Commutative Algebra Seminar

We will examine new and old results concerning lower bounds for Betti numbers of modules over a polynomial ring. We will be centering around several questions inspired by Horrocks' conjecture that the Betti numbers should be bounded below by certain binomial coefficients. Lots of examples will be presented along with exposition of what is currently known about these bounds and what questions could be studied in the future. We will also present at least one nice proof.

In this talk, we will discuss the following question: when are the thick subcategories of the bounded derived category of a commutative ring parameterizable by the specialization closed subsets of a Noetherian scheme. To do this, we will use pointless topology to construct a space and a notion of support for any triangulated category whose thick subcategories form a set. The same construction can also be used to develop support theories which classify certain subcategories in other contexts, such as Serre subcategories of an Abelian category, or resolving subcategories of a exact category with enough projectives.

Let T be a tensor triangulated category, that is, a triangulated category with symmetric monoidal structure. The Balmer spectrum of T is by definition the set of thick tensor ideals of T which are prime. This set has a topology similar to the Zariski topology for the spectrum of a commutative ring. In this talk, we study the Balmer spectrum of the right bounded derived category of finitely generated modules over a commutative Noetherian ring. We also consider classifying thick tensor ideals of the derived category. This talk is based on joint work with Hiroki Matsui.

We will talk about the algebraic fundamental group of a strongly F-regular singularity. More concretely, via a new transformation rule for the F-signature under certain sort of mildly ramified finite extensions of rings, we will discuss the finiteness of these groups, analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero. In contrast, our result and method are effective: we show that the reciprocal of the F-signature of the singularity gives a bound on the size of its fundamental group. As another consequence of these transformation rules, we also obtain purity of the branch locus over rings with mild singularities. Algebraic local fundamental groups will also be reviewed. This is joint work with Karl Schwede and Kevin Tucker.

We will present some old and new results on ideals of joint reduction number zero in a Noetherian ring. Complete ideals in two dimensional rational singularities and zero dimensional monomial ideals in polynomial rings are examples of such ideals. A local cohomological characterization of such ideals will be presented. We show how one can compute the Hilbert polynomial of such ideals and how they produce Cohen-Macaulay Rees algebras. We also point out their connection with an unsolved conjecture of Itoh about the normal reduction number of an m-primary ideal in a three dimensional Gorenstein local ring.

It is not difficult to construct a commutative ring R with R isomorphic to R[x]. But is it possible to have R isomorphic to R[x,y] and yet R not isomorphic to R[x]? Similarly, it is easy to show that if R is a commutative ring, and M,N are two right R-modules with the ascending chain condition on cyclic submodules, then M direct sum N has the same property. Does the same result hold for the ascending chain condition on finitely generated submodules? We discuss approaches to these, and other, questions, using free constructions.

The F-signature of a local ring is a numerical invariant that was formally introduced by Huneke and Leuschke, and it was shown to always exist by Tucker. Roughly speaking, it provides an asymptotic measurement for the number of splittings of the Frobenius endomorphism on a ring of prime characteristic, which has very subtle connections with the theory of singularities. In this talk, we will show how to define the F-signature for rings which are not necessarily local. We will also discuss how to define the F-signature with respect to a Cartier algebra in this generality, as well as how the resulting invariants still provide a measure of the singularities. This talk is based on joint work with Thomas Polstra and Yongwei Yao.

We show that the Castelnuovo-Mumford regularity and related invariants of products of powers of ideals in a standard graded polynomial ring are affine-linear in the exponents if these are large enough, provided that each ideal is generated by elements of constant degree. A counterexample shows that linearity is false without the condition, but the regularity is always given by the maximum of finitely many affine-linear polynomials. This is joint work with Aldo Conca (Genoa).

Hochster and Huneke proved that if R is an excellent local domain of charac- teristic p, then the absolute integral closure R+ of R is a big Cohen-Macaulay module. This result is certainly not true in characteristic zero or mixed characteristic. However, Gennady Lyubeznik suggested an analog which may work in mixed characteristic. He asked, "If P is the radical of pR+ , is R+/P a big Cohen-Macaulay module?" I will discuss some observations directed to answering this question.

We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.

There are well-known lower bounds on the Hilbert-Samuel multiplicity of a Noetherian local ring, depending on what type of ring is under consideration (e.g., complete intersection, Gorenstein, Cohen-Macaulay). Rings for which these lower bounds are achieved are known to have nice properties, including homological ones. In this talk we shall show that many of these desirable homological properties characterize rings with minimal multiplicity.

Boij-Söderberg theory is a structure theory for syzygies of graded modules: a near-classification of the possible Betti tables of such modules (which record the degrees of generators in a minimal free resolution). One of the surprises of the theory was the discovery of a "dual" classification of sheaf cohomology tables on projective space. I'll tell part of this story, then describe some recent extensions of it to the setting of Grassmannians. Here, the algebraic side concerns modules over a polynomial ring in kn variables, thought of as the entries of a k x n matrix. The goal is to classify "GL_k-equivariant Betti tables" and relate them to sheaf cohomology tables on the Grassmannian Gr(k,n). This work is joint with Nic Ford and Steven Sam.