Commutative Algebra Seminar
Spring 2015, Friday 3:10–4:00, LCB 222
Date  Speaker  Title — click for abstract (if available) 
January 16  Madhusudan Manjunath
UC Berkeley 
Minimal free resolutions of toppling ideals
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 FrankOlaf Schreyer and John Wilmes.

January 30  Nick Switala
University of Minnesota 
Matlis duality for Dmodules and de Rham cohomology
Let R be a formal power series ring over a field k of characteristic zero, and let D be the
ring of klinear differential operators on R. By interpreting the Matlis dual of an Rmodule M in
terms of certain klinear maps from M to k (following SGA2), we show that given any left Dmodule,
there is a natural structure of left Dmodule on its Matlis dual, whose de Rham cohomology is kdual
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 Hodgede Rham spectral sequences
(beginning with their E_2 terms) are independent of the embedding used in their definition and
consist of finitedimensional kspaces.

February 20  Tony Se
University of Kansas 
Finite Ftype and Fabundant modules
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 Ftype. They include reflexive ideals representing torsion
elements in the divisor class group. The second class is what we call Fabundant 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 PatakfalviSchwede, Yao and Watanabe. They also
afford a deeper understanding of these categories, including complete classifications in
many cases of interest.

March 6  Peder Thompson
University of Nebraska, Lincoln 
Stable local cohomology
Let R be a Gorenstein local ring, I an ideal in R, and M an Rmodule. The local cohohomology of M supported at I can be computed by applying the Itorsion 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 Itorsion 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 nonzero 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.

March 27  Michael Brown
University of Nebraska, Lincoln 
Knoerrer periodicity and Bott periodicity
The goal of my talk will be to describe a precise sense in which Knoerrer periodicity, a property of maximal CohenMacaulay modules over certain hypersurface rings, is a manifestation of Bott periodicity in topological Ktheory. Along the way, I will introduce an 8periodic 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 Ktheory of its Milnor fiber.

March 27  Luchezar L. Avramov
University of Nebraska, Lincoln 
Polynomial growth of Betti sequences over local rings 
April 3  Hema Srinivasan
University of Missouri, Columbia 
Unmodality of Hilbert functions of graded Artinian algebras 
April 3  Dale Cutkosky
University of Missouri, Columbia 
Stable forms of generically finite morphisms along a valuation in arbitrary characteristic
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).

April 10  Kevin Tucker
University of Illinois, Chicago 
On the limit of the Fsignature function in characteristic zero
The Fsignature of a local ring in positive characteristic gives a measure of singularities by
analyzing the asymptotic behavior of the number of splittings (Fsplittings) 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 Fsignature 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 HilbertKunz multiplicities of diagonal hypersurfaces.

April 10  Florian Enescu
Georgia State University 
The Frobenius complexity of a local ring
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.

April 17  Luis NúñezBetancourt
University of Virginia 
Lyubeznik numbers and injective dimension of local cohomology in mixed characteristic
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.

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