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Rellek
Rellek
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Let $A = k[x_1 , \dots , x_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x_1 , \dots , x_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset of variables $(x_{i_1} , \dots , x_{i_\ell})$ also has a linear minimal free resolution?

The answer seems to be yes. Indeed, it seems like the resolution of the subset of variables is obtained as a direct summand of the Priddy (generalized Koszul) complex, which is acyclic by the Koszul assumption on $A$. Probably this subcomplex is realized as a Tate construction, and I was looking for a reference (or a quick proof/counterexample) of the question in the title.

Edit: As provided by Hailong, there is indeed a counterexample to the above claim. I would like to mention that there is a notion of "strongly Koszul" introduced by Herzog, Hibi, and Restuccia. This imposes the additional assumption that all colon ideals of the form $(x_{i_1} , \dots , x_{i_k} ) : x_{i_{k+1}}$ (for $i_1 < \cdots < i_{k+1}$) are generated by a subset of the variables. It turns out that being strongly Koszul does imply that the ideal generated by any subset of the variables has linear resolution.

Let $A = k[x_1 , \dots , x_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x_1 , \dots , x_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset of variables $(x_{i_1} , \dots , x_{i_\ell})$ also has a linear minimal free resolution?

The answer seems to be yes. Indeed, it seems like the resolution of the subset of variables is obtained as a direct summand of the Priddy (generalized Koszul) complex, which is acyclic by the Koszul assumption on $A$. Probably this subcomplex is realized as a Tate construction, and I was looking for a reference (or a quick proof/counterexample) of the question in the title.

Let $A = k[x_1 , \dots , x_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x_1 , \dots , x_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset of variables $(x_{i_1} , \dots , x_{i_\ell})$ also has a linear minimal free resolution?

The answer seems to be yes. Indeed, it seems like the resolution of the subset of variables is obtained as a direct summand of the Priddy (generalized Koszul) complex, which is acyclic by the Koszul assumption on $A$. Probably this subcomplex is realized as a Tate construction, and I was looking for a reference (or a quick proof/counterexample) of the question in the title.

Edit: As provided by Hailong, there is indeed a counterexample to the above claim. I would like to mention that there is a notion of "strongly Koszul" introduced by Herzog, Hibi, and Restuccia. This imposes the additional assumption that all colon ideals of the form $(x_{i_1} , \dots , x_{i_k} ) : x_{i_{k+1}}$ (for $i_1 < \cdots < i_{k+1}$) are generated by a subset of the variables. It turns out that being strongly Koszul does imply that the ideal generated by any subset of the variables has linear resolution.

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Rellek
Rellek
  • 553
  • 2
  • 9

In a commutative Koszul algebra, does every ideal generated by a subset of variables have linear resolution?

Let $A = k[x_1 , \dots , x_n] / I$ be a commutative Koszul algebra; that is, the ideal $(x_1 , \dots , x_n)$ has linear minimal free resolution. Does it follow that the ideal generated by any subset of variables $(x_{i_1} , \dots , x_{i_\ell})$ also has a linear minimal free resolution?

The answer seems to be yes. Indeed, it seems like the resolution of the subset of variables is obtained as a direct summand of the Priddy (generalized Koszul) complex, which is acyclic by the Koszul assumption on $A$. Probably this subcomplex is realized as a Tate construction, and I was looking for a reference (or a quick proof/counterexample) of the question in the title.