CARS
Solomon Akesseh
Ideal Containments under Flat Extensions
December 8
The relative point of view in geometry roughly states that if an object has a property, then that object can be placed in a family depending on parameters such that every object in that family has the property. In this talk we will see an example of this point of view. Ein, Lazarsfeld and Smith showed that if I ⊆ k[ℙn] is the ideal of a reduced 0-dimensional subscheme of ℙn over an algebraically closed field of characteristic 0, then I(mn) ⊆ Im for all m. Hochster and Huneke would shortly thereafter extend this result to all ideals I ⊆ k[ℙn] over fields k of arbitrary characterisitc. Harbourne and Huneke conjectured that if n=2, then I(3) ⊆ I2 for homogeneous ideals I ⊆ k[ℙn]. The first counterexample to this containment in k[ℙ2] was given by Dumnicki, Szemberg and Tutaj-Gasinska and it is the ideal of the Fermat configuration in the plane which is just the dual to the Hesse configuration which in turn is just the 9 flex points of an irreducible plane cubic together with the 12 lines each of which goes through 3 of the flex points. Our goal in this talk is to obtain other counterexamples and we shall do that by considering fibres of morphisms f : ℙ2 → ℙ2 defined by regular sequences. The fact that from the example of the Fermat configuration, we can obtain a family of configurations parametrized by regular sequences in k[ℙ2] that are counterexamples to the containment I(3) ⊆ I2 is the relative point of view mentioned above.
Josh Pollitz
Complete Intersection Defects and a Six-Term Exact Sequence
December 1
We will define the complete intersection defect of a noetherian local ring, which vanishes exactly when your ring is a complete intersection. The main goal of the talk is to establish a relationship between the complete intersection defects of the "base", "total space" and "fibre" of a flat local homomorphism. From this we can deduce that the localization of a complete intersection is a complete intersection. The talk will involve some DG algebra techniques that establish a certain six term exact sequence and a formula relating the deviations of the rings in a flat local homomorphism. In turn, this will allow us to obtain the desired relationship of complete intersection defects.
Chris Evans
Prime Filtrations of the Powers of an Ideal
November 17
It is well known that finitely generated modules over a Noetherian ring always admit a prime filtration. Summarizing recent results of Huneke and Smirnov, I will (hopefully) prove that there is a finite set of primes such that for all n, M/ (In M) admits a prime filtration which uses only primes from that finite set using some results on superficial elements.
Brittney Falahola
Characterizing Gorenstein Rings using Frobenius
November 10
Throughout the years, the Frobenius functor has been used to characterize classical properties of local rings of prime characteristic. One of the most significant of these was the result of Kunz in 1969 which states that a ring is regular if and only if the Frobenius functor is exact. In this talk, we'll explore a way to characterize Gorenstein rings using the Frobenius functor and its behavior with injective dimension.
Doug Dailey
Matlis Reflexive Modules over a Noetherian Ring - Part II
October 27
Last time, I showed a structure theorem regarding Matlis reflexive modules from Bashir, Enochs, and Garcia-Rozas. This time, I will discuss a "change of rings" lemma for Matlis reflexivity from their paper. The lemma states that if M is an R-S-module where S is a multiplicatively closed subset of R, then M is R-reflexive if and only if it is R-S-reflexive. Their proof of this lemma is not valid, and the converse of the lemma does not hold in general. I will give examples as to why. I will go on to give a valid proof of the converse in an important case. Tools required for the proof include DVRs and Henselian rings. If time allows, I will show that the forward direction holds in generality.
Doug Dailey
Matlis Reflexive Modules over a Noetherian Ring - Part I
October 21
One property of finitely generated and Artinian modules over a complete local ring is that they are Matlis reflexive, meaning that the natural injection M<\em> → Mvv is an isomorphism (where (-)v = HomR(-,E(R/m))). For (non-local, non-complete) Noetherian rings, there is a general notion of Matlis reflexivity, which we will discuss in this talk. As I will indicate, the existence of Matlis reflexive modules dramatically influences the structure of the ring. I will also demonstrate the confusion that can arise when things one expects to be true are, in fact, false.
John Myers
Resolutions over Very Non-Commutative Rings
October 6
One of the most familiar routines in local algebra is the construction of a surjection from a free module on to a minimal generating set of a finitely generated module. Iteration of this routine constructs the minimal free resolution of the module, an important object from which many invariants of the module are obtained. In this talk, we'll see that this routine can be viewed in a broader categorical context, and from this viewpoint it will be clear how to define minimal projective resolutions of modules over other classes of rings. One such class of rings are the bound quiver algebras, which are non-commutative finite dimensional algebras obtained from bound quivers. Many of the questions that a commutative algebraist asks about minimal free resolutions can also be asked for resolutions over these algebras -- for example, we'll see that Poincaré series can be defined, and we'll ask when these series are rational. The talk will be a mix of the abstract and the concrete: classical theorems such as Artin-Wedderburn and Krull-Schmidt provide the foundation for many of our definitions, but at the end of the day one of the most attractive parts of the theory is that resolutions over certain bound quiver algebras are easily computable. Hopefully this means that the talk will be interesting to experts and beginners alike.
Neil Steinburg
The Depth Formula and Vanishing of Tor
September 29
In general, computing the depth of a tensor product is very non-trivial. In this talk, we will find that under certain conditions, we can compute the depth of a tensor product using only the depth of the modules in the tensor and the depth of the ring. The main condition for this to hold, however, is that the modules be Tor-independent. So, we will also look at some conditions that force the Tor modules to vanish and allow us to use the depth formula.
Seth Lindokken
Canonical Modules and MCM Approximations
September 22
If V is a finite dimensional vector space over a field k, there is a natural isomorphism between V and its double dual. We saw last week that this phenomenon does not extend in general to finitely generated modules over commutative rings, but there is an extent to which this duality can be recovered in certain cases. Having established Ischebeck’s equality and Matlis duality, we are close to being able to state exactly when this is possible. Upon doing so, we will explore some applications of these duality results by showing the existence of MCM approximations for modules over "nice" rings, which we will use to give an alternate proof of a classical result concerning modules over Gorenstein rings.
Seth Lindokken
Matlis Duality and Injective Dimension
September 15
Let V be a finite-dimensional k-vector space. Letting V* denote Hom(V,k), the dual space of V, it turns out that V and V* are isomorphic. After iterating we see that V will also be isomorphic to V**, the double dual of V. But in this case there is a "natural" isomorphism from V to V** obtained without any mention of a basis. In general, given a ring R and a finitely-generated R-module M one can form the same "natural map," but it is often nowhere near an isomorphism. However, we will show that for sufficiently nice rings and modules, there are still plenty of isomorphisms to be had. Much of this first talk will be devoted to introducing the machinery needed to tackle the problem, and should be accessible to those at all ends of the algebraic spectrum. Next week we will deliver the punchline and see an application of these results to Maximal Cohen-Macaulay Approximations.
Andrew Windle
Homological Algebra Modulo Regular Elements and Exact Zero Divisors
September 8
If S is a commutative ring and I is an ideal of S, how does the homological algebra of S/I compare to the homological algebra of S? This is the motivating question behind Eisenbud's 1980 paper "Homological Algebra on a Complete Intersection with an Application to Group Representations". In general, this question is difficult to answer. In this talk, we will explore one of the cases that is well known: when I is generated by a regular sequence. After, we will look at recent results when a similar process is applied to the case that the ideal I is generated by an element x satisfying that (x,x) forms an exact pair of zero divisors in S. In particular, we will develop conditions on when a module M over S/(x) can be lifted to an S-module.
Eric Canton
An Application of Regularity to Frobenius Actions on Local Cohomology
September 1
The Frobenius action on local cohomology of a local ring R of positive prime characteristic gives an interesting way to study the singularities of that local ring. For example, if R is a regular ring, then this action on local cohomology is always injective. If we instead assume that R has an isolated singularity, then the kernel of this Frobenius action has finite length. When R is a graded complete intersection quotient of the polynomial ring S which has an isolated singularity, we compute the degrees in which the kernel of the Frobenius action is nonzero. The regularity comes up.
Eric Canton
Basic Properties of Castelnuovo-Mumford Regularity
August 25
I'll give some basic properties of Castelnuovo-Mumford Regularity.
