Commutative Algebra Seminar

Fall 2017, Friday 2:30 - 3:20, LCB 215

Date Speaker Title — click for abstract
September 1 Aaron Bertam
University of Utah
Unstable syzygies
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.
September 8 Pham Hung Quy
FPT University, Vietnam
Applications of filter regular sequences to some positive characteristic problems
We use the notion of filter regular sequence to study some singularities and invariant defined by Frobenius action on local cohomology.
September 15 Roger Wiegand
University of Nebraska
Betti tables over short Gorenstein algebras
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.
September 22 Henning Krause
Bielefeld University, Germany
The what, where, and why of endofinite modules
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).
October 6 TBD
October 19
4pm-5pm Nonstandard day
Jonathan Montano
University of Kansas
Local cohomology of powers of ideals and modules
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.
October 20 Alex Tchernev
SUNY, Albany
Determinantal hypersurfaces and representations of Coxeter groups
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.
November 3 Benjamin Briggs
University of Toronto
Long exact sequences for the homotopy Lie algebra and the L.S. category of a homomorphism
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.
November 3
3:30 - 4:20 PM
JWB 333
Vincent Galinas
University of Toronto
The A-infinity structure on the Ext algebra of a commutative ring
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.
November 10 Ilya Smirnov
University of Michigan
December 8 Lukas Katthaen
University of Minnesota
January 12 Florian Enescu
Georgia State University
February 9 Akhil Mathew
University of Chicago

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