Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In joint work with Iyengar we prove that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.
We discuss a homological method for transferring algebra structure on
complexes along suitably nice homotopy equivalences. As an application
which motivated this project, we discuss how to use this method to
build a concrete permutation invariant differential graded algebra
structure on a well-known resolution. This is joint work with Claudia
Miller
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The graded deviations $\varepsilon_{ij}(R)$ of a graded ring $R$
record the vector space dimensions of the graded pieces of a certain
Lie algebra attached to the minimal resolution of the quotient of $R$
by its homogeneous maximal ideal. Vanishing of deviations encodes
properties of the ring: for example, $\varepsilon_{ij}(R)= 0$ for
$i\geq3$ if and only if $R$ is complete intersection and, provided $R$
is standard graded, $\varepsilon_{ij}(R)$ whenever $i$ is not equal to
$j$ implies R is Koszul. We extend this fact by showing that if
$\varepsilon_{ij}(R)=0$ whenever $j$ and $i\geq3$, then $R$ is a
quotient of a Koszul algebra by a regular sequence. This answers a
conjecture by Ferraro.
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The Koszul Algebra structure for codepth 3 commutative local
rings has been studied significantly in recent years. Avramov
compiled a comprehensive list of properties for these structures using
their respective Poincaré and Bass series. He showed that codepth 3
local rings could be described with 5 distinct classifications;
$\textrm{C}(c)$, $\textrm{T}$,
$\textrm{B}$, $\textrm{G}(r)$, and
$\textrm{H}(p,q)$. This led to many other studies
of these structures, with $\textrm{H}(p,q)$ being
the most diverse of the given classes. Christensen, Veliche, and
Weyman determined strong restrictions for the values $p$ and $q$ could
take on with respect to the rank of the first and last free modules in
the minimal free resolution of the given ring. For this talk we
further restrict the bounds for $p$ and $q$ when the ring $R/I$ has
Koszul algebra structure $\textrm{H}(p,q)$ and $I$
is an artinian monomial ideal in $R=\Bbbk[x,y,z]$ where $\Bbbk$ is a
field. These bounds will depend only on the number of minimal
generators for the given monomial ideal and fixed values of either $p$
or $q$.
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Fekete polynomials play an important role in the study of special
values of L-functions. While their analytic properties are
well-studied in the literature, little is known about their
arithmetics. In this talk, we will discuss some surprising
arithmetical properties of these polynomials. In particular, we will
see that Fekete polynomials contain some rich information about the
class numbers of quadratic fields. This is based on joint work with
Jan Minac and Nguyen Duy Tan.
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We investigate a pair of surjective local ring maps $S_1\leftarrow
R\to S_2$ between local commutative rings and their relation to the
canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are
Tor-independent over $R$. The main result asserts a structural
connection between the homotopy Lie algebra of $S$, denoted $\pi(S)$,
in terms of those of $R,S_1$ and $S_2$, where $S=S_1\otimes_R
S_2$. Namely, $\pi(S)$ is the pullback of (restricted) Lie algebras
along the maps $\pi(S_i)\to \pi(R)$ in a wide variety cases, including
when the maps above have residual characteristic zero. Consequences to
the main theorem include structural results on Andr\'{e}--Quillen
cohomology, stable cohomology, and Tor algebras, as well as an
equality relating the Poincar\'{e} series of the common residue field
of $R,S_1,S_2$ and $S$, and that the map $R\to S$ can never be Golod.
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