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.