Commutative Algebra Seminar
Fall 09 & Spring 10


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

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Seminars


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



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