Commutative Algebra Seminar

I will give an introduction to Oliver Lorscheid's theory of ordered blueprints - one of the more successful approaches to 'the field of one element' - and sketch its relationship to Berkovich spaces, tropical geometry, Tits models for algebraic groups, and moduli spaces of matroids. The basic idea for the latter two applications is quite simple: given a scheme over Z defined by equations with coefficients in {0,1,-1}, there is a corresponding 'blue model' whose K-points (where K is the Krasner hyperfield) sometimes correspond to interesting combinatorial structures. For example, taking K-points of a suitable blue model for a split reductive group scheme G over Z gives the Weyl group of G, and taking K-points of a suitable blue model for the Grassmannian G(r,n) gives the set of matroids of rank r on {1,...,n}. Similarly, the Berkovich analytification of a scheme X over a valued field k coincides, as a topological space, with the set of T-points of X, considered as an ordered blue scheme over k. Here T is the tropical hyperfield, and T-points are defined using the observation that a (height 1) valuation on k is nothing other than a homomorphism to T.

We will indicate how, using the Riemann-Hilbert correspondence, one can recover several results on local cohomology modules of polynomial rings.

F-signature plays a crucial role when measuring singularities of varieties in positive characteristics. For example, if R is a local ring, s(R) = 1 implies R is regular, and 0 < s(R) < 1 implies R is strongly F-regular which is a char p analog of klt singularities. For a globally F-regular variety X, the F-signature of an ample invertible sheaf is defined as the F-signature of the section along the invertible sheaf over X. In this talk, we will discuss the F-signature is well-defined and is (locally Lipschitz) continuous on the ample cone, and how the signature extends to the boundary of the cone. This is joint work with Swaraj Pande.

This is a joint work in progress with Tomohiro Okuma and Ken-ichi Yoshida. Let (A, m) be a 2-dimensional excellent normal local ring. Let I be an integrally closed m-primary ideal, and let f : X -> Spec(A) be a resolution of singularity such that I O_X = O_X(-Z) is invertible. Also let Q be a minimal reduction of I. We define \bar{r}(I) = min {r | \bar{I^{n+1}} = Q bar{I^n} for all n \ge r}, where \bar{I^n} is the integral closure of I^n. If A is a rational singularity, then H^1(X, O_X(-Z)) = 0 and \bar{I^2} = QI for every integrally closed ideal I, \bar{r}(I) = 1. We call I an elliptic ideal if \bar{r}(I) = 2. This naming comes from the fact that if A is an elliptic singularity, then \bar{r}(I) \le 2 for every integrally closed ideal. Today we discuss about the following topics: 1. Let \bar{G}(I) denote the associated graded ring of the filtration {\bar{I^n}}. For what I, is \bar{G}(I) Gorenstein? If I is an elliptic ideal, then \bar{G}(I) is Cohen-Macaulay but Gorenstein in very limited cases. We discuss about the condition for \bar{G}(I) to be Gorenstein. 2. We give a formula for Core(I) for elliptic ideals.

We will discuss the notions of splinters and birational derived splinters. They are related to various interesting notions of singularities, but less are known about them. We will discuss some basic properties of those notions including behavior under limit and etale extensions. Then we will discuss some more advanced properties, one of which involves ultrapower to prove.

In this talk we focus on Rees algebras and symbolic Rees algebras of determinantal ideals. In particular, we will show that they have mild singularities in prime characteristic. We will also discuss consequences for numerical invariants and initial ideals of the symbolic powers of these ideals. This is joint work with Alessandro De Stefani and Jonathan Montaño.

Work on infinite resolutions beyond the cases of complete intersections and Golod rings has tended to focus on the sequence of Betti numbers. Hai Long Dao and I have recently begun to study a question of a different kind, and I will report on this joint work: Let R = S/I be an Artinian quotient of a regular local ring S, with residue field k. When does it happen that k is a direct summand of a syzygy module in the R-free resolution of k, or indeed in the R-free resolution of every module? We were surprised by what we found experimentally, and were able to prove a little of what we observed.

We investigate the lengths of certain local cohomology modules over polynomial rings. By fixing the degree component, and using the fact that the length of an Artinian ring is the same as that of its injective hull, we transform this into a question about rings of the form $k[x_1,\ldots,x_n]/(x_1^k,\ldots,x_n^k)$, and the annihilator of $x_1 + \cdots + x_n$ therein. We in particular use refinements of functions introduced by Han and Monsky. This was motivated by questions about behavior of the length of local cohomology with support in the maximal ideal of thickenings, that is, $R/I^t$ as $t$ grows. This project is joint work with Mel Hochster.

In this talk, we will discuss a conjecture of R. Berger dating back to around 1963. He conjectured that in a reduced one dimensional local complete algebra over a perfect field k, the torsion submodule of the module of differentials should capture information about the regularity of the algebra. We will talk about some recent partial developments in this direction. These recent results were obtained in joint work with Craig Huneke and Vivek Mukundan.

Resolving a monomial ideal over the polynomial ring is an easy task, but resolving it minimally isn't. In this talk we discuss Barile-Macchia resolutions, which is a new construction of free resolutions for all monomial ideals using discrete Morse theory, and point out important classes of ideals for which these resolutions are minimal. This project is joint work with Selvi Kara.

Given a noetherian ring of positive characteristic, Kunz proved that the Frobenius endomorphism is flat if and only if the ring is regular. Radu and André established an analogue for homomorphisms of noetherian rings of positive characteristic, proving that such a map regular if and only if the relative Frobenius map is flat. In this talk, we present more general results of this ilk, relating the homological properties of the fibers of a ring homomorphism to those of the relative Frobenius map.

Rings of differential operators and their module theory have a number of important applications in Commutative Algebra. Much of the power of D-module theory stems from the fact that many "large" modules over polynomial rings exhibit striking finiteness properties as D-modules. The fountain of finiteness is a fundamental result called Bernstein's inequality; this Bernstein inequality also readily implies the existence of the well-studied Bernstein-Sato polynomial. Motivated by the study of Bernstein's inequality on singular rings, we introduce a two-sided or "sandwich" analogue of the Bernstein-Sato polynomial, which has a closer connection to Bernstein's inequality. We will discuss this new notion and its connections to simplicity of rings of differential operators and Bernstein's inequality. If time permits, we will encounter some odd numerical F-invariants along the way. This is based on joint work with David Lieberman (University of Nebraska).