Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
While proving a Lucas sequence contains infinitely many primes is an
ongoing problem, Mark Broderius and John Greene explore two families
of Lucas sequences containing finitely many primes and provide a
conjecture classifying all Lucas sequences that contain infinitely
many primes.
Let \(R\) be a commutative Noetherian local ring of prime characteristic
\(p\). For an nonnegative integer \(e\), let \(R^e\) be the ring \(R\) viewed as a
module over itself via the \(e\)th iteration of the Frobenius
endomorphism; i.e., for \(r\in R\) and \(s\in R^e\), \(r\cdot s := r^{p^e}s\). In
the early 1970s, Peskine and Szpiro proved that for a finitely
generated module \(M\) of finite projective dimension over \(R\),
\(\mathrm{Tor}_i^R(R^e,M)=0\) for all \(i,e>0\). Shortly afterward, Herzog proved the
converse. Subsequently, several authors established stronger converse
statements under certain hypotheses, the strongest being in the case \(R\)
is a complete intersection. In this situation, Avramov and Miller
(2001) proved that for a finitely generated module \(M\), if
\(\mathrm{Tor}_i^R(R^e,M)=0\) for some positive \(i\) and \(e\), then \(M\) has finite
projective dimension. Whether or not that statement holds over an
arbitrary local ring is (to the best of my knowledge) an open
question. In this talk, I will discuss the history of this problem as
well as some recent progress in the case \(R\) is Cohen-Macaulay and also
when the module \(M\) is not assumed to be finitely generated.
The derived category \(D(A)\) of a ring \(A\) is an important example of a
triangulated category in algebra; indeed, this category is the right
setting for the study of complexes "up to homology".
A complex can be viewed as a representation of a certain quiver with
relations, \(Q^{cpx}\). The vertices in this quiver are the integers,
there is an arrow \(q \to q-1\) for each integer \(q\), and the relations are
that consecutive arrows compose to 0. Hence the (classic) derived
category \(D(A)\) can be viewed as a category of representations of
\(Q^{cpx}\).
It is an insight of Iyama and Minamoto that the reason \(D(A)\) is well
behaved is that \(Q^{cpx}\) has a so-called Serre functor. We prove that
if \(Q\) is any quiver with relations that has a Serre functor, then the
category of \(A\)-module valued representations of \(Q\) has a "derived
category" \(D_Q(A)\), which we call the \(Q\)-shaped derived category of the
ring \(A\). For \(Q\) equal to \(Q^{cpx}\) the category \(D_Q(A)\) coincides with the
classic derived category \(D(A)\). In the talk we will demonstrate that
the \(Q\)-shaped derived category \(D_Q(A)\) has many similarities with the
classic derived category \(D(A)\) and the theory will be illustrated by
several concrete examples.
Follow the talk via this Zoom link
Meeting ID: 979 3375 1447
Passcode: 136297
Ideals in the algebra of power series in three variables can be
classified based on algebra structures on their minimal free
resolutions. The classification is incomplete in the sense that it
remains open which algebra structures actually occur; this
realizability question was formally raised by Avramov in 2012. We
discuss the outcomes of an experiment performed to shed light on this
question: Using the computer algebra system Macaulay2, we classify a
billion randomly generated ideals and build a database with examples
of ideals of all classes realized in the experiment. Based on the
outcomes, we discuss the status of recent conjectures that relate to
the realizability question.
Staged tree models are discrete statistical models, generalizing
Bayesian networks, and are described by algebraic varieties, that is
the zero set of a collection of polynomials. A particularly nice class
of varieties for computational purposes are the toric varieties. If
the staged tree model variety is toric, it offers a Markov basis,
helpful for hypothesis testing, and facilitates the study of maximum
likelihood estimates. In 2021 G\"orgen, Maraj, and Nicklasson showed
that the variety for staged tree models with one color having at most
3 children at each vertex, and depth at most 3, are toric. We showed
that their conjecture fails in depth 4, using methods from geometry
and representation theory. In this talk I'm going to introduce staged
tree models, the associated variety to the model, define and explain
how to compute its symmetry Lie algebra, and will introduce an
algorithmic way to prove or disprove if certain ideals can be
generated by binomials. This is a joint work with Aida Maraj.
Two classical problems in commutative algebra are the classification
of the (perfect) ideals of a (regular local) ring in terms of their
minimal free resolutions, and the description of their linkage
classes. In particular one would like to describe all the ideals in
the linkage class of a complete intersection. In this seminar we deal
with these two problems for perfect ideals of codimension 3. The main
tools that we use are sequences of linear maps that can be defined
over an exact complex of length 3 and that generalize the well-known
multiplicative structure (Joint work with Xianglong Ni and Jerzy
Weyman).