Research

My main research interests lie in Commutative Algebra. Most of my recent efforts are directed towards understanding singularities related to the Frobenius endomorphism for rings in prime characteristic. More specifically, I focus my attention on numerical invariants such as F-thresholds, Hilbert-Kunz multiplicities and F-signature. Outside the world of positive characteristic, I like to think about Golod rings and strongly Golod ideals, especially in the monomial case. I also worked on Auslander's delta invariant and the index of a Cohen-Macaulay local ring, mainly in relation with Ding's conjecture. Before starting my Ph.D, I studied properties of Artinian algebras over a field k that can be detected by Macaulay's inverse system. These include Hilbert functions and isomorphism classes of k-algebras.

List of publications

1. “On the existence of F-thresholds and related limits.” (with L. Núñez-Betancourt and F. Pérez).
   To appear in Transactions of the AMS. arXiv:1605.03264
   • Description:

     We prove the existence of F-thresholds in full generality. We study limits of F-thresholds that converge to F-pure thresholds, inspired by a method due to Hernández. When the ring is standard graded, we study the equality between F-pure thresholds and F-thresolds at the irrelevant maximal ideal: if the two invariants coincide, and the ring is Gorenstein, then it is strongly F-regular. We produce new bounds for the a-invariants and the Castelnuovo-Mumford regularity of Frobenius powers in terms of F-thresholds, with specific focus on asymptotic behavior.

2. “F-thresholds of graded rings.” (with L. Núñez-Betancourt).
   To appear in Nagoya Mathematical Journal. arXiv:1507.05459
   • Description:

     We prove most of a conjecture made by Hirose, Watanabe and Yoshida, relating the a-invariant, the F-pure threshold and the diagonal F-threshold of an F-finite standard graded algebra. The conjecture was originally stated for strongly F-regular rings, but we prove it, much more generally, for F-pure rings. We introduce the notion of F-pure regular sequence and show that, when the ring is Gorenstein, the maximal length of such sequence is equal to the F-pure threshold. Furthermore, when the base field is infinite and the algebra is Gorenstein, we prove that one can always find an F-pure regular sequence of maximal length. We also prove some analogous statements in characteristic zero.

3. “A counterexample to a conjecture of Ding”
   J. Algebra 452 (2016), 324–337. arXiv:1506.09168
   • Description:

     We exhibit several one dimensional complete intersections that are counterexamples to a conjecture posed by Songqing Ding. This relates the index and the generalized Löewy length of Gorenstein local rings. The conjecture was known to be true for rings whose associated graded ring is Cohen-Macaulay, and the residue field is infinite.

4. “Products of ideals may not be Golod”
   J. Pure Appl. Algebra 220 (2016), no. 6, 2289–2306. arXiv:1506.09129
   • Description:

     We give an example of a product of monomial ideals in a polynomial ring over a field such that the residue class ring is not Golod. We study the strongly Golod property for rational powers of monomial ideals and introduce the notion of lcm-strongly Golod ideal. Throughout the article we list several open problems about Golod rings.

5. “Frobenius Betti numbers and modules of finite projective dimension” (with C. Huneke and L. Núñez-Betancourt).
   To appear in Journal of Commutative Algebra. arXiv:1412.4266
   • Description:

     We study Frobenius Betti numbers, which were introduced by Li and previously studied by Aberbach and Li. We study some of their properties, many of which resemble the behavior of the Hilbert-Kunz multiplicity. We try to relate the vanishing of Frobenius Betti numbers to the projective dimension of a module. We obtain results mainly for one-dimensional rings. In particular, in dimension one, we characterize the vanishing of a Frobenius Betti number in terms of the Krull dimension of a certain syzygy module. This leads to seemingly unrelated questions about asymptotic dimension of syzygies. In a few cases, we are able to conclude that a module of finite length cannot have a syzygy of finite length, passed a certain homological degree.

6. “A sufficient condition for strong F-regularity” (with L. Núñez-Betancourt).
   Proc. Amer. Math. Soc. 144 (2016), no. 1, 21–29. arXiv:1411.7078
   • Description:

     We show that, if an S2 local ring R of positive characteristic has a canonical ideal I such that the top local cohomology module of R/I is simple as an R{F}-module, then R is strongly F-regular. This extends a result of Enescu, by dropping the Cohen-Macaulay assumption on R and some regularity conditions on the ideal I.

7. “An algorithm for constructing certain differential operators in positive characteristic” (with A. F. Boix and D. Vanzo).
   Matematiche (Catania) 70 (2015), no. 1, 239–271. arXiv:1503.01419
   • Description:

     We exhibit an algorithm to compute a differential operator that, given a non zero polynomial f in a polynomial ring R of characteristic p>0, raises 1/f to its p-th power. We also describe a Macaulay2 code that implements such an algorithm. We give a complete description of the case when f is a monomial, and study some families of polynomials for which the differential operator can be chosen to be R^p-linear. Finally, we give a characterization of supersingular elliptic curves in the projective plane in terms of the associated differential operator.

8. “Artinian Level Algebras of Low Socle Degree”
   Comm. Algebra 42 (2014), no. 2, 729–754. arXiv:1208.3506
   • Description:

     We find a characterization of Hilbert functions (1,m,n,t) which are admissible for an Artinian level local K-algebra, where K is a field of characteristic zero. Level, in this case, means that the dimension of the socle is precisely t. We also show that level local algebras such that the Hilbert function (1,m,n,t) is maximal (such algebras are called compressed) are in fact isomorphic to their associated graded ring. In particular, they are standard graded.

Submitted to journal and work in progress

1. “Globalizing F-invariants.” (with T. Polstra and Y. Yao). arXiv:1608.08580
   • Description:

     In this paper we define and study the global Hilbert-Kunz multiplicity and the global F-signature of prime characteristic rings which are not necessarily local. Our techniques are made meaningful by extending many known theorems about Hilbert-Kunz multiplicity and F-signature to the non-local case.

2. “Generalizing Serre’s Splitting Theorem and Bass’s Cancellation Theorem via free-basic elements.” (with T. Polstra and Y. Yao). arXiv:1608.08591
   • Description:

     We give new proofs of two results of Stafford, which generalize two famous Theorems of Serre and Bass regarding projective modules. Our techniques are inspired by the theory of basic elements. Using these methods we further generalize Serre's Splitting Theorem by imposing a condition to the splitting maps, which has an application to the case of Cartier algebras.

3. Appendix to: “On the behavior of singularities at the F-pure threshold.” by E. Canton, D. Hernández, K. Schwede, and E. Witt (with J. Jeffries, Z. Kadyrsizova, R. Walker, and G. Whelan). arXiv:1508.05427
   • Description:

     In this short appendix we exhibit a family of elements in a polynomial ring over a field of characteristic 2 for which the the log canonical threshold coincides with the F-pure threshold, but the corresponding test ideal is not radical.