Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
Regular sequences are a fundamental tool in commutative algebra. In
this talk, we introduce a notion of regular sequences in \(R\)-linear
triangulated categories, where \(R\) is a graded-commutative ring acting
centrally. As applications of this definition, we show that the length
of a regular sequence provides lower bounds on levels and on the
Rouquier dimension. This is joint work with Janina C. Letz and Marc
Stephan.
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In this talk, we discuss the almost complete intersection ring \(R\)
defined by \(n+1\) general quadrics in a polynomial ring in \(n\)
variables over a field \(\Bbbk\) and a corresponding linked Gorenstein
ring \(A\). Although these rings are not Koszul (except for some small
values of \(n\)), they have homological properties that extend those of
Koszul rings. This is joint work with Rachel Diethorn, Sema Gunturkun,
Pinar Mete, Liana Sega, Ola Sobieska, and Oana Veliche.
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Meeting ID: 937 0527 6265
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Serre defined and studied an intersection multiplicity for finitely
generated modules over a regular local ring by using the Euler
characteristic, and showed it satisfies many properties that one would
expect from an intersection theory. In this talk we discuss a new
notion of lifting modules over a noetherian local ring to a regular
local ring. We show how it can be used to prove a new case of Serre's
long standing conjecture on the positivity of the Euler
characteristic, and then provide characterizations of these liftable
modules. This is joint work with Nawaj KC, and with Benjamin Katz,
Nawaj KC, Kesavan Mohana Sundaram, and Ryan Watson.
Let \(R\) be a commutative noetherian ring. In the derived category of
\(R\), the level of a bounded \(R\)-complex \(M\) with respect to a
collection of objects \(C\) (often referred to as the \(C\)-level of
\(M\)) is the fewest number of mapping cones involving objects in
\(C\) needed to obtain \(M\). When \(C\) is a nice collection of
objects in \(D(R)\) (such as the projective modules), the \(C\)-level
of a complex can give a wealth of information about that complex and
the ring itself. For example, if \(R\) is local, then \(R\) is regular
if and only if the projective level of all bounded complexes is
finite. Recently, Christensen, Kekkou, Lyle, and Soto Levins have
found optimal upper bounds for the Gorenstein projective level of
bounded complexes with finitely generated homology. In my talk, I'll
show how to improve their result to find optimal upper bounds for the
projective, injective, flat, Gorenstein projective, Gorenstein
injective, and Gorenstein flat levels of all bounded R-complexes. As
an application of my results, I'll prove a version of the Bass Formula
for injective levels and for Gorenstein injective levels.
A wealth of homological data about a commutative noetherian local ring
is encoded in two sequences of numbers that record the ranks of
certain (co)homology modules. The generating functions for these
series are known as the Poincar\'e and Bass series of the ring. For
rings "small enough" they are known to rational functions, and I will
provide a status update on a project to describe exactly which
rational functions can be realized in this way.