Abstract

Let Λ be an artin algebra and let ${\mathcal{P}}_{\Lambda }^{<\infty }$ the category of finitely generated right Λ-modules of finite projective dimension. We show that ${\mathcal{P}}_{\Lambda }^{<\infty }$ is contravariantly finite in if and only if the direct sum E of the indecomposable Ext $\mathrm{mod}\hspace{0.5em}\Lambda$-injective modules in ${\mathcal{P}}_{\Lambda }^{<\infty }$ form a tilting module in $\mathrm{mod}\hspace{0.5em}\Lambda .$ Moreover, we show that in this case E coincides with the direct sum of the minimal right ${\mathcal{P}}_{\Lambda }^{<\infty }$-approximations of the indecomposable Λ-injective modules and that the projective dimension of E equal to the finitistic dimension of Λ.

中文翻译：

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单射模和有限维的近似

摘要

令 Λ 成为艺术代数，并令 ${\mathcal{磷}}_{\Lambda }^{<\infty }$有限投影维的有限生成右 Λ 模的范畴。我们证明${\mathcal{磷}}_{\Lambda }^{<\infty }$在当且仅当不可分解 Ext的直和E是逆变有限的$\mathrm{模组}\hspace{0.5em}\Lambda$- 注入模块 ${\mathcal{磷}}_{\Lambda }^{<\infty }$ 形成一个倾斜模块 $\mathrm{模组}\hspace{0.5em}\Lambda .$此外，我们表明，在这种情况下，E与最小权利的直接和重合${\mathcal{磷}}_{\Lambda }^{<\infty }$-不可分解的 Λ 注入模块的近似值，并且E的射影维数等于 Λ 的有限维数。