Commutative Algebra Seminar
Fall 09 & Spring 10

6th May (Thursday)
    N. Sthanumoorthy 
(University of Madras, Chennai, India)
    An outline on different classes of root systems in finite and infinite dimensional Lie(super)algebras

5th May (Wednesday)
Alex Dugas (Virginia Commonwealth)
    Periodic projective resolutions

Abstract: It is well-known that maximal CM-modules over hypersurfaces have periodic projective resolutions of period 2.  We will demonstrate how this fact can be used to explain the existence of periodic projective resolutions over certain endomorphism rings (reviewing and extending work of Auslander, Reiten and Buchweitz).  The periodicity of projective resolutions in these cases provides a nice homological parallel that suggests these endomorphism rings may be viewed as noncommutative analogues of hypersurfaces.

28th and 29th April (Wednesday and Thursday)
   Ragnar-Olaf Buchweitz
(University of Toronto, Canada)
   Hochschild-Tate Cohomology for Gorenstein Algebras

Abstract: This is work in progress (partly with Flenner) and finds, among other things, that the characteristic class of a commutative Gorenstein algebra is  a universal annihilator for Ext^i_R(M,N), for i \neq 0, with M,N MCMs over R Gorenstein. It also gives yet another characterization of smoothness, even in the non-commutative case. The picture is particularly pretty for c.i.s, where one can use old results of Kersken for an explicit description. Along the way, I generalize the previous work with Avramov (C.I.s in codim 2)  to exhibit a universal complete resolution over any c.i.

22nd April (Thursday)
    Roger Wiegand
    Totally reflexive modules

22nd April (Thursday), at 10 AM
Liana Sega (Univeristy of Missouri - Kansas city)

21st April  (Wednesday)
    Ines Henriques (UNL)
    Quasi-complete intersections  with applications to Poincare series of modules over Artinian rings

14th and 15th April (Wednesday and Thursday)
    Srikanth Iyengar (University of Nebraska, Lincoln)

    Growth of Betti numbers over contracting endomorphisms

8th April  (Thursday)
    Dan Katz (University of Kansas, Lawrence)
    Multiplicities and Rees valuations

Abstract: Let (R,m) be a local ring and let I an ideal with maximal analytic spread.  We discuss joint work with J. Validashti which shows that the j-multiplicity of I is determined by the Rees valuations of I that are centered on m.

7th April (Wednesday)
    Rest cure

30th and 31st March (Wednesday and Thursday)
    Justin DeVries (UNL)
Lower bounds on Betti numbers of multi-graded differential modules

25th March (Thursday)
    Sean Sather-Wagstaff (North Dakota State University, Fargo)

    Extension and Torsion Functors for Artinian Modules

Abstract: Let R be a commutative noetherian ring. It is well known that if N and N' are noetherian R-modules, then the modules Ext^i_R(N,N') and Tor^R_i(N,N') are also noetherian. Similarly, if N is a noetherian R-module and A is an artinian R-module, then the modules Ext^i_R(N,A) and Tor^R_i(N,A) are artinian. We will discuss the properties of Ext and Tor modules when applied to other combinations of noetherian modules, artinian modules, and Matlis reflexive modules. This is joint work with Bethany Kubik (NDSU) and Micah Leamer (UNL).   

24th March (Wednesday)
    Henning Krause (University of Paderborn, Germany)
    Cosupport and colocalizing subcategories of modules and complexes over commutative rings

11th March (Thursday)
    Yongwei Yao (Georgia State University, Atlanta)
    Uniform test exponents over rings of FFRT

Abstract: Let R be a Noetherian reduced ring of prime characteristic p. Then, for every q = p^e, the extension ring R^{1/q} is naturally a module over R. We say R has finite F-representation type (FFRT for short) if there exist finitely many finitely generated R-modules that are enough to represent R^{1/p^e} for all e via direct sum. For example, polynomial rings of finitely many variables over F-finite (e.g., perfect of prime characteristic) fields have FFRT. It is known that if R has FFRT, then tight closure commutes with localization. In this talk, we show that, under the FFRT assumption as above, there are uniform test exponents (for tight closure) for all R-modules.

10th March (Wednesday)
    Yongwei Yao (Georgia State University, Atlanta)
    An embedding theorem for modules of finite (G-)projective dimension

Abstract: Assume R is a commutative Noetherian ring. Then the embedding theorem states that every finitely generated R-module of finite projective dimension embeds into a finite direct sum of cyclic R-modules each of which is the quotient of R by an ideal generated by an R-regular sequence. In fact, this embedding theorem applies to all finitely generated R-modules of finite G-dimension. This is joint work with Mel Hochster.

3rd and 4th March (Wednesday and Thursday)
    Hailong Dao (University of Kansas, Lawrence)
    Non-commutative crepant resolutions, cluster tilting and Ext vanishing over Cohen-Macaulay rings

Abstract:  Let R be a  normal domain. A resolution of singularity X-> Y=Spec(R) is called crepant if  the inverse image  of the canonical sheaf on Y is the canonical sheaf on X. A conjecture by Bondal-Orlov states that two such resolutions of Y have the same derived category (this can be seen as a generalization of the classical McKay correspondence). A proof of this conjecture in dimension 3 led Van den Bergh to define the notion of non-commutative crepant resolutions, which can be stated purely in terms of homological algebra.  We will begin by discussing  the history and background of this fascinating topic, leading up to recent results on the existence of Van den Bergh's resolutions. In the second talk, we shall describe a recent joint project with Craig Huneke, where we improve certain results by Burban-Iyama-Keller-Reiten on connections of crepant resolutions to special objects in the category  of Cohen-Macaulay modules  over hypersurfaces. Despite all the fancy words, it will be our main priority to keep the talks very elementary.

25th February (Thursday)
    Christine Berkesch (Purdue University, West Lafayette)

    The rank of a hypergeometric system

Abstract: An A-hypergeometric system is a system of PDEs determined by a toric ideal and certain homogeneity parameters. The dimension of its solution space, called its rank, is constant for generic parameters. I will discuss the combinatorial nature of this rank at non-generic parameters and its ties to the local cohomology of the toric algebra with support in the maximal ideal.

24th February (Wednesday)
    Christine Berkesch (Purdue University, West Lafayette)
    A-graded methods for monomial ideals

Abstract: Over the years, several families of simplicial complexes have been associated to an arbitrary monomial ideal to capture various aspects of Stanley--Reisner theory. We consider another such family, showing that the "exponent complexes" of a monomial ideal topologically detect if it is Cohen--Macaulay. We also derive a formula for the vector space dimension of the quotient of the polynomial ring by a monomial ideal and a linear system of parameters.

18th February (Thursday)
    Griff Elder (University of Nebraska-Omaha)

    Valuation criterion for normal basis generators and other topics in local Galois module theory

Abstract: The Normal Basis Theorem holds for any finite Galois extension of fields. If you spent your life focused on one particular type of field (like I have focused on local fields), you might wonder whether something stronger can be said. Local field have a notion of valuation, and so in the case of local fields, it is reasonable to ask about the valuations of those elements that generate normal bases. This, it turns out, is a good question (i.e. has an interesting answer). Furthermore, the valuation defines a valuation ring (or ring of integers). Thus the question that was once asked on the field level in the Normal Basis Theorem can now be asked on the integral (or ring) level. This gives rise to the subject known as local Galois module theory (or additive Galois module structure). I will discuss some recent developments.

17th February (Wednesday)
    Srikanth Iyengar (University of Nebraska, Lincoln)

    Detecting flatness over smooth bases, II

11th February (Thursday) [joint with Algebraic Geometry Seminar]
    Greg Smith (Queen's University, Canada)

    Tangential schemes of determinantal varieties

Abstract: The n-th jet scheme of a variety encodes the n-th order Taylor  expansions of functions on the variety.  The jet schemes associated to the varieties of matrices of a given rank are cut out by a relatively simple and explicit collection of polynomials.  In this talk, I give an overview of the geometric properties of these jet schemes and describe the minimal free resolution for the coordinate ring of some 1-st jet schemes.

10th February (Wednesday) No seminar. [Greg Smith's colloquium]

3rd and 4th February (Wednesday)
    Roger Wiegand (University of Nebraska, Lincoln)

    MCM approximations and FID (finite injective dimension) hulls

28th January (Thursday)
    Luchezar Avramov (University of Nebraska, Lincoln)

    Detecting flatness over smooth bases, I
(Part II on 17th February)

27th January (Wednesday) No seminar.

20th and 21st January 2010 (Wednesday & Thursday)
    Marc Chardin (University of Paris VI)

    Torsion in symmetric algebras

9th  December (Wednesday)
    Sylvia Wiegand (UNL)

    Prime ideals in power series rings

We discuss the structure of the set of prime ideals in certain two-dimensional Noetherian integral domains of power series. In particular

(1) In work with Roger Wiegand we  use methods from a recent paper by Oman and Kearnes to give more precise cardinalities for the prime spectrum of R[[x]], where R is a one-dimensional Noetherian domain. This completes a characterization given in  a 2006 article with Heinzer and Rotthaus. We also give a better reason for some cardinalities than given before. (The 2006 paper had referred to a flawed paper by Shah.)

(2) We discuss results of Eubanks-Turner, Luckas and Saydam that for R a Noetherian one-dimensional domain characterize the set of prime ideals of R[[x]][g/f], where f,g is a generalized R[[x]]sequence (that is, f,g is an R[[x]] sequence or (f,g)=(1), but f isn't 0).

10th December (Thursday) No seminar.

2nd December (Wednesday)
    Dave Jorgensen (University of Texas, Arlington)

    Pinched homological algebra and Tate cohomology

3rd December (Thursday)
    Dave Jorgensen (University of Texas, Arlington)

    The depth formula revisited

18th  November (Wednesday)
    Silvia Saccon (UNL)

    Direct-sum behaviour over one-dimensional rings of infinite Cohen-Macaulay type - II

19th  November (Thursday) No seminar.

11th  November (Wednesday)
    Silvia Saccon (UNL)

    Direct-sum behaviour over one-dimensional rings of infinite Cohen-Macaulay type - I

    Abstract:Let (R,m) be a one-dimensional Noetherian local ring whose m-adic completion is reduced. The monoid C(R) of isomorphism classes of maximal Cohen-Macaulay modules carries information about the direct-sum behavior of modules in C(R). The key to describing the monoid C(R) is to determine the possible ranks of indecomposable maximal Cohen-Macaulay modules over the completion. I will discuss the structure of C(R) when all the analytic branches of R have infinite Cohen-Macaulay type.

12th  November (Thursday) No seminar.

4th and 5th November (Wednesday & Thursday)
    Brian Harbourne (UNL)

    Results of Waldschmidt, Skoda and Chudnovsky, with asymptotic applications to ideals in polynomial rings

    Abstract: Motivated by work of Schneider, Lang, Baker and Bombieri in transcendence theory and complex variables, Waldschmidt, Skoda and Chudnovsky studied asymptotic invariants for ideals of points in affine space. This work turns out to be related to recent results of Ein-Lazarsfeld-Smith and Hochster-Huneke on symbolic powers of ideals.

27th October (Tuesday)
    Janusz Adamus (University of Western Ontario, London, Canada)
    Geometric criterion for flatness of analytic and polynomial mappings-I

    Abstract:I n his seminal '61 paper "Modules over unramified regular local rings", Auslander gave a beautiful criterion for freeness of a finitely generated module over a regular local ring in terms of torsion-freeness of tensor powers of the module. In '97 Vasconcelos conjectured that this could be generalized to a flatness criterion for arbitrary algebras essentially of finite type over a regular ring. The conjecture was only recently proved (jointly with E. Bierstone and P.D. Milman) in the geometric setting; i.e., for morphisms of schemes of finite type with a regular target. In the first talk, I will sketch in general terms the complex-analytic approach to the conjecture, via the so-called vertical components of Galligo and Kwiecinski. The second talk will be devoted to a more detailed and technical exposition of the generalization of Auslander's homological apparatus, and a kind of reduction of fibre dimension technique, which lie behind our prove of Vasconcelos' conjecture.

28th October (Wednesday)
N. Saradha (Tata Institute of Fundamental Research, Mumbai, India)
    Irreducibility of polynomials via Newton Polygons

    Abstract: In 1929, Schur proved the irreducibility of truncated exponential polynomials with some possible variations in the coefficients using prime ideals in algebraic number fields. In 1987, Coleman and later Filaseta gave a proof of Schur’s result via Newton polygons. They used an old result of Dumas in 1906 on Newton Polygons. Since then irreducibility of several orthogonal polynomials were proved using Newton Polygons. In this talk we shall present some of the recent results in this area.

29th October (Thursday)
    Janusz Adamus (University of Western Ontario, London, Canada)
    Geometric criterion for flatness of analytic and polynomial mappings-II

21st and 22nd October (Wednesday & Thursday)
    Ananth Hariharan (UNL)

    Applications of n-standardness and three-standardness of the maximal ideal

    Abstract: These talks are a tale of two halves. In the first half, we will talk about a notion called n-standardness (defined by M. E. Rossi) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring. We will discuss some of its consequences which are related to a result of T. Marley. In the second half, we will investigate conditions under which the maximal ideal is three-standard, first proving results when the residue field is of prime characteristic and then use the method of reduction to prime characteristic to extend the results to the characteristic zero case. As an application, we extend a result due to T. Puthenpurakal and show that a certain length associated to a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.

14th and 15th September, No seminar.

7th and 8th October (Wednesday & Thursday)
    Mark Walker (UNL)

    Hochster's theta invariant and the Hodge-Riemann bilinear relations

30th September and 1st October (Wednesday & Thursday)
    Jesse Burke (UNL)

    Vanishing of cohomology over complete intersections

24th September (Thursday)
    Daniel Murfet (Bonn)
    Complete injective resolutions and duality for singularity categories

23rd September (Wednesday)
    William Sanders (UNL)
    Irreducible representations of metacyclic groups

17th September (Thursday)
    Olgur Celikbas (UNL)

    Vanishing of Tor over complete intersection rings

16th September (Wednesday)
    Louiza Fouli (New Mexico State University,
Las Cruces)
    Systems of Parameters in Non Cohen-Macaulay Rings
    Abstract: Let R be a Noetherian local ring of dimension d.  Let x_1,...,x_d be a system of parameters. Let y_1,...,y_d be a sequence such that the ideal (y_1,...y_d) is contained in
(x_1,...x_d) and let A be a matrix such that [y_1,..,y_d]=[x_1,...,x_d]A.  Dutta and Roberts proved that when R is Cohen-Macaulay  y_1,...y_d is also a system of parameters if and only if the map R/(x_1,...x_d)-->R/(y_1,...y_d) induced by multiplication with det A is injective.  We will discuss necessary and sufficient conditions for when the sequence y_1,...y_d is a system of parameters without the assumption that the ring is Cohen-Macaulay. This is joint work with Craig Huneke.

10th September (Thursday)
    Hans-Christian Herbig (Griefswald)

    On deformation quantization of singular symplectic quotient spaces

9th  September (Wednesday), No seminar.

2nd and 3rd September (Wednesday & Thursday)
    Srikanth Iyengar (UNL)

    Homological dimensions and regular rings

Visitors in Fall 2009 and Spring 2010  

Past seminars

Spring 2009   Fall 2008    Spring 2008    Fall 2007    Spring 2007    Fall 2006    Spring 2006    Fall 2005    2004

Maintained by Srikanth Iyengar

Back to the top