http://www.math.utah.edu/caseminar/

This is joint work with Claudiu Raicu and Jerzy Weyman. Consider the polynomial ring over a field of characteristic zero whose entries come from a matrix of indeterminates. The ideal generated by the maximal minors of the matrix has provided useful examples in many areas of mathematics; however, the structure of the local cohomology modules of the ring with support in this ideal has remained mysterious until recently. We give an explicit (GL-equivariant) description of these local cohomology modules. Note that the study of local cohomology with support in more general determinantal ideals (i.e., ideals of non-maximal minors) is much better understood from recent work of Lyubeznik, Singh, and Walther, and of Raicu and Weyman.

In 1961 Auslander proved that over any regular local ring R, if the n-fold tensor power of a finitely generated R-module M is torsion-free for some n larger than or equal to the dimension of R, then M is free. In my lectures, I will discuss some recent developments in this line of research, including work of C. Huneke and R. Wiegand, and L. Avramov and myself.

We show that if (X, Theta ) is a PPAV over an F-finite field of characteristic p and D in |m*Theta|, then (X,(1/m)D) is a limit of strongly F-regular pairs and in particular mult_x(D) is at most m*dim X for any x in X.

Motivated by hyperplane arrangements, I'll talk about freeness of divisors. In particular, I'll discuss a recent theorem that describes conditions that imply freeness, based on the Jacobian ideal. Lots of the talk should be easily understandable.

We discuss the Boij-SÃ¶derberg conjectures, which were proved by Eisenbud-Schreyer in the Cohen-Macaulay case, and subsequently by Boij in the non-CM case. These classify which Betti diagrams arise, up to scalar multiple, as the Betti table of a module over the standard graded polynomial ring S=k[x_1, ..., x_n]. Moreover, the (proven) conjectures guarantee a unique decomposition of each Betti diagram into a positive sum of "pure" diagrams. We will discuss the main applications of this theory, and if time permits, we will also discuss decompositions of the Betti diagrams of complete intersections.

This is a continuation in a series of expository talks by various speakers.

This is a continuation in a series of expository talks by various speakers.

F-pure thresholds are certain invariants of singularities in characteristic p that are supposed to mimic the behaviour of log canonical thresholds in characteristic 0. In particular, there exists theorems and deep conjectures relating the two numbers when one specialises from characteristic 0 to positive characteristic, provided the prime p is allowed to suitably large. In this talk, we will introduce these numbers, and discuss some calculations in a fixed characteristic p. Contrary to the conjectural situation where the prime p is allowed to vary, we will show that when p is fixed, the behaviour of these numbers is closely related to the arithmetic of the singularities. The main example discussed will be the cone on an elliptic curve, where the invariant is completely determined with the aid of some geometry (especially, the deformation theory) of elliptic curves; this answers a question raised by M. Mustata. Time permitting, we will discuss some higher dimensional calculations.

This is a continuation in a series of expository talks by various speakers.

The Eisenbud-Goto conjecture has been open for over 25 years and even for simplicial affine semigroup rings the conjecture is widely open. We will discuss the conjecture and the already known results, before we turn our attention to the simplicial case. I will explain why the simplicial case is special and why I care about monomial ideals. Moreover, I will provide examples where the conjecture is far away to be sharp. Finally, we will discuss the latest results on the conjecture in the seminormal simplicial case. Time permitting, I will talk a little about monomial curves.

This is a continuation in a series of expository talks by various speakers.

I will talk about the Bertini-type theorem for normal local rings in mixed characteristic, which improves Flenner's version in a strong form. As applications, we discuss characteristic ideals and the Iwasawa main conjecture. This is a report on joint work with T. Ochiai.

This will be the first in a series of expository talks by various speakers to go through Lyubeznik's

The F-signature of a positive characteristic local ring is a numerical measure of singularities defined by asymptotically counting the number of splittings of the iterates of Frobenius. In this talk, we shall briefly review a proof of the existence of F-signature and subsequently consider how this invariant behaves after a module-finite ring extension. As an application, we give an explicit computation of the F-signature of an arbitrary finite quotient singularity.

The most fundamental complexes of free modules over a commutative ring are the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and the Eagon--Northcott and the Buchsbaum--Rim complexes, which are constructed from a matrix (i.e., a 2-tensor). I will discuss a multilinear generalization of these complexes, which we construct from an arbitrary higher tensor. Our construction provides detailed new examples of minimal free resolutions, as well as a unifying view on several well-known examples. This is joint work with Berkesch, Kummini, and Sam.

Let R be a reduced ring of prime characteristic. For each positive integer e, set F_*^e R to be the e-th iterate Frobenius pushfoward of R, that is R as a set with module structure twisted by the eth iterate of the Frobenius map. Given an R-linear map from F_*^e R back to R, such a map always extends to a corresponding map on the normalization R^N. In this talk, we show under some tameness assumptions on R, surjectivity of the extended map on the normalization implies surjectivity of the original map. This is joint work with Karl Schwede.

Let R be a commutative ring and I an ideal of R. Huneke asked if the local cohomology modules H_I^k(R) have only finitely many associated prime ideals. Singh constructed a counterexample with R a hypersurface. Singh showed that the local cohomology module H^3_(x,y,z)(R) has p-torsion. Equivalently, it contains an isomorphic copy of the abelian group Z/pZ for every prime integer p. Under certain conditions several authors have shown that the set of associated primes is finite. I show how to embed any finitely generated abelian group into a graded component of the local cohomology module.

The talk will raise a question on the lattice of Frobenius invariant submodules of the highest local cohomology module of a local ring with support in its maximal ideal and answer it in some special cases.

F-jumping coefficients are characteristic p analogues of jumping coefficients defined via multiplier ideals in characteristic zero. For a polynomial f with an isolated singularity in characteristic zero there is an upper bound on the number of its jumping coefficients in terms of the Jacobian of f. We describe a characteristic p analog of this result. An interesting feature of the proof is the use of differential operators (in characteristic p!) even though the statement of the result has nothing to do with differential operators. This is joint work with Mordechai Katzman and Wenliang Zhang.

The operation of t-closure on the ideals of an integral domain is a useful tool for studying certain classes of integral domains, such as the UFD's, Krull domains, and PVMD's. In this talk I will introduce the t-closure operation and give some applications to the theory of integral domains and in particular to the study of integer-valued polynomial rings.

This talk will build upon that of the previous week, where F-signature of a local ring in positive characteristic was introduced. Using a well-known duality argument, we present a generalization of F-signature to divisor pairs. As before, we show this generalization may be used to test for (strong) F-regularity. Furthermore, these techniques allow us to verify an open question of I. Aberbach and F. Enescu: the splitting-dimension (or s-dimension) of a positive characteristic local ring is the same as the dimension of the splitting prime. The work presented herein is joint with M. Blickle and K. Schwede.

In positive characteristic, the existence of splittings of the Frobenius map and its iterates has strong algebraic and geometric consequences. In this talk, we will sketch a proof of the existence of a numerical invariant, first formally defined by C. Huneke and G. Leuschke, which gives an asymptotic measure of the number of Frobenius splittings. Explicitly, suppose R is a d-dimensional complete Noetherian local ring with prime characteristic p > 0 and perfect residue field. Let q = p^e for e in N, and R^{1/q} the ring of q-th roots of elements of R. Denote by a_e the maximal rank of a free R-module appearing in a direct sum decomposition of R^{1/q}. The F-signature - which can be extended to arbitrary local rings in positive characteristic - is by definition the limit s(R) := lim_e a_e/(q^d) and has previously been shown to exist only in special cases. The proof of the main result is based on certain uniform Hilbert-Kunz estimates of independent interest.

The classical Witt vectors are a functorial construction which takes perfect fields of characteristic p to p-adically complete domains of characteristic 0. This functor was generalized by Dress and Siebeneicher to a family of functors parameterized by profinite groups. Witt's original functor corresponds to the p-adic integers as an additive pro-p group. In this talk, I will explore some examples of these functors corresponding to other pro-p groups taken over fields of characterisitc p. We will see some properties that are surprising when compared to the classical case.

The finite unramified extensions of Q_p (in a fixed algebraic closure) are in one-to-one correspondence with the finite fields of characteristic p. In this talk I will describe a functorial construction, due to Witt, which turns finite fields of characteristic p into unramified extensions of Q_p. This talk will be an expository preparation for the next seminar and will be accessible for graduate students who have taken abstract algebra.

A Frobenius near splitting of a commutative ring R of prime characteristic p is an additive map f : R -> R with the property that f(r^p a) = r f(a). Given a near splitting f, we call an ideal I f-compatible if f(I) is contained in I. In this talk I show a method for producing all prime compatible ideal recently discovered by Karl Schwede and myself. If time permits, I will further discuss a generalization of this method and its applications.

Let M be a reflexive module over a Cohen-Macaulay local ring R and A = Hom(M,M). If A has finite global dimension, then it has been proposed to serve as a non-commutative analogue of desingularizations of Spec(R). In this talk, we will survey what is known about when such module M exists, and discuss the connections with birational geometry and representation theory of commutative rings. Part of the new results are joint work with Craig Huneke.