**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.**