Algebra and Number Theory
Department of Mathematics and Statistics
Texas Tech University
Using homological algebra, Serre defined and studied an intersection multiplicity for finitely generated modules over a regular local ring. He did this by using the Euler characteristic, and showed that it satisfied many properties one would expect with an intersection theory. The goal of this talk is to give an introduction to the partial Euler characteristics and share a result about an Ext analog of the partial Euler characteristics.
I this talk we will discuss the notion of Weak and Strong Lefschetz
properties for graded ring. We will then connect these properties to
the study of strongly stable ideals and generic initial
ideals. Finally, we will discuss almost complete intersection and
hyperplane arrangements from the point of view of Lefschetz
properties. This is partially based on a collaboration with
S. Marchesi and E. Palezzato.
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Meeting ID: 944 0908 4315
Passcode: 015259
Over an algebraically closed field, the {\it double point
interpolation} problem asks for the vector space dimension of the
projective hypersurfaces of degree $d$ singular at a given set of
points. After being open for 90 years, a series of papers by
J. Alexander and A. Hirschowitz in 1992--1995 settled this question in
what is referred to as the Alexander-Hirschowitz theorem. In this
talk, we use commutative algebra to prove analogous statements in the
{\it weighted projective space}, a natural generalization of the
projective space. I will also introduce an inductive procedure for
weighted projective planes, similar to that originally due to
A. Terracini from 1915, to demonstrate the only example of a weighted
projective plane, with mild assumptions, where the analogue of the
Alexander-Hirschowitz theorem holds {\it without any
exceptions}. Furthermore, I will give interpolation bounds for an
infinite family of weighted projective planes.
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Meeting ID: 944 0908 4315
Passcode: 015259
Recall that a commutative noetherian ring $R$ is Gorenstein if
and only if every finitely generated $R$-module has finite Gorenstein
projective dimension. In papers from 2011 (Murfet and Salarian), 2017
(Estrada, Fu, and Iacob), and 2018 (Christensen and Kato) Gorenstein
rings are characterized in terms of qualitative properties of their
complexes---but always under some additional assumption on \(R\) such as
finite Krull dimension or finite projective dimension of flat
\(R\)-modules. These assumptions are, we now know, superfluous, and I
will sketch a proof of the equivalence of the following conditions:
\(R\) is Gorenstein.
Every acyclic complex of injective \(R\)-modules is totally
acyclic.
Every acyclic complex of flat \(R\)-modules is totally acyclic.
Every acyclic complex of projective \(R\)-modules is totally
acyclic.
For every acyclic complex \(P\) of projective \(R\)-modules the complex
\(\mathrm{Hom}_R(P,R)\) is acyclic.
For every acyclic complex \(P\) of projective \(R\)-modules and every
injective \(R\)-module \(E\) the complex \(E \otimes_R P\) is acyclic.
A differential graded (DG) algebra is the confluence of a graded
associative algebra and a complex. An example of a prototypical
DG-algebra is the Koszul complex. The algebra structure of the Koszul
complex is inherited by its homology, making both the Koszul complex
and its homology an incredibly useful tool in algebra. There is much
that can be learned about the properties of a ring from its resulting
DG-algebra resolution. In this talk we will examine a spectrum of
properties that can be gleaned from a DG-algebra resolution,
specifically the Koszul complex. We will use this to create a basic
framework for the purpose of examining the relationship between a ring
and the algebra structure of its resulting DG-algebra resolution.
Follow the talk via this Zoom link
Meeting ID: 944 0908 4315
Passcode: 015259