Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
The Auslander bound of a module can be thought of as a generalization
of projective dimension. For a finitely generated module \(M\) with finite
projective dimension over a Noetherian local ring \(R\),
\(Ext_{R}^{n}(M,N)=0\) for \(n>pdim_{R}M\) and \(Ext_{R}^{pdim_{R}M}(M,N)\neq 0\)
for all finitely generated modules \(N\). We say that the Auslander bound
of \(M\) is finite if for all finitely generated modules \(N\), there exists
an integer \(b\) that only depends on \(M\) so that \(Ext_{R}^{n}(M,N)=0\) for \(n>b\)
whenever \(Ext_{R}^{n}(M,N)=0\) for \(n\gg 0\). The Auslander bound is the
least such \(b\). In this talk we give an introduction to the Auslander
bound of a module and share some related results. We will then see
that we can define an Auslander bound for complexes and extend the
module results to complexes.
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In this talk, we discuss the elements of non-abelian homological
algebra in the framework of Quillen's model category theory. We
briefly explain a direct algebraic approach to the model structure of
simplicial commutative algebras. Then we describe the construction of
the cotangent complex and Andre-Quillen (co)homology. As for
applications, we recall the characterizations of some classes of rings
and ring homomorphisms by Andre-Quillen (co)homology. Finally, we
discuss exceptional complete intersection maps and some recent results
on detecting them by bounded homological dimensions where
Andre-Quillen homology is exploited implicitly.
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Let \((\mathcal{A},\mathcal{E})\) be an exact category. We establish
basic results that allow one to identify sub(bi)functors of
\(\operatorname{Ext}_{\mathcal{E}}(-,-)\) using additivity of numerical
functions and restriction to subcategories. We also study a small
number of these new functors over commutative local rings in detail
and find a range of applications from detecting regularity to
understanding Ulrich modules. Time permitting, we also see how one can
define the notion of Ulrich Sprit rings naturally arising from a class
of such subfunctors and demonstrate some basic properties of Ulrich
Split rings.
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A semidualizing module is a generalization of Grothendieck's dualizing
module. For a local Cohen-Macaulay ring \(R\), the ring itself and its
canonical module are always realized as (trivial) semidualizing
modules. In this talk, we discuss the existence of nontrivial
semidualizing modules over numerical semigroup rings. We review the
techniques that allow us to completely classify which of these rings,
up to multiplicity 9, possess a nontrivial semidualizing module. We
will end with a construction that allows us to extend our results to
higher multiplicity. This is joint work with Ela Celikbas (West
Virginia University) and Toshinori Kobayashi (Meiji University).
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A commutative noetherian ring \(R\) is regular if every finitely
generated \(R\)-module has finite projective dimension. In a paper from
2009, Iacob and Iyengar characterize regularity of \(R\) in terms of
properties of (unbounded) \(R\)-complexes. Their proofs build on results
of Jorgensen, Krause, and Neeman on compact generation of the
homotopy categories of complexes of projective modules and complexes
of injective modules.
In the commutative case
these results can be reproved with standard homological methods from
local algebra. I will illustrate how this is done by proving that the
following conditions are equivalent for a commutative noetherian ring \(R\):
(1) \(R\) is regular.
(2) Every complex of finitely generated projective \(R\)-modules is
semi-projective.
(3) Every complex of projective \(R\)-modules is
semi-projective.
(4) Every acyclic complex of projective \(R\)-modules is contractible.
There are similar characterizations in terms of complexes of flat
modules and complexes of injective modules.
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Meeting ID: 913 0074 4693
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