CARS

Jason Hardin

Yoneda's Ext and the Yoneda Splice

April 29

The modules Ext_R(M,N) are typically defined by computing the cohomology of a complex formed using a projective resolution of M or an injective resolution of N. Yoneda provided an alternative description of Ext_R(M,N) in terms of equivalence classes of exact sequences which avoids the use of projective and injective modules. We'll describe Yoneda's Ext and discuss it's equivalence with classical Ext. If time permits, we'll see how Yoneda's Ext gives a concrete description of the classical Yoneda pairing.

Marcus Webb

The Frobenius Functor and Homological Dimensions - Part II

April 22

For a local ring R of positive characteristic we can define the Frobenius endomorphism f: R → R by f(r)=r^p. This map allows us to consider the ring as an algebra over itself in a non-trivial way. Such a construction was used by Peskine and Szpiro in their proof of the New Intersection Theorem and has led to many important results in homological algebra. We'll discuss the basics of the Frobenius functor of Peskine and Szpiro, including several examples, and, time permitting, present some recent applications.

Marcus Webb

The Frobenius Functor and Homological Dimensions - Part I

April 15

For a local ring R of positive characteristic we can define the Frobenius endomorphism f: R → R by f(r)=r^p. This map allows us to consider the ring as an algebra over itself in a non-trivial way. Such a construction was used by Peskine and Szpiro in their proof of the New Intersection Theorem and has led to many important results in homological algebra. We'll discuss the basics of the Frobenius functor of Peskine and Szpiro, including several examples, and, time permitting, present some recent applications.

Becky Egg

Cohen-Macaulay dimension for coherent rings

April 8

When faced with a non-Noetherian ring, one has several options, including but not limited to

Weeping, because the idea of a non-Noetherian ring makes you highly uncomfortable
Trying to adapt some beloved concepts of Noetherian rings, in an effort to ease your discomfort

In this talk, we'll mainly focus on the latter. In particular, we'll define notion Cohen-Macaulay (CM) dimension for modules over non-Noetherian rings, based on work of A. Gerko in the Noetherian case. We can then define a quasi-local coherent ring to be CM if every finitely presented module has finite CM dimension. We will look at some properties of CM rings, compared to both the usual notion of Cohen-Macaulayness and to other generalizations to non-Noetherian rings.

Luigi Ferraro

The Homotopy Lie Algebra of a Local Ring - Part II

March 18

I'll show the construction of the homotopy Lie algebra of a local ring, this is the plan: I'll start by talking about DG algebras and how to construct resolutions of the residue field that are DG algebras, in particular I'll talk about Tate resolutions and acyclic closures; I'll use acyclic closures to prove 2 theorems on the Poincare' series of a local ring. Then I'll move to derivations, I'll define what are the gamma derivations and give some properties; then I'll define what is a graded Lie algebra and I will use the derivations of the acyclic closure to define the homotopy Lie algebra of a local ring. At the end I will use the homotopy Lie algebra to prove a theorem of Martsinkovsky about bounded maps of resolutions of the residue field that has an application to stable cohomology.

Luigi Ferraro

The Homotopy Lie Algebra of a Local Ring - Part I

March 11

I'll show the construction of the homotopy Lie algebra of a local ring, this is the plan: I'll start by talking about DG algebras and how to construct resolutions of the residue field that are DG algebras, in particular I'll talk about Tate resolutions and acyclic closures; I'll use acyclic closures to prove 2 theorems on the Poincare' series of a local ring. Then I'll move to derivations, I'll define what are the gamma derivations and give some properties; then I'll define what is a graded Lie algebra and I will use the derivations of the acyclic closure to define the homotopy Lie algebra of a local ring. At the end I will use the homotopy Lie algebra to prove a theorem of Martsinkovsky about bounded maps of resolutions of the residue field that has an application to stable cohomology.

Solomon Akeeseh

Affine Schemes - Part II

March 4

Affine Schemes are the fundamental spaces in which geometry is done. In this talk, we will develop intuition and feel for affine schemes and how they relate to classical varieties through examples and appropriate hand waving. We will follow the treatment laid out in Chapter's 3 and 4 of Ravi Vakil's algebraic geometry notes. We start by looking at the underlying set of affine schemes, the Zariski topology and then structure sheaf of "functions" on the space. We will see why "functions" is written with quotes and conclude with the role of nilpotents in modern geometry by looking at some canonical examples.

Solomon Akeeseh

Affine Schemes - Part I

February 25

Affine Schemes are the fundamental spaces in which geometry is done. In this talk, we will develop intuition and feel for affine schemes and how they relate to classical varieties through examples and appropriate hand waving. We will follow the treatment laid out in Chapter's 3 and 4 of Ravi Vakil's algebraic geometry notes. We start by looking at the underlying set of affine schemes, the Zariski topology and then structure sheaf of "functions" on the space. We will see why "functions" is written with quotes and conclude with the role of nilpotents in modern geometry by looking at some canonical examples.

Peder Thompson

Local Cohomology, Duality, and a proof of HLVT Part - II

February 18

Let (R,𝔐) be a d-dimensional complete local Gorenstein ring, with E the injective hull of R/𝔐, and M any finitely generated R-module. Then we have Ext _Rⁱ(M,R)≌Hom_R(H_𝔐^d^-ⁱ(M),E), where H_𝔐ⁱ(M) are the local cohomology modules with support in 𝔐. Applying these ideas, we will prove a variant of the Hartshorne-Lichtenbaum Vanishing Theorem (HLVT), an important result on the vanishing of local cohomology modules with support not 𝔐-primary.

Peder Thompson

Local Cohomology, Duality, and a proof of HLVT Part - I

February 11

We will outline the basics of local cohomology (which will first require a brief treatment of derived functors). As a first application of local cohomology, we will examine local duality: Let (R,𝔐) be a d-dimensional complete local Gorenstein ring, with E the injective hull of R/𝔐, and M any finitely generated R-module. Then we have Ext _Rⁱ(M,R)≌Hom_R(H_𝔐^d^-ⁱ(M),E), where H_𝔐ⁱ(M) are the local cohomology modules with support in 𝔐. Applying these ideas, we will prove a variant of the Hartshorne-Lichtenbaum Vanishing Theorem (HLVT), an important result on the vanishing of local cohomology modules with support not 𝔐-primary.

Jason Lutz

Symmetry in the Vanishing of Ext Part - II

February 4

We'll continue the discussion of AB rings, defined by Huneke and Jorgensen, with the goal of showing that they, like complete intersections, exhibit a surprising symmetry in the vanishing of Ext. We'll also discuss whether AB rings live strictly between the classes of Gorenstein rings and complete intersection rings.

Jason Lutz

Symmetry in the Vanishing of Ext Part - I

January 28

We'll explore classes of rings that exhibit a surprising symmetry in the vanishing of Ext. In Part 1, we'll discuss the motivating result of Avramov and Buchweitz over complete intersections, and build tools to understand the class Gorenstein of rings later defined by Huneke and Jorgensen.

Kat Shultis

Principal Systems

January 21

We'll discuss some of the results in the paper "Principal Systems" by Northcott and Rees.