Commutative Algebra Seminar
Spring 2007



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


25th April (Wednesday).  Cristiano Bocci, University of Milan, Italy.


Title: Secant varieties in phylogenetics - II



Special seminar: 23rd April (Monday). Ezra Miller, University of Minnesota.


Time and place: 10:30 ??


Title:   Multiplier ideals of sums via cellular resolutions



19th April (Thursday). Mike Mandell, Indiana University.


Title: A Localization Sequence for the Algebraic K-Theory of Topological K-theory


Abstract:


While the K-theory of the integers is closely connected with number theory, the K-theory

of the "sphere spectrum" is closely connected with geometry.  Waldhausen's "chromatic

tower" in K-theory interpolates between these.  Conjectures of Rognes relate the layers in

this tower to the K-theory of ring spectra that are familiar to topologists, and at the

bottom is topological K-theory.  This talk will describe this picture and explain what is

currently known.



18th April (Wednesday). Cristiano Bocci, University of Milan, Italy.


Title: Secant varieties in phylogenetics - I


Abstract:


I will talk about a recent result for phylogenetics which permits one to extend the result

of Lansberg and Manivel about the ideal of secant varieties of Segre varieties.


Iin the first one I will introduce some concepts of algebraic statistics, explain why

statistical models are algebraic varieties and then introduce general markov tree models.


This provides all the background needed for the second talk, on the particular subfield

(of algebraic statistics) of phylogenetic algebraic geometry.



12nd April (Thursday). Alexander Berglund.


Title: T.B.A.



11th April (Wednesday). Neil Epstein, University of Michigan. 


Title: T.B.A.



Special seminar: 6th April (Friday). Dumitru Stamate.


Time and place: 10:30 AM, Avery 351.


Title: Koszul rings via sequentially Cohen-Macaulay posets


Koszul rings are graded algebras satisfying certain nice homological properties. In

particular, they are quadratic. Woodcock and Polo proved that the incidence algebra of a

finite graded poset P is Koszul if and only if each open interval in P is Cohen-Macaulay.

We study a possible extension of this result to arbitrary finite posets. We expect that in

this case, the associated graded ring of the incidence algebra of the poset play an

important role. We can describe the posets for which this associated ring is quadratic and

we'll present progress towards the conjectured result.  This is joint work with Vic

Reiner, Univ. of Minnesota.


I'll start with Woodcock's approach on Koszul rings and introduce the incidence algebra of

a poset. All other notions will be introduced during the talk.  Attending the previous

talk could be useful.



5th April (Thursday). Giuglio Cavaglia, Purdue University.


Title: How to compute generic initial ideals and extremal Betti numbers.



4th April (Wednesday). Dumitru Stamate.


Title: Sequentially Cohen-Macaulay modules and combinatorics


Abstract:


A module over a noetherian ring is called sequentially Cohen-Macaulay (seq-CM) if it has a

finite filtration of submodules whose quotients are CM modules of increasing Krull

dimension. The notion was introduced by Stanley starting from combinatorial

considerations, and it proved useful in commutative algebra. To begin with, I will survey

several characterizations and properties of seq-CM modules due to Schenzel, Peskine,

Herzog, Popescu,and Sbarra among others.  Using the Stanley-Reisner correspondence, we

restrict to the case of face rings of simplicial complexes and give combinatorial and

homological characterizations of seq-CM complexes and posets, presenting results of Duval,

Bjorner, Wachs and Welker.  In the end, I will present some important classes of seq-CM

modules.



29th March (Thursday). Liana Sega, University of Missouri, Kansas City.


Title: Cohomology over short local rings



28th March (Wednesday). Lars Winther Christensen, UNL.


Title: The Auslander Condition can be checked on the subcategory of maximal Cohen-Macaulay

modules



22nd March (Thursday). Kiriko Kato, Osaka Prefecture University, Japan.


Title: The quotient category of the homotopy category



21st March (Wednesday). Mustapha Lahyane.


Title: On Basic Rational Surfaces.


Abstract: In 1960, Masayoshi Nagata showed that the monoid of effective divisor classes of

a smooth projective rational surface may fail to be finitely generated. This phenomenon

began to be studied in detail in the 80s, beginning with work by Rosoff in 1980 and

Harbourne in 1985.  The aim of this seminar is to discuss some complementary work

regarding this problem.




14th and 15th March. Spring Break.



8th March (Thursday). Tom Marley, UNL.


Title: Concerning the support of local cohomology modules


Abstract: 


Let R be a Noetherian ring, I an ideal of R, and M a finitely generated R-module.  In

general, it is unknown whether the supports of the local cohomology modules H^i_I(M) are

closed subsets of Spec R.  In this talk, I will present some recent progress (done jointly

with Craig Huneke and Dan Katz) on this question.



Special seminar: 7th March (Wednesday). Hyman Bass, University of Michigan.


Time and place: 1 PM,  Bessey Hall 108. 


Title:  The zeta function of a finite graph


Abstract:


This is a nice topic about a generating function to count "prime closed paths" in a finite

graph.  It is defined as a formal power series, and the main theorem says that it is a

(very explicit) polynomial. This is really a discrete analogue of the Selberg zeta

function on a Riemann surface, but one doesn't need to know any of this to understand the

talk.  In fact the talk should be understandable to grad students and even upper

undergraduates.



1st March (Thursday). White-out.



28th February (Wednesday). No seminar. 



Special seminar: 23rd February (Friday) B. Purnaprajna, University of Kansas.


Time and place: 10:30 AM, Avery 351.


Title: Koszul rings from geometry.


Abstract:


Koszul rings have evoked interest among algebraists and geometers.  To paraphrase Arnold

any homogeneous ring that has a serious reason for being quadratically presented is

Koszul.  There are a large class of quadratic algebras arising from geometry where this

meta-principle always seem to hold.  I will illustrate some vector bundle techniques that

are used to prove the above meta-principle.



22nd February (Thursday). B. Purnaprajna, University of Kansas.


Title: Mapping geography of surfaces of general type


Abstract: 


The canonical map of a curve (that is the map induced by the canonical line bundle) of

genus $g\geq 2$ is well studied; either it gives an embedding or is a 2:1map onto a

rational normal curve. The higher dimensional analogue is much more subtle due to the

existence of higher degree covers, among other things. In this talk we show how the

geometry of canonical covers (that is the covers induced by the canonical map) is

intimately connected to various problems in algebraic and Kahler geometry.  For example,

they appear in the determination of ampleness of a linear series on threefolds such as

Calabi-Yau, they also appear in the determination of generators of the canonical ring of

some varieties of general type. Most importantly, they play a role in mapping geography of

surfaces of general type, that is construction of surfaces of general type with prescribed

invariants such as first chern class, geometric genus and irregularity.



21st February (Wednesday). No seminar.



15th February (Thursday). Anurag Singh, University of Utah.


Title: Magic Squares



14th February (Wednesday). No seminar.



8th February (Thursday). S. Iyengar. UNL.


Title: Modules with prescribed cohomological varieties-II.



7th February (Wednesday). Lucho Avramov. UNL.


Title: Modules with prescribed cohomological varieties-I.



1st February (Thursday). Diana White. UNL.


Title: (Lack of) balance - II.



31 January (Wednesday). No seminar. 



25th January (Thursday). Diana White. UNL


Title: (Lack of) balance - I.



24th January (Wednesday). No seminar. 




18th January (Thursday). Frank Moore, UNL.


Title:   Cohomology of fiber products of local rings.


Abstract:


The title says  it all.



17th January (Wednesday) Henrik Holm, University of Copenhagen


Title: Aspects of precovering classes.


Abstract:


Informally speaking, a class of modules F over some ring is precovering (or

contravariantly finite) if every module admits a nice resolution consisting of modules

from F.


In general it is not easy to decide whether a given class is precovering or not. For

example, it was conjectured by Enochs in 1981 that for any ring R the class of flat

R-modules is (pre)covering. This conjecture remained open for twenty years until it was

finally settled by Bican, El Bashir, and Enochs in 2001.


The first part of the talk presents sources and examples of (pre)covering classes. In the

second part of the talk we look at some aspects of precovering classes in relative

homological algebra. More precisely, we relate the vanishing of Ext functors with the

vanishing of resolutions and Schanuel maps relative to a given precovering class.


This talk summarizes work of the speaker and of other people (proper references will be

given during the talk).