Schedule and Abstracts

All talks will take place in Avery Hall 106, with refreshments in Avery Hall 13B. (Avery Hall Floorplan)

Saturday, April 26, 2015
10:30 – 10:50 Registration
10:50 – 11:00 Opening Remarks
11:00 – 11:40 Brooke Ullery
Constructing ideals with high Castelnuovo-Mumford regularity
11:50 – 12:30 Daniel Hern├índez
Roots of Bernstein-Sato polynomials via reduction to characteristic $p>0$
12:30 – 2:00 Lunch
2:00 – 2:20 Rebecca R.G.
Closures that give big Cohen-Macaulay modules and algebras, and smallest closures
2:30 – 3:10 Luigi Ferraro
Regularity of Tor for weakly stable ideals
3:20 – 3:40 Kat Shultis
Systems of parameters of modules and the Cohen-Macaulay property
3:40 – 4:00 20 Minute Break
4:00 – 4:40 Hannah Altmann
Semidualizing DG modules over tensor products
4:50 – 5:10 Aaron Feickert
Gorenstein injective filtrations
Sunday, April 27, 2015
9:30 – 10:10 Thomas Polstra
Lower Semi-Continuity of the F-signature
10:20 – 10:40 Alessio Sammartano
Deviations of graded algebras
10:50 – 11:30 William Sanders
Cohomological supports and the tensor product
11:40 – 12:00 Peder Thompson
Stable local cohomology
12:00 Closing Remarks

Abstracts

Speaker: Hannah Altmann
Title: Semidualizing DG modules over tensor products
Abstract: Let $R$ be a commutative, noetherian ring with identity. A finitely generated $R$-module $C$ is semidualizing if the homothety map $\chi_C^R:R\rightarrow \operatorname{Hom}_R(C,C)$ is an isomorphism and $\operatorname{Ext}_R^i(C,C)=0$ for all $i>0$. For example, $R$ is semidualizing over $R$, as is a dualizing module, if R has one. In some sense the number of semidualizing modules measures the severity of the singularity of $R$. We are interested in that number. We can extend this idea to semidualizing complexes of $R$ and generalize even further over Differential Graded (DG) algebras. We will discuss constructing semidualizing DG modules over tensor products of DG algebras over a field. In particular, this gives us a lower bound on the number of semidualizing DG modules over the tensor product.

Speaker: Luigi Ferraro
Title: Regularity of Tor for weakly stable ideals
Abstract: In this talk we prove that if $I$ and $J$ are weakly stable ideals in a standard graded polynomial ring $R=k[x_1,...,x_n]$ then the regularity of $\mathrm{Tor}^R_i(R/I,R/J)$ has the expected upper bound. We also give a bound for the regularity of $\mathrm{Ext}^i_R(R/I,R)$ where $I$ is a weakly stable ideal.

Speaker: Aaron Feickert
Title: Gorenstein injective filtrations
Abstract: Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit filtrations analogous to those for injective modules. This extends results of Enochs and Huang over Gorenstein rings.

Speaker: Daniel Hernández
Title: Roots of Bernstein-Sato polynomials via reduction to characteristic $p>0$.
Abstract: The Bernstein-Sato polynomial associated to a polynomial over the complex numbers is a subtle and important invariant that measures the singularities of the corresponding complex hypersurface. Unfortunately, these invariants are notoriously difficult to compute, and are not very well understood. In this talk, I will recall an approach for determining roots of Bernstein-Sato polynomials (due to Mustata, Takagi, and Watanabe) via reduction to characteristic $p>0$, and will report on (ongoing) recent progress obtained jointly with Emily Witt.

Speaker: Thomas Polstra
Title: Lower Semi-Continuity of the F-signature
Abstract: The F-signature of a local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$, denoted $s(R)$, is a numerical invariant describing the asymptotic growth of the number of splittings admitted by powers of the Frobenius endomorphism. Introduced by Huneke and Leuschke as the limit of normalized splitting numbers in 2002 and shown to exist in full generality by Tucker in 2012, the F-signature provides a way to detect subtle information about singularities of local rings. One can then look at the F-signature function defined on a not necessarily local ring which maps a prime $P\mapsto s(R_P)$, the F-signature of the localized ring $R_P$. This function is seen to be the limit function of lower semi-continuous functions, namely the normalized Frobenius splitting number functions. Therefore it is natural to ask whether the F-signature function is lower semi-continuous. In this talk, we will actually discuss why the F-signature function is the uniform limit of the normalized Frobenius splitting number functions, hence must be lower semi-continuous itself.

Speaker: Rebecca R.G.
Title: Closures that give big Cohen-Macaulay modules and algebras, and smallest closures
Abstract: Geoffrey Dietz has introduced a set of axioms for a closure operation on a complete local domain $R$ so that if such a closure operation exists, the ring is guaranteed to have a balanced big Cohen-Macaulay module. These are called Dietz closures. In characteristic $p>0$, solid closure, tight closure, and plus closure all satisfy the axioms. I will discuss a new axiom that, together with the Dietz axioms, is equivalent to the existence of a big Cohen-Macaulay algebra. This new axiom holds for large classes of closures, including those listed above. Further, any Dietz closure is trivial on regular rings. I will also discuss the existence of smallest closure operations satisfying certain sets of axioms, including the Dietz axioms. There are many open questions about the nature of various smallest closures.

Speaker: Alessio Sammartano
Title: Deviations of graded algebras
Abstract: The deviations of a standard graded $k$-algebra are a sequence of numbers that determine its Poincare series and arise as the number of generators of certain DG algebras. We study extremal deviations among those of algebras with a fixed Hilbert series. We prove that deviations do not decrease when passing to initial and lex-segment ideals. We also prove that deviations grow exponentially for Golod rings and for algebras presented by certain edge ideals.

Speaker: William Sanders
Title: Cohomological supports and the tensor product
Abstract: This work is joint with Hailong Dao. Developed by Avramov and Buchweitz, the theory of cohomological supports over a complete intersection ring encodes important homological information about a module into a geometric object. Cohomological supports are often called support varieties. In this talk, we investigate the cohomological support of the tensor product of two modules using the geometry of the cohomological support of the original modules. Furthermore, we pose questions regarding the asymptotic behavior of cohomological supports of Tor modules.

Speaker: Kat Shultis
Title: Systems of parameters of modules and the Cohen-Macaulay property.
Abstract: Let $R$ be a local ring and $M$ a finitely generated $R$-module. In 1956, Rees showed that if $M$ is Cohen-Macaulay, then $\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b}M)\cong M/\mathfrak{a}M$ for all parameter ideals $\mathfrak{b}\subseteq\mathfrak{a}$. When $M=R$ this is a free $R/\mathfrak{a}$-module of rank one, and hence is indecomposable. Recently, K. Bahmanpour and R. Naghipour proved the converse for $M=R$. Namely, if $R$ is not Cohen-Macaulay, then there exist parameter ideals $\mathfrak{b}\subseteq\mathfrak{a}$ such that $\operatorname{Hom}_R(R/\mathfrak{a},R/\mathfrak{b})\not\cong R/\mathfrak{a}$. In this talk, we'll explore what the structure of $\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b}M)$ is when $M$ is not Cohen-Macaulay. In particular, we'll focus on the properties of indecomposability and free-ness.

Speaker: Peder Thompson
Title: Stable local cohomology
Abstract: Let $R$ be a Gorenstein ring, $I$ an ideal of $R$, and $M$ any $R$-module. Local cohomology of $M$ with support in $I$ is defined by applying the $I$-torsion functor to an injective resolution of $M$ (and taking cohomology). In a Gorenstein ring, every module has a complete injective resolution, so a natural question to consider is what one obtains by applying the $I$-torsion functor to a complete injective resolution of $M$. It turns out the resulting complex is exact in every degree (due to Lipman), so we consider a syzygy rather than take cohomology. We define the stable local cohomology module of $M$ to be the $0$th syzygy of this complex, and as a main result give a tight connection relating it to classical local cohomology in the case of one non-zero local cohomology module. Along the way, we show that this process yields a functor to the stable category of Gorenstein injective modules, and much of the behavior of this functor mirrors results in local cohomology.

Speaker: Brooke Ullery
Title: Constructing ideals with high Castelnuovo-Mumford regularity.
Abstract: The Castelnuovo-Mumford regularity of a module is a homological invariant that roughly measures complexity. Though straight-forward to define, it is difficult to find ideals in polynomial rings with high Castelnuovo-Mumford regularity. I will demonstrate a method that takes as input well-understood modules and outputs ideals which cut out schemes supported on linear spaces with high Castelnuovo-Mumford regularity and other desirable homological properties.

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