Let \$(R,rak{m})\$ be a standard graded \$K\$-algebra over a field \$K\$. Then \$R\$ can be written as \$S/I\$, where \$Isubseteq(x_1,ldots,x_n)^2\$ is a graded ideal of a polynomial ring \$S=K[x_1,ldots,x_n]\$. Assume that \$ngeq3\$ and \$I\$ is a strongly stable monomial ideal. We study the symmetric algebra \${ m Sym}_R({ m Syz}_1(rak{m}))\$ of the first syzygy module \${ m Syz}_1(rak{m})\$ of \$rak{m}\$. When the minimal generators of \$I\$ are all of degree 2, the dimension of \${ m Sym}_R({ m Syz}_1(rak{m}))\$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

### On the symmetric algebra of certain first syzygy modules

#### Abstract

Let \$(R,rak{m})\$ be a standard graded \$K\$-algebra over a field \$K\$. Then \$R\$ can be written as \$S/I\$, where \$Isubseteq(x_1,ldots,x_n)^2\$ is a graded ideal of a polynomial ring \$S=K[x_1,ldots,x_n]\$. Assume that \$ngeq3\$ and \$I\$ is a strongly stable monomial ideal. We study the symmetric algebra \${ m Sym}_R({ m Syz}_1(rak{m}))\$ of the first syzygy module \${ m Syz}_1(rak{m})\$ of \$rak{m}\$. When the minimal generators of \$I\$ are all of degree 2, the dimension of \${ m Sym}_R({ m Syz}_1(rak{m}))\$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11570/3212166`
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