article
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Using Grothendieck's semicontinuity theorem for half-exact functors, we derive two semicontinuity results on Hochschild cohomology. We apply these to show that the first Hochschild cohomogy group of the mesh algebra of a translation quiver over a domain vanishes if and only if the translation quiver is simply connected. We then establish an exact s...
Article
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We investigate how to compare Hochschild cohomology of algebras related by a Morita context. Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent. Along the way, we show that...
Article
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Translation quivers appear naturally in the representation theory of finite dimensional algebras; see, for example, Bongartz and Gabriel (Bongartz, K., Gabriel, P., (1982). Covering spaces in representation theory. Invent. Math. 65:331–378.). A translation quiver defines a mesh algebra over any field. A natural question arises as to whether or not...
Article
Let (R,m,k) be a commutative noetherian local ring with dualizing complex DR, normalized by . Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative) k-algebras of finite rank, we conjecture that if for all n>0, then R is Gorenstein, and prove this in several significant cases.
Article
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We construct families of Artin algebras over fields of arbitrary characteristic that contain a loop in their ordinary quiver but admit no nontrivial outer derivation. This refuses the long-held belief that such algebras should not exist.
Article
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this paper we develop geometric methods for the study of nite modules over a
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This paper contains the details and complete proofs of our earlier announcement in math.AG/9907004 . We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture....
Article
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We construct a general semiregularity map for cycles on a complex analytic or algebraic manifold and show that such semiregularity map can be obtained from the classical tool of the Atiyah-Chern character. The first part of the paper is fairly detailed, deducing the existence and explicit form of a generalized semiregularity map from known results...
Article
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We construct Artin algebras with vanishing first Hochschild Cohomology but a loop in their ordinary quiver. Some authors guessed earlier that such examples could not exist.
Article
. Let M be a finite module over a ring R obtained from a commutative ring Q by factoring out an ideal generated by a regular sequence. The homological properties M over R and over Q are intimately related. Their links are analyzed here from the point of view of differential graded homological algebra over a Koszul complex that resolves R over Q. On...
Article
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. Let M and N be finite graded modules over a graded commutative ring generated over a field K = R0 by homogeneous elements x1 ; : : : ; xe of positive degrees d1 ; : : : ; de . By the Hilbert-Serre Theorem, the Hilbert series P n2Z (rank K Mn )t n is the Laurent expansion around t = 0 of a rational function HM (t) = q M (t)= Q e i=1 (1 Gamma t d i...
Article
We determine the Hilbert-Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this completes the list of Hilbert-Kunz functions of plane cubics. Combining these results with the calculation of the...
Article
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In this paper, we outline the construction of obstructions tok-formality wherek is a field of arbitrary characteristic. We show that the obstructions to intrinsick-formality of a graded algebraH * lie in the Hochschild cohomologyH H m,1(H */k, H *). We calculate explicitly the Hochschild cohomology of the algebraH(ℂP ∞/ℂP n ;k) and deduce that the...
Article
In this paper we show how to compute the parameter space X for the versal deformation of an isolated singularity (V,0) - whose existence was shown by Grauert in 1972, - under the assumptions dim V ≥ 4, depth{o} V ≥ 3, from the CR-structure on a link M of the singularity. We do this by showing that the space X is isomorphic to the space (denoted her...
Article
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Although there has been a lot of work and success lately in the theory of such modules, of which this conference witnessed, it has remained mysterious -at least to the present author -why these modules provide such a powerful tool in studying the algebra and geometry of singula-rities for example. We try to give one answer here, at least for the ca...
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Contents lntroduction 241 1. The Milnor Number 1.1 Definition of μ . . . . . . . . 243 1.2. Some Consequences . . . . . 245 2. Coherence of the Hypercohomology 2.1. Statement of the Theorem . . . . 247 2.2. Proof of the Coherence 249 3. Investigation of the Hypercohomology 3.1. A Hypercohomological Gysin Sequence 251 3.2. Freeness of the Hype...
Article
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For basic material on singularities we will consider, [15, 19, 25] are excellent ref-erences. The fundamentals of matrix factorizations and maximal Cohen–Macaulay modules are well presented in [40]. The monographs [20, 38] contain more than enough to cover the homological algebra used. The lectures are meant as a survey and intended to wet your app...

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