# On the derived category of Grassmannians in arbitrary characteristic

Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh

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.

Notes:
This is a complete revision of the original note, now four times as long with a lot of new material. Rather than folding it into part 2 of the determinantal varieties paper, we decided to publish it on its own.

# Non-commutative desingularization of determinantal varieties, II: Arbitrary minors

Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh. DOI: 10.1093/imrn/rnv207

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 existence of non-commutative resolutions in the general case in a manner which is still characteristic free, and carry out the explicit description by generators and relations in characteristic zero. 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.

Notes:

Watch this space.

# Brauer-Thrall theory for maximal Cohen-Macaulay modules

Joint work with Roger Wiegand.

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 existence of lots of indecomposable modules of arbitrarily large k-dimension. These conjectures have natural interpretations in the context of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings. This is a survey of progress on these transplanted conjectures.

Notes:

Much of this material appears in our book, but here it is collected in one place, and explained a bit more clearly. The paper also lists some open questions.

Bibtex code:

@incollection {MR3051386,
AUTHOR = {Leuschke, Graham J. and Wiegand, Roger},
TITLE = {Brauer-{T}hrall theory for maximal {C}ohen-{M}acaulay modules},
BOOKTITLE = {Commutative algebra},
PAGES = {577--592},
PUBLISHER = {Springer},
YEAR = {2013},
MRCLASS = {13C14},
MRNUMBER = {3051386},
DOI = {10.1007/978-1-4614-5292-8_18},
URL = {http://dx.doi.org/10.1007/978-1-4614-5292-8_18},
}

# Non-commutative crepant resolutions: scenes from categorical geometry

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 program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh’s definition within these contexts and describe some of the current research in the area.

Notes:

I was invited to write this expository paper for the proceedings that was put together for three AMS special sessions held in Boca Raton in 2009. I now think I was just too ambitious — it would have been better if I’d made some hard decisions at the beginning, about what to assume and what to say. As it happened, the thing just kept growing and growing. Also, I don’t remember why I included “scenes” in the title; that should have been changed, since it’s basically a non sequitur.

Bibtex code:

@incollection {MR2932589,
AUTHOR = {Leuschke, Graham J.},
TITLE = {Non-commutative crepant resolutions: scenes from categorical
geometry},
BOOKTITLE = {Progress in commutative algebra 1},
PAGES = {293--361},
PUBLISHER = {de Gruyter},
YEAR = {2012},
MRCLASS = {14A22 (14E15 14F05 16E35 16S38)},
MRNUMBER = {2932589},
}

# Presentations of rings with non-trivial semidualizing modules

Joint work with David A. Jorgensen and Sean Sather-Wagstaff.

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 self-orthogonal module C satisfying Hom_R(C,C) \cong R if and only if R is the homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a non-trivial way, forcing significant structural requirements on the ring R.

Notes:

We started working this out at the conference for Mel Hochster in Ann Arbor in 2008. I think the paper contains a very satisfying answer to a natural question. It’s a bit technical, but still quite pretty in my opinion.

BibTeX code:

@article {MR2909823,
AUTHOR = {Jorgensen, David A. and Leuschke, Graham J. and
Sather-Wagstaff, Sean},
TITLE = {Presentations of rings with non-trivial semidualizing modules},
JOURNAL = {Collect. Math.},
FJOURNAL = {Collectanea Mathematica},
VOLUME = {63},
YEAR = {2012},
NUMBER = {2},
PAGES = {165--180},
ISSN = {0010-0757},
MRCLASS = {13D07 (13H10)},
MRNUMBER = {2909823},
MRREVIEWER = {Lars Winther Christensen},
DOI = {10.1007/s13348-010-0024-6},
URL = {http://dx.doi.org/10.1007/s13348-010-0024-6},
}

# Wild Hypersurfaces

Joint work with Andrew Crabbe.

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 spent the most time on. It began as an effort to understand Bondarenko’s result that two-dimensional hypersurfaces of multiplicity four or more have wild representation type. In working through his proof, we found some simplifications that allowed the argument to actually work for all dimensions at least two. For the higher dimensions, the result seems to be “morally” known thanks to earlier work of Buchweitz-Greuel-Schreyer, but we thought it was worth writing down explicitly. Among other things, it lets us introduce the idea of tameness and wildness to a commutative-algebraist audience, which I’ve wanted to do for a while.

Bibtex code:

@article {MR2811571,
AUTHOR = {Crabbe, Andrew and Leuschke, Graham J.},
TITLE = {Wild hypersurfaces},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {215},
YEAR = {2011},
NUMBER = {12},
PAGES = {2884--2891},
ISSN = {0022-4049},
CODEN = {JPAAA2},
MRCLASS = {13C14 (13H10 16G60)},
MRNUMBER = {2811571 (2012e:13018)},
MRREVIEWER = {Geoffrey D. Dietz},
DOI = {10.1016/j.jpaa.2011.04.009},
URL = {http://dx.doi.org/10.1016/j.jpaa.2011.04.009},
}

# Non-commutative desingularization of determinantal varieties, I: Maximal minors

Joint work with Ragnar-Olaf Buchweitz and Michel Van den Bergh

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:
This paper grew out of some observations Buchweitz and I made while working on our previous paper. Michel showed us how to verify those observations, and then much more followed. It took over five years for us to finish the paper, mostly because we kept improving the results. Finally it was declared closed, with “, I” added to the end so that we could continue with a part II.

BibTeX code:

 @article {MR2672281,
AUTHOR = {Buchweitz, Ragnar-Olaf and Leuschke, Graham J. and Van den
Bergh, Michel},
TITLE = {Non-commutative desingularization of determinantal varieties
{I}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {182},
YEAR = {2010},
NUMBER = {1},
PAGES = {47–115},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {13C14 (14A22 14E15 14M12 16E10 16S38)},
MRNUMBER = {2672281},
DOI = {10.1007/s00222-010-0258-7},
URL = {http://dx.doi.org/10.1007/s00222-010-0258-7}
}


# Factoring the adjoint and MCM modules over the generic determinant

Joint work with Ragnar-Olaf Buchweitz.

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 Cohen–Macaulay (MCM) modules; we construct and classify all of these, as well as the corresponding factorizations of the adjoint. The classification shows that even in rank two, the MCM-representation theory of the generic determinant is quite wild. We also describe completely the Ext-algebra of the rank-one maximal Cohen–Macaulay modules.

Notes:
Of all my research, this paper is the stuff that it’s most fun to give talks about. I gave colloquia and seminar talks about factoring the adjoint for nearly 4 years. I imagine people were starting to get tired of it, but it was just so much fun. The fact that it brought in “combing the hairy ball”, elementary matrix stuff that undergrads can follow, and concrete, explicit constructions with abstract-seeming things like Ext and free resolutions, made it a real pleasure to tell people about.

Bibtex code:

 @article {MR2343380,
AUTHOR = {Buchweitz, Ragnar-Olaf and Leuschke, Graham J.},
TITLE = {Factoring the adjoint and maximal {C}ohen-{M}acaulay modules
over the generic determinant},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {129},
YEAR = {2007},
NUMBER = {4},
PAGES = {943–981},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {13C14},
MRNUMBER = {MR2343380},
}

# On the growth of the Betti sequence of the canonical module

Joint work with David Jorgensen.

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 ago, when I was a postdoc at Kansas, Dave Jorgensen spent a couple of summers visiting Lawrence to work with Craig Huneke and Dan Katz. Somewhere in there, we heard from Craig the main question this paper tries to answer: If the Betti numbers of the canonical module are bounded, must it have finite projective dimension? I believe it was motivated by some work that Craig had been doing with Doug Hanes. The question has stuck in our craws ever since: it’s so easy to state, but we’ve never been able to get a good clear answer to it. Many hundreds of examples lead us to believe it’s true. After three or four years of batting it back and forth intermittently, we felt like we had made enough progress for a short paper.

Erratum:
Three results appearing in Section 2 are incorrect as stated. The second part of Lemma 2.1, which assumes that $\Ext^i_R(M,N^\vee)=0$ for all $i$ in a certain range andconcludes an inequality on the Betti numbers of $N$, is not true for the stated range of $\Ext$-vanishing. The correction forces changes in the statements of Theorems 2.2 and 2.4 as well. Here is an erratum explaining and repairing the mistake.

The erratum above also fixes a bit of sloppy writing in the same Lemma 2.1, where the assumptions on n weren’t exactly clear. Thanks to Roger Wiegand for pointing this out to us.

Bibtex code:

 @article {MR2299575,
AUTHOR = {Jorgensen, David A. and Leuschke, Graham J.},
TITLE = {On the growth of the {B}etti sequence of the canonical module},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {256},
YEAR = {2007},
NUMBER = {3},
PAGES = {647–659},
ISSN = {0025-5874},
CODEN = {MAZEAX},
MRCLASS = {13Dxx},
MRNUMBER = {MR2299575},
}

# Endomorphism rings of finite global dimension

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 which is maximal Cohen–Macaulay over R (a “noncommutative crepant resolution of singularities”). We produce algebras Λ = EndR(M) having finite global dimension in two contexts: when R is a reduced one-dimensional complete local ring, or when R is a Cohen-Macaulay local ring of finite Cohen-Macaulay type. If in the latter case R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.

Notes:
I started working on this paper at MSRI in Spring 2003, after a series of conversations with Ragnar-Olaf Buchweitz. We spent hours talking about Auslander’s work on representation dimension, the McKay correspondence and particularly the work of Ito-Nakamura and Bridgeland-King-Reid, Van den Bergh’s recent preprints on noncommutative crepant resolutions of singularities, Lipman’s work on Arf rings, and a whole circle of ideas I hadn’t encountered before. In the end it turned out that Osamu Iyama had beaten me to the punch on one of the main results of the paper (the case of dimension one) in “Rejective subcategories of artin algebras and orders“, though my result gives information that his doesn’t (and I understand mine) The other main result probably raises more questions than it answers, but it is a nice bridge from the topic that I’ve been working on for the last several years to several new ones. This is probably the paper that I’ve learned the most from writing.

Errata: There are several small-to-medium errors in the paper; thanks to Hailong Dao, Igor Burban, and Lucho Avramov for pointing them out.

The proof of Proposition 8 does not work for complete local rings of mixed characteristic that are not domains. Such a ring need not contain a regular local ring over which it is a finitely generated module. Since finite CM type implies isolated singularity, as long as the dimension is at least two this is no problem. For dimension one, on the other hand, this argument fails. (Question: if $R$ is a complete $p$-ring and $A = R[[x]]/(px)$, does $A$ have finite CM type? It’s known that $A$ contains no complete RLR with the same residue field as $A$, over which it is module-finite.) Luckily, Iyama’s results in dimension one are more than strong enough to rescue the statement of Prop. 8 in that case.

In the discussion in section 3, I falsely assert that “Theorem 6 implies that if R is Gorenstein of finite CM type and dimension 2 or 3, then R has a non-commutative crepant resolution”. In dimension 2, this is correct, but fails in dimension 3. A counterexample is already in Van den Bergh’s paper “Three-dimensional flops and noncommutative rings”: the 3-dimensional A1 singularity has no non-commutative crepant resolution. It is apparently well-known to geometers that, among the 3-dimensional simple singularities, only A2n+1 and D2n have crepant resolutions.

Bibtex code:

 @article {MR2310620,
AUTHOR = {Leuschke, Graham J.},
TITLE = {Endomorphism rings of finite global dimension},
FJOURNAL = {Canadian Journal of Mathematics.
VOLUME = {59},
YEAR = {2007},
NUMBER = {2},
PAGES = {332–342},
ISSN = {0008-414X},
CODEN = {CJMAAB},
MRCLASS = {16G60},
MRNUMBER = {MR2310620},
}

# Appendix: Some Examples in Tight Closure

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 session at the Introductory Workshop that MSRI had at the beginning of the 2002-2003 Special Year in Commutative Algebra. The workshop consisted of series of lectures by Mel Hochster, Mark Haiman, Dave Benson, Rob Lazarsfeld, David Eisenbud, and Bernard Teissier, and each of them had a “helper”, who gave a single help session. The idea of the help session was that senior mathematicians were not allowed, and participation from the younger participants was explicitly encouraged. I was Mel Hochster’s “helper”, and rambled on for an hour about how hard it is to compute examples in tight closure. I’ve never worked in tight closure, so it was a fun session to prepare and a very hard paper to write.

Bibtex code:

 @incollection {MR2132652,
AUTHOR = {Hochster, Melvin},
TITLE = {Tight closure theory and characteristic {$p$} methods},
BOOKTITLE = {Trends in commutative algebra},
SERIES = {Math. Sci. Res. Inst. Publ.},
VOLUME = {51},
PAGES = {181–210},
NOTE = {With an appendix by Graham J. Leuschke},
PUBLISHER = {Cambridge Univ. Press},
YEAR = {2004},
MRCLASS = {13A35},
MRNUMBER = {MR2132652},
}

# Local rings of bounded Cohen-Macaulay type

Joint work with Roger Wiegand.

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 CM type. We classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer-Thrall theorem for excellent rings.

Notes:
This is the second of a pair of papers I wrote with Roger Wiegand, completely settling the question of which equicharacteristic one-dimensional rings have a bound on the ranks of indecomposable torsion-free modules. It’s a little surprising that the answer is almost completely contained in the first paper (Hypersurfaces of bounded Cohen-Macaulay type): in addition to the two hypersurfaces, there is only one additional isomorphism class. The second section, on ascent to and descent from the completion, is very satisfying.

Bibtex code:

 @article {MR2162283,
AUTHOR = {Leuschke, Graham J. and Wiegand, Roger},
TITLE = {Local rings of bounded {C}ohen-{M}acaulay type},
JOURNAL = {Algebr. Represent. Theory},
FJOURNAL = {Algebras and Representation Theory},
VOLUME = {8},
YEAR = {2005},
NUMBER = {2},
PAGES = {225–238},
ISSN = {1386-923X},
CODEN = {ARTHF4},
MRCLASS = {13C14 (13H10 13H15)},
MRNUMBER = {MR2162283 (2006c:13013)},
MRREVIEWER = {Geoffrey D. Dietz},
}

# Hypersurfaces of bounded Cohen-Macaulay type

Joint work with Roger Wiegand.

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[[x_0,x_1]]/(g) has bounded Cohen-Macaulay type. Up to analytic isomorphism, only two of the form k[[x_0,x_1]]/(g) have bounded Cohen-Macaulay type:g=x_1^2 and g=x_0x_1^2.

Notes:
This is the first of a pair of papers that I wrote with Roger Wiegand in 2002. There were two particularly fun parts: the general construction that shows that every hypersurface of bounded Cohen-Macaulay type comes from one of dimension one, and the delicate calculations that rule out most candidates in dimension one. Roger and I passed this manuscript back and forth by email for months (changing it from LaTeX to AMSTeX and back again each time).

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.

Erratum: There is an error in the proof of the classification in dimensions higher than one. While the statement of the main theorem 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.

Bibtex code:

 @article {MR2158755,
AUTHOR = {Leuschke, Graham J. and Wiegand, Roger},
TITLE = {Hypersurfaces of bounded {C}ohen-{M}acaulay type},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {201},
YEAR = {2005},
NUMBER = {1-3},
PAGES = {204–217},
ISSN = {0022-4049},
CODEN = {JPAAA2},
MRCLASS = {13C14 (13C05 13H10 13H15)},
MRNUMBER = {MR2158755 (2006c:13014)},
MRREVIEWER = {Geoffrey D. Dietz},
}

# On a conjecture of Auslander and Reiten

Joint work with Craig Huneke.

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 Theorem 1.3, we say that the case d=1 follows from results of Auslander-Ding-Solberg. To apply the ADS results, however, we need one additional Ext to vanish: we have to assume both Ext1(M,M) and Ext2(M,M) are zero. Thus the statement of Theorem 1.3 should include this assumption in the case d=1. Of course, in the context of the Auslander-Reiten conjecture, this is not a major problem.

Notes:
The first version of this paper was an utter failure. I had an argument using spectral sequences (which I learned about from Foxby’s paper “On the μi…”) that I thought gave the full result. I spoke on the theorem several places (including Illinois, when I visited there around Thanksgiving 2001), but always swept the details under the rug (because who wants to see a spectral-sequence argument?). Finally Lucho Avramov sat me down and asked to see the details, and immediately pointed out where I had gone wrong. I was mortified. Luckily, Craig got interested in the question, and convinced me to keep thinking about it. (In fact, I’d originally heard the question from him, and he’d learned it from Ragnar-Olaf Buchweitz.) We were able to put together a very nice argument to answer almost all cases. The assumptions that the ring be a quotient of a normal domain by a regular sequence, and/or contain a field, are still quite frustrating, though, since they obviously have nothing to do with the question.

Bibtex code:

 @article {MR2052636,
AUTHOR = {Huneke, Craig and Leuschke, Graham J.},
TITLE = {On a conjecture of {A}uslander and {R}eiten},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {275},
YEAR = {2004},
NUMBER = {2},
PAGES = {781–790},
ISSN = {0021-8693},
CODEN = {JALGA 4},
MRCLASS = {13D07 (13H10)},
MRNUMBER = {MR2052636 (2005a:13033)},
MRREVIEWER = {Juan Ramon Garc{\'{\i}}a Rozas},
}

# The F-signature and strong F-regularity

Joint work with Ian Aberbach.

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 have nothing to do with F-rationality). It’s caught on, which is immensely satisfying. Before the paper even appeared, Ian Aberbach emailed me to say he had some ideas about how to improve one of the theorems about the F-signature. He invited me to Missouri to give a talk and to chat about it. We worked out the details in a very swanky coffeehouse in Columbia.

Bibtex code:

 @article {MR1960123,
AUTHOR = {Aberbach, Ian M. and Leuschke, Graham J.},
TITLE = {The {$F$}-signature and strong {$F$}-regularity},
JOURNAL = {Math. Res. Lett.},
FJOURNAL = {Mathematical Research Letters},
VOLUME = {10},
YEAR = {2003},
NUMBER = {1},
PAGES = {51–56},
ISSN = {1073-2780},
MRCLASS = {13A35},
MRNUMBER = {MR1960123 (2004b:13003)},
MRREVIEWER = {Karen E. Smith},
}

# Local rings of countable Cohen-Macaulay type

Joint work with Craig Huneke.

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 countable CM type is again of countable CM type.

Notes:
This is one of a series of papers that I wrote with Craig Huneke while I was at Kansas. Schreyer’s 1987 survey article on finite and countable CM type has been a wellspring of ideas for me (and, I think, others who think about these things). Craig and I were mostly focused (as I recall) on proving things for countable CM type that were already known for finite CM type, like descent from the completion, etc. We were especially interested in applying the techniques we’d used to prove Auslander’s theorem (in Two theorems about maximal Cohen-Macaulay modules) to the more general situation. It didn’t turn out that way, but we were able to give what I think are very nice arguments, one using prime avoidance and a pigeonhole argument, and one using CM approximations. Even more interesting, we gave a couple of quite tempting conjectures about the relationship between the singular locus, countable type, and finite type. Morally speaking, the only thing that distinguishes between finite and countable type should be the singular locus.

Bibtex code:

 @article {MR1993205,
AUTHOR = {Huneke, Craig and Leuschke, Graham J.},
TITLE = {Local rings of countable {C}ohen-{M}acaulay type},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {131},
YEAR = {2003},
NUMBER = {10},
PAGES = {3003–3007 (electronic)},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {13C14 (13C05 13H10)},
MRNUMBER = {MR1993205 (2005a:13021)},
MRREVIEWER = {Sunsook Noh},
}

# Two theorems about maximal Cohen-Macaulay modules

Joint work with Craig Huneke.

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 ranks up to the sum of the ranks of M and N. This has several corollaries. In particular it proves that a Cohen-Macaulay local ring of finite Cohen-Macaulay type has an isolated singularity. A well-known theorem of Auslander gives the same conclusion but requires that the ring be Henselian. Other corollaries of our result include statements concerning when a ring is Gorenstein or a complete intersection on the punctured spectrum, and the recent theorem of Leuschke and Wiegand that the completion of an excellent Cohen-Macaulay local ring of finite Cohen-Macaulay type is again of finite Cohen-Macaulay type. The second theorem proves that a complete local Gorenstein domain of positive characteristic p and dimension d is F-rational if and only if the number of copies of R splitting out of R1/p^e divided by pde has a positive limit. This result generalizes work of Smith and Van den Bergh. We call this limit the F-signature of the ring and give some of its properties.
Notes:

This was the first paper that Craig Huneke and I wrote together, shortly after I got to Kansas. It’s got two completely separate sections, one about Ext groups and one about rings of prime characteristic. I thought of the first section as being essentially about rings of finite Cohen-Macaulay type, but Craig saw that it was really a theorem about how Exts behave. Like all of his best theorems (and like our result in On a Conjecture of Auslander and Reiten), it’s an effective result, in that it gives more precise numerical data than is a priori needed. In retrospect, I almost wish we’d published this first part separately – I think it’s important enough to stand on its own, and even to get placed in a very good journal. The second section was my first introduction to thinking about tight closure, and I’m still very pleased to have been able to bring something to that theory.

Bibtex code:

 @article {MR1933863,
AUTHOR = {Huneke, Craig and Leuschke, Graham J.},
TITLE = {Two theorems about maximal {C}ohen-{M}acaulay modules},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {324},
YEAR = {2002},
NUMBER = {2},
PAGES = {391–404},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {13C14 (13A35 13D07 13H10)},
MRNUMBER = {MR1933863 (2003j:13011)},
MRREVIEWER = {{\=I}. {\=I}. Burban},
}

# Gorenstein modules, finite index and finite Cohen-Macaulay type

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 spectrum. We use this to show that a Cohen-Macaulay local ring of finite Cohen-Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is a Zariski-open set.

Notes:
My favorite memory about this paper is that I realized the key to the proof of the main theorem while driving from Indiana to Pennsylvania to visit my friend Nick. About halfway from Lafayette to Pittsburgh, somewhere in rural Ohio (which is very pretty in March), I suddenly had the proof in my mind, clear as day. I got off the highway at the next exit, which happened to be a park with a reservoir called Senecaville Lake, and wrote it down using the hood of my car as a desk. At my dissertation defense, I credited the theorem to “– and Senecaville Lake”, which seemed really witty at the time.

Bibtex code:

 @article {MR1894057,
AUTHOR = {Leuschke, Graham J.},
TITLE = {Gorenstein modules, finite index, and finite
{C}ohen-{M}acaulay type},
JOURNAL = {Comm. Algebra},
FJOURNAL = {Communications in Algebra},
VOLUME = {30},
YEAR = {2002},
NUMBER = {4},
PAGES = {2023–2035},
ISSN = {0092-7872},
CODEN = {COALDM},
MRCLASS = {13C14 (13D02 13D05 13H10)},
MRNUMBER = {MR1894057 (2003a:13010)},
MRREVIEWER = {Liam O'Carroll},
}

# Mixed-characteristic hypersurfaces of finite Cohen-Macaulay type

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 also have finite Cohen-Macaulay type.

Notes:
This was the final third of my dissertation. It’s one of those classic things that end up in dissertations because nobody but a graduate student would ever spend the time checking the details. There’s very little of interest here for most people, except perhaps the statement of the main theorem. For me personally, though, it was a great learning experience, if only because I had to work carefully through the matrix operations that are hinted at in Yoshino’s book.

Bibtex code:

 @article {MR1874543,
AUTHOR = {Leuschke, Graham J.},
TITLE = {Mixed characteristic hypersurfaces of finite
{C}ohen-{M}acaulay type},
JOURNAL = {J. Pure Appl. Algebra},
FJOURNAL = {Journal of Pure and Applied Algebra},
VOLUME = {167},
YEAR = {2002},
NUMBER = {2-3},
PAGES = {225–257},
ISSN = {0022-4049},
CODEN = {JPAAA2},
MRCLASS = {13C14 (13H10 14B05)},
MRNUMBER = {MR1874543 (2002k:13019)},
MRREVIEWER = {Marcel Morales},
}

# Ascent of finite Cohen-Macaulay type

Joint work with Roger Wiegand.

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 can fail in general.

Notes:
This was the first paper I wrote, and it’s joint work with Roger Wiegand. Learning to write a paper (which I must admit I’m still doing) is incredibly difficult, and this one took me a very long time. I’m quite proud of how it turned out, though. One bit I particularly like is the ephemeral concept of “finite syzygy type”, which immediately turns out to be the same thing as “finite Cohen–Macaulay type” for CM rings. One of these days I’ll come back to this idea and see how badly it fails for non-CM rings (assumption: infinitely badly).

Bibtex code:

 @article {MR1764587,
AUTHOR = {Leuschke, Graham and Wiegand, Roger},
TITLE = {Ascent of finite {C}ohen-{M}acaulay type},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {228},
YEAR = {2000},
NUMBER = {2},
PAGES = {674–681},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {13H10 (13C14)},
MRNUMBER = {MR1764587 (2001k:13035)},
MRREVIEWER = {Sunsook Noh},
}