
On liftable and weakly liftable modules
J. Algebra, 318 (2007), 723736.
comments
Mel Hochster formulated two lifting questions which, if true, would settle Serre's Positivity Conjecture. I showed that the
first one has a negative answer, and the second one quite likely so. But the work grows to general criteria for
weak liftability, which allows us to give simple and concrete anwers to whether one can weakly lift a module.
Interestingly, there is necessary condition for an ideal to be annihilator of a weakly liftable module. This is actually my first thesis
problem and was done during my third year. I never included this in my thesis out of sheer laziness and the fact that the results seemed discouraging
(to try to attack the Positivity question this way). After graduation, I came back and realized that this has more applications to the other general
questions on lifting and decided to write it up properly. Well, and one needs papers to get a job!
 Some observations on local and projective hypersurfaces
Math. Res. Lett., 15 (2008), 207219.
comments
This is a continuation of 8! Specifically, I made a conjecture (I think it is better to make conjectures early, since you have no reputation
to lose) that Hochster's theta function would always vanish when R is a hypersurface with isolated singularity and even dimension. In simple terms it says that the for any pair of finitely generated modules over R, the Tor_i(M,N) modules will eventually have constant length when i>>0.
When R is the local ring at the vertex of a smooth projective hypersurfaces, this would follow from a conjecture by Hartshorne and even grander ones
from KTheory (surely made me feel better about making it). As a consequence, over such hypersurfaces, any module would be decent and rigid, much like
the regular case. After coming to Utah, I was able to prove the conjecture and its consequences for certain cases, using Tate and ladic cohomologies.
Such cases are enough to extend some theorems by Auslander and others about Hom(M,M) over regular local rings to local hypersurfaces with isolated
singularity and even dimension. Another nice consequence is a splitting criterion for vector bundles over projective hypersurfaces. UPDATE: the graded, characteristic 0 case has recently been settled in a beautiful paper by Moore, Piepmeyer, Spiroff and Walker. UPDATE 2: the characteristic 0 case now follows from work by PolyshchukVaintrob using dg categories, and the local analytic case in char 0 was recently proved by BuchweitzVan Straten using topological methods.
 On injectivity of maps between Grothendieck groups induced by completion
Mich. Math. J, 57 (2008), 195199.
comments
At my only second conference during grad school, I went to LipmanFest with Mel and heard him talked about some (thirteen) open
questions in commutative algebra. At the end of that talk, he mentioned an example of a local ring R whose map from the Grothendieck
group of R to that of R^ is not injective. Of course, at that point I had no idea what Grothendieck groups are, so I completely forgot
about it. Several years later, I went to Utah for a job talk, and Anurag Singh told me about the problem of finding a normal example
(Mel's example was not normal, and he predicted there would be a normal one). By now, I had much more appreciation for Grothendieck groups,
so I was completely hooked. After several failures, while reading
Swan's paper on Ktheory of hypersurfaces, I realized it should give me what I needed. The example also raised some interesting
questions about the kernel of the map between the Grothendieck groups of a ring and its completion, which should be settled someday.
 Comparing complexities of pairs of modules
(with Oana Veliche)
J. Algebra, 322 (2009), 30473062.
comments
This is my first collaborative paper. I had approached several people about working together since grad school, but with
no luck! My luck changed after becoming a postdoc. Actually, it was just that Oana is too nice a person to say no, even to me. So we
started on this project. We came from quite different point of views. Oana was mainly interested in AuslanderReiten Conjecture (which I admit I still
did not understand the motivations for). I was more excited about the asymptotic behavior of Ext and Tor, to me it seems like a fundamental
problem, especially to define the sort of functions similar to Serre's multiplicity as in my number 11. We finally found some common ground, proved
some modest results, and managed to submit before Oana's forth child was due! She is my hero, I can never understand how she manages to take care of her
(very lovely) children and do Math at the same time. Some people just have it!
 Remarks on noncommutative crepant resolutions over complete intersections
Advances in Math. 224 (2010), 10211030.
comments
I observe certain obstructions to existence of noncommutative crepant resolutions, in the sense of Van den Bergh, over complete intersections. They mostly agree with our intuition from birational geometry.
 On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings
(with Jinjia Li and Claudia Miller)
Algebra and Number Theory, 4:8 (2010), 10391053 comments
We show that Torrigidity of Frobenius implies nonexistence of ptorsion elements in the class group. A wellknown consequence is that for local complete intersections in dimension 3, the Picard group of the punctured spectrum is torsionfree. We also give many examples when Torrigidity of Frobenius fails to hold for Gorenstein, isolated singularities.
 Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension
(with Olgur Celikbas)
Math. Z., 269 (2011), 10051020.
comments
After several chats with Olgur during my visits to Nebraska (where he was a student of Roger Wiegand), I realized he knew more about my thesis than I did! So we decided to work together on expanding what we both knew, and this is the first of hopefully several papers in that direction. In this one we imitated the idea in [11] to define an asymptotic formula on Ext modules of a pair (M,N) and used that to study vanishing of Ext when one of the modules have finite competeintersection dimension (a notion which include both modules over complete intersections and modules of finite projective dimensions over arbitrary rings).
 Decent intersection and Torrigidity for modules over local hypersurfaces
Transactions of the AMS, 365 (2013), no. 6, 28032821.
comments
Decency is the following wellknown property (except no one gave it a name!) : over a local ring $R$
a module M is decent if for any module N such that Supp(M) and Supp(N) intersect only at the maximal ideal,
then dim M + dim N <= dim R. Serre showed that if R is regular, any module is decent. To determine whether a
given module over a singular R is decent is difficult, and Mel showed that the Direct Summand Conjecture could
be formulated as a question whether a particular module is decent over a particular unramified (i.e good)
hypersurface. Totally by a freak accident, I discovered that this is related to
another wellknown question known as "rigidity of Tor", through a "theta" function defined by Mel to attack the decency problem.
Turns out this is very useful for understanding rigidity over hypersurfaces, if you also aplly some Intersection Theory (whick makes sense,
since decency is an Intersection Theory problem).
 Picard groups of punctured spectra of dimension three local hypersurfaces are torsionfree
Compositio Math, 148 (2012), 145152.
comments
I show that the Picard group of the punctured spectrum of a dimension 3 local hypersurface is torsion free, using only homological methods. This was motivated by the conjectures made by Ofer Gabber in his Oberwolfach report (see Conjecture 3, page 1975). The main result also complements (resolving the ramified case, and indeed was motivated by) the main theorem in number 5.
 Vanishing of Ext, clustertilting modules and finite global dimension of endomorphism rings
(with Craig Huneke)
American J. Math., 135 (2013), no. 2, 561578.
comments
This is a continuation of number 5. We were trying to understand when does the endomorphism ring End(M) has finite global dimension for a maximal CohenMacaulay module over a CohenMacaulay ring R. This has been shown by works of Iyama to be intimately connected to clustertilting objects in the category of maximal CohenMacaulay modules. We are able to recover and strengthen a result by BurbanIyamaKellerReiten on such objects over reduced onedimensional hypersurface. This in turns gives characterization of hypersurfaces of form xyf(u,v) which admit noncommutative crepant resolutions (in characteristic 0, they coincide with the ones admitting a projective crepant resolution). Our results work in both zero and positive characteristics.
 Asymptotic behavior of Tor and applications
preprint. comments
This is the second half or my thesis (the first was basically number 8). The main point is to generalize Mel Hochster's theta function to
complete intersections. First, one needs to show that the lengths of Tor modules have wellbehaved polynomial growth. Avramov and Buchweitz
have proved the same thing for Ext, unfortunately, their paper did not address Tor. So I basically dualize what's in their paper, fixing
some technical problems, and then define an asymptoctic version of theta function. The second half deals with some applications, but there are
a lot more to be done.
Warning: this preprint contains many typos, although I believe the main results are correct. I will clean it up when I find enough motivation to work on my thesis again!
 Necessary conditions for Auslander's depth formula
(with Olgur Celikbas)
J. Pure and
Applied Algebra, 218 (2014), no.3 522530.
comments
 Bounds on the HilbertKunz Multiplicity
(with Olgur Celikbas, Craig Huneke and Yi Zhang)
Nagoya Math. Journal, 205 (2012), 149165. comments
In this paper we give new lower bounds on the HilbertKunz multiplicity of unmixed nonregular local rings, bounding them uniformly away from one. Our results improve previous work of Aberbach and Enescu.
 Bounds on the regularity and projective dimension of ideals associated to graphs
(with Craig Huneke and Jay Schweig)
Journal of Algebraic Combinatorics, 38 (2013), no.1, 3755. comments
This paper is aimed at answering the following question: if a quotient R= S/I of a polynomial ring satisfies
Serre condition S_2, how close it is to being CohenMacaulay? When I is a squarefree monomial ideal of codimension 2, we obtain a surprising (at least to us) answer: it is quite close, roughly speaking the depth is about log(n) from being CohenMacaulay, where n is the number of variables!
 Projective Dimension, Graph Domination Parameters, and Independence Complex Homology
(with Jay Schweig)
Journal of Combinatorial Theory A, 120 (2013), no. 2, 453469. comments
This is a continuation of the last one with Craig, who was initially a coauthor but had to withdraw prematurely due to too much committee work (poor Craig wasn't too happy about it)! We construct several bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs; in particular we draw heavily from the topic of dominating sets (this is an insight that has been used by people working in combinatorial topology). Through Hochster's Formula, these bounds recover and strengthen existing results on the homological connectivity of graph independence complexes. We had a lot of fun during this project, not least because Jay makes awesome espresso! He also taught me a lot about graphs.
 Modules that detect finite homological dimensions
(with Olgur Celikbas and Ryo
Takahashi)
Kyoto Journal of
Mathematics, 54 (2014), no.2, 295310. comments
We study homological properties of test modules that are, in principle, modules that detect finite homological dimensions. The main outcome of our results is a generalization of a classical theorem of Auslander and Bridger: we prove that, if a commutative Noetherian complete local ring R admits a test module of finite Gorenstein dimension, then R is Gorenstein.
 Some Homological Properties of Modules over
a Complete Intersection, with Applications
Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday, Springer, 2013.
comments
The title is of course a rather unsubtle reference to David's very influential paper.
 The radius of a subcategory of modules
(with Ryo Takahashi)
Algebra and Number Theory, 8 (2014), no. 1, 141172.
comments
In May 2011 Ryo Takahashi (and several others) visited Lawrence to speak at Homological Days 2011. The speakers introduced me and the graduate students to a bunch of new ideas, especially the exciting developments on triangulated categories. I raised a slightly provocative question: Can we play the same game with abelian categories? Ryo and me started talking and our collaboration soon yielded dividents. In this paper, we introduced a new invariant on a subcategory X of mod(R), inspired by Rouquier's notion of dimension of a triangulated subcategory. Roughly speaking, it measures the number of maximal extensions one needs to build an arbitrary object in X from a single object (which may not be in X) together with unlimited number of taking syzygies, finite direct sums and summands.
Our most significant result is that the category of maximal CohenMacaulay modules has finite radius when R is a CohenMacaulay complete local ring with perfect coefficient field. We conjecture a converse, namely that a resolving category of mod(R) with finite radius has to consist of maximal CohenMacaulay modules only. We could only prove our conjecture for complete intersections, however. The results raised a few interesting questions about subcategories of mod(R), for example see Stevenson's recent work which answered one of them.
 The dimension of a subcategory of modules
(with Ryo Takahashi)
Forum
Math. Sigma, accepted.
comments
This is closely related to the previous paper (but the invariants turn our to have quite different properties, that's why they are separated into two papers). We define the dimension of a subcategory X of mod(R) (or in general any abelian category with enough projectives). The definition is similar to that of radius (see previous paper), except that the generating object is now required to be in X.
This slight change turn our to make a rather big difference. For example, we show that when R is local CohenMacaulay, under a mild assumption finiteness of the dimension of the full subcategory consisting of maximal CohenMacaulay modules which are locally free on the punctured spectrum is equivalent to R having an isolated singularity (recall that by our previous paper, the radius of any subcategory of maximal CohenMacaulay modules is finite). This vastly generalizes a classical result by Auslander (which was proved in full generalities by HunekeLeuschke and Wiegand): if R has only finitely many indecomposable MCM modules than R has an isolated singularity.
To my mind such result indicates that the dimension might be of a more geometric nature. Several other reults which we point out seem to support such statement. For example, over a rational surface singularity the dimension of maximal CohenMacaulay modules is at most one, and it is zero precisely for quotient singularities.
 Classification of resolving subcategories and grade consistent functions
(with Ryo Takahashi)
International Math. Res. Notices, (2015), no. 1, 119149.
comments
Let R be a commutative ring. A resolving subcategory X of mod(R) is a nonempty one closed under taking syzygies and extensions; note that this immediately implies X contains all projectives in mod(R). We classify certain resolving subcategories of finitely generated modules over a commutative noetherian ring R by using integervalued functions on Spec R. Another key result is a categorical version of a classical theorem by AuslanderBuchweitz: over a complete intersection, a resolving subcategory is completely determined by its subcategories of maximal CohenMacaulay modules and finite projective dimensions modules. As an application one can completely classify all resolving subcategories when R is a locally complete intersection (using recent results of Stevenson). This is quite surprising as mod(R) seems be too big for such classification.
One key theorem of the paper, the classification of resolving subcategories of finite projective dimensions modules is announced at virtually the same time by Hügel, Pospisil, Stovicek and Trlifaj. The amusing part is I was actually thanked in that paper, essentially for answerings David Pospisil's questions on Mathoverflow (which was only used in their section on Hochster's conjecture). Of course, at that time I had no idea what David was onto (: This also shows clearly they started well before I talked with Ryo, so I would gladly acknowledge their priority in this matter.
 Upper bounds for dimensions of singularity categories
(with Ryo Takahashi)
Comptes
rendus Mathematique, 353 (2015), no. 4, 297301.
comments
This is a spinoff of the previous works. We give upper bounds for the dimension of the singularity category of a CohenMacaulay local ring with an isolated singularity. One of them recovers an upper bound given by Ballard, Favero and Katzarkov in the case of a hypersurface. Over Gorenstein rings, the singulariy category is very close to the stable category of maximal CohenMacaulay modules, so it is not surprising that the techniques in [17] can be adapted here, even for CohenMacaulay rings.
 Noncommutative resolutions and Grothendieck groups
(with Osamu Iyama, Ryo Takahashi and
Charles Vial)
Journal of
Noncommutative Geometry, 9 (2015), no. 1,
2134.
comments
Let R be a noetherian normal domain. We investigate when R admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a noncommutative desingularization of Spec(R). We show that the existence of such modules forces stringent conditions on the Grothendieck group of finitely generated modules over R. In some cases those conditions are enough to imply that Spec(R) has only rational singularities. This extends in certain cases a result by StaffordVan den Bergh for Gorenstein algebras.
 Hochster's theta pairing and numerical equivalence
(with Kazuhiko Kurano)
Journal of
KTheory, 14 (2014), no. 3, 495525. comments
This is something that I wanted to prove since my thesis day, that is
the theta function should vanish on
numerically trivial elements in the
Grothendieck group. Kazu is of course the
perfect person to prove that! One the
outcomes, which generalizes the famous
DuttaHochsterMacLaughlin counterexample, is
very emotionally satisfying for me as it was
one of my starting points in graduate
school. We also managed to prove that theta
is positive/negative semidefinite in the
small dimension cases, and raise a lot of
conjectures about higher dimensions. It will
appear in the last ever issue of Journal of Ktheory!
 Bounding Projective Dimensions of squarefree Monomial Ideals using domination parameters for clutters
(with Jay Schweig)
Proceedings of
AMS, 143 (2015), no. 2, 555565. comments
We introduce the concept of edgewise domination in clutters, and use it to provide an upper bound for the projective dimension of any squarefree monomial ideal. We compute domination parameters for certain classes of these clutters.
 On generalized HilbertKunz function and multiplicity
(with Ilya Smirnov)
submitted. comments
This was one rare happy outcome of Craig Huneke's departure from KU! His graduate students stayed behind for one semester, so Ilya started coming to my office to chat ...
 Noncommutative desingularizations and the global spectrum of a commutative ring
(with Eleonore Faber and Colin Ingalls)
Algebra and Representation
Theory, 18 (2015), no. 3, 633664. comments
This paper started when we were all members of the MSRI in Spring
2013. It took more than a year (and many
workshops) to finish, but I am rather
satisfied with the end product. We introduce a notion of noncommutative desingularization using
endomorphism algebras. We show that it
satisfies a number of basic properties similar
to the commutative desingularizations. We also
compute a lot of examples.
 Further applications of clutter domination parameters to projective dimension
(with Jay Schweig)
Journal of
Algebra, 432 (2015), 111. comments
We tie a few loose ends in number 24. The paper was finished in the
beautiful Halifax, during this. We
thank Sara Faridi for being such an excellent host!
 Boundary and shape of CohenMacaulay cone
(with Kazuhiko Kurano)
Mathematische Annalen, accepted. comments
The CohenMacaulay cone is a fascinating object defined by Kazu and
Jean Chan. Their insight is that maximal
CohenMacaulay modules can be viewed as numerically
positive objects, via generalized Serre's
intersection multiplicity formula. Computing these cones, however,
is not easy, which is understandable given how
little our knowledge of Serre's formula still
is. In this paper we managed to compute it for
a large class of hypersurfaces in dimension 3,
those that are Knorrerequivalent to a
onedimensional hypersurface.
 On the relationship between depth and
cohomological dimension
(with Shunsuke Takagi)
Compositio
Math., accepted comments
This paper was inspired by this
wonderful result from Matteo. Together with Matteo in Genoa in
September 2014, we
guessed that the cohomological in depth 3
would depend on the local Picard groups, but
later learned that Shunsuke already proved it!
When I visited Japan later that year, we
chatted a bit about the paper and Shunsuke graciously proposed
that I become his coauthor. Being a huge fan
of Japanese collaborators, I would not say
no (:. I love the clean statements we have,
and strongly believe there are a lot more to
do in this direction.
 Some computations of higher HilbertKunz multiplicities
(with Keiichi Watanabe)
Proceedings of
AMS, accepted comments
This project was mostly done when Keiichi and I shared an office in
MSRI in 2013. The paper provided strong evidence that
higher HilbertKunz multiplicities are very reasonable, in
the sense that they can be computed in most
nice situations. In fact, as the work of
Brenner showed, sometimes they are quite
a bit more
accessible than the classical HilbertKunz,
and one can use them to construct irrational
examples of the former.
 Representation schemes and rigid maximal CohenMacaulay modules
(with Ian Shipman)
Selecta
Mathematica, accepted comments
I first met Ian at this
inspiring Banff workshop in summer 2012
(where I also heard Dale's talk for the first time
about his epsilon multiplicity result, which
lead to number [25]). We hit
off pretty well, aided by the fact that we were the only ones who
stayed an extra afternoon after the conference, and so we did some
hiking together. As fate would have it, Ian and
I also spent a semester together at MSRI in
Fall 2013. We often talked math, about things
like Ulrich modules and bundles. However, we only started
collaborating after his visit to Lawrence in
Spring 2014. The topic was actually fairly easy to pick after
all those chats, and it looks like we have
quite a bit more to talk about.
 Cohomological support and the geometric join
(with Billy Sanders)
submitted. comments
Billy is my first PhD student, and a really fun person to be
around. He loves math and life equally, a
quality I like in people. We worked on
our first paper in Kansas and during various
math trips to Halifax, Berkeley, Barcelona, but
always had lots of fun while doing so! The
first main
result did not take too much time to prove
once we realized what to prove,
but I really like it, since it links two
completely different topics: cohomological
supports à la AvramovBuchweitz, and joins of
varieties.
 On the associated primes of local cohomology
(with Pham Hung Quy)
submitted. comments
This is my first paper done entirely via Facebook! I knew about Quy as
one of the very fine group of young
commutative algebraists working in
Vietnam, and he often sent me his preprints
and papers, which I appreciated but never have
a chance to think much about them. One day, we
became Facebook friend, and he sent me his
preprint about associated primes of local
cohomology modules, a very classical and important
question in this area. The main idea of using
NagelSchenzel isomorphism to prove finiteness
result over rings with FFRT is his. Somehow
our FB chats became really interesting, and a
few days later this paper emerged. We learned
shortly after that Mel Hochster and Luis
NúñezBetancourt have proved the same results
in unpublished work,
surely before I even started thinking about
this problem. Still, it made me appreciate
this question a lot more, and perhaps we will come
back to this in the future.
 Finite Ftype and Fabundant modules
(with Tony Se)
submitted. comments
This is Tony's third preprint, and the second I wrote with a
student. We spent many evenings on this
rather technical project, and it is a big relief to finish it
off. The idea of finite Ftype is of course borrowed
from representation theory, and it turned out
to be quite useful for proving various
things. For example, it turns out to be
convenient for proving that torsion divisor
classes over nice singularities must be
CohenMacaulay, a topic I am intrigued about.
 Noncommutative resolutions using syzygies
(with Osamu Iyama, Srikanth B. Iyengar, Ryo Takahashi, Michael Wemyss, Yuji Yoshino)
submitted. comments
This is a first preprint from the AIM SQuaRe program with the five people above. We had a lot of fun, but it is always tricky to write a paper of six authors. We argued
vigorously about the use of "while" versus "whilst" until the very end. The simplest, least technical consequence of our main Theorem (which was stated in a very scary way)
is the following: take a polynomial ring R over a field K. Take M to be a direct sum of R and arbitrary syzygies of K. Then End(M) has finite global dimension. This
generalizes a result by Buchweitz and Pham, who proved it for direct sum of all syzygies with more geometric method.
 Gorenstein modifications and QGorenstein rings
(with Osamu Iyama, Ryo Takahashi, Michael Wemyss)
submitted. comments
This is a second preprint from the AIM SQuaRe program of the six people in 37. Roughly speaking, we prove that one can descend and ascend noncommutative crepant resolutions (NCCRs)
along cyclic covers. This helps to reduce the distance with (classical, commutative) algebraic geometry significantly. Several consequences follow: nonGorenstein quotient singularities by
connected reductive groups cannot admit an NCCR, the centre of any NCCR is logterminal, and the AuslanderEsnault classification of twodimensional CMfinite algebras can be deduced from
BuchweitzGreuelSchreyer. One interesting feature: it turned out that topic of cyclic covers has not been studied enough for our purposes, so a lot of effort was spent on foundational
issues, which perhaps made the paper much more technical than originally thought.