Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension
3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and
Eisenbud proved that there is a skew-symmetric matrix of odd size such
that $I$ is generated by the sub-maximal pfaffians of this matrix. Let
$J$ be the ideal obtained by multiplying some of the pfaffian
generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of
$I$. In this paper we construct an explicit free resolution of $R/J$
with a DG algebra structure. Our work builds upon a recent paper of
Vandebogert. We use our DG algebra resolution to prove that recent
conjectures of Christensen, Veliche and Weyman on ideals of class
$\mathbf{G}$ hold true in our context.
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In the 1980's, Greene introduced $_nF_{n-1}$ $\text{hypergeometric
functions}$ over finite fields using normalized $\text{Jacobi
sums}$. The structure of his theory provides that these functions
possess many properties that are analogous to those of the classical
hypergeometric series studied by Gauss, Kummer and others. These
functions have played important roles in the study of Apery-style
supercongruences, the Eichler-Selberg trace formula, Galois
representations, and zeta-functions of arithmetic varieties. In this
talk we discuss the value distributions of simplest families of these
functions. For the $_2F_1$ functions, the limiting distribution is
semicircular, whereas the distribution for the $_3F_2$ functions is
$\text{Batman}$ distribution. This is a joint work with Ken Ono and Hasan
Saad.
The cotangent complex is an important but difficult to understand
object in commutative algebra. For a homomorphism $\varphi: R\to S$ of
commutative noetherian rings, this is a complex $L_{\varphi} =
L_0\rightarrow L_1\rightarrow \cdots $ of free $S$-modules. To start
with I'll connect this back to some more familiar commutative algebra
invariants by explaining how you can see the module of differentials,
the conormal module, and the Koszul homology as syzygies inside
$L_{\varphi}$.
When the cotangent complex was introduced by Quillen, he conjectured
(for maps of finite flat dimension) that if $\varphi$ is not complete
intersection then $L_{\varphi}$ must go on forever. This was proven by
Avramov in 1999. I will explain how to get a new proof (of a stronger
result) by paying attention to the $syzygies of L_{\varphi}$. This is
all joint work with Srikanth Iyengar.
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Let $\mathfrak{p}$ be a prime ideal in a commutative noetherian ring
$R$. We show that if an $R$-module $M$ satisfies
$\mathrm{Ext}_R^{n+1}(k(\mathfrak{p}),M)=0$ for some
$n\geq\mathrm{dim}\;R$, where $k(\mathfrak{p})$ is the residue field
at $\mathfrak{p}$, then $\mathrm{Ext}^i_R(k(\mathfrak{p}),M)=0$ holds
for all $i>n$. This is an improvement of a result of Christensen,
Iyengar and Marley. Similar improvements concerning homological
dimensions and the rigidity of Tors are proved. The main tool that we
use to provide these improvements is the existence of minimal
semi-flat-cotorsion replacements.
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It is well-known in number theory that some of the deepest results
come in connecting complex analysis in the form of $L$-functions with
algebra/geometry in the form of Galois representations/motives. In
this talk we will consider this for a particular case. Let $f$ be a
newform of weight $k$ and full level. Associated to $f$ one has the
adjoint Galois representation and the symmetric square $L$-function.
The Bloch-Kato conjecture predicts a precise relationship between
special values of the symmetric square $L$-function of $f$ with size
of the Selmer groups of twists of the adjoint Galois representation.
We will outline a result providing evidence for this conjecture by
lifting $f$ to a Klingen Eisenstein series and producing a congruence
between the Klingen Eisenstein series and a Siegel cusp form with
irreducible Galois representation. This is joint work with Kris
Klosin.
Differential modules are modules equipped with the additional data of
a square-zero endomorphism. As such, they form a natural
generalization of complexes and have recently seen a more focused
study for the novel perspective they provide on older problems in
commutative algebra, algebraic geometry, and representation theory. In
full generality, the theory of differential modules (and their
resolutions) can stray wildly from the classical case, but recent work
of Brown and Erman has shown that the classical theory of minimal free
resolutions still plays an important role in understanding the
homological properties of differential modules. In this talk, we will
explore differential modules whose homology is generated by a regular
sequence, comparing and contrasting the classical theory of minimal
free resolutions of complete intersections (which is quite simple)
with the analogous generalization to differential modules (which turns
out to be much more subtle). This work is joint with Maya Banks.