Department of Mathematical Sciences
Florida Atlantic University
Boca Raton, Florida 33431-0991

Boca Raton Algebra Seminar

Fall 2000

All lectures take place in Science and Engineering, room 215.  Everybody is invited to participate.



Tuesday, November 28, 1:00 p.m. SE 215

Tameness of Unsplit Dedekind-like Rings

Part IV: Examples

Lee Klingler

See the abstract for the November 7 lecture.



Tuesday, November 21, 1:00 p.m. SE 215
 

Tameness of Unsplit Dedekind-like Rings, Part III

Lee Klingler

See the abstract for the November 7 lecture.



Tuesday, November 14, 1:00 p.m. SE 215

Tameness of Unsplit Dedekind-like Rings, Part II

Lee Klingler

This is a continuation of the November 7 lecture.



Tuesday, November 7, 1:00 p.m. SE 215
 

Tameness of Unsplit Dedekind-like Rings

Lee Klingler

Finitely generated modules over split Dedekind-like rings can be classified as either strings or bands, just as for finitely generated modules over certain finite-dimensional algebras (the string algebras).

In this talk, I shall define strings and bands, and show how to use a change of scalar argument to classify modules over  unsplit Dedekind-like rings (provided a certain separability condition is satisfied).  Each finitely generated module over an unsplit Dedekind-like ring will correspond to either a self-conjugate string or band, or to a pair of conjugate strings or conjugate bands.



Tuesday, October 31, 1:00 p.m., SE 215
 

Divisor Classes, the General Cubic Surface, and Generators of Fat Point Ideals, Part II

Stephanie Fitchett

This is a continuation of the previous lecture.



Tuesday, October 24, 1:00 p.m., SE 215
 

Divisor Classes, the General Cubic Surface, and Generators of Fat Point Ideals

Stephanie Fitchett

Geometrically, a fat point ideal of k[x,y,z] defines a zero-dimensional subscheme of the projective plane.  If points in the support of a fat point ideal are "blown up", some divisor classes on the resulting surface correspond to graded components of the fat point ideal.  We can use geometric tools to discern information about maps between linear systems of divisors, which ultimately gives information about generators for fat point ideals.

I'll explain the correspondence between the algebra of the ideal and the geometry of the blow-up surface, and show that for a fat point ideal supported at six of fewer general points of the projective plane, one can completely determine the structure of a minimal generating set for the ideal. The new project involves removing the qualification in the last sentence that the points be "general".



Tuesday, 17 October, 1:00, S&E 215
 

A Tame/Wild Theorem for Commutative Noetherian Rings, Part II

Lee Klingler

Our tame/wild dichotomy theorem for commutative noetherian rings is based on the following dichotomy theorem.  Let R be a complete local commutative noetherian ring.  Then either R has, as homomorphic image, one of two types of "minimal" wild rings (which we call artinian triads and Drozd rings), or R is a homomorphic image of one of two types of "maximal" tame rings (which we call Klein rings and Dedekind-like rings).  In this talk I shall give precise definitions of these four types of rings and sketch a proof of the dichotomy theorem.



Tuesday, 10 October, 1:00, S&E 299
 

A Tame/Wild Theorem for Commutative Noetherian Rings

Lee Klingler

Our tame/wild dichotomy theorem for commutative noetherian rings is based on the following dichotomy theorem.  Let R be a complete local commutative noetherian ring.  Then either R has, as homomorphic image, one of two types of "minimal" wild rings (which we call artinian triads and Drozd rings), or R is a homomorphic image of one of two types of "maximal" tame rings (which we call Klein rings and Dedekind-like rings).  In this talk I shall give precise definitions of these four types of rings and sketch a proof of the dichotomy theorem.



Tuesday, October 3, 1 p.m.

Submodules of Finitely Generated k[T]/T^6 - Modules,  Part IV

Markus Schmidmeier

We are going to recover the (known) submodules of  k[T]/T^5 - modules. Using our techniques from the previous lectures it will be an easy task to describe first the corresponding representations of the double  infinite quiver.  Then we will adopt a result from covering theory to obtain the corresponding list of submodules of  k[T]/T^n - modules, for  n=5 .  This result also works well for  n=6 , completing our construction of the submodules of finite length  k[T]/T^6 - modules.



Tuesday, September 26, 1 p.m.
 

Submodules of Finitely Generated k[T]/T^6-Modules

Part III:  The Boundary Modules

Markus Schmidmeier

In this talk we study in detail certain submodules of finitely geneated  k[T]/T^n - modules, the socalled Boundary Modules, which are related to the projective  and the injective submodules.  We will use them, in case  n>6, to construct indecomposable representations of arbitrary large support.



Tuesday, 19 September, 1 p.m.
 

Submodules of Finitely Generated k[T]/T^6-Modules, Part II

Markus Schmidmeier
This lecture is a continuation of --- but independent of ---  last weeks Algebra Seminar talk on joint work with C. M. Ringel.

We will recall the representation theory of the algebra given by the quiver
 

                            a      a
                       2'  <-- 3' <--   4'

                        |       |       |
                        | b     | b     | b
                        v       v       v

        0  <--  1  <--  2  <--  3  <--  4 <--  5  <--  6
            c       c       c       c       c       c
 

with relations  ba = cb  and  c^6 = 0 .

As a consequence we obtain a full list of the indecomposable representations of the quiver in the abstract of the previous talk.

This gives us the desired list of submodules of finitely generated  k[T]/T^6-modules (up to a result from covering theory).


Tuesday, 5 September, 1:30, S&E 215

  Submodules of Finitely Generated  k[T]/T^6 - Modules

 Markus Schmidmeier

This will be the start of a series of talks on this topic.

Let  R  be a uniserial ring, e. g.  R = k[T]/T^n  or  R = Z/p^n , Z  the integers,  p  a prime number and  n  a positive  nteger. We denote by  S(R)  the category of pairs  (M,U)  where  M  is a finitely generated  R-module and  U  a submodule of  M ;  a map  f: (M,U) ->  (M',U')  in  S(R)  is just an  R-linear map  f: M -> M' such that f(U)  is contained in  U' .

The case of  R = Z/p^n  has attracted a lot of interest since the categories  S(Z/p^n)  describe the possible subgroups of finite  p^n  -bounded abelian groups.  Recall that the classification of the indecomposable objects in  S(Z/p^5)  by F. Richman and E. A. Walker has been presented to this Algebra Seminar last fall.

In this series of talks we report on joint work with C. M. Ringel, in which we consider the corresponding case  R = k[T]/T^n . We are going to show that  S(k[T]/T^n)  has finitely many indecomposables for  n <= 5 , is tame for  n = 6  and wild for  n >= 7 . In particular, we present a complete classification of the indecomposable objects in S(k[T]/T^n)  for  n <= 6 .

In the first talk we are going to study the finite dimensional representations of the double infinite quiver
 

             a_0       a_1       a_2        a_3        a_4
        ...  <----  0' <----  1' <----   2' <----   3' <----  ...

                    |         |          |          |
                    | b_0     | b_1      | b_2      | b_3
                    v         v          v          v

        ...  <----  0  <----  1  <----   2  <----   3  <----  ...
              c_0       c_1       c_2        c_3        c_4
 

which satisfy the commutativity relations  ba = cb , the nilpotence relations  a^6 = 0 = c^6  (we omit the indices of the a_i, b_i, c_i ), and for which the maps  b_i  are monomorphisms.


For nostalgic reasons you can consult the Previous Programs of this seminar:  Fall 1999Spring 2000, Current Seminar



Last modified: November 14, 2000, by Markus Schmidmeier