Commutative Algebra Seminar
Spring 2024, Friday 2:00–3:00 pm, LCB 215
Date  Speaker  Title — click for abstract 
January 12th  Trevor Arrigoni University of Kansas 
Finvariants of simple algebroid plane branches
Frobenius thresholds are a family of invariants associated to singularities in positive characteristic. Though originally defined in terms of the splitting properties of Frobenius,
Mustaţă, Takagi, and Watanabe showed in 2005 that Frobenius thresholds are closely related to the BernsteinSato polynomial and other invariants of singularity in characteristic zero. In this
talk, we will present algorithms to compute Frobenius thresholds of a natural family of irreducible power series in two variables. These algorithms look at computing Frobenius thresholds as an
integer programming problem. The integer programs discussed here build upon those recently developed by Hernández and Witt to compute the roots of the BernsteinSato polynomial for this family
of curves.

January 19th  No seminar 
TBA

January 26th  No seminar 
TBA

February 2nd @2:30, (note unusual time) 
Souvik Dey Charles University 
Finitistic dimension and singularity categories
Let A be a Noetherian ring (not necessarily commutative). When is there a uniform
upper bound on the projective dimensions of all (left) Amodules of finite projective
dimension? When A is commutative, it follows from the works of Bass and GrusonRaynaud, that
this is the case if and only if A has finite Krull dimension. The question of whether such a
uniform upper bound exists for Artin algebras, even when restricted to finitely generated
modules only, was first publicized by Bass in the 1960s. This question, since known as the
finitistic dimension conjecture, remains open even after half a century. In this talk, based on
ongoing joint work with Jan Stovicek, we will present some criteria for the existence of such
uniform upper bounds in terms of certain form of generation in singularity categories. One
ingredient of our approach is based on a generalization of the "delooping level" of Gélinas.

February 9th  Nursel Erey Gebze Technical University 
Regularity and Normalized Depth Function of Squarefree Powers
Let I be a monomial ideal. The k'th squarefree power of $I$ is the ideal generated by the squarefree monomials in I^k. In this talk, we investigate the regularity and depth
function of squarefree powers and consider the question of when such powers have linear resolution.

February 16th  Hunter Simper University of Utah 
Annihilators of local cohomology of determinantal thickenings
Let I be the maximal minors of generic matrix X in R=\mathbb{C}[X]. In this talk I will discuss the module
structure, in particular that annihilators, of the Rmodules H^i_\mathfrac{m}(R/I^t) and Ext^i(R/I^t,R).

February 23rd  Austyn Simpson University of Michigan 
Weakly Fregular rings can be noncatenary
Weakly Fregular rings (i.e. Noetherian rings in which all ideals are tightly closed) are among the mildest singularity types in prime characteristic commutative algebra. They are always normal, and under excellent hypotheses are CohenMacaulay. In this talk, I will explain how in the absence of excellence, weakly Fregular rings need not be CohenMacaulay or even catenary. As a consequence, we extend a theorem of Ogoma concerning the existence of noncatenary splinters of equal characteristic zero to prime characteristic. This is joint work with Susan Loepp.

March 1st  Daniel McCormick University of Utah 
Ghost Maps and AndréQuillen Homology
A map in the derived category is ghost if it induces the zero map on homology. Ghost
maps play an important role in understanding the homological behavior of modules. In this talk, we shift our perspective to commutative ring homomorphisms and introduce an analogous class of maps to this setting. The frobenius endomorphism will be a primary example of a ghost map. We begin with an overview of the necessary tools for handling derived functors in the category of rings, and use this framework to define ghost maps. We conclude with an analog of the ghost lemma in this setting and discuss applications to the study of singularities.

March 8th  Spring break No seminar 

March 15th  Prashanth Sridhar Auburn University 
Orlov's theorem for dgalgebras
A landmark theorem of Orlov relates the derived category of coherent sheaves on a projective complete intersection to the singularity category of its affine cone. I'll discuss joint work with Michael K. Brown generalizing this result to dgalgebras.

March 22nd  Rankeya Datta University of Missouri 
Finite generation of split Fregular monoid algebras
Conjectures about the finite generation of various classes of rings in function fields have been instrumental in the development of algebraic geometry and commutative algebra. In this talk we will introduce one such conjecture in prime characteristic using a variant of Fregularity, which is a prime characteristic analog of KLT singularities. We will mention the connection of this problem to other longstanding questions in the theory of Fsingularities. Our main goal is to discuss evidence in favor of the conjecture by considering the class of algebras determined by lattice points inside convex cones of finite dimensional vector spaces. This talk is based on joint work with Karl Schwede and Kevin Tucker.

March 29th  Aryaman Maithani University of Utah 
Linear quotients of connected ideals of graphs
Given a graph G, we define the ideal J_t(G) to be the monomial ideal generated by the
tconnected subsets of the graph. In this talk, I will discuss some homological and combinatorial
properties associated to monomial ideals that are of interest, namely linear resolutions and linear
quotients. The two main results are (i) characterizing when J_t of a tree has linear quotients, and (ii)
noting some necessary and sufficient conditions for arbitrary graphs. This is joint work with H.
Ananthnarayan and Omkar Javadekar.

April 5th  No seminar 
TBA

April 12th  Tim Tribone University of Utah 
Matrix factorizations and Knorrer’s Theorem
A local ring is said to have finite CohenMacaulay type if there are only finitely many indecomposable maximal
CohenMacaulay modules up to isomorphism. The classification of such rings is far from complete; we understand small
dimensions and the case of a hypersurface ring, but that is essentially it. This talk will focus on the key tool used in
proving the hypersurface case (matrix factorizations). We will introduce the basics of the theory of matrix factorizations
and explain how they are used in the classification of hypersurface rings of finite CohenMacaulay type. Along the way, we
will address some natural questions inspired by the classification regarding “dfold” matrix factorizations. This is joint
work with Graham Leuschke.

April 19th  TBA TBA 
TBA
TBA

April 26th Unusual room LCB222 
Devlin Mallory University of Utah 
Finite Frepresentation type for homogeneous coordinate rings
Finite Frepresentation type is an important notion in characteristicp commutative algebra, and
is closely connected to the behavior of differential operators. Despite this, explicit examples of
varieties with or without this property are few. We prove that a large class of homogeneous coordinate
rings (essentially, those of CalabiYau or generaltype varieties) will fail to have finite
Frepresentation type, via an analysis of their rings of differential operators. This illustrates a
connection between the commutativealgebraic property of FFRT, and the algebrogeometric properties of
positivity/negativity of tangent sheaves of varieties. This also provides instructive examples of the
structure of the ring of differential operators for nonFpure rings, which to this point have largely
been unexplored. We will also discuss the case of Fano varieties: Recent work has provided nontoric
smooth Fano varieties that do have FFRT (Grassmannians Gr(2,n) and the quintic del Pezzo surface).
However, it seems unlikely that this will be true for all Fano varieties; we will present conjectural
evidence that “in general” smooth Fano varieties will often fail to have FFRT.

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