On the symmetric algebra of certain first syzygy modules

Restuccia, Gaetana, Tang, Zhongming, and Utano, Rosanna. "On the symmetric algebra of certain first syzygy modules." Czechoslovak Mathematical Journal 72.2 (2022): 391-409. <http://eudml.org/doc/298317>.

@article{Restuccia2022,
abstract = {Let $(R,\mathfrak \{m\})$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $I\subseteq (x_1,\ldots ,x_n)^2$ is a graded ideal of a polynomial ring $S=K[x_1,\ldots ,x_n]$. Assume that $n\ge 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra $\{\rm Sym\}_R(\{\rm Syz\}_1(\mathfrak \{m\}))$ of the first syzygy module $\{\rm Syz\}_1(\mathfrak \{m\})$ of $\mathfrak \{m\}$. When the minimal generators of $I$ are all of degree 2, the dimension of $\{\rm Sym\}_R(\{\rm Syz\}_1(\mathfrak \{m\}))$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.},
author = {Restuccia, Gaetana, Tang, Zhongming, Utano, Rosanna},
journal = {Czechoslovak Mathematical Journal},
keywords = {symmetric algebra; syzygy; dimension; depth},
language = {eng},
number = {2},
pages = {391-409},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the symmetric algebra of certain first syzygy modules},
url = {http://eudml.org/doc/298317},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Restuccia, Gaetana
AU - Tang, Zhongming
AU - Utano, Rosanna
TI - On the symmetric algebra of certain first syzygy modules
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 2
SP - 391
EP - 409
AB - Let $(R,\mathfrak {m})$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $I\subseteq (x_1,\ldots ,x_n)^2$ is a graded ideal of a polynomial ring $S=K[x_1,\ldots ,x_n]$. Assume that $n\ge 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\rm Sym}_R({\rm Syz}_1(\mathfrak {m}))$ of the first syzygy module ${\rm Syz}_1(\mathfrak {m})$ of $\mathfrak {m}$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\rm Sym}_R({\rm Syz}_1(\mathfrak {m}))$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.
LA - eng
KW - symmetric algebra; syzygy; dimension; depth
UR - http://eudml.org/doc/298317
ER -