Commutative Algebra Seminar

TBA

A Cartier subalgebra of a prime characteristic commutative ring R is an associated non-commutative 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 F-graded systems. One special subclass of F-graded system is called a p-family; these appear in numerical applications such as the Hilbert-Kunz multiplicity and the F-signature. In this talk, I will discuss how to characterize some properties of a Cartier subalgebra in terms of its F-graded system. I will further present a way to construct, for an arbitrary F-graded system, a closely related p-family with especially nice properties.

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.

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,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.

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 Tor-independent modules. This talk is based on joint work with Kaito Kimura, Yuya Otake, and Ryo Takahashi.

TBA

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 Carvajal-Rojas

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.