Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
In this presentation we will talk about three problems and how these
are related to the Grassmann variety \(G_{n,m}\). The first problem
concerns the intersection of \(_{n,m}\) with a linear variety and the
corresponding low-rank decomposition problem. The second problem is
derived from the first one and studies separately the degree of
mappings connected to \(G_{n,m}\). The third problem is about dual and
reflexive projective varieties defined over fields of positive
characteristic. The Grassmann variety is used in the Gauss map for
determining duality and reflexivity criteria, where we will see how
to construct a theory where biduality and reflexity hold in finite
characteristic as well.
There are similar characterizations in terms of complexes of flat
modules and complexes of injective modules.
Follow the talk via this Zoom link
Meeting ID: 910 1221 3761
Passcode: 095533
In this talk, we will go through what closure and interior
operations are and some ways they are used, what that has to do with
inner product spaces, and how that leads us to consider pair
operations. Expanding on the work of Kemp, Ratliff and Shah, for any
closure cl defined on a class of modules over a Noetherian ring, we
develop the theory of cl-prereductions of submodules and the dual
notion of i-postexpansions. Pair operations can be endowed with
specific properties: we will consider how they behave under Matlis
duality, how combining them can force the operation to become the
constant or the identity operation, and how pair operations can be
constructed in a multitude of ways to preserve properties.
There are similar characterizations in terms of complexes of flat
modules and complexes of injective modules.
Follow the talk via this Zoom link
Meeting ID: 913 0074 4693
Passcode: 586188
Topology furnishes us with many commutative rings associated to finite
groups. These include the complex representation ring, the Burnside
ring, and the G-equivariant K-theory of a space. Often, these admit
additional structure in the form of natural operations on the ring,
such as power operations, symmetric powers, and Adams operations. We
will discuss two ways of constructing Adams operations. The goal of
the talk is to understand these in the case of the Burnside ring.
There are similar characterizations in terms of complexes of flat
modules and complexes of injective modules.
Follow the talk via this Zoom link
Meeting ID: 910 1221 3761
Passcode: 095533
In a paper from 1968, Golod proved that the Betti sequence of the
residue field of a local ring attains the upper bound given by Serre
if and only if the homology algebra of the Koszul complex of the ring
has trivial multiplications and trivial Massey operations. This is the
origin of the notion of Golod ring. Using the Koszul complex
components as building blocks Golod also constructed a minimal free
resolution of the residue field of a Golod ring. With Van Nguyen, we
extend this construction for an arbitrary local ring, up to
homological degree five, and explicitly show how the multiplicative
structure of the homology of the Koszul algebra is involved, including
the triple Massey products. The talk will illustrate this construction
and various consequences of it.
There are similar characterizations in terms of complexes of flat
modules and complexes of injective modules.
Tor-persistence is the claim that Tor of a module with itself is only
zero if the module has finite projective dimension. Work done by
Avramov, Iyengar, Nasseh, Sather-Wagstaff, and various other authors
have proven modules over certain rings are Tor-persistent. In this
work, we study Tor-persistence for the generic module over
determinantal rings, specifically for the hypersurface R defined by
the determinant of a generic matrix. We give an explicit proof that
$Tor^R_2(M,M)$ is never zero, where M is the cokernel of the generic
matrix. We will also discuss the more general determinantal rings.
Every module over a Noetherian ring \(R\) has a minimal injective
resolution that is unique up to isomorphism. Since every injective
module can be written uniquely (up to isomorphism) as a direct sum of
indecomposable injective modules, the number of summands isomorphic to
the injective hull of \(R/\mathfrak{q}\) for a given prime
\(\mathfrak{q}\) that appears in i-th stage of such a resolution is
well defined. This number is called the i-th Bass number of the module
with respect to the prime ideal \(\mathfrak{q}\). By work of Huneke
and Sharp and of Lyubeznik, the Bass numbers of the various local
cohomology modules of the ring are all finite provided \(R\) is a regular
local ring containing a field. In this talk we compute some of the
Bass number of the first nonzero local cohomology module of \(R\).