## Commutative Algebra SeminarFall 10 & Spring 11

### Seminars

27th April (Wednesday)

Lee  Klingler (Florida Atlantic University)
A Stone-Weierstrass Theorem for Even Integer-valued Polynomials

26th April (Tuesday)
Gerry Schwarz (Brandeis)

Reduced Invariant Sets

Abstract: Let K be a compact Lie group, W a K-module and  X a subset of X that is real algebraic and K-stable. Let I(X) be the ideal of X in R[W] and let I_K(X) be generated by I(X)\cap\R[W]^K. For which X do we have that I(X)=I_K(X)? (We say that X is K-reduced.)\ Complexifying one is forced to consider the following question: Let G be complex reductive and V a G-module. The null cone N(V) is the set of all v in V such that f(v)=f(0) for all f in C[V]^G. Let I_G(N(V)) be generated by I(N(V)) \cap\C[V]^G. When do we have that I(N(V))=I_G(N(V))? (We say that V is coreduced.). We give criteria for X to be K-reduced and  classifications of V which are coreduced.

20th April (Wednesday)

Rest cure

19th April (Tuesday), Non-standard time: 4 - 5 PM
Micah Leamer (UNL)

TBA

13th April (Wednesday)
Justin DeVries (UNL)

12th April (Tuesday) -- Joint with Operator Algebras seminar

The uncanny resemblance between Leavitt path algebras and graph C*-algebras

Abstract: In the fundamental 1977 paper Simple C*-algebras generated by isometries", Cuntz described a class of C*-algebras, the now-so-called, Cuntz algebras,  O_n.  These algebras were generalized a few years later to the Cuntz-Krieger algebras  O_A, and subsequently in the early 2000s to the graph C*-algebras C*(E).

In this talk we will define these graph C*-algebras, and describe some of their structural results.  We then compare and contrast these results with analogous results for the Leavitt path algebras L_C(E) having coefficients in the complex numbers (introduced in Monday's colloquium).     While the similarities  between the results for C*(E) and for L_C(E) in terms of the graph E are more than striking, there is currently no known way of passing these results directly from the analytic side to the algebraic, nor vice-versa.

We will present two of the ongoing efforts at trying to understand various aspects of this phenomenon.   We will describe the Isomorphism Conjecture  (joint work with Mark Tomforde, U. Houston), and the search for an algebraic analog in the context of Leavitt path algebras of the Kirchberg-Phillips Theorem for graph C*-algebras (joint work with Adel Louly, Enrique Pardo, and Chris Smith).

6th April (Wednesday)

Dave Jorgensen (University of Texas at Arlington)
Products in Tate cohomology

5th April (Tuesday)
Peter Symonds  (University of Manchester)

Regularity of polynomial invariants of finite groups

30th March (Wednesday)

Rest cure

29th March (Tuesday)
Ryo Takahashi (UNL)

Dimensions of derived categories of commutative rings

Abstract: Several years ago Rouquier introduced the notion of the dimension of a triangulated category, and proved that the bounded derived category of
coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension. In this talk, we study the dimension of the bounded derived category of finitely generated modules over a commutative noetherian ring. We show that it is finite when the base ring is a complete local ring.

22nd and 23rd March (Tuesday and Wednesday)
Spring break

16th March (Wednesday)
Bill Heinzer (Purdue University, West Lafayette)

Examples of non-Noetherian domains inside power series rings

15th March (Tuesday)
Andrei Zelevinsky (Northeastern University, Boston)

Quiver Grassmannians and their Euler characteristics

Abstract: The aim of this talk is to advertise quiver Grassmannians, a very interesting family of projective algebraic varieties which are a far-reaching generalization of ordinary Grassmannians. Namely, the quiver Grassmannian associated to a quiver representation M and a nonnegative integer vector e is the variety of subrepresentations of M with the dimension vector e. The Euler characteristics of  these varieties have an unexpected application (due to Caldero, Chapoton and Keller) to cluster algebras. We discuss some examples of quiver Grassmannians and some of their general properties (sufficient
conditions for smoothness and for the positivity of the Euler characteristic).

8th and 9th March (Tuesday and Wednesday)
Shunsuke Takagi (Kyushu University, Japan)

A correspondence between F-purity and log canonicity

Abstract: Let R be a normal Q-Gorenstein domain essentially of finite type over a field of characteristic zero. It is then conjectured that Spec R has only log canonical singularities if and only if its module p reduction R_p is  F-pure for infinitely many p. I will discuss recent progress on this conjecture.

2nd March (Wednesday)
Wenliang Zhang (University of Michigan, Ann Arbor)

Local cohomology invariants of projective schemes

Abstract: We will discuss a set of numerical invariants of projective schemes arising from local cohomology.

1st March (Tuesday)
Wenliang Zhang (University of Michigan, Ann Arbor)

F-jumping coefficients

Abstract: F-jumping coefficients consist of an increasing sequence of positive rational numbers; they encode interesting geometric and algebraic information of singularities in positive characteristic. After an introduction to those numbers, I will discuss some recent developments.

23rd February  (Wednesday)
Kristen Beck (University of Texas, Arlington)

Asymmetric Linear Complete Resolutions over Short Local Rings

15th and 16th February (Tuesday and Wednesday)
Rest cure

9th February (Wednesday)
Ananth Hariharan (UNL)

When is the bowtie Gorenstein?

8th February (Tuesday)
Micah Leamer (UNL)

Asymptotic behavior of dimensions of syzygies

Abstract: Let R be a noetherian local ring and M a finitely generated R-module. It is a well known result that the depth of an nth syzygy module of M equals depth(R) for n > max{0,depth(R)-depth(M)}.  However, the asymptotic behavior of the dimension of syzygies of M is not known in general.  We show that when R is equidimensional, pd (M) is infinity and the Betti numbers of M are eventually non-decreasing, the dimension of any sufficiently high syzygy of M equals dim(R).

2nd February (Wednesday)

Srikanth Iyengar  (UNL)
TBA

25th and 26th January (Tuesday and Wednesday)
Rest cure

18th and 19th January 2011 (Tuesday and Wednesday)
Manoj Kummini  (Purdue University, W. Lafayette)

On a conjecture of Lyubeznik on etale cohomological dimension

Abstract: We will describe a counter-example to a conjecture of G. Lyubeznik relating etale and quasi-coherent cohomological dimensions of schemes. This is joint work with U. Walther.

12th January 2011 (Wednesday)

Kamran Divaani-Aazar (IPM, Iran)
Local homology and Gorenstein flat modules

17th November (Wednesday)

Ela Celikbas (UNL)
The projective line over the integers

21st September (Tuesday)
Lori McDonnell (UNL)

The second Hilbert coefficient of a parameter ideal in an unmixed ring

Abstract: In recent work, Ghezzi, Goto, Hong, Ozeki, Phuong, and Vasconcelos showed that an unmixed ring R is Cohen-Macaulay if and only if the ﬁrst Hilbert coefficient satisﬁes  e_1(Q) = 0  for some parameter ideal  Q  of  R. Inspired by this result, we will look at what conclusions can be drawn in an unmixed ring when the second Hilbert coefficient, e_2 (Q), is zero for a parameter ideal  Q.  We will also show that  e_2 (Q) \leq 0.

9th and 10th November (Tuesday and Wednesday)
Emily Witt  (University of Michigan, Ann Arbor)

Local cohomology with support in ideals of maximal minors

Abstract: Suppose that k is a field of characteristic zero, X is an r x r matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X.  Let I be the ideal of R generated by the maximal (r x r) minors of X.  Using the structure induced on the local cohomology modules H^i_I(R) by the action of SL_r(k) on R, as well as Lyubeznik's theory of local cohomology modules as D-modules, we provide information about the modules H^i_I(R).  In particular, we find their associated primes, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r) +1, identify the indices i for which these modules vanish, and characterize the nonzero ones as submodules of certain indecomposable injective modules.  Moreover, these results are consequences of a more general theorem regarding local cohomology modules with actions of linearly reductive groups.

2nd and 3rd November (Tuesday and Wednesday)
Rest cure

26th and 27th October (Tuesday and Wednesday)
Brian Johnson  (UNL)

Commutative rings graded by abelian groups

19th and 20th October (Tuesday and Wednesday)
Fall Break

12th and 13th October (Tuesday and Wednesday)
Micah Leamer  (UNL)

Torsion in tensor products

5th and 6th October (Tuesday and Wednesday)
Jesse Burke  (UNL)

Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

28th and 29th September (Tuesday and Wednesday)
Srikanth Iyengar  (UNL)

Hopkins' theorem on perfect complexes over commutative rings

22nd September (Wednesday)

No seminar

21st September (Tuesday)
Henning Krause (Univeristy of Bielefeld, Germany)

An axiomatic description of coherent sheaves on weighted projective lines

Abstract: This talk gives an elementary introduction to weighted projective lines, as they play an important role in representation theory of algebras. I present a simple formalism which describes the abelian categories arising as categories of coherent sheaves on weighted projective lines.

15th September (Wednesday)
Courtney Gibbons  (UNL)

Boij-Soederberg Theory II:  Betti diagrams over  k[x,y]/(x^2)

Abstract: One might hope to have a Boij-Soederberg type of decomposition theory for Betti tables of modules over other graded rings.  Hypersurface rings are a natural place to start this investigation. For the ring k[x,y]/(x2), we provide characterizations of the cone of Betti
diagrams in terms of extremal rays generating the cone, functionals determining the halfspaces cutting out the cone, and a simplicial fan
structure guaranteeing uniqueness of the decomposition.  In this talk, I'll describe these characterizations and outline a proof of the theorem.  This is joint work with Christine Berkesch, Jesse Burke, and Dan Erman.

14th  September (Tuesday)
Courtney Gibbons  (UNL)

Boij-Soederberg Theory I:  Betti diagrams over polynomial rings

Abstract: The Boij-Soederberg conjectures (now theorems, proved by Eisenbud and Schreyer in 2008) state that, given a graded, finitely generated Cohen-Macaulay module over a polynomial ring, its Betti diagram can be written uniquely as a postive rational linear combination of "special" Betti diagrams.  These results allow us to use convex geometry tools to identify possible (or impossible) Betti diagrams up to rational multiples.  In this talk, I'll describe the "special" Betti diagrams and an outline of the proof of the conjectures in the two variable case.

9th September (Thursday) 3:00 - 3:50 PM, Burnett Hall 121
Lutz Hille (University of Muenster, Germany)

Tilting Bundles on Rational Surfaces  and Quasi-Hereditary Algebras

Abstract: Here we use recent results in a joint work with Markus Perling to construct tilting bundles on any rational surface. The construction in general produces from any exceptional sequence a tilting complex, whose endomorphism algebra is quasi-hereditary. In case of a surface we can choose the exceptional sequence in such a way, that the tilting complex is even a vector bundle.

8th September (Wednesday)

Ryo Takahashi (Shinsu Univeristy, Japan)
Resolving subcategories and the punctured spectrum

Abstract: Let R be a Cohen-Macaulay local ring.  In this talk, we consider classifying resolving subcategories of the category of finitely generated R-modules in terms of specialization-closed subsets of Spec(R) when R is
- locally a hypersurface,
- locally with minimal multiplicity, or
- locally of finite Cohen-Macaulay representation type on the punctured spectrum.

7th September (Tuesday)

No seminar

1st September (Wednesday)
Luchezar Avramov  (UNL)

Connected Sums of Gorenstein Rings  II

31st August (Tuesday)
Ananth Hariharan (UNL)

Connected Sums of Gorenstein Rings I

### Visitors in Fall 2010 and Spring 2011

• Gerry Scharwz, Brandeis, 22nd - 28th April 2011
• Gene Abrams, University of Colorado at Colorado Springs, 10th - 12th April 2011
• Dave Jorgensen, University of Texas, Arlington,  31st March - 8th April 2011
• Petter Bergh, University of Trondheim, Norway, 2nd April  - 7th May 2011
• Jesse Burke, University of Bielefeld, Germany, 31st March - 17th April 2011
• Peter Symonds,  University of Manchester, U.K., 31st March - 20th April 2011
• Bill Heinzer, Purdue University, West Lafayette,  14th - 19th March 2011
• Olgur Celikbas, University of Kansas, Lawrence,  14th - 18th March 2011
• Andrei Zelevinsky, Northeastern University, Boston,  13th - 16th March 2011
• Ryan Karr, University of Central Florida,  7th - 11th March 2011
• Wenliang Zhang, University of Michigan, Ann Arbor  27th February - 3rd March 2011
• Kristen Beck, University of Texas, Arlington, week of February 23rd. 2011
• Hans-Christian Herbig, University of Aarhus, Denmark,  2nd February - 30th April 2011
• Manoj Kummini, Purdue University, West Lafayette, 16th - 20th January 2011
• Kosmas Diveris, University of Syracuse, Syracuse,  1st - 12th December 2010
• Emily Witt, University of Michigan, Ann Arbor,  8th - 11th November 2010
• Henning Krause,  University of Bielefeld, Germany,  18th - 22nd September 2010
• Lutz Hille, University of Muenster, Germany,  8th - 10th September 2010
• Ryo Takahashi, Shinsu Univeristy, Japan, 1st - 12th September 2010
• Hans-Christian Herbig, University of Aarhus, Denmark,  29th August - 11th September 2010

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