Commutative Algebra Seminar

Fall 2004

2nd September (Thursday). Roger Wiegand, UNL.

Indecomposable finitely generated modules of large rank over local rings

8th and 9th September. Nick Baeth, UNL. Representation theory of one-dimensional equicharacteristic Cohen-Macaulay local rings of finite representation type. I and II. Abstract: Let $(R,\m,k)$ be an equicharacteristic one-dimensional Cohen-Macaulay local ring of finite representation type. For each ring with $k$ perfect of characteristic not 2, 3, or 5, we determine the monoid $C(R)$ of isomorphism classes of finitely generated torsion-free $R$-modules. When $R$ is complete and $k$ is algebraically closed of characteristic zero this classification is well-known. To complete this study, we analyze maps $C(R) \to C(\hat R)$ and $C(R) \to C(S)$ where $(S,\n,l)$ is a flat local extension of $R$ and $l/k$ is a separable algebraic extension. Using the monoid $C(R)$ we are able to determine which rings have the Krull-Schmidt uniqueness property and which rings have the weaker property that any two representations of a torsion-free module have the same number of indecomposable summands.
16th September (Thursday). Lee Klingler, UNL.
22nd and 23rd September. Sean Sather-Wagstaff, UNL. Semi-dualizing modules Abstract: A finitely generated module C over a Noetherian ring R is semidualizing if the natural homothety homomorphism R \to Hom_R(C,C) is bijective and Ext_R^i(C,C)=0 for each i>0. The set of isomorphism classes of such modules is denoted S_0(R). We will describe basic properties of these modules, our motivation for their study, some open questions about the structure of S_0(R), and some recent progress on these questions.
30th September (Thursday). Griff Elder, University of Nebraska-Omaha. Galois Structure, Ramification and Truncated Exponentiation Abstract: This talk will begin with an introduction to additive Galois structure in local number fields, a topic which lies on the boundary between Number Theory and Integral Representation Theory. After observing that additive Galois structure is dependent upon Hilbert's ramification filtration, we will specialize to fully ramified $p$-extensions of local number fields (the totally wild extensions). And then we will specialize further to elementary abelian extensions. Here Hilbert's ramification filtration does not provide us with a rich enough source of invariants (to determine additive Galois structure), and we are forced to look for a richer source. We find that truncated exponentiation (a particular truncation of the binomial series) allows the residue field to act upon the elementary abelian Galois group. The result is an interesting generalization of the group and a new ramification filtration. The new breaks are necessary (and sometimes sufficient) for additive Galois structure. This is joint work with Nigel Byott.
6th October (Wednesday). Lars Chirstensen, UNL. Stability of G-dimensions Abstract: Auslander's 1966 definition of the so-called G-dimension for finitely generated modules over a commutative, noetherian ring marked the start of the study of Gorenstein homological dimensions. In commutative algebra, these homological invariants are used to characterize Gorenstein rings in much the same way the usual homological dimensions characterize regular rings. This talk will center around some questions on Gorenstein dimensions, I have been working on recently. It will only be a survey; the technical details will be deferred to a later occasion, but I will give some tangible examples to shed light on these questions.
7th October (Thursday). Janet Striuli, Universit of Kansas. Short exact sequences Abstract: We prove a main theorem which gives a property of the short exact sequences in $I\Ext^{1}_{R}(M,N)$. As a corollary, we will give a extension of Miyata's theorem, which characterizes when a short exact sequence is split exact. In particular, we will show that given two short exact sequences $\alpha$ and $\beta$ in $\Ext^{1}_{R}(M,N)$, with isomorphic middle modules, if $\alpha \in I\Ext^{1}_{R}(M,N)$ then tensoring by R/I makes $\beta$ split exact. We will give some applications of the techniques used in proving the main theorem. In particular we will study the structure of $\Ext^{1}(M,N)$, when $R$ is a ring of finite Cohen-Macaulay type and $M$ and $N$ are maximal Cohen-Macaulay modules.In this setting, we will be able to give a simple proof of the fact that $\Ext^{1}_{R}(M,N)$ is a module of finite length .The peculiarities of such $\Ext$ modules will lead to the definition of sparse modules, which have a finite number of submodules of the form $xM$ for $x$ being a non-zero-divisor in the maximal ideal of a local ring. We will give some properties of sparse modules.
13th and 14th October. Greg Piepmeyer, UNL. Making complexes act like cycles Abstract: These talks will discuss, primarily via example, how one may construct a perfect complex with homologies of finite length which acts on a Chow group in the same manner as a cycle in the kernel of the hyperplane section on the Chow group does. These perfect complexes can be compressed into modules of finite length and finite projective dimension when the base ring is Cohen--Macaulay; in particular, examples like the Dutta--Hochster--McLaughlin and Miller--Singh examples can be constructed. The construction has geometric motivation.
27th October (Wednesday). Part II of a working seminar based on Modules of finite length and finite projective dimension, by Paul Roberts and V. Srinivas.
28th October (Thursday). Neil Epstein. University of Kansas. T. B. A.
11th November (Thursday). Alex Martsinkovsky, Northeastern University. T. B. A.
Maintained by Srikanth Iyengar