http://www.leuschke.org/Research/RecentChanges Research.Recent Changes Wed, 01 Sep 2010 16:41:50 GMT pmwiki-2.2.0-beta31 http://www.leuschke.org/Research/FactoringTheAdjointAndMaximalCohen-MacaulayModulesOverTheGenericDeterminant http://www.leuschke.org/Research/FactoringTheAdjointAndMaximalCohen-MacaulayModulesOverTheGenericDeterminant 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 factorizations in terms of extensions of maximal Cohen—Macaulay modules over the hypersurface ring defined by the generic determinant. Specifically, a nontrivial factorization wherein one of the factors has determinant equal to the generic determinant gives an extension of the rank-one maximal ... http://www.leuschke.org/Research/Non-commutativeDesingularizationOfDeterminantalVarietiesI http://www.leuschke.org/Research/Non-commutativeDesingularizationOfDeterminantalVarietiesI Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh, Invent. Math. 182 (2010), no. 1, pp. 47—115, DOI: 10.1007/s00222–010–0258–7 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 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. Notes: Watch this space. Up: ... http://www.leuschke.org/Research/WildHypersurfaces http://www.leuschke.org/Research/WildHypersurfaces Joint work with Andrew Crabbe. Download Paper Abstract: Complete hypersurfaces of dimension at least 2 and multiplicity at least 4 have wild Cohen-Macaulay type. Notes: Watch this space. Up: Papers index Previous: A Characteristic Free Tilting Bundle For Grassmannians ... http://www.leuschke.org/Research/ACharacteristicFreeTiltingBundleForGrassmannians http://www.leuschke.org/Research/ACharacteristicFreeTiltingBundleForGrassmannians Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh. Download Paper Abstract: We give a characteristic free tilting bundle for Grassmannians. Notes: This version is more up-to-date than the arXiv copy: it both contains the Acknowledgments that an earlier draft lacked, and has the embarrassing typo fixed. Up: Papers index Previous: Non-commutative desingularization of determinantal varieties, I Next: Wild Hypersurfaces ... http://www.leuschke.org/Research/MCMBook http://www.leuschke.org/Research/MCMBook We (Roger Wiegand and Graham Leuschke) are writing a book about maximal Cohen—Macaulay modules over Cohen—Macaulay local rings. We’ll be putting draft chapters up here as we get them to a presentable state, hoping that friendly passersby will read them and give us feedback. We appreciate any feedback you’re willing to provide: read a chapter, or a section, or some pages, or some collection of results, or just make sure we’re not omitting an important reference. Contact either one of us: rwiegand@math.unl.edu gjleusch@math.syr.edu Of course, we’d also ... http://www.leuschke.org/Research/PresentationsOfRingsWithNon-trivialSelf-orthogonalModules http://www.leuschke.org/Research/PresentationsOfRingsWithNon-trivialSelf-orthogonalModules Joint work with David A. Jorgensen and Sean Sather-Wagstaff. Download Paper (revised version of 16 April 2010, to appear in Collect. Math.) Download Paper (revised version of 23 Nov 2009) Download Paper (initial version of 5 May 2009) 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 local Gorenstein ring Q. We establish an analogous result by showing that such a ring R having a dualizing module admits a non-trivial finitely generated ... http://www.leuschke.org/Research/EndomorphismRingsOfFiniteGlobalDimension http://www.leuschke.org/Research/EndomorphismRingsOfFiniteGlobalDimension Download (PDF) Abstract: For a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = EndR(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, a resolution of singularities of Spec R. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal C-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra Λ with finite global dimension and ... http://www.leuschke.org/Research/Editing http://www.leuschke.org/Research/Editing I am an Associate Editor for the new Journal of Commutative Algebra. Electronic submission is strongly preferred, specifically PDF files, directly via email to gjleusch@math.syr.edu. The submission should be a single file, with a distinctive name. The body of the email message should include the authors’ full names and the full title of the paper. Please see the author instructions for JCA for additional information. ... http://www.leuschke.org/Research/24HoursOfLocalCohomology http://www.leuschke.org/Research/24HoursOfLocalCohomology Joint work with Srikanth Iyengar, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther. Since the book has now appeared (Amazon, AMS), I don’t have a download link here. You can still view a (very, very) preliminary version at the web page for the conference. Abstract: This book is aimed to provide an introduction to local cohomology which takes cognizance of the breadth of its interactions with other areas of mathematics. It covers topics such as the number of defining equations of algebraic sets, connectedness properties of algebraic sets, connections to ... http://www.leuschke.org/Research/Talks http://www.leuschke.org/Research/Talks These are slides and notes from various talks I’ve given. I generally prefer to use a blackboard if there’s one available, and if not I usually hand-write my slides, so this represents only a fraction of the talks I’ve given. Files are in PDF format, and more or less reverse-chronologically ordered. Semidualizing modules and Gorenstein presentations, based on joint work with Dave Jorgensen and Sean Sather-Wagstaff, given at the Winter Meeting of the Canadian Mathematical Society in Ottawa, December 2008. Here’s another version of the same talk, given at the Spring ... http://www.leuschke.org/Research/Papers http://www.leuschke.org/Research/Papers For each paper below, you can either download the PDF version directly from the link provided, or click on the title to go to that paper’s own little page, which has the abstract, MR reference information, and some notes about the paper. Most of these are also available from the arxiv. Wild Hypersurfaces, with Andrew Crabbe. (pdf) A characteristic free tilting bundle for Grassmannians, with Ragnar-Olaf Buchweitz and Michel Van den Bergh. (pdf) Non-commutative Desingularization of Determinantal Varieties I, with Ragnar-Olaf Buchweitz and Michel Van den Bergh, Invent. Math. 182 ... http://www.leuschke.org/Research/HypersurfacesOfBoundedCohen-MacaulayType http://www.leuschke.org/Research/HypersurfacesOfBoundedCohen-MacaulayType 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 modules. As with finite Cohen-Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen-Macaulay type if and only if R \cong k[[x_0,\dots,x_d]]/(g+x_2^2+\dots+x_d^2), where g \in k[[x_0,x_1]] and ... http://www.leuschke.org/Research/OnTheGrowthOfTheBettiSequenceOfTheCanonicalModule http://www.leuschke.org/Research/OnTheGrowthOfTheBettiSequenceOfTheCanonicalModule Joint work with David Jorgensen. Download Paper Download Erratum (revised June 25) 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. We answer the question of exponential growth affirmatively for a large class of rings, and prove that the growth is in general not extremal. As an application of growth, we give criteria for a Cohen—Macaulay ring possessing a canonical module to be Gorenstein. Notes: Several years ... http://www.leuschke.org/Research/CoauthorsAndColleagues http://www.leuschke.org/Research/CoauthorsAndColleagues The biggest, best, most illustrious and highest-rank list of commutative algebraists is the one at commalg.org. Here, I just list (some of!) my collaborators, friends, and other folks whose mathematics I want to be able to have at my fingertips. Collaborators Roger Wiegand Craig Huneke Ian Aberbach Ragnar-Olaf Buchweitz Dave Jorgensen Michel Van den Bergh Colleagues, etc. Florian Enescu [Link requires approval](approve) Srikanth Iyengar [Link requires approval](approve) Moira McDermott Claudia Miller Ezra Miller Idun Reiten [Link requires approval](approve) [Link requires approval](approve) ... http://www.leuschke.org/Research/OnAConjectureOfAuslanderAndReiten http://www.leuschke.org/Research/OnAConjectureOfAuslanderAndReiten 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 ExtiR(M \oplus R , M \oplus R)=0 for all i > 0 then M is projective. We establish the conjecture for excellent Cohen-Macaulay normal domains containing the rational numbers, and slightly more generally. Erratum: There is a small mistake in the proof of the main theorem, which necessitates a small change in the statement. At the beginning of the proof of ... http://www.leuschke.org/Research/ChalabiGenealogy http://www.leuschke.org/Research/ChalabiGenealogy Hm. Looks pretty awful at the moment. I’ll have to work on the formatting some. Chalabi | | Glauberman | | Bruck | | Brauer / / Schmidt Schur | | \— | | Hilbert Fuchs Frobenius | | \ | \---— | | \ | Lindemann \ Kummer Weierstrass | \ \ | | \ \ | ... http://www.leuschke.org/Research/Etc http://www.leuschke.org/Research/Etc Probably a good place for Chalabi Number stuff (though of course that’s all obsolete now that the AMS computes collaboration distances for you), maybe my genealogy, and so on. How about Chalabi’s genealogy? Maybe lecture notes if I ever get around to typing any of them up. ... http://www.leuschke.org/Research/TheF-SignatureAndStrongF-Regularity http://www.leuschke.org/Research/TheF-SignatureAndStrongF-Regularity 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 modules called the F-signature of a local ring of characteristic p (originally we called it the “rational signature” — Craig wanted to get the word ‘signature’ in there somewhere — but it’s just as well we didn’t use that, since the concept turns out to ... http://www.leuschke.org/Research/Research http://www.leuschke.org/Research/Research Analysis is the study of estimates; Topology is the study of nearness; Combinatorics is the study of counting; Algebra is the study of tautologies. Executive Summary: My research interests are in commutative algebra, with leanings toward the homological. My current research is in the representation theory of local rings. In practice, this means thinking about relationships between the structure of a ring and the structure of the category of maximal Cohen-Macaulay modules over the ring. My work is funded by the National Science Foundation and has been funded in the past by the National ... http://www.leuschke.org/Research/PoorPurePercyP http://www.leuschke.org/Research/PoorPurePercyP I first saw this poem outside some professor’s door at the University of Nebraska (who? Gary Meisters, maybe?) around 1995. I don’t know who the original author is. Percy P was a mathematicianwhose “pureness” was never denied.But he found one day, to his sorrow,that his theorems had been applied!He had used all the standard precautions;his papers were pointedly dry!But his own esoteric notationhad been solved by a physicist spy!The colloquium buzzed with the gossip;he could offer no valid excuse.Percy P was a traitor of traitors,for his work was of PRACTICAL ...