## CHAMPS academic year 2020‐2021 speakers

Pinches Dirnfield, University of Utah

Title: Base Change Along the Frobenius Endomorphism And The Gorenstein Property

Abstract:

Let \(R\) be a local ring of positive characteristic and \(X\) a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of \(X\) via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then \(R\) is Gorenstein.

Mohsen Gheibi, University Of Texas At Arlington

Some criteria for detecting large, small, and Golod homomorphisms

Abstract:

This talk consists of two parts. In the first part, I will focus on large homomorphisms introduced by Levin in 1978. I will address some known examples and characterization of large homomorphisms in terms of Koszul homologies over complete intersection and Golod local rings. Next, I will discuss large homomorphisms from or to a Koszul algebra and its consequences. In the second part, I will focus on a special class of local rings namely minimal intersections. Recall that a local ring \(R\) is a minimal intersection if R (or the completion of R) has a presentation \(R=Q/(I+J)\) with Q a local ring and \(Q/I\) and \(Q/J\) Tor-independent as \(Q\)-modules. I will discuss the large and smallness of the natural map from \(Q\) to \(Q/I\) and its consequences on computing the Poincaré series of \(R\).

Joseph Skelton, Tulane University

Koszul property of symbolic powers cover ideals

Abstract:

For cover ideals we are motivated by the results of Villarreal showing that whiskering a graph results in a Cohen-Macaulay graph which, in turn, implies the cover ideal of the whiskered graph has linear resolution. Later it was shown that whiskering \(S\subsetneq V(G)\) resulted in the cover ideal of the graph whiskered at \(S\), \(J(G\cup W(S))\), being sequentially Cohen-Macaulay and therefore Koszul. In '16, Fakhari introduced a graph construction \(G_k\) that corresponds to the symbolic power of the cover ideal \(J(G)^{(k)}\). Using this construction and the whiskering technique we will establish conditions on \(S\) such that \(J(G\cup W(S))^{(k)}\) is Koszul for all \(k\).

Tyler Anway, University of Texas at Arlington

Classification of Totally Acyclic Complexes over Local Gorenstein Rings

Abstract:

Let \(Q\) be a commutative local ring to which we associate the subcategory \(\textrm{Ktac}(Q)\) of the homotopy category of \(Q\)-complexes, consisting of the totally acyclic complexes. Assume further that \(Q\) is a Henselian Gorenstein ring with a surjective ring homomorphism \(Q \xrightarrow{\phi} R\) and \(\textrm{pd}_Q(R) < \infty\). We will use the indecomposable objects of \(Q\), and the idea of approximations to classify totally acyclic complexes over \(R\) using our defined notion of Arnold-tuples.

Sankhaneel Bisui, Tulane University

Stable Harbourne—Huneke Containment and Bounds on Waldschmidt Constant

Abstract: The study of the degree of a homogeneous polynomial vanishing on given points with multiplicities is always intriguing. Nagata raised the following fundamental question:

Q: Given a finite set of points \(X= \{P_1,\dots P_s \} \subset \mathbf{P}_{\mathbb{C}}^N \) what is the minimal degree, \(\alpha_x(m) \) of a hyper-surface that passes through the points with multiplicity at least \(m\)?

Chudnovsky provided a conjectural answer to the above question which was generalized by Demailly. Both conjectures have equivalent statements involving a lower bound of the Waldschmidt constant of the ideal defining points. Harbourne and Huneke gave containment conjectures involving the symbolic and the ordinary powers of the ideals, which implies Chudnovsky's conjecture and Demailly's conjecture respectively.

We study the stable versions of the containment conjectures and consequently, we prove Chudnovsky and Demailly's conjecture for a large number of general points. In this talk, I will introduce all these conjectures and the tools that we used. I will also present the results from our joint work with Eloísa Grifo, Tài Huy Hà, and Thái Thành Nguyễn.

Prashanth Sridhar, University of Kansas

Finding maximal Cohen Macaulay and reflexive modules

Abstract:

Maximal Cohen Macaulay and Reflexive modules are both very classical objects and their properties have been studied extensively. However, their existence and/or ubiquity is far from clear. In this talk, I will consider these questions and provide answers in certain settings.

Ashley Wheeler, Mount Holyoke College

Geometric equations for matroid varieties

Abstract:

Each point \(x\) in Gr\((r,n)\) corresponds to an \(r \times n\) matrix \(A_x\) which gives rise to a matroid \(M_x\) on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets \(\left\{y \in \right.\)Gr\(\left.(r,n)\, |\, M_y = M_x\right\}\) form a stratification of Gr\((r,n)\) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals \(I_x\) of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of \(I_x\) geometrically when the combinatorics of the matroid is sufficiently rich.

Rankeya Datta, University of Illinois at Chicago

How valuation rings behave like non-noetherian regular rings

Abstract:

Valuation rings have wide-ranging applications in algebra, arithmetic and geometry. These highly non-noetherian objects became popular in commutative algebra and algebraic geometry when Zariski used them to study normality and resolution of singularities. In recent years there has been a renewed interest in valuation rings because of their foundational roles in various approaches to rigid geometry such as Berkovich and Perfectoid spaces as well as their applications in K-stability and moduli problems. In this talk we will examine commutative and homological algebraic properties of valuation rings. We will show that despite being non-noetherian, valuation rings share striking similarities with regular rings. This talk is based on joint works with Karen Smith and Benjamin Antieau.

Nick Cox-Steib, University of Missouri

Perturbing Ideals in Arbitrary Noetherian Local Rings and the \(\mathfrak{m}\)-adic Continuity of Hilbert-Kunz Multiplicity

Abstract:

In 2018, Polstra and Smirnov showed that the Hilbert-Kunz multiplicity of F-finite CM local rings exhibits a remarkable kind of \(\mathfrak{m}\)-adic continuity. In this talk I will discuss techniques that can be used to extend their result to arbitrary F-finite local rings. These methods also have applications to Hilbert-Samuel multiplicity and general questions of \(\mathfrak{m}\)-adic stability in equal characteristic Noetherian local rings.

## Academic year 2020‐2021 schedule

Pinches Dirnfield, University of Utah

Title: Base Change Along the Frobenius Endomorphism And The Gorenstein Property

Abstract:

Let \(R\) be a local ring of positive characteristic and \(X\) a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of \(X\) via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then \(R\) is Gorenstein.

Mohsen Gheibi, University Of Texas At Arlington

Some criteria for detecting large, small, and Golod homomorphisms

Abstract:

This talk consists of two parts. In the first part, I will focus on large homomorphisms introduced by Levin in 1978. I will address some known examples and characterization of large homomorphisms in terms of Koszul homologies over complete intersection and Golod local rings. Next, I will discuss large homomorphisms from or to a Koszul algebra and its consequences. In the second part, I will focus on a special class of local rings namely minimal intersections. Recall that a local ring \(R\) is a minimal intersection if R (or the completion of R) has a presentation \(R=Q/(I+J)\) with Q a local ring and \(Q/I\) and \(Q/J\) Tor-independent as \(Q\)-modules. I will discuss the large and smallness of the natural map from \(Q\) to \(Q/I\) and its consequences on computing the Poincaré series of \(R\).

Joseph Skelton, Tulane University

Koszul property of symbolic powers cover ideals

Abstract:

For cover ideals we are motivated by the results of Villarreal showing that whiskering a graph results in a Cohen-Macaulay graph which, in turn, implies the cover ideal of the whiskered graph has linear resolution. Later it was shown that whiskering \(S\subsetneq V(G)\) resulted in the cover ideal of the graph whiskered at \(S\), \(J(G\cup W(S))\), being sequentially Cohen-Macaulay and therefore Koszul. In '16, Fakhari introduced a graph construction \(G_k\) that corresponds to the symbolic power of the cover ideal \(J(G)^{(k)}\). Using this construction and the whiskering technique we will establish conditions on \(S\) such that \(J(G\cup W(S))^{(k)}\) is Koszul for all \(k\).

Tyler Anway, University of Texas at Arlington

Classification of Totally Acyclic Complexes over Local Gorenstein Rings

Abstract:

Let \(Q\) be a commutative local ring to which we associate the subcategory \(\textrm{Ktac}(Q)\) of the homotopy category of \(Q\)-complexes, consisting of the totally acyclic complexes. Assume further that \(Q\) is a Henselian Gorenstein ring with a surjective ring homomorphism \(Q \xrightarrow{\phi} R\) and \(\textrm{pd}_Q(R) < \infty\). We will use the indecomposable objects of \(Q\), and the idea of approximations to classify totally acyclic complexes over \(R\) using our defined notion of Arnold-tuples.

Sankhaneel Bisui, Tulane University

Stable Harbourne—Huneke Containment and Bounds on Waldschmidt Constant

Abstract: The study of the degree of a homogeneous polynomial vanishing on given points with multiplicities is always intriguing. Nagata raised the following fundamental question:

Q: Given a finite set of points \(X= \{P_1,\dots P_s \} \subset \mathbf{P}_{\mathbb{C}}^N \) what is the minimal degree, \(\alpha_x(m) \) of a hyper-surface that passes through the points with multiplicity at least \(m\)?

Chudnovsky provided a conjectural answer to the above question which was generalized by Demailly. Both conjectures have equivalent statements involving a lower bound of the Waldschmidt constant of the ideal defining points. Harbourne and Huneke gave containment conjectures involving the symbolic and the ordinary powers of the ideals, which implies Chudnovsky's conjecture and Demailly's conjecture respectively.

We study the stable versions of the containment conjectures and consequently, we prove Chudnovsky and Demailly's conjecture for a large number of general points. In this talk, I will introduce all these conjectures and the tools that we used. I will also present the results from our joint work with Eloísa Grifo, Tài Huy Hà, and Thái Thành Nguyễn.

Prashanth Sridhar, University of Kansas

Finding maximal Cohen Macaulay and reflexive modules

Abstract:

Maximal Cohen Macaulay and Reflexive modules are both very classical objects and their properties have been studied extensively. However, their existence and/or ubiquity is far from clear. In this talk, I will consider these questions and provide answers in certain settings.

Ashley Wheeler, Mount Holyoke College

Geometric equations for matroid varieties

Abstract:

Each point \(x\) in Gr\((r,n)\) corresponds to an \(r \times n\) matrix \(A_x\) which gives rise to a matroid \(M_x\) on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets \(\left\{y \in \right.\)Gr\(\left.(r,n)\, |\, M_y = M_x\right\}\) form a stratification of Gr\((r,n)\) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals \(I_x\) of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of \(I_x\) geometrically when the combinatorics of the matroid is sufficiently rich.

Rankeya Datta, University of Illinois at Chicago

How valuation rings behave like non-noetherian regular rings

Abstract:

Valuation rings have wide-ranging applications in algebra, arithmetic and geometry. These highly non-noetherian objects became popular in commutative algebra and algebraic geometry when Zariski used them to study normality and resolution of singularities. In recent years there has been a renewed interest in valuation rings because of their foundational roles in various approaches to rigid geometry such as Berkovich and Perfectoid spaces as well as their applications in K-stability and moduli problems. In this talk we will examine commutative and homological algebraic properties of valuation rings. We will show that despite being non-noetherian, valuation rings share striking similarities with regular rings. This talk is based on joint works with Karen Smith and Benjamin Antieau.

Nick Cox-Steib, University of Missouri

Perturbing Ideals in Arbitrary Noetherian Local Rings and the \(\mathfrak{m}\)-adic Continuity of Hilbert-Kunz Multiplicity

Abstract:

In 2018, Polstra and Smirnov showed that the Hilbert-Kunz multiplicity of F-finite CM local rings exhibits a remarkable kind of \(\mathfrak{m}\)-adic continuity. In this talk I will discuss techniques that can be used to extend their result to arbitrary F-finite local rings. These methods also have applications to Hilbert-Samuel multiplicity and general questions of \(\mathfrak{m}\)-adic stability in equal characteristic Noetherian local rings.

Beihui Yuan, Cornell University

Splines and a counter-example to the Schenck-Stiller ''\(2r+1\)" conjecture.

Abstract:

To approximate a function over a region, it is useful to consider a subdivision of the region and then approximate the function by a piecewise polynomial. In this talk, I would like to talk about what we know about splines, commutative algebra tools we use to study this subject, conjectures on splines and a counter-example to the Schenck-Stiller ''\(2r+1\)" conjecture. This talk is based on joint work with Mike Stillman and Hal Schenck.

Alessandra Costantini, University of California, Riverside

Cohen-Macaulay property of the fiber cone of modules

Abstract:

Let R be a Noetherian local ring and let E be a finite R-module. The fiber cone of E is the graded algebra F(E) defined by tensoring the Rees algebra R(E) with the residue field of R. In 2003 Simis, Ulrich and Vasconcelos showed that the study of the Cohen-Macaulay property of the Rees algebra R(E) can be reduced to the case of Rees algebras of ideals, by means of the so called generic Bourbaki ideals. The Cohen-Macaulay property of Rees algebras and fiber cones are usually unrelated. However, in this talk I will show that sometimes generic Bourbaki ideals can effectively be used in order to study the Cohen-Macaulay property of the fiber cone F(E) as well, and provide classes of modules whose fiber cone is Cohen-Macaulay. The talk is based on a preprint available at https://arxiv.org/abs/2011.08453.

Benjamin Briggs, University of Utah

Complete intersections and the cotangent complex

Abstract: The cotangent complex is an important but difficult to understand object in commutative algebra. For a homomorphism \(\varphi: R\to S\) of commutative noetherian rings, this is a complex \(L_{\varphi} = L_0\leftarrow L_1\leftarrow \cdots\) of free \(S\)-modules. Inside it you can find the Kähler differentials, the conormal module, the Koszul homology, and it has a lot to say about deformation theory. For complete intersection maps one can completely write down \(L_{\varphi}\), but otherwise it's extremely complicated. When it was introduced by Quillen, he conjectured (for maps of finite flat dimension) that if \(\varphi\) is not complete intersection then \(L_{\varphi}\) must go on forever. This was proven by Avramov in 1999. I will explain how to exploit the connection between the cotangent complex and Hochschild cohomology to get a new proof, and how to simultaneously prove a conjecture of Vasconcelos on the conormal module. This is joint work with Srikanth Iyengar.

Thomas Polstra, University of Virginia

F-purity deforms in \(\mathbb{Q}\)-Gorenstein rings

Abstract:

We positively settle a long-standing question in the theory of prime characteristic singularities: If \( (R,\mathfrak{m},k) \) is a local \( \mathbb{Q} \)-Gorenstein ring of prime characteristic \(p>0\) which admits a non-zero-divisor \(f\) so that \(R/(f)\) is normal and \(F\)-pure, is \(R\) necessarily \(F\)-pure? The origins of the question date back to work of Fedder in the early 1980's where it was shown that \(F\)-purity deforms in Gorenstein rings but fails to deform in rings which are not \(\mathbb{Q}\)-Gorenstein. This talk is based on joint work with Austyn Simpson.

Peder Thompson, Norwegian University of Science and Technology

Using totally acyclic complexes to extend work of Buchweitz into a non-affine setting

Abstract:

In the 1950s, Auslander, Buchsbaum, and Serre set the stage for systematically using homological dimensions to understand ring structure with their elegant characterization of regular local rings in terms of projective dimension. This was followed by the characterization of Gorenstein local rings in terms of “G-dimension” by Auslander and Bridger, which in turn inspired the notions of Gorenstein projective, injective, and flat modules. Such modules are defined in terms of "totally acyclic" complexes. In this talk I will outline recent joint work with Christensen and Estrada where we propose a general notion of total acyclicity that unifies these classic notions and introduces the new “Gorenstein flat cotorsion” modules. This abstraction allows us to extend work of Buchweitz—involving equivalences between homotopy, stable, and singularity categories—into the non-affine setting, thus completing a project initiated by Murfet and Salarian.

Rebekah Aduddell, University of Texas at Arlington

The Critical and Cocritical Degrees of a Totally Acyclic Complex

Abstract:

It is widely known that minimal free resolutions of a module over a complete intersection ring have nice patterns that arise in their betti sequences. In the late 1990's Avramov, Gasharov and Peeva defined a new class of \(R\)-modules (those with finite CI dimension) that would exhibit similar patterns in their free resolutions. In doing so, they additionally defined the notion of critical degree for an \(R\)-module, which describes exactly when such patterns arise in the betti sequence. In this talk, I will define a ``naive'' extension of critical degree to the category of totally acyclic complexes, \(\mathbf{K_{tac}}(R)\), where \(R\) is a commutative local ring. After discussing some basic properties, we will then explore an alternative extension and investigate the relationship between the two definitions.

Jay Yang, University of Minnesota

Random Monomial Ideals

Abstract:

Random and Probabilistic techniques have a long history across a variety of fields, but their use in commutative algebra has been comparatively limited. Random monomial ideals, as inspired by results in random graphs and random simplicial complexes, are a unique perspective that allows us to study the asymptotic behavior of ideals. I discuss a pair of models for random monomial ideals, and a collection of results for these models including work with Caytlin Booms and Daniel Erman as well as work with Lily Silverstein and Dane Wilburne.

Eamon Quinlan-Gallego, University of Michigan

Bernstein-Sato polynomials on singular rings in positive characteristic

Abstract:

Given a smooth \(\mathbb{C}\)-algebra \(R\) and an ideal \(\mathfrak{a} \subseteq R\), an invariant from \(D\)-module theory known as the Bernstein-Sato polynomial of \(\mathfrak{a}\) quantifies the singularities of the pair \( (R, \mathfrak{a}) \). Recently, two generalizations of this construction were given: in one direction, Àlvarez-Montaner, Huneke, and Núñez-Betancourt showed that Bernstein-Sato polynomials can still be defined in some settings where \(R\) has singularities; in the other direction, work of Mustaţă, Bitoun and myself shows that we can also define Bernstein-Sato polynomials in positive characteristic. In this talk, I present joint work with J. Jeffries and L. Núñez-Betancourt in which we show that the combined generalization is possible. Namely, we show that Bernstein-Sato polynomials exist in positive characteristic when \(R\) has mild singularities (direct summand or graded F-finite representation type).

James Gossell, Clemson University

Characterizing Cohen-Macaulay power edge ideals of trees

Abstract:

Every electric power system can be modeled by a graph \(G\) whose vertices represent electrical buses and whose edges represent power lines. A

*phasor measurement unit*(PMU) is a monitor that can be placed at a bus to observe the voltage at that bus as well as the current and phase through all incident power lines. The problem of monitoring an entire electric power system using the fewest number of PMUs is closely related to the well-known vertex covering and dominating set problems in graph theory.

In this talk, we will give an overview of the PMU placement problem and its connections to commutative ring theory. Specifically, we will define the

*power edge ideal*\(I^P_G\) of a graph \(G\) as a way to generate polynomial rings with desired algebraic properties using graphs of electric power grids. Finally, we will classify the trees \(G\) for which \(I^P_G\) is Cohen-Macaulay and prove that every such ideal is also a complete intersection.

This project is joint work with Michael Cowen, Alan Hahn, Frank Moore, and Sean Sather-Wagstaff.

Sarasij Maitra, University of Virginia

A Study of Colength in Dimension One

Abstract:

We define and study an invariant of any module over a local one dimensional analytically unramified Noetherian domain whose integral closure is a DVR. We shall discuss a key property of this invarant. If time permits, we will briefly venture into trace ideals and into reflexive ideals.

Monica Lewis, University of Michigan

The closed support problem over a complete intersection ring

Abstract:

Local cohomology modules are (typically) very large algebraic objects that encode rich geometric information about the structure of a commutative ring. These modules are rarely finitely generated, but sometimes still exhibit remarkable finiteness properties. For example, the local cohomology of a smooth algebra over a field will always have a finite set of associated primes. This property can fail for complete intersection rings (even in codimension 1), but independent results of Hochster and Núñez-Betancourt (2017) or Katzman and Zhang (2017) have shown that at least in characteristic \(p>0\), the local cohomology of a hypersurface ring will still have Zariski closed support. It remains open whether this property holds in arbitrary codimension. In this talk, I will present my results on the local cohomology of a parameter ideal illustrating an obstruction to straightforwardly generalizing existing hypersurface strategies. I will then present joint work with Eric Canton on a possible alternative route of attack in higher codimension, involving a novel Frobenius-compatible simplicial complex of local cohomology modules.

Zhan Jiang, University of Michigan

The "size" of an ideal

Abstract:

Hochster and Huneke defined quasilength for any \(I\)-torsion module, generalizing the notion of length to any non-maximal ideal \(I\). Based on quasilength, we develop a new numerical invariant for ideals, called "size". It is invariant up to taking radicals and bounded between the arithmetic rank and height of the ideal. We will present some results in low dimensions and discuss a lot of open questions related to "size" and asymptotic behaviors of quasilength.

Farrah Yhee, University of Michigan

Ulrich modules do not always exist

Abstract: Ulrich modules were introduced by Bernd Ulrich in 1984 and has since been a very active area of research. The existence of Ulrich modules for complete local domains have powerful applications. For example, existence implies Lech's conjecture: given a flat local map of local rings from R to S, the Hilbert-Samuel multiplicity of S is at least the Hilbert-Samuel multiplicity of R. Until recently, it was unknown if there were any counterexamples to the existence of Ulrich modules for (complete) local domains. In this talk, I will introduce the notion of Ulrich modules and discuss implications of their existence. Then I will give a counterexample to the existence of Ulrich modules for (complete) local domains.

Hugh Geller, Clemson University

DG-Structures on Minimal Free Resolutions of Fiber Products

Abstract:

A construction of Tate shows that every algebra over a ring \(R\) possess a DG-algebra resolution over \(R\). These resolutions are not always minimal and Avramov even shows that certain algebras cannot have a minimal resolution with a DG-algebra structure. In the first half of this talk, I give an explicit construction of the minimal resolution of the fiber product \(k[\underline{x}]/\mathcal{I} \times_k k[\underline{y}]/\mathcal{J}\) over \(k[\underline{x},\underline{y}]\) where \(\mathcal{I} \subseteq \langle \underline{x} \rangle^2\) and \(\mathcal{J} \subseteq \langle \underline{y} \rangle^2\). In the second half, I show how to put different DG-structures on these minimal free resolutions.

Ayah Almousa, Cornell University

Title: Polarizations of Powers of Graded Maximal Ideals

Abstract:

Given a monomial ideal, one can "polarize" it to a square-free monomial ideal that has all of the same homological invariants as the original one. Many commutative algebraists are familiar with the use of the "standard" polarization, but the first use of a nonstandard polarization was by Nagel and Reiner in the 2000s, who used the "box polarization" to produce a minimal cellular resolution for strongly stable ideals. This leads to the natural question: what other ways are there to polarize a monomial ideal, and what other applications might there be for these non-standard polarizations? In this talk, I will give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal in a polynomial ring. I will also give a combinatorial description of their Alexander duals and discuss applications of polarizations to commutative algebra, algebraic geometry, and combinatorics. This is joint work with Gunnar Fløystad and Henning Lohne.

Keller VandeBogert, University of South Carolina

Tate-like Complexes and Their Applications to DG Algebras

Abstract:

In this talk, I will talk about so-called Tate-like complexes. These are defined as a quotient of the tensor product complex and encompass more well-known complexes such as the "symmetric square" complex, commonly used to induce DG-products on resolutions (associative up to homotopy). After introducing necessary notation and terminology, I will talk about how these complexes have made a tacit appearance in previous work of others, and give a brief overview on how these complexes can be used to endow the length 4 "big from small" construction of Kustin and Miller with the structure of an associative DG algebra.