A note on cohomological dimension over Cohen-Macaulay rings
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Let $(R,\m)$ be a Noetherian local Cohen-Macaulay ring and $I$ be a proper ideal of $R$. Assume that $\beta_R(I,R)$ denotes the constant value of ${\rm depth}_R(R/I^{n})$ for $n\gg0$. In this paper we introduce the new notion $\gamma_R(I,R)$ and then we prove the following inequalities: $$\beta_R(I,R)\leq\gamma_R(I,R)\leq \dim R - \cd(I,R)\leq \dim R/I.$$ Also, some applications of these inequalities will be included.
Abstract
Let $(R,\m)$ be a Noetherian local Cohen-Macaulay ring and $I$ be a proper ideal of $R$. Assume that $\beta_R(I,R)$ denotes the constant value of ${\rm depth}_R(R/I^{n})$ for $n\gg0$. In this paper we introduce the new notion $\gamma_R(I,R)$ and then we prove the following inequalities: $$\beta_R(I,R)\leq\gamma_R(I,R)\leq \dim R - \cd(I,R)\leq \dim R/I.$$ Also, some applications of these inequalities will be included.
참고문헌 (15)
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