Research Interests

• My interests lie in Commutative Algebra. Specifically, I am interested in the homological questions. This group of conjectures and theorems provide very surprising connections between the homological (and sometimes non-homological) properties of modules over local rings. The connections are mostly expressed through the Ext and Tor functors. All of the questions in this area can be stated very concretely and understood by any beginning graduate student. Unfortunately, their answers are most often not so simple! In my thesis I focus on two questions : "rigidity of Tor" and "decent intersections". Using a function defined by Hochster, I studied when these properties hold for pair of modules over hypersurfaces. The answers are quite surprising and require some Intersection Theory, but they make applications a lot easier . I also generalized Hochster's function for complete intersections, using the ideas of homology operators by Eisenbud, Avramov and Buchweitz. My research so far indicates that for these questions, in many situations one can replace a commonly used assumption of a module "having finite projective dimension" to the much weaker "having isomorphism class equal to 0 in the reduced Grothendieck group with rational coefficients".

Papers

1. "On liftable and weakly liftable modules" , to appear in Journal of Algebra.
Comments: Mel Hochster formulated 2 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 paper to get a job!
2. "Decency and rigidity over hypersurfaces", submitted.
Comments: 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. That's why I worked on this! 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).
3. "Some observations on local and projective hypersurfaces" , to appear in Math. Res. Lett.
Comments: This is a continuation of 2. 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. When R is the local ring at the vertex of a smooth projective hypersurfaces, this would follow from a conjecture by Hartshorne and even more grand 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. In particular, one could 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.
4. "Asymptotic behavior of Tor and applications" , preprint.
Comments: This is the second half or my thesis (the first was basically number 2). The main point is to generalize Mel'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.
5. "On injectivity of maps between Grothendieck groups induced by completion" ,to appear in Michigan Math Journal.
Comments: In my only second conference during grad school, I went to LipmanFest with Mel and heard him talk 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 told me about the problem of finding a normal example (Mel's example was not normal, and he predicted the 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 pretty interesting questions about the kernel of the map between the Grothendieck groups of a ring and its completion, which should be settled someday.
6. "Comparing complexities of pairs of modules" ,with Oana Veliche, in preparation.
Comments: This is my first collaborative attempt. I have 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 start on this project. We came from different point of views. Oana was only interested in Auslander-Reiten Conjecture (which I admit I still did not understand the motivation 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 4. We finally found some common ground, proved some modest results, and manage to submit before Oana's forth child is due! She is my hero, I can never understand how she manage to take care of her (very lovely) children and do Math. Some people just have it!