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!
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 K-Theory (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 l-adic 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 Polyshchuk-Vaintrob using dg categories, and the local analytic case in char 0 was recently proved by Buchweitz-Van Straten using topological methods.
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 K-theory 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.
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 Auslander-Reiten 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!
I observe certain obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over complete intersections. They mostly agree with our intuition from birational geometry.
We show that Tor-rigidity of Frobenius implies non-existence of p-torsion elements in the class group. A well-known consequence is that for local complete intersections in dimension 3, the Picard group of the punctured spectrum is torsion-free. We also give many examples when Tor-rigidity of Frobenius fails to hold for Gorenstein, isolated singularities.
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  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 compete-intersection dimension (a notion which include both modules over complete intersections and modules of finite projective dimensions over arbitrary rings).
Decency is the following well-known 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 well-known 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).
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.
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 Cohen-Macaulay module over a Cohen-Macaulay ring R. This has been shown by works of Iyama to be intimately connected to cluster-tilting objects in the category of maximal Cohen-Macaulay modules. We are able to recover and strengthen a result by Burban-Iyama-Keller-Reiten on such objects over reduced one-dimensional hypersurface. This in turns gives characterization of hypersurfaces of form xy-f(u,v) which admit non-commutative crepant resolutions (in characteristic 0, they coincide with the ones admitting a projective crepant resolution). Our results work in both zero and positive characteristics.
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 well-behaved 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!
See number 7.
In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed non-regular local rings, bounding them uniformly away from one. Our results improve previous work of Aberbach and Enescu.
This is a continuation of the last one with Craig, who was initially a co-author 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.
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.
The title is of course a rather unsubtle reference to David's very influential paper.
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 Cohen-Macaulay modules has finite radius when R is a Cohen-Macaulay 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 Cohen-Macaulay 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.
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 Cohen-Macaulay, under a mild assumption finiteness of the dimension of the full subcategory consisting of maximal Cohen-Macaulay 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 Cohen-Macaulay modules is finite). This vastly generalizes a classical result by Auslander (which was proved in full generalities by Huneke-Leuschke 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 Cohen-Macaulay modules is at most one, and it is zero precisely for quotient singularities.
Let R be a commutative ring. A resolving subcategory X of mod(R) is a non-empty 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 integer-valued functions on Spec R. Another key result is a categorical version of a classical theorem by Auslander-Buchweitz: over a complete intersection, a resolving subcategory is completely determined by its subcategories of maximal Cohen-Macaulay 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.
This is a spin-off of the previous works. We give upper bounds for the dimension of the singularity category of a Cohen-Macaulay 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 Cohen-Macaulay modules, so it is not surprising that the techniques in  can be adapted here, even for Cohen-Macaulay rings.
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 non-commutative 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 Stafford-Van den Bergh for Gorenstein algebras.
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 Dutta-Hochster-MacLaughlin 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 semi-definite 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 K-theory!
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.
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 ...
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.
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!
The Cohen-Macaulay cone is a fascinating object defined by Kazu and Jean Chan. Their insight is that maximal Cohen-Macaulay 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 Knorrer-equivalent to a one-dimensional hypersurface.
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 co-author. 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.
This project was mostly done when Kei-ichi and I shared an office in MSRI in 2013. The paper provided strong evidence that higher Hilbert-Kunz 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 Hilbert-Kunz, and one can use them to construct irrational examples of the former.
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 ). 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.
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 Avramov-Buchweitz, and joins of varieties.
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 Nagel-Schenzel 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 Núñez-Betancourt 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.
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 F-type 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 Cohen-Macaulay, a topic I am intrigued about.
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.
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 non-commutative crepant resolutions (NCCRs) along cyclic covers. This helps to reduce the distance with (classical, commutative) algebraic geometry significantly. Several consequences follow: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of any NCCR is log-terminal, and the Auslander-Esnault classification of two-dimensional CM-finite algebras can be deduced from Buchweitz-Greuel-Schreyer. 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.
This grows out of Scot's Master's thesis at KU. Coming soon!
Hopefully coming soon!