Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
Levels are a numerical invariant of complexes that are defined by
using the triangulated structure of the derived category. Levels with
respect to a ring have been studied extensively, and have been used to
show that a commutative noetherian ring R is regular if and only if
there is an upper bound on the R-level of perfect R-complexes. In this
talk we discuss a Gorenstein version of this fact. This is joint work
with Lars Winther Christensen, Antonia Kekkou, and Justin Lyle.
 | Wednesday
Sep. 3
|
| No Seminar
Please attend the departmental colloquium
|
| Wednesday
Sep. 10
|
| No Seminar
Please attend the departmental colloquium
|
The Kauffman bracket skein module of a lens space was determined by
Hoste and Przytycki. It can be obtained as the tensor product of two
skein modules of solid tori over the algebra of the cylinder over the
torus. We examine this tensor product structure in detail.
Let \(R\) be a commutative noetherian ring. The G-level of an
\(R\)-complex with bounded and degreewise finitely generated homology
counts the number of mapping cone constructions it takes to build the
complex from the collection of finitely generated Gorenstein
projective \(R\)-modules. We prove that if \(d+1\) is an upper bound for
the $G$-level of perfect \(R\)-complexes, then \(R\) is Gorenstein of
Krull dimension at most $d$. Further, for a Gorenstein ring of Krull
dimension \(d\), we show that the G-level of an \(R\)-complex with bounded
and degreewise finitely generated homology is at most
\(\max\{2,d + 1\}\). This improves the bound of \(2(d + 1)\)
obtained by Awadalla and Marley a few years ago and aligns with the
bound on \(R\)-levels in case \(R\) is regular. The talk is based on joint
work with Kekkou, Lyle, and Soto Levins.
| Wednesday
Oct. 29
|
| No Seminar
Please attend the departmental colloquium
|
| Wednesday
Dec. 3
3:00 PM
MATH 012
|
| TBA
Ryan Watson
Department of Mathematics, University of Nebraska-Lincoln
|