### CARS Spring 2015

We meet 3:30-4:30 on Tuesdays in Avery 111. To receive our
announcements, please visit the
mailing list subscription page. You will need to
register in order to send mail to the members of CARS. CARS is
currently organized by Kat Shultis and Peder Thompson. Kat is
located in Avery 315 and Peder can be found in Avery 336.

#### Line up:

###### Updated Wednesday 21 April 2015 9:10am

- January 13:
No Seminar;
*JMM*

**Abstract:**N/A - January 20:
No Seminar;
*Job Candidate*

**Abstract:**N/A - January 27:
Peder Thompson;
*Complete Intersection Dimension*

**Abstract:**We will highlight portions of the paper "Complete Intersection Dimension by Avramov, Gasharov, and Peeva from 1997. This paper introduced a new homological dimension that has since gained attention. We'll take a look at the basic definitions, some motivation, and examples, which all work toward producing a class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions. Along the way we'll see how CI-dimension (complete intersection dimension) interacts with G-dimension and projective dimension. The paper can be found at: http://www.math.cornell.edu/~irena/papers/ci.pdf - February 3:
No Seminar;
*Job Candidate*

**Abstract:**N/A - February 10:
Doug Dailey;
*On the Decomposition of Flat, Cotorsion Modules*

**Abstract:**The decomposition of injective modules over Notherian rings is well-known. In this talk, I plan to cover a "dual" notion of this phenomenon, namely the decomposition of flat, cotorsion modules. The topic offers many simple, but interesting, results, and I hope to provide a solid introduction to the topic. I will be citing and following some of the results in Edgar Enochs's paper "Flat Covers and Cotorsion Modules" during this talk. Anyone who took Tom's class should be very comfortable with the material. - February 17:
Andrew Windle;
*Hyman Bass and Ubiquity: Gorenstein Rings*

**Abstract:**In this talk, we will discuss the history and the ubiquity of Gorenstein rings, as presented by Craig Huneke in his paper "Hyman Bass and Ubiquity: Gorenstein Rings". In particular, we will outline the history of the study of Gorenstein rings, starting with Gorenstein, and culminating in Bass's famous paper "On the Ubiquity of Gorenstein Rings". Along the way we will survey why Gorenstein Rings are ubiquitous and their basic properties, which appear throughout commutative algebra.

- February 24:
Seth Lindokken;
*On the Support of Local Cohomology*

**Abstract:**This talk will center on the following question: If R is a commutative Noetherian ring, I an n-generated ideal, and M a finitely generated R-module, is the support of the n'th local cohomology module of M with respect to I a closed subset of Spec(R)? In particular, we will give an affirmative answer for the case n=2, which comes to us from the 2007 paper "On the Support of Local Cohomology" by Huneke, Katz, and Marley. - March 3:
Jason Lutz;
*An introduction to quasi-complete intersection ideals*

**Abstract:**Quasi-complete intersection ideals were first introduced by Rodicio in 1996, and are defined as ideals with "free exterior Koszul homology". We'll discuss this class of ideals and its relationship to the (two-step) Tate complex. - March 10:
Brittney Falahola;
*Applications of Dualizing Complexes*

**Abstract:**A well-known result of Bass states that Cohen-Macaulay local rings of type 1 are Gorenstein. Vasconcelos conjectured that one only needs the ring to be of type 1 to be Gorenstein. In 1977, Foxby proved this was indeed the case for rings which contain a field using a version of the Intersection Theorem. In 1980, Paul Roberts proved the more general statement that type 1 local rings are Gorenstein using dualizing complexes. In this talk, I will develop some of the background on dualizing complexes that is necessary in understanding the proof that local domains of type 1 are Gorenstein. - March 17:
No Seminar;
*Hopkins Week*

**Abstract:**N/A - March 24:
No Seminar;
*Spring Break*

**Abstract:**N/A - March 31:
Josh Pollitz;
*Gorenstein Dimension*

**Abstract:**In 1969, Auslander and Bridger introduced a homological dimension for finitely generated modules called Gorenstein dimension (or G-dimension). In today’s talk, we will discuss G-dimension and see how this is a refinement of projective dimension. In particular, we will sketch a proof of the Auslander-Bridger formula, which is an analogue of the Auslander-Buchsbaum formula, and see how G-dimension gives a characterization of a Gorenstein local ring.

- April 7:
Eric Canton;
*Uniform convergence of F-signature of pairs*

**Abstract:**The F-signature of a local ring R (of characteristic p>0) is a number between 0 and 1 which gives an asymptotic measure of the number of free summands of R considered as a module over itself via restriction of scalars along Frobenius. This information can be twisted by some element f \in R and a real parameter t >=0, where now we require the projection mappings to be pre-multiplied "by" f^t. Whereas before the F-signature of R gave a measure of the singularities of R, the F-signature of the pair (R, f^t) measures both the singularities of R and of f. As it turns out, t |---> s(R, f^t) gives an almost-everywhere differentiable function from [0, 1] to [0, 1]; assuming f is an integer polynomial, we can consider the mod-p reduction f_p in the power series ring F_p[[x_1, \dots, x_d]], where F_p = Z/pZ. When f is a product of r >=2 linear polynomials, we will show that the sequence of functions {s(R_p, f_p^t)} converges uniformly on [0, 2/r] to the quadratic polynomial (rt/2 - 1)^2.

Time permitting, I'll relate this circle of ideas to log canonical singularities arising in the characteristic 0 classification of algebraic varieties up to birational equivalence. - Thursday,
April 16, 2:30-3:30, Avery 351 : Michael Brown;
*Matrix Factorizations and Clifford Algebras*

**Abstract:**I will talk about a 1987 theorem of Buchweitz-Eisenbud-Herzog that relates matrix factorizations of quadratic polynomials to modules over Clifford algebras. I will carefully define what a Clifford algebra is along the way. My goal is to make the talk understandable to anyone who has been attending Mark Walker's hypersurfaces course. - April 21:
John Myers;
*You could have discovered Golod rings*

**Abstract:**A Golod ring is a type of commutative noetherian local ring that is encountered in the theory of infinite free resolutions. While the precise definition is not difficult to state (they are defined by an extremal growth condition on the Poincar\'e series of the residue field), the motivation leading up to the definition can be difficult to understand. The goal of this talk is to convince the audience that the definition is actually quite natural -- so natural, in fact, that they could have discovered the class of Golod rings for themselves. We will start from the basics, assuming no background in the theory of infinite free resolutions, and then progress through the main definitions and a few examples to a description of Golod’s main contribution to the theory, which is a characterization of the class of Golod rings in terms of “higher homology operations.” We will end with a discussion of how the class of Golod rings relates (or doesn’t) to the more familiar classes of commutative noetherian local rings, e.g., Cohen-Macaulay rings and complete intersections. - April 28:
Mohsen Gheibi;
*Introduction to theory of Linkage*

**Abstract:**Linkage of algebraic varieties first appeared in the work of Noether, Halperin and Severi in late 19th century and early 20th century. They used linkage to study the curves in $P^3$. The idea of linkage is to pass from a given curve to another one which is simpler, without losing certain properties of the original curve. Iterating the procedure gives a series of curves which are in the same "linkage class". Peskine and Szpiro in 1974 reduced general linkage to certain questions on ideals over commutative algebras and after then, many works have been done to develop this theory in commutative algebra and algebraic geometry. In this talk I will try to address some important results in linkage of ideals and generalization of them for modules.

#### Fall 2014 talks:

- September 2: Eric Canton;
*Test Ideals*

**Abstract:**Test ideals first arose in the theory of tight closure. In the last twenty years, they have assumed a prominent role in characteristic p commutative algebra due to their link with tight closure, and in algebraic geometry through analogy with multiplier ideals, some of the most interesting and useful objects in complex algebraic geometry. In this talk, we will define and explore test ideals as algebraic geometers use them, though we stay with the language of commutative algebra. - September 9: Eric Canton;
*Test Ideals - Part II*

**Abstract:**Test ideals first arose in the theory of tight closure. In the last twenty years, they have assumed a prominent role in characteristic p commutative algebra due to their link with tight closure, and in algebraic geometry through analogy with multiplier ideals, some of the most interesting and useful objects in complex algebraic geometry. In this talk, we will define and explore test ideals as algebraic geometers use them, though we stay with the language of commutative algebra. - September 16: Doug Dailey;
*Ramblin' about Artinian Modules and Cofinite Modules (and more?)*

**Abstract:**I will prove a nice characterization of Artinian modules over Noetherian local rings as Tom mentioned in class. My motivation for proving this result comes from a few of Tom's papers where this characterization is the first statement in the paper. I will also discuss the result of one of Tom's papers which involves cofinite modules. As time permits, and if I have not rambled myself out the door or my audience to sleep, I may shift gears entirely and talk about a neat application of the Cohen Structure Theorem in local cohomology. All of the elements of this talk should be accessible to those taking Tom's course, and it will be most interesting for beginners in the area. - September 23: Marcus Webb;
*On a Conjecture of Serre*

**Abstract:**We'll be looking at some basics of intersection theory and discussing Serre's multiplicity conjectures. I'll start off by computing some examples and special cases, and, time permitting, give a proof in the graded case. - September 30: Marcus Webb;
*On a Conjecture of Serre, Part 2*

**Abstract:**We'll be looking at some basics of intersection theory and discussing Serre's multiplicity conjectures. I'll start off by computing some examples and special cases, and, time permitting, give a proof in the graded case. - October 7: Luigi Ferraro;
*Local cohomology as a D-module*

**Abstract:**We are going to study the theory of modules over the Weyl algebra. We are going to use this theory to prove that if R is a polynomial ring of characteristic zero then the set of associated primes of local cohomology modules Ass_R H^i_a(R) is always finite for every i and every ideal a. - October 14: Luigi Ferraro;
*Local cohomology as a D-module, part 2*

**Abstract:**We are going to study the theory of modules over the Weyl algebra. We are going to use this theory to prove that if R is a polynomial ring of characteristic zero then the set of associated primes of local cohomology modules Ass_R H^i_a(R) is always finite for every i and every ideal a. - October 21: No Seminar;
*Enjoy Fall Break!*

**Abstract:**N/A - October 28: John Meyers;
*Gulliksen's characterization of local complete intersections*

**Abstract:**In 1980, Gulliksen gave the following extension of the Auslander-Buchsbaum-Serre Theorem: Local complete intersections are exactly the (noetherian local) rings over which all finitely generated modules have finite complexity, or equivalenty, they are the rings for which the residue field has finite complexity. Just as one uses the Auslander-Buchsbaum-Serre Theorem to prove regularity localizes, an immediate corollary of Gulliksen's result is that the complete intersection property localizes -- a fact originally proved by Avramov in 1975 using different techniques. The proof of Gulliksen's result exploits a characterization of local complete intersections as those rings for which certain invariants of the ring (called deviations) vanish. This talk will have three parts: (1) We shall quickly review the main cast of characters, including DG algebras, Tate's method of killing cycles, and acyclic closures. (2) We will then define deviations and survey the characterization of complete intersections in terms of them. (3) We will end the talk with a complete proof of Gulliksen's result, and then show the complete intersection property localizes. - November 4: Jason Lutz;
*Hochschild homology for complete intersections*

**Abstract:**We'll discuss Hochschild homology and its applications to detecting regular homomorphisms of Noetherian rings, and give two constructions of complexes that compute Hochschild homology for complete intersections. Then, we'll apply these constructions to note a connection between smoothness and the vanishing of Hochschild homology. - November 11: Jason Lutz;
*Hochschild homology for complete intersections, Part II*

**Abstract:**In Part I, we defined Hochschild homology and used it to prove a theorem about regular homomorphisms. Today, in Part II, we'll construct two complexes that compute Hochschild homology. We'll use one construction to show that vanishing of Hochschild homology in two degrees of opposite parity implies smoothness. - November 18: No Seminar;
*Hopkins Week*

**Abstract:**N/A - November 25: No Seminar;
*Day before Thanksgiving break!*

**Abstract:**N/A - December 2: Seminar Cancelled

**Abstract:**N/A - December 9: Brittney Falahola;
*Examples of Gorenstein Rings*

**Abstract:**The standard definition of a Gorenstein local ring involving injective dimension, while simple, does not give one great insight into how to come up with examples of such rings. Thus, the desire for an equivalent characterization of Gorenstein rings that leads to examples manifests itself. In this talk, I will give two such characterizations: one which is true for noetherian local rings, and another pertaining to numerical semigroup rings. As the title of the talk suggests, we will use these characterizations to discuss several examples (and non-examples) of Gorenstein rings.

#### Spring 2014 talks:

- January 21: Kat Shultis;
*Principal Systems*

**Abstract:**We'll discuss some of the results in the paper "Principal Systems" by Northcott and Rees. - January 28: Jason Lutz;
*Symmetry in the Vanishing of Ext, Part 1*

**Abstract:**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. - February 4: Jason Lutz;
*Symmetry in the Vanishing of Ext, Part 2*

**Abstract:**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. - February 11: Peder Thompson;
*Local Cohomology, Duality, and a proof of HLVT*

**Abstract:**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 $(R,\frak{m})$ be a $d$-dimensional complete local Gorenstein ring, with $E$ the injective hull of $R/\frak{m}$, and $M$ any finitely generated $R$-module. Then we have $\Ext_R^i(M,R)\cong \Hom_R(H_\frak{m}^{d-i}(M),E)$, where $H_\frak{m}^i(M)$ are the local cohomology modules with support in $\frak{m}$. 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 $\frak{m}$-primary. - February 18: Peder Thompson;
*Local Cohomology, Duality, and a proof of HLVT - Part II*

**Abstract:**This is the second part of last week's talk.

- February 25: Solomon Akesseh;
*Affine Schemes*

**Abstract:**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. - March 4: Solomon Akesseh;
*Affine Schemes*

**Abstract:**This is the second part of last week's talk.

- March 11: Luigi Ferraro;
*The homotopy Lie algebra of a local ring*

**Abstract:**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. - March 18: Luigi Ferraro;
*The homotopy Lie algebra of a local ring - Part II*

**Abstract:**This is the second part of last week's talk.

- April 1: No Seminar; N/A

**Abstract:**N/A - April 8: Becky Egg;
*Cohen-Macaulay dimension for coherent rings*

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

1. Weeping, because the idea of a non-Noetherian ring makes you highly uncomfortable

2. 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.- April 15: Marcus Webb;
*The Frobenius Functor and Homological Dimensions*

**Abstract:**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. - April 22: Marcus Webb;
*The Frobenius Functor and Homological Dimensions - Part II*

**Abstract:**This is the second part of last week's talk.

- April 29: Jason Hardin;
*Yoneda's Ext and the Yoneda Splice*

**Abstract:**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.

#### Fall 2013 talks:

- August 27: Haydee Lindo and
Kat Shultis;
*Organizational Meeting*

**Abstract:**We'll plan for the semester. - September 3 & 10: Haydee
Lindo and Kat Shultis;
*The Tools from Croll's 2013 Paper "Periodic Modules over Gorenstein Local Rings"*

**Abstract:**In this two part talk, we will present the tools and techniques used by Amanda Croll in her thesis work. In the first talk, we will provide some historical and mathematical background along with the major results of her thesis. In the second talk, we will discuss some of the well-known results Amanda used repeatedly and proved for lack of adequate references.

**Paper:***Periodic Modules over Gorenstein Local Rings*by Amanda Croll - September 17: Michael Brown;
*What Do Combinatorics, Semigroups, and Toric Varieties Have in Common?*

**Abstract:**I'll talk about the connection between toric varieties, affine semigroups, and convex polyhedra. (References: Polytopes, Rings, and K-Theory, a text by Bruns and Gubeladze; Normal Semigroup Rings, an expository paper by Bruns)

**Paper:***Normal Semigroup Rings*by Winfried Bruns - September 24:
*Affine Semigroup Rings: Part 2*

**Abstract:**When we left off, we were in the middle of proving the following theorem: - Theorem (Hochster): If M is a positive, normal, affine semigroup ring, and k is any field, k[M] is Cohen-Macaulay.
- I'll quickly finish this proof, and then sketch another proof of the same fact using a completely different (but hopefully illustrative) approach involving local cohomology.
- October 1: Marcus Webb;
*Hilbert Functions*

**Abstract:**We'll talk about some classical and recent results concerning Hilbert Functions.

- October 8 & 15: Eric Canton;
*Morphisms between Projective Spaces OR An exercise out of Hartshorne*

**Abstract:**The leading question of these talks will be to understand the statement and solution of the following problem (Ex. II.7.3 in Hartshorne) Let f: \P_k^n \to \P_k^m be a morphism of schemes. If m < n, then f(\P_k^n) = pt (i.e. f is constant if m < n).

I will try to give a general feel for the algebraic geometry associated to this question, focusing on how the objects and structures are used rather than burdening the audience with technical baggage. Along the way, we'll see some old friends like Spec of a commutative ring, as well as meet some new friends like Proj of a graded ring and invertible sheaves. I will be assuming a basic familiarity with rings, graded rings, and modules, as well as some point-set topology. - October 29: Douglas Dailey;
*The Artin-Rees Lemma*

**Abstract:**The Artin-Rees Lemma is an extremely useful theorem in commutative algebra. I will give a nice proof of the Artin-Rees Lemma that does not get bogged down in technical mumbo-jumbo and is easily understandable. Anyone who understands the basics of Noetherian rings will understand the majority of this talk. - November 5: Brittney Turner;
*Applications of the Artin-Rees Lemma*

**Abstract:**Last week, we saw the statement and a proof of the Artin-Rees Lemma. This week, I will discuss some applications of the lemma that highlight its usefulness.

- November 12 & 19:
Christopher Evans and Nick Packauskas;
*Multiplicity of Regular Local Rings*

**Abstract:**We will quickly define multiplicity of a module and prove a well-known Theorem of Nagata concerning multiplicity and regular local rings. This proof will include an application of the result about superficial elements which was proved in the previous seminar.