Morgantown Algebra Days 2023

Poster Titles & Abstracts

Heath Camphire

(George Mason University)

Title: Determining the Betti numbers of $R/(x^{p^e}, y^{p^e}, z^{p^e})$ for most even degree hypersurfaces in odd characteristic

Abstract: Fix a homogeneous polynomial $f$ in $k[x,y,z]$ with $k$ a field of characteristic $p$, set $R=k[x,y,z]/(f)$, and let ${\mathfrak m} = (x,y,z)$. If $f$ has the property link-$p^e$-compressed, then the graded Betti numbers of $R/{\mathfrak m}^{[p^e]}$ depend only on $p^e$ and the degree of $f$, with the high graded Betti numbers only depending on $d$ up to a constant shift. My results show that most even degree choices of $f$ are link-$p^e$-compressed when $p$ is odd.

Andy Jiang

(University of Michigan)

Title: Grothendieck duality without compactifications

Abstract:  Following a formula found in the Avramov, Iyengar, Lipman, Nayak (2010) and ideas of Neeman and Khusyairi, we indicate that Grothendieck duality for finite tor-amplitude maps can be developed from scratch via the formula $f^! := \delta^*\pi_1^{\times}f^*$. Our strategy centers on the subcategory $\Gamma_{\Delta}(\mathrm{QC}({X \times X}))$ of quasicoherent sheaves on $X \times X$ supported on the diagonal. By exclusively using this subcategory instead of the full category $\mathrm{QC}({X \times X})$ we give systematic categorical proofs of results in Grothendieck duality and reprove many formulas found in Neeman (2017). We also relate some results in Grothendieck duality with properties of the sheaf of (derived) Grothendieck differential operators.

 Adam LaClair

(Purdue University)

Title: Castelnuovo—Mumford regularity of binomial edge ideals

Abstract: In 2010, Herzog et al. and independently Ohtani, associated to any simple graph $G$ the binomial edge ideal $J_{G}$. Many people have sought to relate the combinatorial structure of $G$ with the algebraic structure of $J_{G}$. In this poster, I present research where I introduce a new invariant $\nu(G)$ associated to a graph and discuss its connections to the Castelnuovo—Mumford regularity of the binomial edge ideal, $\text{reg}(R/J_{G})$. For example, I show that $\nu(G) \leq \text{reg}(R/J_{G})$ and that $\ell(G) \leq \nu(G)$ for any graph $G$ which recovers a result of Matsuda and Murai. Furthermore, I show that $\nu(G) = \text{reg}(R/J_{G})$ when $G$ is closed (which recovers a result of Ene and Zarojanu), or if $G$ is a block graph (which answers a question of Herzog and Rinaldo).

Brian Laverty

(West Virginia University)

Title: Depth formula for modules of finite reducing projective dimension

Abstract: Auslander introduced the depth formula $\text{depth} M + \text{depth}   N = \text{depth} R+ \text{depth}(M\otimes_RN)$, where he showed that this depth equality holds when $M$ and $N$ are Tor-independent and one of $M$ or $N$ has finite projective dimension. The depth formula was further studied by Foxby, Araya-Yoshino, Huneke-Wiegand, and others. More recently Bergh and Jorgensen proved that, under mild conditions, the depth formula holds for Tor-independent modules, either of which has reducible complexity. We generalize their result and study the depth formula for modules of finite reducing projective dimension. The definition of reducing projective dimension is similar to that of reducible complexity, but the new advantage we have is that a module of finite reducing projective dimension can have infinite complexity, whereas modules that have reducible complexity have, by definition, finite complexity, that is, have polynomial growth on their Betti numbers. This is based on the ongoing joint work with Olgur Celikbas, Toshinori Kobayashi, and Hiroki Matsui.

Uyen (ENNI) Le

(West Virginia University)

Title: Remarks on a conjecture of Huneke and Wiegand

Abstract: Unlike vector spaces over a field, not all modules over a ring have a basis. Using an operation known as the tensor product, the Huneke-Wiegand Conjecture (HWC) seeks conditions on a module $M$ over a one-dimensional local ring such that $M$ has a basis. This research shows (HWC) is affirmative for 2-periodic (joint work with O. Celikbas, Matsui, and Sadeghi) and 4-periodic modules.

David Lieberman

(University of Nebraska - Lincoln)

Title: Generalizing Bernstein's inequality for D-modules

Abstract: Bernstein's inequality is a classic result in the realm of D-modules. The inequality puts a lower bound on the dimension of a D-module over polynomial rings. More generally, when the inequality holds modules that achieve this lower bound enjoy nice properties. Our overall goal is to show the inequality holds in certain non-polynomial ring settings. This has previously been shown in the case of Veronese rings. In this poster we present a new proof of the same fact that we hope may be useful in showing other singular rings satisfy Bernstein's inequality.

Cheng Meng

(Purdue University)

Title: Multiplicities in flat local extensions

Abstract: We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if $(R, \mathfrak m)$ and $(S,\mathfrak n)$ are Noetherian local rings of the same dimension, $S$ is a flat local extension of $R$, and up to completion $S$ is standard graded over a field and $I=\mathfrak mS$ is homogeneous, then the multiplicity of $R$ is no greater than that of $S$.

Hunter Simper  

(Purdue University)

Title: Local cohomology of certain determinantal thickenings

Abstract:   Let $R=\mathbb{C}[\{x_{ij}\}]$ be the ring of polynomial functions in $n(n-1)$ variables and $X$ the $n\times (n-1)$ matrix in these variables. Set $I=I_{n-1}(X)$ to be the ideal of maximal minors of $X$. This poster discusses the $R$-modules structure of certain Ext and local cohomology modules involving $R/I^t$ for $t\geq 1$. To do this we describe a map between the Koszul complex of the $t$-powers of the maximal minors and a free resolution of $R/I^t$, the latter of which is obtained via linear strands of a resolution of the Rees algebra of $I$.