Commutative Algebra Seminar

A stability condition gives a measure of the instability of a complex of vector bundles via its Harder-Narasimhan filtration. There is a canonical Euler stability condition on projective space that we can use in this way to measure the instability of free resolutions of Gorenstein rings. This has more information than the Betti tables (and Hilbert functions), and leads to questions about Gorenstein rings with socle in degree 3 that we would love know more about. This is joint work with Brooke Ullery.

We use the notion of filter regular sequence to study some singularities and invariant defined by Frobenius action on local cohomology.

Let k be a field and R a short, standard-graded Gorenstein k-algebra with embedding dimension e > 2. (Thus the Hilbert series of R is 1+es+s^2 .) The category of finitely generated graded R-modules has wild representation type, but much of the representation theory of the category is revealed by the Betti tables of modules. The additive semigroup B of all Betti tables of R-modules is atomic but very far from being factorial. We will show how the atoms of B arise as Betti tables of cosyzygies of ideals of R and describe some specific relations among these atoms. This is joint work with Lucho Avramov and Courtney Gibbons.

The notion of an endofinite module was introduced some 25 years ago by Crawley-Boevey. In my talk I'll explain what it is (a natural finiteness condition), where it comes from (representation theory of algebras), and why it is useful also in many other contexts (e.g. in homotopy theory or the study of derived categories).

Let R be a Noetherian local ring of dimension d. In this work, our first goal is to study the behavior of the sequence of lengths of local cohomology modules of powers of ideals. For homogeneous ideals, we are able to show that after restricting the lower degrees to a linear bound, the sequence does not grow faster than n^d. Combining this result with Kodaira-like vanishing theorems, we obtain that the sequence of lengths grows as expected for several broad classes of ideals. Moreover, we study similar vanishing results for powers of modules. This is joint work with Hailong Dao.

Let ( A_1, .... , A_m ) be a tuple of n by n matrices with complex coefficients. We consider the hypersurface in complex affine m-space given by the equation det( - I + x_1 A_1 + ... + x_m A_m ) = 0 where I is the identity matrix. Motivated by questions arising from functional analysis of operators on Hilbert spaces, we investigate how the geometry of this determinantal hypersurface reflects the mutual behavior of our matrices. The applications we will discuss in this talk are to the case when V is a complex n-dimensional unitary representation of a Coxeter group G with Coxeter generators g_1, ... , g_m , and our matrices represent the action of the generators on V. This is joint work with Michael Stessin.

The minimal model of a local map $\phi: R\ to S$ presents a graded Lie algebra $\pi^*(\phi)$, known as the homotopy Lie algebra of $\phi$. While $\pi^*(\phi)$ produces sensible results from the perspective of Koszul duality, it has poor formal properties. In particular, one would like a long exact sequence of homotopy Lie algebras from a fibre sequence, analogous to the topological situation. Andr\'e-Quillen cohomology repairs these formal defects, but produces strange results from the the Koszul duality perspective. I will present a variation on $\pi^*$, namely $\lambda^*$, which enjoys many of the formal properties of Andr\'e-Quillen cohomology while also producing the expected Koszul Duality results. In particular, $\lambda^*$ possesses a Jacobi-Zariski long exact sequence in all situations, and vanishing of $\lambda^*$ characterises regular, complete intersection, and quasi-complete intersection homomorphisms.

Let R be a Noetherian k-algebra, and for simplicity assume that R is augmented of finite type over k of characteristic zero. The minimal semifree model of R is given by the Chevalley-Eilenberg construction of a Lie(-infinity) coalgebra whose dual is the homotopy Lie algebra $\pi$ of R. Taking the universal enveloping algebra of the latter gives rise to the Ext algebra of k. In this talk we will show how to calculate the higher products on Ext out of the differential on the minimal semifree model of R. This comes as a byproduct of a formula for A-infinity PBW multiplication on the universal envelope of a Lie-infinity algebra. This is especially tractable when R is a complete intersection, since then the homotopy Lie algebra is so short. This is joint work with Ben Briggs.

In 1960 Lech proved a simple inequality relating the multiplicity and the colength of an m-primary ideal in a local ring. His proof also shows that the inequality is never sharp if dimension is at least two. I will describe two possible ways to refine Lech's inequality and fix this problem. This is a joint work with Craig Huneke and Javid Validashti.

Assume that G is a finite group and that k is a field of characteristic p. A p-divisible module is one whose absolutely indecomposable direct summands all have dimension divisible by p and whose support variety equals the full spectrum of the cohomology ring. These modules (tensor) generate the stable module category, but not as directly as one would like. There are several interesting problems that involve this class of modules.

The minimal free resolution of a quotient of the polynomial ring admits a (generally non-associative) multiplication which satisfies the Leibniz rule. This multiplication is far from being unique and in favorable cases it can be chosen to be associative, which turns the resolution into a DGA. In this talk, I consider these multiplicative structures in the setting of monomial ideals. On the one hand, I will present some structure theorems about these multiplications, and on the other hand, I will show that the presence of an associative multiplication has implications on the possible Betti numbers of the ideal.

In 2000, Ein, Lazarsfeld, and Smith found a uniform bound for the size of symbolic powers of ideals in a regular ring. Their original proof, in characteristic zero, uses multiplier ideals, though the same proof works in positive characteristic using test ideals instead. The main reason for the regularity assumption is so that these test ideals satisfy the subadditivity property: the test ideal of a product is contained in the product of test ideals.

In this talk, we'll discuss a new subaddivity formula that works for non-regular rings. This formula will use the formalism of Cartier algebras. Along the way, we'll see some constructions in the toric case and applications to bounding symbolic powers of ideals.