# Morgantown Algebra Days

## Speakers

### Jürgen Herzog

University of Duisburg-Essen, Germany

### Haydee Lindo

Harvey Mudd College, U.S.
Personal Website

### Linquan Ma

Purdue University, U.S.
Personal Website

### Sarasij Maitra

University of Utah, U.S.
Sarasij Maitra Personal Website

### Takumi Murayama

Purdue University, U.S.
Personal Website

### Rebecca R.G.

George Mason University, U.S.
Personal Website

### Liana Şega

University of Missouri-Kansas City, U.S.
Personal Website

## TALK TITLES AND ABSTRACTS

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

### JÜrgen herzog

#### Rings of Teter type and the trace of the canonical module and the Teter number of determinantal rings

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 {\em 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.

Coming soon...

### Linquan Ma

#### Vanishing and non-negativity of the first normal Hilbert coefficient

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.

Coming soon...

Coming soon...

Coming soon...

Coming soon...