**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**** **