Commutative Algebra Seminar

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The seminar meets Fridays, 3:00--4:00pm, in LCB 222.

February 3, 2006:

Hailong Dao (University of Michigan), Vanishing of Chow groups and intersection properties of modules over hypersurfaces.

Abstract:  Let (R,m,k) be a local hypersurface and M be an R module. We consider the following properties:
(1) For another R-module N, if Tor_i(M,N)=0 then Tor_j(M,N)= 0 for j>i. (2) If N is such that M_tensor N has finite length, then dim M + dim N <= dim R.  These  properties have been studied over the years, with the classical view that they "should" hold when M has finite projective dimension. We will present some recent results that show a connection between the two, as well as some K-theoretic sufficient conditions for them. We will also discuss some open questions, which are linked to conjectures about Chow group of smooth projective hypersurfaces in P^n.


February 17, 2006:  (Joint seminar with Number Theory)

Ravi Ramakrishna (Cornell),  Deforming global galois representations.

Abstract:  I'll talk about galois theoretic techniques that allow one to deform mod p representations to characteristic zero. While many of these results have been surpassed by techniques of Khare-Wintenberger and Khare, there are still some applications to obtain modularity of the original residual representation. This is joint work with S. Hamblen.


February 24, 2006:

Joro Todorov, A-infinity algebras and superpotentials.


March 3, 2006:

Erin Chamberlain, The Monomial Conjecture and Cohen-Macaulay canonical modules--a paper by Le Thanh Nhan.


March 6, 2006:   (Special Monday seminar in JWB 208 at 2pm)

Luchezar Avramov (University of Nebraska), Length of homology of finite free complexes.


March 10, 2006:  

Erin Chamberlain, The Monomial Conjecture and Cohen-Macaulay canonical modules--a paper by Le Thanh Nhan, Part II.


March 24, 2006:  

Greg Piepmeyer (University of Nebraska), Tame and wild type--last case.

Abstract:  Levy and Klingler have studied which rings can have the structure of all indecomposable modules and their direct sum behaviour described without having to describe the module structure of all indecomposable modules over all finitely generated algebras at the same time.  There is one open case in their theory, and Klingler understands this as a canonical form matrix problem.  This is the problem of the current work.  I'll try to motivate it, describe it, and describe the canonical form.

March 31, 2006:  

Kazuma Shimomoto, Some calculations of Fontaine rings.


April 7, 2006:  

Paul Roberts, The commutativity of intersection with divisors.


April 14, 2006:  

Oana Veliche, Modules over local rings with m^3=0.


April 21, 2006:

Esben Halvorsen (Københavns Universitet), How to make complexes smaller within Grothendieck groups.


Past Seminars



September 2, 2005:

Paul Roberts, Fontaine rings of Noetherian rings.


September 9, 2005:

Sandra Spiroff, Some results on divisor class groups.


September 16, 2005:

No Seminar.


September 23, 2005

Kazumo Shimomoto, Perfect rings in mixed characteristic.

Abstract: Based on the Gabber and Ramero's almost ring theory I will discuss the class of rings in mixed characteristic for which the Frobenius map is surjective modulo $p$. This talk will also contain discussion on the behavior of the Frobenius map under $\acute{e}$tale extensions of rings in some weak sense.


September 30, 2005

Kazumo Shimomoto, Perfect rings in mixed characteristic, Part II.


October 14, 2005

Oana Veliche, Acyclicity over local rings with m^3=0.


October 21, 2005

No seminar.


October 28, 2005

Trung Dinh, Approximation rings and Artin's functions.


November 4, 2005

Anurag Singh, Big Cohen-Macaulay algebras: a new proof by Huneke & Lyubeznik.


November 11, 2005

Anurag Singh, Big Cohen-Macaulay algebras: a new proof by Huneke & Lyubeznik, Part II.


November 18, 2005:

TBA


December 2, 2005:

Christine Cumming (Purdue University),  Cohen-Macaulayness of residual intersections.


December 9, 2005:

Brent Strunk (Tulane University), Castelnuovo-Mumford regularity, reduction numbers, and postulation numbers.


Monday, December 12, 2005 at 1pm in LCB 323:

Lars Christensen (University of Nebraska), When are evaluation homomorphisms invertible?

Abstract: Certain standard maps between modules, 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 survey 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.


January 21, 2005:

Kazuma Shimomoto, Frobenius Action in Mixed Characteristic.

Abstract: I will start with a basic set-up and define a length function for certain modules. In the first talk, I will mention some basic properties of the length function with an application to modules with the Frobenius action in mixed characteristic. In the second talk, I will relate the almost vanishing of modules to the length zero modules that appear in homological conjectures.

January 28, 2005:

Anurag Singh (Georgia Institute of Technology), A connectedness theorem in positive characteristic.

Abstract: Lyubeznik recently proved a criterion for the connectedness of the punctured spectrum of a local ring of positive characteristic. His proof uses a vanishing theorem of Peskine and Szpiro. We give a simple proof of this criterion, and also present a variation which determines the precise number of connected components. This is joint work with Uli Walther.

February 4, 2005:

No Seminar.


February 11, 2005:

Kazuma Shimomoto, Frobenius Action in Mixed Characteristic, Part II.


February 18, 2005:

Oana Veliche, Stable Cohomology of Local Rings, Part I.


February 25, 2005:

Claudia Miller (Syracuse University), Extremal Algebras.


March 4, 2005:

Peter Trapa, Geometry of Primitive Ideals in Enveloping Algebras.


March 11, 2005:

No Seminar.


March 18, 2005:

Spring Break.


March 25, 2005:

Lars Christensen (University of Nebraska-Lincoln),  Stability of Gorenstein dimensions.

Abstract: Auslander and Bridger initiated the study of Gorenstein dimensions in 1966. In
commutative algebra, these homological dimensions are used to characterize
Gorenstein rings in much the same way the usual homological dimensions
characterize regular rings.

The study of Gorenstein dimensions takes cues from the theory of classical
homological dimensions. However, while the results may be parallel, the proofs
are sometimes very different. The talk will illustrate this by considering some
questions on stability of Gorenstein dimensions.

April 1, 2005:

Oana Veliche, Stable Cohomology of Local Rings, Part II.


April 8, 2005:

Kazuma Shimomoto, Almost Ring Theory.

Abstract: I will start with definitions of almost modules and maps in the non-Notherian category.  The main focus is to discuss the finiteness condition of local cohomology modules of a certain non-Noetherian ring.  The key point is the Lipman-Sathaye Jacobian theorem.

April 15, 2005:

Erin Chamberlain, Modules of Finite Length and Finite Projective Dimension-a paper by P. Roberts and V. Srinivas.

Abstract: The vanishing conjecture states that if A is a local ring and M and
N are modules of finite length and finite projective dimension, and if
dim (M) + dim (N) < d = dim (A), then the intersection multiplicity of M and N = 0.

Much work has been done to try to determine when the vanishing conjecture
holds.  Roberts and Srinivas determined correspondences between
intersection properties of modules of finite length and finite projective
dimension and the Chow group of the ring in their paper entitled Modules
of finite length and finite projective dimension.

April 22, 2005:

TBA.


September 3, 2004:

Sandra Spiroff, Growth of Ideals.


September 10, 2004:

Paul Roberts, Constructing Modules from Cycles, Part I.


September 24, 2004:

Paul Roberts, Constructing Modules from Cycles, Part II.


October 1, 2004:

Kazuma Shimomoto, on the Minimality of Hilbert-Kunz Multiplicity.


October 22, 2004:

Oana Veliche, Gorenstein Dimension for Modules.


October 29, 2004:

Oana Veliche, Gorenstein Dimension for Modules, Part II.


November 12, 2004:

Bo-Hae Im, Hilbert Irreducibility and its Application.


November 19, 2004

Tyler Evans (Westminster)


August 29, 2003:

Florian Enescu, Rings with small Hilbert-Kunz multiplicity.

Abstract: We will investigate the structure of rings of positive characteristic that have small Hilbert-Kunz multiplicity. We will present some bounds for the Hilbert-Kunz multiplicity of a ring R that imply that make the ring R F-rational. The new results improve similar estimates that are currently in the literature. The work is joint with Ian M. Aberbach.

September 5, 2003:

Paul Roberts, Killing local cohomology with elements of small valuation (I).


September 12, 2003:

Paul Roberts, Killing local cohomology with elements of small valuation (II).


September 19, 2003:

Indranath Sengupta, Affine monomial curves.

Abstract: We will discuss about the syzygies of certain affine monomial curves followed by some open problems on their blow-up and symbolic blow-up.

September 26, 2003:

Kazuma Shimomoto, Approximately Gorenstein rings (I).

Abstract: The talk will present the notion of approximately Gorenstein rings introduced by Hochster in the late seventies.

October 10, 2003:

Kazuma Shimomoto, Approximately Gorenstein rings (II).


October 24, 2003:

Robert Lazarsfeld (University of Michigan), Asymptotic invariants of linear series. (LCB 222, this week only).



October 31, 2003:

Sandra Spiroff, The behavior of the restriction of divisor classes to sequences of hypersurfaces .

Abstract: Let (A,m) be a normal, local domain and let f be a prime element such that A/fA is a normal hypersurface. Then there is a group homomorphism j*: Cl(A) -> Cl(A/fA). It is well-known that j* need not be injective. However, consider a sequence {f_n} of primes such that A/f_nA is normal and lim f_n = 0 in the m-adic topology. We consider the two questions: 1. Can a non-trivial divisor class be in every kernel of Cl(A) -> Cl(A/f_nA)? 2. Are there situations where an integer N>0 exists such that Cl(A) -> Cl(A/f_nA) is monic if f_n is in m^N?

November 7, 2003:

Sandra Spiroff, The behavior of the restriction of divisor classes of sequences of hypersurfaces (II).


November 21, 2003:

Florian Enescu, The structure of F-pure rings.

Abstract: We will discuss the notions of splitting dimension and Frobenius ratio of a local ring of positive characteristic. This is joint work with Ian M. Aberbach.