Commutative Algebra Seminar
Fall 2023, Friday 2:00–3:00 pm, LCB 323
Date  Speaker  Title — click for abstract 
October 6th  Jon Carlson University of Georgia 
TBA
TBA

October 27th  Anna Brosowsky University of Michigan 
Cartier algebras through the lens of pfamilies
A Cartier subalgebra of a prime characteristic commutative ring R is an associated noncommutative ring of operators on R that play nicely with the Frobenius map.
When R is regular, its Cartier subalgebras correspond exactly with sequences of ideals called Fgraded systems. One special subclass of Fgraded system is called a pfamily; these appear in numerical applications such as the HilbertKunz multiplicity and the Fsignature. In this talk, I will discuss how to characterize some properties of a Cartier subalgebra in terms of its Fgraded system. I will further present a way to construct, for an arbitrary Fgraded system, a closely related pfamily with especially nice properties.

November 3rd  Janina Letz University Bielefeld 
Koszul homomorphisms and universal resolutions in local algebra
Abstract: I will define a Koszul property for a homomorphism of local
rings $\varphi \colon Q \to R$. Koszul homomorphisms have good
homological properties. Using $\mathrm{A}_\infty$structures one can
construct universal free resolutions of $R$modules from free
resolutions over $Q$, generalizing the classical construction by
Priddy. This recovers the resolutions of Shamash and Eisenbund for
complete intersection homomorphisms and the resolutions of Iyengar and
Burke for Golod homomorphisms. This is based on work with Ben Briggs,
James Cameron and Josh Pollitz.

November 10th  Cheng Meng Purdue 
Multiplicities in flat local extensions
We introduce the notion of strongly Lechindependent ideals as a generalization of Lechindependent ideals defined by Lech and Hanes, and use this notion to derive inequalities on
multiplicities of ideals. In particular we prove that if (R,m) and (S,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=mS is homogeneous, then the multiplicity of R is no greater than that of S.

November 17rd  Justin Lyle University of Arkansas 
Counterexamples for Several Open Problems on the Vanishing of Ext and Tor
Let R be a commutative Noetherian local ring. We discuss several properties R may satisfy that behave well under certain operations, and provide a general construction for producing
local rings with some nice local behavior that satisfy one such property but not another. Through our methods we provide counterexamples for several open problems, for instance an open question
of Araya on the vanishing of Ext^i(M,R), work of Yoshino on independence of totally reflexive conditions, and a famous open question on the depth of tensor products of Torindependent modules.
This talk is based on joint work with Kaito Kimura, Yuya Otake, and Ryo Takahashi.

November 24th  No Seminar Thanksgiving break 
TBA

December 1st  Anne Fayolle University of Utah 
Tame ramification and compatible ideals
We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves. However, when applied to the Frobenius map, it naturally yields the notion of compatible ideals, which lets us describe how these behave under finite covers it all comes down to a transitivity property for tame ramification in towers. This is joint work with Javier CarvajalRojas

December 8th  Henning Krause University Bielefeld 
Representations of hereditary algebras
Hereditary algebras and their representations became popular when Gabriel introduced quivers and their representations in the 1970s. My talk will present some of the highlights of this theory, without restricting to algebras arising from quivers.

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