Schedule and Abstracts

All talks are expected to take place in Avery Hall. (Avery Hall Floorplan)

 Saturday, April 23, 2016 10:00 – 10:20 Registration 10:20 – 10:30 Opening Remarks 10:30 – 11:10 William Taylor F-singularities of the generic link 11:15 – 11:35 Basanti Poudyal Existence of exact zero divisors in Artinian Gorenstein rings 11:40 – 12:20 Jason Lutz Homological characterizations of quasi-complete intersections 12:30 – 2:00 Lunch 2:00 – 2:20 John Myers Multiplicity and the homotopy Lie algebra 2:30 – 3:10 Tony Se On Affine Semigroup Rings of Dimension 2 3:20 – 3:40 Nathan Steele Support and Rank Varieties of Totally Acyclic Complexes 3:40 – 4:00 20 Minute Break 4:00 – 4:40 Thomas Polstra Global Hilbert-Kunz Multiplicity 4:50 – 5:10 Brittney Falahola Characterizing Gorenstein Rings Using Frobenius
 Sunday, April 24, 2016 9:30 – 10:10 Alessandro De Stefani Frobenius Betti Numbers 10:20 – 10:40 Shalan Alkarni Three Dimensional Jacobian Derivations And Divisor Class Groups 10:50 – 11:30 Patricia Klein Associated Primes of Local Cohomology Modules 11:40 – 12:00 Brent Holmes On Simplicial Complexes with Serre Property $S_2$ 12:00 Closing Remarks

## Abstracts

Speaker: Shalan Alkarni
Title: Three Dimensional Jacobian Derivations And Divisor Class Groups
Abstract: We use P. Samuel’s purely inseparable descent methods to investigate the divisor class groups of the intersections of pairs of hypersurfaces of the form $w_1 = f$, $w_2 = g$ in affine 5-space with $f , g \in A = k[x, y, z]$; $k$ is an algebraically closed field of characteristic $p > 0$. This corresponds to studying the divisor class group of the kernels of three dimensional Jacobian derivations on $A$ that are regular in codimension one. Our computations focus primarily on pairs where $f , g$ are quadratic forms. We find results concerning the order and the type of these groups. We show that the divisor class group is a direct sum of up to three copies of $\mathbb{Z}_p$, is never trivial, and is generated by those hyperplane sections whose forms are factors of linear combinations of $f$ and $g$.

Speaker: Alessandro De Stefani
Title: Frobenius Betti Numbers
Abstract: We study properties of some numerical invariants that can be associated to a module $M$ of finite length over a local ring $R$ of prime characteristic. In a way, these generalize the Hilbert-Kunz multiplicity and, when $M$ is the residue field of $R$, they can be viewed as some kind of "asymptotic Betti numbers". We aim at a relation between the vanishing of these invariants and projective dimension. Time permitting, we will also discuss some related problems about Krull dimension of syzygy modules of finite length. This is joint work with Craig Huneke and Luis Nuñez-Betancourt.

Speaker: Brittney Falahola
Title: Characterizing Gorenstein Rings Using Frobenius
Abstract: We will give two characterizations of Gorenstein rings of prime characteristic using the Frobenius functor in various ways. The first characterization, which only applies for rings which possess a canonical module, states that a ring is Gorenstein if and only if the Frobenius functor preserves the injective dimension of the canonical module. As a result of this characterization, we'll see that in rings possessing a canonical module, the canonical module serves as a test module for when the Frobenius functor preserves injective dimension. The second characterization depends on the existence of a certain finitely generated module $M$ of finite injective dimension for which the Frobenius functor “plays well” with an injective resolution of $M$.

Speaker: Brent Holmes
Title: On Simplicial Complexes with Serre Property $S_2$
Abstract: Given an ideal $I$ in a polynomial ring, $S$, one can form a graph from the minimal prime ideals of $R = S/I$, where the vertices of the graph are the minimal prime ideals of $R$ and an edge connects two vertices, $v_1, v_2$, if and only if $\mathrm{height}(v_1+v_2) = 1$. This graph is known as the Hochster-Huneke or dual graph of $R$. The $S_2$ property of $R$ implies the connectedness of this graph. We will discuss lower bounds and upper bounds for the diameter of the dual graph in the case that $R$ is $S_2$ and $I$ is a square free monomial ideal.

Speaker: Patricia Klein
Title: Associated Primes of Local Cohomology Modules
Abstract: We will define and briefly discuss some motivation for studying local cohomology. We will then turn to the question of which rings $R$ have the property that $H^i_I(R)$ has finitely many associated primes for all $i$ and $I$. We will survey some results from the past 15 years on rings that do have this property and then share some examples due to Singh and Swanson of hypersurface domains that do not. Lastly, we will state some vanishing theorems for local cohomology and discuss how those results restrict the class of graded hypersurface domains of dimension 4 that may have local cohomology modules with infinitely many associated primes.

Speaker: Jason Lutz
Title: Homological characterizations of quasi-complete intersections
Abstract: Let $(R,\mathfrak m, \Bbbk)$ be a commutative, local, Noetherian ring. Let $I = (\boldsymbol f)$ be a proper, non-zero ideal of $R$ and let $S$ denote $R/I$. Consider the Koszul complex $E = K(\boldsymbol f; R)$; its homology $H(E)$ has the structure of an $S$-algebra. When $H_1(E)$ is free as an $S$-module and $H(E)$ is the exterior algebra $\wedge^SH_1(E)$, we say that $I$ is a quasi-complete intersection (q.c.i) ideal . We'll explore the ideal-theoretic properties q.c.i.$\!$ ideals, develop homological characterizations using Tate's "adjunction of variables", and discuss the relationship between the q.c.i.$\!$ property of ideals and the complete intersection property of rings.

Speaker: John Myers
Title: Multiplicity and the homotopy Lie algebra
Abstract: If $R$ is a commutative noetherian local ring with residue field $k$, then the graded vector space $\operatorname{Ext}_R{(k,k)} = \bigoplus_{n\geq 0} \operatorname{Ext}^n_R{(k,k)}$ has a product making it a graded connected $k$-algebra. It is known that this algebra is the universal enveloping algebra of a (graded) Lie algebra $\pi(R)$ called the homotopy Lie algebra of $R$. In this talk we will explain how rings with minimal Hilbert-Samuel multiplicity can be detected by examining the structure of $\pi(R)$ and its Lie subalgebras.

Speaker: Thomas Polstra
Title: Global Hilbert-Kunz Multiplicity
Abstract: This talk will be a report on some results from a joint work in progress with Alessandro De Stefani and Yongwei Yao. Hilbert-Kunz multiplicity is a classical numerical invariant of study attached to local rings of prime characteristic. In this talk we will discuss how to naturally extend this numerical invariant to rings which are not local and F-finite in a meaningful way.

Speaker: Basanti Poudyal
Title: Existence of exact zero divisors in Artinian Gorenstein rings
Abstract: Let $A$ be a local ring with maximal ideal $m$. We say that $(a,b)$ is an exact pair of zero divisors if there exists a pair of elements $a,b$ in $m$ such that $(0:a)=(b)$ and $(0:b)=(a)$. It is known that a generic Artinian Gorenstein ring of socle degree 3 contains exact pairs of zero divisors. We are interested in the existence of exact zero divisors in the case of socle degree $d>3$. In this talk, I will discuss the conditions when an Artinian Gorenstein ring of socle degree $d>3$ contains linear pairs of exact zero divisors.

Speaker: Tony Se
Title: On Affine Semigroup Rings of Dimension 2
Abstract: Let $k$ be a field. We consider a subring $R = k[x^a, x^{p_1} y^{q_1}, \dots, x^{p_t} y^{q_t}, y^b]$ of the polynomial ring $k[x,y]$. We will give simple numerical criteria for the ring $R$ to be Cohen-Macaulay when $t=2$. We will also give an answer to related questions about $R$, such as finding the Hilbert polynomial for the ideal $(x^a, y^b)$ in $R$ and finding a simple algorithm that generates the monomial $k$-basis of $R/(x^a, y^b)$. This is joint work with Grant Serio.

Speaker: Nathan Steele
Title: Support and Rank Varieties of Totally Acyclic Complexes
Abstract: Support and rank varieties of modules over a group algebra of an elementary abelian p-group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. Avramov and Buchweitz proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. We study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent. We then show that any homogeneous affine variety is realizable as the support of some totally acyclic complex.

Speaker: William Taylor
Title: F-singularities of the generic link
Abstract: In a recent paper of Niu, a strong connection was shown between the multiplier ideal of a pair $(R,I^c)$, where $R$ is a ring of characteristic 0, $I$ an ideal in $R$, $c$ is the codimension of $I$, and the singularities of a generic link $J$ of $I$. In the same paper, a comparison is given between the log canonical thresholds of $I$ and $J$. In this talk we will give a brief introduction to linkage theory and Frobenius singularities. Then we will show analogous positive characteristic statements for the test ideal and the F-pure threshold. Of initial interest will be a proof of the positive characteristic version of (one direction of) Ein's lemma [Niu 2014], with a partial converse.