Commutative Algebra Seminar

Fall 2005



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


31st November (Thursday). Hamid Rahmati. UNL.


Free resolutions of parameters ideals for some rings with finite local cohomology


Abstract:


Let R be a local noetherian ring. We consider the following question: Does there exist an

integer `n' such that all ideals generated by systems of parameters contained in the nth

power of the maximal ideal of R have the same Poincar\'e series? We obtain a positive

answer for some rings with finite local cohomology.



23rd and 24th November. Thanksgiving break.



17th November (Thursday). Anders Frankild. University of Copenhangen.


The Frobenius homomorphism revisited 


Abstract:


Recently, Foxby and the speaker showed that if a commutative, local,

noetherian ring admits  a cyclic module of finite Gorenstein injective

dimension, then the ring must be Gorenstein - provided it admits a

dualizing complex.


As an immediate corollary, we obtain the classical result due to

Peskine and Szpiro (from 73) stating that if the cyclic module has finite

injective dimension, then the ring must (also) be Gorenstein.


We will discuss how the Gorenstein version of Peskine and Szpiro's

result yields a nice result on the Frobenius homomorphisms ability to

characterize Gorenstein rings using Gorenstein injective

dimension. Other results in this context will be presented as well. This is all joint

work with Foxby.



 Special seminar 

15th November (Tuesday). 2:30 PM. Avery 110. Gennady Lyubeznik. University of Minnesota.


A simple proof of the Hochster-Huneke theorem that R^+ is Cohen-Macaulay in characteristic p>0.


Abstract: 


A complete proof will be given. This is joint work with Craig Huneke.



2nd and 3rd November (Wednesday and Thursday). Srikanth Iyengar. UNL.


Matrices over commutative rings: Parts I and II 


Abstract:


Let S be a commutative noetherian ring S containing a field.  Given a square-zero lower

triangular matrix A with coefficients in S, lower bounds are obtained on the number of

sub-diagonal blocks of A, and on the size of these blocks, in terms of the S-module

Ker(A)/Image(A). The bound on the number of blocks contain the New Intersection Theorem

for rings containing fields, and that on the size of the blocks is in the spirit of the

Buchsbaum-Eisenbud-Horrocks conjecture on the Betti numbers of modules of finite

projective dimension.  This is part of a on-going project, in collaboration with Lucho

Avramov, Ragnar-Olaf Buchweitz, and Claudia Miller.



27th October (Thursday). Hailong Dao. University of Michigan.


Asymptotic multiplicity and rigidity over complete intersections


Abstract:


A pair of modules (M,N) over a local ring R is called c-rigid if the vanishing of c

consecutive Tor_i(M,N) modules implies the vanishing of all higher Tor. We show that

c-rigidity is tied to a function defined on (M,N). Our function can be viewed as an

asymptotic version of Serre's definition of intersection multiplicity, as well as of

Hochster's "theta" function for hypersurfaces. We apply this observation to extend certain

results on tensor products by Auslander, Huneke, Jorgensen, and R. Wiegand.



26th October (Wednesday). Hailong Dao. University of Michigan.


Complexities of Ext and Tor over complete intersections


Abstract:


For a pair of modules (M,N) such that all the high Tor(M,N) modules have finite length, we

study the behaviour of the sequence beta_i(M,N) = l(Tor_i(M,N)). This lead us to define

the length complexity of (M,N) to be the degree of the polynomial growth of {beta_i}. We

investigate how this relates to the notion of complexity given by Avramov. One interesting

question is when the "complexities" of Ext and Tor are equal,inspired by the work of

Avramov and Buchweitz. We answer this question in some cases.



19th and 20th October. Lars Christensen. UNL.


Acyclicity over local rings with m^3=0; in two parts.

 

Abstract:


Let (R,m,k) be a local ring with m^3=0. Existence of an acyclic complex (an exact doubly

infinite complex of f.g. projective modules) has structural implications for the ring.  In

the first talk I shall focus on motivations and main results.  In the second talk I

will discuss proofs and techniques.



12th and 13rd October. Rest cure.



6th October (Thursday). Lars Christensen. UNL.


Invertible Evaluation Homomorphisms


Abstract:


Certain standard maps, sometimes called evaluation homomorphisms, come up repeatedly in

commutative algebra. Sufficient conditions for invertibility are easy to establish,

necessary ones are harder to come by. I will address this problem and show how

invertibility of these maps can be used to detect Gorenstein and regular rings. For the

latter case one needs a test module with betti numbers of amenable proportions.

 


5th October (Thursday). Srikanth Iyengar. UNL.


Recognizing the residue field of a local ring


Abstract:


I will discuss the statement and a proof of a new criterion, based on finite free

complexes, for detecting when a local ring is regular.  This is part of work in progress

in collaboration with Tom Bridgeland, and is motivated by his work on equivalences of

derived categories of coherent sheaves.



28th and 29th September (Wednesday and Thursday). Tom Marley. UNL.


Parameter ideals and Cohen-Macaulayness for non-Noetherian rings


Abstract:  


In 1992, Sarah Glaz asked whether there was a definition of Cohen-Macaulay for

non-Noetherian rings such that 1) The definition agreed with the usual definition for

Noetherian rings and 2) every coherent regular ring is Cohen-Macaulay.  (A ring is regular

if every finitely generated ideal has finite projective dimension.  A ring is coherent if

ever finitely generated module is finitely presented.)  In joint work with Tracy Hamilton,

we give a solution to this problem.  The key idea is to come up with the right extension

of "parameter ideal", which we define using Cech cohomology.



22nd September (Thursday).  Hans-Bjoern Foxby. University of Copenhagen.


Flat dimensions of non-finitely generated modules


Abstract:


Complete intersection dimension, respectively, Cohen-Macaulay dimension,

of finitely generated modules were introduced by Avramov, Gasharov, and

Peeva, respectively, by Gerko, and it was studied for non-finitely

generated modules by Sharif. The talk will present relations between these

dimensions and related ones.  This is joint work with Frankild,

Sather-Wagstaff, Taylor, and Yassemi.



 

21st September (Wednesday). Kiriko Kato. Osaka Women's University.


Cohen-Macaulay approximation using stable module theory 


Abstract: 


We show the following: every maximal Cohen-Macaulay module is a sygyzy of some

Cohen-Macaulay module with codimension one if and only if the ring is an integral

domain. Our viewpoint is that Cohen-Macaulay approximations are closely related to stable

module theory. And there is an analogy of a stable module category to a homotopy

category. Using this analogy, we characterize Cohen-Macaulay approximation as a

equivalence of stable categories.



15th September (Thursday). Tim Roemer. University of Osnabruck.


On seminormal monoid rings


Abstract:


Given a seminormal affine monoid M we consider several monoid properties of M and their

connections to ring properties of the associated affine monoid ring K[M] over a field

K. We characterize when K[M] satisfies Serre's condition (S_2) and analyze the local

cohomology of K[M]. As an application we present criteria which imply that K[M] is

Cohen-Macaulay and we give lower bounds for the depth of K[M].



 

14th September (Wednesday). Tim Roemer. University of Osnabruck.


h-vectors of Gorenstein polytopes


Abstract:


We show that the Ehrhart h-vector of an integer Gorenstein polytope with a unimodular

triangulation satisfies McMullen's g-theorem; in particular it is unimodal. This result

generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed

polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids

M: one can factor K[M] (K a field) by a ``long'' regular sequence in such a way that the

quotient is still a normal affine monoid algebra. In the case of a polytopal Gorenstein

normal monoid E(P), this technique reduces all questions about the Ehrhart h-vector to a

normal Gorenstein polytope Q with exactly one interior lattice point.  If P has a

unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides

with the h-vector of the boundary complex of a simplicial polytope, and the g-theorem

applies.



8th September (Thursday). Rest cure.



 

7th September (Wednesday). Janet Striuli. UNL.


Extensions of module II

 

Abstract:


I will go over the Yoneda's correspondence and prove a main theorem of short 

exact sequences.



1st September (Thursday). Janet Striuli. UNL.


Extensions of module I

 

Abstract:


I will go over the Yoneda's correspondence and prove a main theorem of short 

exact sequences.



 

31st August (Wednesday). Janet Striuli. UNL.


Freeness of the endomorphism ring

 

Abstract:


I will present a simplified proof of a theorem of Braun which answers a question of

Auslander.



24th August (Wednesday). Henrik Holm. University of Aarhus.


Cohen-Macaulay injective dimension.


Abstract:


Gerko (2001) has introduced the Cohen-Macaulay dimension for finitely generated modules

over a local ring, and this homological dimension characterizes Cohen-Macaulay rings in a

way one could hope for.


Gerkos CM-dimension satisfies an Auslander-Buchsbaum formula, which suggests that it s a

"projective CM-dimension". In the talk we shall see how to construct an "injective

CM-dimension" which works for all (not necessarily finitely generated) modules, and we

will look at some of its fundamental properties.


The material to be presented in the talk is joint work with Peter Jorgensen, University of

Leeds.



Maintained by  Srikanth Iyengar