# Morgantown Algebra Days 2023

April 22-23, 2023

April 22-23, 2023

Harvey Mudd College, U.S.

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Purdue University, U.S.

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University of Utah, U.S.

Sarasij Maitra Personal Website

Sarasij Maitra Personal Website

Purdue University, U.S.

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George Mason University, U.S.

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University of Missouri-Kansas City, U.S.

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Read MoreTalks will be at G15 Life Sciences Building (First Floor).
View the Full Schedule.

A 0-dimensional local ring $(R, \mathfrak{m})$ is called a *Teter ring*, if there exists a local Gorenstein ring $G$ such that $R\simeq
G/(0:\mathfrak{m}_G)$. This class of rings was introduced in 1974 by William Teter.
It has been shown by Huneke, Vraciu and others that $R$ is a Teter ring if and
only if there exists an epimorphism $\varphi: \omega_R\to \mathfrak{m}$. We call
$R$ of *Teter type*, if there exists an epimorphism $\omega_R\to \mbox{tr}(\omega_R)$,
and we define the *Teter number *of $R$ to be the smallest number $s$ for which
there exist $R$-module homomorphisms $\varphi_i: \omega_R\to R$ such that $\mbox{tr}(\omega_R)=\sum_{i=1}^s\varphi_i(\omega_R)$.

Various characterizations of rings of Teter type are given and families of rings of Teter type are presented. We determine the canonical trace of generic determinantal rings and compute its Teter numbers. As an application we determine the canonical trace of a Cohen-Macaulay ring $R$ of codimension two, which is generically Gorenstein. It is shown that if the defining ideal $I$ of $R$ is generated by $n$ elements, then $\mbox{tr}(\omega_R)$ is generated by the (n-2)-minors of the Hilbert-Burch matrix of $I$.

This is joint work with Gasanova, Hibi and Moradi, and with Ficarra, Stamate and Trivedi.

Arf rings originate from Cahit Arf’s classification of certain singular points of plane curves and such rings have a well-established history within singularity theory. Lipman classically characterized Arf rings as one-dimensional Cohen-Macaulay rings in which every integrally closed ideal is stable. In this talk we will explore the intersection of trace ideals and stable ideals, that is, ideals that are stable under homomorphisms to the ring and ideals that are isomorphic to their endomorphism rings, respectively. We apply our results to the study of Arf rings and arrive at a new characterization. Along the way, we make precise the relationship between the various notions of closure that coincide in the Arf setting: reflexive, integrally closed, trace, and stable.

This is joint work with Hailong Dao.

Let $(R,m)$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that the first normal Hilbert-coefficient $\overline{e_1}(Q)$ vanishes, then $R$ is regular and $\nu(m/Q)\leq 1$. This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal. We also give a new proof of the non-negativity of the first normal Hilbert-coefficient (which also shows the non-negativity of the first tight Hilbert-coefficient in positive characteristic).

This is joint work with Pham Hung Quy.

Let $R$ be a one dimensional analytic $k$-algebra which is a domain and where $k$ is algebraically closed of characteristic $0$. In a recent work, Huneke-Maitra-Mukundan introduced the notion of *reduced type* $s(R)$ in such a ring and used it to study a long-standing open question concerning the torsion of the module of differentials of $R$. In this work, we further study the behavior of this new invariant drawing comparisons with the Cohen-Macaulay type of $R$. We shall discuss how this study can be changed to the study of the numerical semigroup ring generated by the valuation semi-group of $R$. This enables us to utilize the rich literature on numerical semi-group rings to establish various results about the extremal behavior of $s(R)$. The far-flung Gorenstein numerical semi-group rings which were recently introduced by Herzog-Kumashiro-Stamate, pose an interesting class of numerical semi-group rings in our study.

This is joint work with Vivek Mukundan.

The containment problem asks: For a fixed ideal $I$, which symbolic powers of $I$ are contained in an ordinary power of $I$? We present a closure-theoretic proof of the theorem which says that for ideals $I$ in regular rings $R$, there is a uniform containment of symbolic powers of $I$ in ordinary powers of $I$.

Test ideals arise in the study of singularities in commutative algebra, particularly the tight closure test ideals. Trace ideals come up in a more representation-theoretic context, such as in the work of Haydee Lindo. In joint work with Felipe Pérez, I proved that trace ideals are the test ideals for closure operations defined by tensoring with a module. This makes sense of the connection between trace ideals and singularities discussed in the work of Jürgen Herzog and others. In this talk I will discuss the connection between test and trace ideals and its applications, particularly how it can be used to understand the test ideals of new closure operations in mixed characteristic.

Parts of this work are joint with Neil Epstein, Zhan Jiang, Felipe Pérez, and Janet Vassilev.

Let $R$ be a polynomial ring in $n$ variables over a field of characteristic $0$, and $f$ a degree $d$ homogeneous polynomial. The symmetric group $\mathfrak{S}_n$ acts on $f$ by permuting the variables. The principal symmetric ideal $I$ generated by $f$ is the ideal generated by the polynomials $\sigma\cdot f$, with $\sigma\in \mathfrak{S}_n$. We parametrize the coefficients of $f$ by a projective space, and we say that a generic principal ideal satisfies a property $(P)$ if there exists a Zariski open set such that $(P)$ holds whenever the coefficients of $f$ are in this set. In this talk, we describe the minimal free resolution of a generic symmetric principal ideal, when $n\gg 0$.

This is joint work with Megumi Harada and Alexandra Seceleanu.