Schedule and Abstracts

All talks took place in Avery Hall 110.
Avery Hall Floorplan

The tentative schedule is:
 Saturday, April 28, 2018 10:30 – 10:50 Registration 10:50 – 11:00 Opening Remarks *11:00 – 11:40 Brent Holmes Higher Nerves, Serre's Condition, and Balanced Simplicial Complexes 11:50 – 12:10 Eric Canton Frobenius splittings and DVRs 12:10 – 1:50 Lunch *1:50 – 2:30 Carly Matson Algebraic techniques in the study of elliptic curves 2:40 – 3:00 Liliam Carsava Merighe On classical integral closure and integral closure relative to an Artinian module *3:10 – 3:50 Seth Lindokken Resolutions of Finite Length Modules over Complete Intersections 3:50 – 4:10 20 minute break *4:10 – 4:50 Michael Perlman Regularity of Pfaffian thickenings 5:00 – 5:20 Justin Lyle Hom and Ext, Revisited
 Sunday, April 29, 2018 8:45 – 9:20 Coffee and Snacks *9:30 – 10:10 Eloísa Grifo Homological algebra vs symbolic powers 10:20 – 10:40 Alessandra Costantini Cohen-Macaulayness of Rees algebras of modules. *10:50 – 11:30 Ben Briggs The homotopy Lie algebra and LS category 11:40 – 12:00 William Taylor Finite Generation of Rings of Toric Frobenius Operators 12:10 Closing Remarks
*40 Minute Talk

• Brent Holmes; Higher Nerves, Serre's Condition, and Balanced Simplicial Complexes
Abstract: We investigate generalized notions of the nerve complex for the facets of a simplicial complex, including how the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[\Delta]$ as well as the $f$-vector and $h$-vector of $\Delta$. We then establish relationships between simplicial complexes satisfying Serre's condition $(S_{\ell})$ and the vanishing of reduced homologies of their higher nerve complexes. We examine the behavior of rank selected subcomplexes of balanced $(S_{\ell})$ complexes, and, generalizing results of Stanley and Hibi, we prove that these subcomplexes retain $(S_{\ell})$.​

• Eric Canton; Frobenius splittings and DVRs
Abstract: Suppose $R$ is a normal integral domain of finite type over a perfect field of characteristic $p > 0$. In this talk, I'll give an overview of an interplay between elements whose $p$-th roots can be made into free $R$-summands of $R^{1/p}$ and their corresponding expressions at height one primes (all of whose local rings are DVRs, since $R$ is normal!). I hope to make this inviting and interesting for people of many Frobenius familiarity levels.

• Carly Matson; Algebraic techniques in the study of elliptic curves
Abstract: One of the oldest areas of number theory is the study of Diophantine equations, or the search for rational solutions of polynomials. Eighteen centuries later, we're still searching. In this talk we will focus on how the group structure of an elliptic curve varies as we take solutions in different fields and will introduce some of the modern techniques involved in this study, including Galois cohomology and the attempt to extend "local" behavior to draw "global" conclusions.

• Liliam Carsava Merighe; On classical integral closure and integral closure relative to an Artinian module
Abstract: Let $(R,\mathfrak{m})$ be a commutative Noetherian complete local ring. Motivated by a Rees' question, in this work we study which is the relationship between $\overline{\mathfrak{b}}$, the classical Northcott-Rees integral closure of $\mathfrak{b}$, and $\mathfrak{b}^{*(H)}$, the integral closure of $\mathfrak{b}$ relative to an Artinian $R$-module $H$. We conclude they are equal when every minimal prime ideal of $R$ belongs to $\mathrm{Att}_R(H)$. As application, we show what happens when $H$ is a generalized local cohomology module.

• Seth Lindokken; Resolutions of Finite Length Modules over Complete Intersections Abstract: The structure of free resolutions of finite length modules over regular local rings has long been a topic of interest in commutative algebra. Conjectures of Buchsbaum-Eisenbud-Horrocks and Avramov-Buchweitz predict that the minimal free resolution of the residue field should give, in some sense, the smallest possible free resolution of a finite length R-module when R is regular. Results of Tate and Shamash suggest analogous lower bounds for the size of certain free resolutions over hypersurface rings. After describing some cases in which these lower bounds hold, we use the theory of higher matrix factorizations developed by Eisenbud-Peeva and recent results of Iyengar-Walker to show that they fail in general.

• Michael Perlman; Regularity of Pfaffian thickenings
Abstract: Let $S$ be the ring of polynomial functions on the space of $n \times n$ complex skew-symmetric matrices. This ring has a natural action of the group $GL(n)$. For every invariant ideal $I$ in $S$, we compute the modules $\text{Ext}^i(S/I,S)$, and as a consequence we obtain formulas for the Castelnuovo-Mumford regularity of powers and symbolic powers of ideals of Pfaffians. This allows us to characterize when powers of ideals of Pfaffians have linear minimal free resolution.

• Justin Lyle; Hom and Ext, Revisited
Abstract: Let $R$ be a commutative Noetherian local ring and $M,N$ be finitely generated $R$-modules. We prove a number of results of the form: if $\text{Hom}_R(M,N)$ has some nice properties and $\text{Ext}^{1 \leq i \leq n}_R(M,N)=0$ for some $n$, then $M$ (and sometimes $N$) must be close to free.​ Using these results, we are able to extend and unify a number of results in the literature.

• Eloísa Grifo; Homological algebra vs symbolic powers
Abstract: The containment problem for symbolic and ordinary powers of ideals asks when the containment $I^{(a)} \subseteq I^b$ holds. Under nice enough conditions, we can replace this question by a purely homological one: whether or not a certain map between Ext modules vanishes. We will answer this question for certain classes of ideals $I$, and long the way compute free resolutions for all $I^n$ using Rees Algebra techniques.

• Alessandra Costantini; Cohen-Macaulayness of Rees algebras of modules.
Abstract: Rees algebras of ideals and modules arise in Algebraic Geometry as homogeneous coordinate rings of blow up or as graphs of rational maps. The Cohen-Macaulayness of the Rees algebra of an ideal I is well-understood in connection with the Cohen-Macaulayness of the associated graded ring of I, thanks to results of Huneke, Trung and Ikeda. However, there is no module analogue for the associated graded ring, so the study of Cohen-Macaulayness of Rees algebras of modules is in general more complicated. In this talk we will present the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos, and use it to provide sufficient conditions for the Rees algebra of a module to be Cohen-Macaulay. Our results generalize results of Johnson and Ulrich, and of Goto, Nakamura and Nishida.

• Ben Briggs; The homotopy Lie algebra and LS category
Abstract: Since the early 80s a glorious and productive friendship has existed between rational homotopy theory and local commutative algebra, centring on the so-called homotopy Lie algebra (and becoming increasingly dormant...). I will discuss models for the homotopy Lie algebra of a local ring (or homomorphism), slightly generalising some classical theorems of Avramov. Then I will talk about LS category, an invariant for local rings and homomorphisms imported from rational homotopy theory. After going over some properties of this invariant, I'll explain how to compute it using the Mapping Theorem. The rational homotopy theory version of the Mapping Theorem was proven by Felix and Halperin in the 80s, but it has been missing from the commutative algebra side for some time.

• William Taylor; Finite Generation of Rings of Toric Frobenius Operators
Abstract: In this talk we will consider Frobenius operators, that is, $p^e$-linear maps, on toric rings of dimension 2. We will give a set of geometric conditions guaranteeing that a certain graded subring of these operators is finitely generated as an algebra over its zeroth piece. The proof will involve representing $p^e$-linear maps as points in space, understanding the convex geometry of the toric structures, and seeing how the compositions of $p^e$ and $p^{-e}$-linear maps behave. This talk is based on work in progress with Lance Miller.