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On the symmetric algebra of certain first syzygy modules
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 391-409
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Let $(R,\frak {m})$ 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 $n\geq 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\rm Sym}_R({\rm Syz}_1(\frak {m}))$ of the first syzygy module ${\rm Syz}_1(\frak {m})$ of $\frak {m}$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\rm Sym}_R({\rm Syz}_1(\frak {m}))$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.\looseness -1
Let $(R,\frak {m})$ 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 $n\geq 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\rm Sym}_R({\rm Syz}_1(\frak {m}))$ of the first syzygy module ${\rm Syz}_1(\frak {m})$ of $\frak {m}$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\rm Sym}_R({\rm Syz}_1(\frak {m}))$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.\looseness -1
DOI : 10.21136/CMJ.2021.0508-20
Classification : 13C15, 13D02
Keywords: symmetric algebra; syzygy; dimension; depth 
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Restuccia, Gaetana; Tang, Zhongming; Utano, Rosanna. On the symmetric algebra of certain first syzygy modules. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 391-409. doi: 10.21136/CMJ.2021.0508-20

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