Commutative Algebra Seminar

The Laplacian lattice of an undirected connected graph is the lattice generated by the rows of its Laplacian matrix. We describe the minimal free resolution of the Laplacian lattice ideal of the graph. This is joint work with Frank-Olaf Schreyer and John Wilmes.

Let R be a formal power series ring over a field k of characteristic zero, and let D be the ring of k-linear differential operators on R. By interpreting the Matlis dual of an R-module M in terms of certain k-linear maps from M to k (following SGA2), we show that given any left D-module, there is a natural structure of left D-module on its Matlis dual, whose de Rham cohomology is k-dual to that of the original module in the holonomic case. We then give an application to Hartshorne's theory of de Rham homology and cohomology for R, showing that the associated Hodge-de Rham spectral sequences (beginning with their E_2 terms) are independent of the embedding used in their definition and consist of finite-dimensional k-spaces.

We will introduce and consider basic properties of two types of finitely generated modules over a commutative noetherian ring R of positive prime characteristic. The first is the category of modules of finite F-type. They include reflexive ideals representing torsion elements in the divisor class group. The second class is what we call F-abundant modules. These include, for example, the ring R itself and the canonical module when R has positive splitting dimension. We will discuss some facts about these two categories. Our methods allow us to extend previous results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper understanding of these categories, including complete classifications in many cases of interest.

Let R be a Gorenstein local ring, I an ideal in R, and M an R-module. The local cohohomology of M supported at I can be computed by applying the I-torsion functor to the injective resolution of M. Since R is Gorenstein, M has a complete injective resolution, so it is natural to ask what one gets by applying the I-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this behaves like the usual local cohomology functor. Our main result is that when there is only one non-zero local cohomology module, there is strong connection between that and stable local cohomology; in fact, the latter gives a Gorenstein injective approximation of the former.

The goal of my talk will be to describe a precise sense in which Knoerrer periodicity, a property of maximal Cohen-Macaulay modules over certain hypersurface rings, is a manifestation of Bott periodicity in topological K-theory. Along the way, I will introduce an 8-periodic version of Knoerrer periodicity for real isolated hypersurface singularities. I will also describe a map from the Grothendieck group of the triangulated category of matrix factorizations associated to a real or complex hypersurface into the topological K-theory of its Milnor fiber.

We discuss some theorems showing that in characteristic zero, along a (not necessarily discrete) valuation, generically finite morphisms of nonsingular varieties and of their associated graded rings along the valuation have stably toric forms under blow ups. We also give counterexamples to these statements in positive characteristic, and discuss what is true in positive characteristic. We discuss how the pathological problems in positive characteristic follow from the existence of defect in the extension of valuations (which can only occur for non discrete valuations in positive characteristic).

The F-signature of a local ring in positive characteristic gives a measure of singularities by analyzing the asymptotic behavior of the number of splittings (F-splittings) of large iterates of the Frobenius endomorphism.  One can also incorporate ideal pairs by restricting the set of "allowable" splittings, and varying the coefficient of the ideal gives rise to the F-signature function of the pair.  While for each fixed characteristic p > 0 these functions tend to be extremely complicated, in the few examples that have been computed they tend to limit to a piecewise polynomial function as p tends to infinity. In this talk I will discuss what is known about these functions and their limits, and present a number of new computations for diagonal hypersurfaces.  The new computations (joint with Shideler) build on the techniques of Han and Monsky used to compute the Hilbert-Kunz multiplicities of diagonal hypersurfaces.

The talk will discuss the concept of Frobenius complexity of a local ring of prime characteristic and analyze it in the case of determinantal rings. This is joint work with Yongwei Yao.

The Lyubeznik numbers in mixed characteristic are invariants for local rings that do not contain a field. These invariants are inspired by the numbers that Lyubeznik defined for equal characteristic rings, which are known to have algebraic and geometric interpretations. In this talk, we will give an overview of Lyubeznik numbers and present several properties for the invariants in mixed characteristic. We will also discuss related results on injective dimension of local cohomology over regular rings of mixed characteristic. This is joint work with Daniel J. Hernández, Felipe Pérez and Emily E. Witt.