半对偶模定义的相对Ext群的平衡性
The balance of relative Ext groups defined by semi dualizing modules
摘要：
设$R$是有单位元的交换诺特环，$C$是$R$的半对偶模.设$\mathscr{P}_C(R)$和$\mathscr{I}_C (R)$分别表示$C$ -投射和$C$ -内射$R$ -模类.我们证明了它们的导出Ext双函子$\text{Ext}^i_{\mathscr{P}_C}(-,\sim)$和$\text{Ext}^i_{\mathscr{I}_C}(-,\sim)$对所有$i\geq 0$都是一致的当且仅当$C$是投射的.同时，利用一些特殊的余挠理论，给出了$C$是射影的其它一些判据.
英文摘要：
Let $R$ be a commutative Noetherian ring with identity and $C$ a semidualizing module for $R$. Let $\mathscr{P}_C(R)$ and $\mathscr{I}_C (R)$ denote, respectively, the classes of $C$-projective and $C$-injective $R$-modules. We show that their induced Ext bifunctors $\text{Ext}^i_{\mathscr{P}_C}(-,\sim)$ and $\text{Ext}^i_{\mathscr{I}_C}(-,\sim)$ coincide for all $i\geq 0$ if and only if $C$ is projective. Also, we provide some other criteria for $C$ to be projective by using some special cotorsion theories.
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