Research

My research interests lie in the field of commutative algebra, a branch of mathematics focused on the study of algebraic equations and their solutions, and representation theory. Graham Leuschke's and my book Cohen-Macaulay Representations appeared in 2012, in the American Mathematical Society's Surveys and Monographs series.<br/ >

Below are links to some of my publications and erreta:

• Factorization theory and decompositions of modules, preprint (with Nicholas Baeth).
• Brauer-Thrall for totally reflexive modules, preprint (with L. W. Christensen, D. A. Jorgensen, H. Rahmati and Janet Striuli).
• Prime ideals in Noetherian rings: a survey, in "Ring and Module Theory", T. Albu, G. F. Birkenmeier, A. Erdogan and A. Tercan, eds., Birkhaueser, Boston, MA, 2010, 175-193 (with S. Wiegand).
• Direct-sum behavior of modules over one-dimensional rings, in "Commutative Algebra", M. Fontana, S. Kabbaj, B. Olberding and I. Swanson, eds., Springer-Verlag, 2010 (with R. Karr).
• Extended modules, J. Commutative Algebra 1 (2009), 481-506 (with W. Hassler).
• Semigroups of modules, in "Rings, Modules and Representations" (International Conference on Rings and Things, 2007), Contemp. Math. 480, 335--349 (with S. Wiegand).
• Ascent of module structure, vanishing of Ext, and extended modules, Michigan Math. J. 57 (2008), 321--337 (with A. Frankild and S. Sather-Wagstaff).
• Big indecomposable modules and direct-sum relations, Illinois J. Math. 51 (2007), 99--122 (with W. Hassler, R. Karr and L. Klingler). Comment: This paper extends the results in ``Large indecomposable modules over local rings" and ``Indecomposable modules of large rank over Cohen-Macaulay local rings" and treats the Cohen-Macaulay and non-Cohen-Macaulay cases in a unified way. One consequence of the new approach is a complete set of invariants for the additive monoid of isomorphism classes of finitely generated R-modules, where R is a commutative Noetherian local ring of dimension one. Erratum: Parts (1) and (3) of Theorem 6.3 are not quite correct as stated, since they do not take into account the fact that some one-dimensional rings, e.g., k[[x,y]]/(x^2,xy) can be proper homomorphic images of Dedekind-like rings. To fix this, change "not Dedekind-like" in Theorem 6.3, Part (1), to "not a homomorphic image of a Dedekind-like ring"; and "Dedekind-like, but not" in Theorem 6.3, Part (2), to "a homomorphic image of a Dedekind-like ring, but is not".
• What is ... a syzygy?, Notices Amer. Math. Soc. 53 (2006), 456-457.
• Direct-sum decompositions over one-dimensional Cohen-Macaulay rings, in "Multiplicative Ideal Theory in Commutative Algebra: a tribute to the work of Robert Gilmer", J. Brewer, S. Glaz, W. Heinzer, eds., Springer, 2006 (with A. Facchini, W. Hassler and L. Klingler).
• Large indecomposable modules over local rings, J. Algebra 303 (2006), 202--215 (with W. Hassler, R. Karr and L.Klingler).
• Big indecomposable mixed modules over hypersurface singularities, in "Abelian Groups, Rings, and Modules", P. Goeters and O. Jenda eds., Lecture Notes in Pure and Appl. Math. 249, CRC/Taylor & Francis Books, 2006 (with W. Hassler). Advice: Read this web version rather than the published version, which somehow got a bit mangled.
• Indecomposable modules of large rank over Cohen-Macaulay local rings, Trans. Amer. Math. Soc. 360 (2008), 1391--1406 (with W. Hassler, R. Karr and L. Klingler).
• Direct-sum cancellation for modules over one-dimensional rings, J. Algebra 283 (2005), 93--124 (with W. Hassler).
• Direct-sum decompositions of modules with semilocal endomorphism rings, J. Algebra 274 (2004), 689-707 (with A. Facchini). Erratum: There is a false statement in the introduction to the paper. This statement does not affect the validity of the results in the paper.
• Local rings of bounded Cohen-Macaulay type, Algebr. Represent. Theory 8 (2005), 225 -- 238 (with G. Leuschke). Erratum: The proof of Theorem 1.5 needs a small repair.
• Hypersurfaces of bounded Cohen-Macaulay type, Algebr. Represent. Theory 8 (2005), 225 -- 238 (with G. Leuschke). Comment: Don't miss the Note added in proof just before the references: The normal form needs a little adjustment when the residue field is not algebraically closed. The adjustment is reproduced here in a more user-friendly form. Erratum: While the statement of the main theorem classifying hypersurfaces of bounded CM type is correct, the argument in the paper does not cover the rings k[[x_0, ..., x_d]]/(x_d^2), where d > 1. Indeed, these rings do not have bounded CM type. Here is a proof of a more general result.
• The Tor Game, in "Commutative Ring Theory: Proceedings of the Fourth International Conference", M. Fontana, S.-E. Kabbaj and S. Wiegand, eds., Marcel Dekker, 2002, 289-300 (with C. Huneke).
• Vanishing theorems for complete intersections, J. Algebra 238 (2001), 684-702 (with C. Huneke and D. Jorgensen).
• Direct-sum decompositions over local rings, J. Algebra 240 (2001), 83-97.
• Ascent of finite Cohen-Macaulay type, J. Algebra 228 (2000), 674-681 (with G. Leuschke). Erratum: The proof of Proposition 1.6 needs a small repair.

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Errata:

• The paper "Tensor products of modules and the rigidity of Tor" (by C. Huneke and R. Wiegand, Math. Ann. 299 (1994), 449--476) has errors caused by an incorrect convention concerning the depth of the zero module. These errors do not affect the validity of the main results of the paper, but several proofs need to be modified. See the erratum for details.
• The paper "Bounds for one-dimensional rings with finite Cohen-Macaulay type" (by R. Wiegand and S. Wiegand, J. Pure Appl. Algebra 93 (1994), 311-342) asserts, in Example 3.12, that there is a local one-dimensional ring with finite Cohen-Macaulay type, having a finitely generated indecomposable torsion-free module of rank 4. Nicholas Baeth has shown, in Section 6 of his paper A Krull-Schmidt for one-dimensional rings of finite Cohen-Macaulay Type, MR 2283435, that the module in question is in fact a direct sum of two modules of rank two. In fact, his results imply, at least in the equicharacteristic-zero case, that the sharp bound on the ranks of the indecomposables is three. One of the main results of the Wiegand /Wiegand paper is that, over a one-dimensional ring (not necessarily local) with bounded Cohen-Macaulay type, each indecomposable torsion-free module has rank 1, 2, 3, 4, 5, 6, 8, 9 or 12. In view of Baeth's results, it is likely that the list can be shortened to the following: 1, 2, 3, 4 or 6, though this has not been worked out in full generality. The construction, in Section 5 of the Wiegand/Wiegand paper, of indecomposable modules (in the non-local case) of ranks 5, 7, 8, 9 and 12, is invalid, since it depends on the incorrect assertion in Example 3.12.
• In the paper "Noetherian rings of bounded representation type", MR 1015536, Lemma 2.4 makes the ludicrous claim that the functor F (reduction modulo an ideal) preserves indecomposability of modules over Artinian pairs. (Consider, for example, the Artinian pair (k[t^3,t^7], k[t]), with t^12 = 0. This is the bottom line of the conductor square for the ring k[[T^3,T^7]], which has infinite representation type. Therefore the Artinian pair has indecomposable modules of arbitrarily large rank. If we take I = tk[t] in the statement of Lemma 2.4, the resulting Artinian pair is (k, k), which has only one indecomposable module. Letting (V,W) be an indecomposable module over (k[t^3,t^7], k[t]), with W free of rank >1, we see that F(V,W) must decompose.) Fortunately this claim is never used in the paper.

Meetings and Conferences

The following are links to conferences I have attended recently or expect to attend soon.

• San Antonio; January 10–13, 2015: Joint Mathematics Meetings
• Bellaterra, Spain; February 8–March 8, 2015: IRTATCA
• Sharjah, United Arab Emirates; April 2–5, 2015: AUS-ICMS15
• Kathmandu, Nepal; April 20–25, 2015; Conference and Earthquake!
• Porto, Portugal, Scotland; June 10–13, 2015: AMS-EMS-SPM
• Isle of Skye, Scotland; June 25, 2015: BensonFest
• Columbia, MO; October 24–25, 2015: KUMUNU 2015
• Seattle; January 6–9, 2016: Joint Mathematics Meetings
• San Marcos, TX; February 27–28, 2016: SLAM 2016
• Fargo, ND; April 16–17, 2016: AMS Spring Central Sectional Meeting

Ph.D. Students

• Mohsen Gheibi (Ph.D. 2018, co-advised by Mark Walker; postdoc, University of Texas-Arlington)
• Neil Steinburg (Ph.D. 2018, co-advised by Tom Marley; Visiting Assistant Professor, Drake University)
• Katharine Shultis (Ph.D. 2015, co-advised by Srikanth Iyengar; Assistant Professor, Gonzaga University)
• Courtney Gibbons (Ph.D. 2013, co-advised by Luchezar Avramov; Assistant Professor, Hamilton College)
• Micah Leamer (Ph.D. 2011, co-advised by Srikanth Iyengar; Postdoctoral Scholar 2011-2012, Chennai Mathematical Institute)
• Olgur Celikbas (Ph.D. 2010, co-advised by Mark Walker; Asst Prof, Univ of West Virginia)
• Silvia Saccon (Ph.D. 2010; Dean's Fellow, University of Texas-Dallas)
• Andrew Crabbe (Ph.D. 2008; Postdoctoral Scholar, Syracuse University; currently attending Perelman School of Medicine)
• Nicholas Baeth (Ph.D. 2005; Assoc. Professor, Central Missouri State University)
• Ryan Karr (Ph.D. 2002; Associate Professor, University of Wisconsin-Parkside)
• Karl Kattchee (Ph.D. 2001; Associate Professor, University of Wisconsin-LaCrosse)
• Graham Leuschke (Ph.D. 2000; Professor, Syracuse University)
• Darren Holley (Ph.D. May 1997; Faculty, Omaha North High School)
• David Jorgensen (Ph.D. 1996; Professor, University of Texas-Arlington)
• Kurt Herzinger (Ph.D. 1996; Professor, US Air Force Academy)
• Nuri Cimen (Ph.D. 1994; Associate Professor, Hacettepe University)
• Bao Ping Jia Bao Ping Jia (Ph.D. 1990; Associate Professor, Maryville University)
• William W. Krauter (Ph.D. 1980; Software Engineer, Lockheed Martin)
• Bette Midgarden (Ph.D. 1978; Professor, Minnesota State University-Moorhead)
• Thomas S. Fischer (Ph.D. 1977; Kaiser Permanente (now retired)