Commutative Algebra Seminar
Spring 2006



Seminars will be in  Avery 351, and will be held twice a week:


Wednesdays 3:30 - 4:20 pm


Thursdays  2:30 - 3:20 pm


 

27th April (Thursday). Sylvia Wiegand, UNL.


Title: T.B.A.


Abstract:


26th April (Wednesday). No Seminar.


19th and 20th April (Wednesday and Thursday). Ragnar-Olaf Buchweitz, Univeristy of Toronto

(visiting UNL Spring 2006)


Title: Non-commutative desingularization of the generic determinant



13th April (Thursday). Paul Roberts, University of Utah.


Title: The Commutativity of Intersection with Divisors.


Abstract:


The operation of intersection with divisors is fundamental in Intersection Theory, and one

of its crucial properties is that it is commutative. In the general setting of Algebraic

Geometry both the definition of this intersection and the proof of commutativity require a

considerable amount of machinery (as develpoed, for example in Fulton's Intersection

Theory).  In the setting of ring theory, there is a very simple definition if intersection

with divisors, but the proof of commutativity has still required a considerable amount of

geometric machinery.  In this talk I will discuss why this property is not as trivial as

it may seem and some joint work with Sandra Spiroff on giving a purely algebraic proof.



12th April (Wednesday). Mauricio Velasco, Cornell University.


Title: Some monomial ideals associated to simplicial complexes


Abstract:


This talk is about the structure of minimal free resolutions of monomial ideals in the

ring of polynomials over a field. There are very few classes of ideals for which this

structure is known. The main ones are monomial regular sequences, Borel ideals and Scarf

ideals.


We introduce a new class-Nearly Scarf ideals- and describe their minimal free

resolutions. These resolutions are intimately related with the combinatorial structure of

their Scarf complexes. Using these ideals we:


(1.) Show that the total betti numbers of nearly scarf ideals correspond to the f-vectors

of acyclic simplicial complexes.


(2.) Prove that the total betti numbers of all ideals with a given Scarf complex are

bounded below by those of a nearly scarf ideal (this bound is sharp).


(3.) Describe the minimal free resolutions of all ideals with minimal betti numbers in

their Scarf class.


(4.) Construct monomial ideals whose minimal free resolutions do not admit any cellular

structure (that is, resolutions not supported by any CW complex). The fourth application

answers negatively a question raised by Jollenbeck-Welker of great importance to the

program of "geometrizing" minimal monomial resolutions.


Some of these results are joint work with Irena Peeva. 



Special seminar, from 2:30 - 3:20, in Oldfather 207


11th April (Tuesday). Kiriko Kato, Osaka Prefecture University, Japan.


Title: Triangulated property of stable module category 


Abstract: 


Stable module category is not triangulated in general.  We show that the obstruction is

related to represented-by-monomorphism property of morphisms.



6th April (Thursday). Peter Jorgensen, University of Leeds, U.K.


Title: Amplitude inequalities for Differential Graded modules


Abstract:  


In commutative algebra, amplitude inequalities are a different way of expressing some well

known Intersection Theorems (the New Intersection Theorem and Serre's Intersection

Theorem).


It turns out that one can also prove amplitude inequalities for Differential Graded

modules over Differential Graded Algebras.  One of these states that over a suitable

Differential Graded Algebra R, each Differential Graded module M which is finitely built

from R has amp M >= amp R.  Conversely, if M is "shorter" than R then it cannot be

finitely built from R.


This means that generally, there will lots of "small" Differential Graded modules which

cannot be built finitely from R, in sharp contrast to ring theory.  



5th April (Wednesday). Tom Marley, UNL.


Title: Canonical modules and quasi-Gorenstein rings


Abstract:  


In joint work with Mark Rogers, I will present progress on the following question: Suppose

R is a Noetherian local ring such that inside every power of the maximal ideal there

exists an irreducible system of parameters.  Is R necessarily Gorenstein?  We show that

every such ring is quasi-Gorenstein.  In particular, R_p is Gorenstein for all primes p of

height at most 2.




29th March (Wednesday) and 30th March (Thursday). Mark Walker, UNL.


Title: Cocycles for singular varieties - I and II


Abstract: 


I will start by describing some relatively old collaborative work I did, first with Dan

Grayson concerning what we termed a "geometric model" for the algebraic K-theory spectrum

and then with Eric Friedlander concerning Chern class maps. I will then talk about some

new ideas I have for using this old work to develop a reasonable theory of Chow cohomology

for singular varieties.  



23rd March (Thursday). Gene Abrams, Univeristy of Colorado, Colorado Springs.


Title: The mathematical legacy of William G. Leavitt


Abstract:


The mathematical work of Bill Leavitt spans nearly seven decades, dating all the way back

to the Truman administration, and continuing through the new millennium.


Among his dozens of research articles, two from the early '60s are currently providing the

basis for a substantial, world-wide investigation into an interesting class of rings. The

rings described in these two seminal articles have been appropriately dubbed the Leavitt

algebras; their currently studied generalizations are the Leavitt path algebras. This

notation is now standard throughout the ring theory literature. Most of the rings one

encounters as basic examples have what's known as the Invariant Basis Number property,

namely, for every pair of positive integers m and n, if the free left R-modules Rm and Rn

are isomorphic, then m = n. For instance, in an undergraduate linear algebra course one

shows that the ring of real numbers has this property: all it says is that any two bases

for a finite dimensional vector space over the Reals have the same size! There are,

however, many important classes of rings which do not have this property. While at first

glance such rings might seem pathological, in fact they arise quite naturally in a number

of contexts (e.g. as endomorphism rings of infinite dimensional vector spaces), and

possess a significant (perhaps surprising) amount of structure.


In this presentation we describe a class of such rings, the (now-classical) Leavitt

algebras, and then describe their recently developed generalizations, the Leavitt path

algebras. One of the nice aspects of this subject is that pictorial representations (using

graphs) of the algebras are readily available. In addition, there are strong connections

between these algebraic structures and a class of C-star (the graph C-star algebras), a

connection which is currently the subject of great interest to both algebraists and

analysts.



22nd March (Wednesday).  


Program: Reception for Bill Leavitt, on his 90th birthday.



Special seminar, from 2:30 - 3:20, in Avery 19 


21st March (Tuesday). Graham Denham, University of Western Ontario, Canada.


Title: Massey products and monomial ideals


Abstract:


A construction of Buchstaber and Panov associates to every finite simplicial complex a

finite CW-complex, the "moment-angle complex."  In the case of a triangulated sphere, one

obtains a compact manifold.  The nature of the construction makes the topology (cohomology

ring, homotopy Lie algebra) amenable to study using methods of commutative algebra.  In

particular, one can explicitly construct nontrivial Massey products to show that even the

manifolds arising from triangulations of S^2 are asymptotically "almost never" formal

spaces.



15th and 16th March. Spring break.



9th March (Thursday). Micah Leamer. UNL.


Title: An Introduction to Groebner finite path algebras:


Abstract:


Let K be a field and G a finite directed graph.  I will introduce path algebras KG and

non-commutative Groebner bases. Some basic facts about path algebras.will be introduced,

for instance which ones are Noetherian.  Some tools for simplifying path algebras and

understanding non-commutative Groebner bases will be explored.  In the non-commutative

setting, a finitely generated ideal may have an infinite reduced Groebner basis.  In the

case of path algebras, in general, an algorithm determining whether an ideal has a finite

Groebner basis does not exist.  In fact, a solution to this problem would imply a solution

to the word problem. Instead I define a path algebra to be Groebner finite if all of its

finitely generated ideals have finite Groebner bases for some admissible order.  I provide

a simple criteria to determine when a path algebra is Groebner finite and I offer an

admissible order that woks for all Groebner finite path algebras. In closing I will show

how these results relate to determining whether a ring A/I is Groebner finite given that

A=K and I is an ideal with a finite Groebner basis.



8th March (Wednesday). No seminar.



1st March and 2nd March (Wednesday and Thursday). Julia Bergner, Kansas State University, Manhattan.


Title: Encoding algebraic structures on spaces


Abstract: 


Certain kinds of algebraic structures on spaces can be described via algebraic theories,

or particular diagrams given by free objects.  This approach is useful in that it allows

us to compare strict algebraic structures with those which are only give up to homotopy.

In certain cases, we can replace theories with "simpler" diagrams encoding the same

algebraic information.  However, in this situation we do not necessarily have the same

relationship between strict and homotopy structures.  I'll begin by describing the

algebraic theory approach and then discuss examples of these other kinds of diagrams.



22nd and 23rd February. No seminars.



16th February (Thursday). Sean Sather-Wagstaff, California State Univerity, Dominguez Hills.


Title:  A first approximation of liaison theory for local homomorphisms


Abstract.  


Liaison (or linkage) theory was introduced by Peskine and Szpiro and have proved

particularly good at identifying nice classes of ideals.  I will discuss preliminary work

extending this notion to identify interesting classes of local ring homomorphisms.



15th February (Wednesday). No seminar.



9th February (Thursday). Janet Striuli. UNL.


Title: A uniform property for infinite resolutions.


Abstract:


Let (R,m,k) be a local Noetherian ring, let M be a finitely generated R-module and let I

an m-primary ideal in R. Let F be a free resolution of M. We study the uniform Artin-Rees

property I^nF_i \cap Ker(d_i) \subset I^{n-h}Ker(d_i) with Artin-Rees exponent h that does

not depend on i. We prove that M has this property when dim R is at most two.  We relate

this property to the property of having uniform annihilators for the family of modules

Tor^R_i(M,R/J^n), paramterized by i,n, for some m-primary ideal.



8th February (Wednesday). Jooyoun Hong, Purdue University.


Title: Hyperplane sections and integral closures of ideals


Abstract:


This joint work with B. Ulrich was motivated by the following surprising results on the

integral closures of ideals proved by Itoh and Huneke independently:


Let A be a Noetherian ring and let I be a complete intersection

ideal of A. Then for each non-negative integer n, one has


\overline{I^{n+1}} \cap I^n = \overline{I}I^n


A particular question we are interested in is whether the assertion in the theorem above

holds for more general ideals than ideals generated by regular sequences. While we are

trying to answer this question, we are able to show that the completeness of any ideals of

height at least 2 is compatible with a specialization of generic elements. We use this

compatibility to give a direct proof of the above Theorem under slightly modified

assumptions.



2nd February (Thursday). Lars Christensen, UNL.


Title: Forced isomorphisms and two conjectures of Auslander

 

ABSTRACT:

 

It was a long standing conjecture, due to Auslander, that local rings would have the

property (ac): For any finitely generated R-module M there is an n_M > 0 such that the

following implication holds for all finitely generated R--modules N:


Ext^i (M,N)=0  for i >> 0  implies Ext^i (M,N)=0  for i > n_M


This conjecture was disproved by Jorgensen and Sega in 2003.

 

The class of rings that have the property (ac) is still poorly understood. I will address

a few of their properties that allow us to see that the Auslander-Reiten conjecture holds

over these rings. The writings of Auslander do not appear to address this connection

between the two conjectures, which is somewhat surprising.



1st February (Wednesday). Brian Harbourne, UNL.


Title: Hilbert functions and simple matroids


Abstract: 


I'll describe joint work (with Geramita and Migliore) answering certain questions raised

in a recent preprint of Geramita, Migliore and Sabourin, concerning what functions arise

as hilbert functions of fat points in P2. As a corollary of my work in the mid 90s, we

show, up to Hilbert functions, that there are only finitely many configurations of r

points in P2, if r is either at most 8, or the r points lie on a conic. In the case that r

is at most 8, these configurations include all simple matroids of size r and rank at most

3. If the r points lie on a conic, one configuration corresponds to points on an

irreducible conic, and the rest to the simple matroids of size r of rank either 2 or 3

whose r elements comprise at most two closed proper subsets.



26th January (Thursday). Liana Sega. University of Missouri, Kansas city.


Title: Ext algebras and Koszulness for certain quadratic graded algebras


Abstract:


We will consider quadratic graded algebras with Hilbert series 1+nt+nt^2+t^3, subject to a

generic condition on the coefficients of the defining quadrics. We prove that all such

algebras are Koszul and their Ext algebras have a finite (non-commutative) Groebner basis,

whose initial terms do not depend on the coefficients of the quadrics.



25th January (Wednesday). Sergio Estrada. University of Almeria, Spain.


Title:  Relative homological algebra in categories of representations of quivers


Abstract:


The general theory of covers and envelopes with respect to an arbitrary class F has

developed widely during the last years.  The existence of such covers allows one to define

F-resolutions which are unique up to homotopy so they can be used to compute new

homology and cohomology functors when an additive functor is applied to them. So then we

spend the first part of the talk giving a historic summary of covers and envelopes. This

will lead us to the resolution of the flat cover conjecture that Enochs formulated in

1981. After that we will see some extensions of this conjecture in other categories. We

focus our attention on the category of quasi-coherent sheaves on a scheme. This category

can be understood as a category of representations of a quiver with relations, so we will

study the problem of the existence of covers and envelopes for those categories.

 

In the second part we will study the so called Gorenstein categories. These categories

will generalize (for Grothendieck categories) the behavior of the category of R-modules

whenever R is a Gorenstein ring. In these categories we have Tate cohomological functors

and Avramov-Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute

and the Tate cohomological functors. As an illustrating example we show that the category

of quasi-coherent sheaves on a locally Gorenstein scheme is Gorenstein.

 

We finish our discussion by giving some recent results on categories of quivers and also

giving open problems.



18th and 19th January (Wednesday and Thursday). Roger Wiegand. UNL.


Title: New constructions of big indecomposable modules


Abstract:


A  Drozd  ring is an Artinian local ring (\Lambda,\m) such that

\m^3=(0), \m and \m^2 are minimally generated by two elements (each), and x^2=0

for some x\in \m-\m^2.  Let (R,\m,k) be a Noetherian local ring, and assume either (i)

some power of \m needs three generators, or (ii) R has a Drozd ring as a homomorphic

image. Let \Cal P be a finite collection of non-maximal prime ideals of R.  Then, for

each positive integer n, there is an indecomposable finitely generated R-module M

that is free of rank n at each prime in \Cal P.  I'll prove case (i) on Wednesday,

using a direct and elementary construction.  The proof in the other case will be sketched

on Thursday. A very strong converse to this result has been proved by Klingler and Levy,

under the assumption that k is not imperfect of characteristic two.


This is joint work with Wolfgang Hassler, Ryan Karr and Lee Klingler.