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