Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh Download Paper (published version of 2015-07-22) 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 … Continue Reading ››

Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh. DOI: 10.1093/imrn/rnv207 Download Paper (final version) Download Paper (preliminary version of 2011-06-09) 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 … Continue Reading ››

Joint work with Roger Wiegand. Download Abstract: The Brauer-Thrall Conjectures, now theorems, were originally stated for finitely generated modules over a finite-dimensional k-algebra. They say, roughly speaking, that infinite representation type implies the … Continue Reading ››

Download Paper (final arxiv version of 2011-08-22, updated with bugfixes and adjustments) Abstract: Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model … Continue Reading ››

Joint work with David A. Jorgensen and Sean Sather-Wagstaff. Download Paper (published version) Abstract: A result of Foxby, Reiten and Sharp says that a commutative noetherian local ring R admits a dualizing module if and only if R is Cohen–Macaulay and a homomorphic image of a … Continue Reading ››

Joint work with Andrew Crabbe. Download Paper (revised version of 2011-03-08) Abstract: Complete hypersurfaces of dimension at least 2 and multiplicity at least 4 have wild Cohen-Macaulay type. Notes: Andrew was a postdoc here at Syracuse 2008–2010. We worked on a couple of projects, but this was the one we … Continue Reading ››

Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh Download Paper Abstract: 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 … Continue Reading ››

Joint work with Ragnar-Olaf Buchweitz. Download Abstract: G. M. Bergman recently asked whether the adjoint of the generic square matrix can be factored as a product of square matrices. He showed that in most cases there are no nontrivial factorizations. We recast the existence of such … Continue Reading ››

Joint work with David Jorgensen. Download Paper Download Erratum (revised June 25 2008) Abstract: We study the growth of the Betti sequence of the canonical module of a Cohen–Macaulay local ring. It is an open question whether this sequence grows exponentially whenever the ring is not Gorenstein. … Continue Reading ››

Download Abstract: For a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = End_R(M), where M is a reflexive R-module with nonzero free direct summand. Such algebras Λ of finite global dimension can be viewed as potential substitutes for, or analogues of, … Continue Reading ››

Download Abstract: This is a writeup of the help session that I led at MSRI in September 2003 after Mel Hochster’s lectures on tight closure. Notes: This isn’t really a proper paper. It’s a reworked version of the notes that I used when I ran a help … Continue Reading ››

Joint work with Roger Wiegand. Download here or from A&RT Abstract: It is known that a one-dimensional CM local ring R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite … Continue Reading ››

Joint work with Roger Wiegand. Download Abstract: We investigate which rings of the form R = k[[x_0,\dots,x_d]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring, have bounded Cohen-Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen-Macaulay … Continue Reading ››

Joint work with Craig Huneke. Download Abstract: We prove a conjecture of Auslander and Reiten for a large class of Cohen-Macaulay rings. The conjecture, proposed in 1975, states that if M is a finitely generated R-module such that Extⁱ_R(M \oplus R , M \oplus R)=0 for … Continue Reading ››

Joint work with Ian Aberbach. Download Abstract: We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular. Notes: Craig Huneke and I defined something in Two theorems about maximal Cohen-Macaulay … Continue Reading ››

Joint work with Craig Huneke. Download here or JSTOR Abstract: We prove (the excellent case of) Schreyer’s conjecture that a local ring with countable CM type has at most a one-dimensional singular locus. Furthermore we prove that the localization of a Cohen-Macaulay local ring of … Continue Reading ››

Joint work with Craig Huneke. Download Abstract: This paper contains two theorems concerning the theory of maximal Cohen-Macaulay modules. The first theorem proves that certain Ext groups between maximal Cohen-Macaulay modules M and N must have finite length, provided only finitely many isomorphism classes of maximal Cohen–Macaulay modules exist having … Continue Reading ››

Download Abstract: A Gorenstein module over a local ring R is a maximal Cohen-Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen–Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured … Continue Reading ››

Download Abstract: We define the “mixed ADE singularities”, which are generalizations of the ADE plane curve singularities to the case of mixed characteristic. The ADE plane curve singularities are precisely the equicharacteristic plane curve singularities of finite Cohen-Macaulay type; we show that the mixed ADE singularities … Continue Reading ››

Joint work with Roger Wiegand. Download Abstract: In 1987 F.-O. Schreyer conjectured that a local ring has finite Cohen-Macaulay type if and only if the completion has finite Cohen-Macaulay type. We prove the conjecture for excellent Cohen-Macaulay local rings and also show by example that it … Continue Reading ››