完全余内射对的交集
Intersection of complete cotorsion pairs
AI总结：
本文研究了两个完全余内射对在特定条件下的交集，并构造了新的完全余内射对，同时刻画了相关模类的性质。
摘要：
给定两个（hereditary）完全余内射对$(\mathcal{X}_1,\mathcal{Y}_1)$和$(\mathcal{X}_2,\mathcal{Y}_2)$在一个正合范畴中，且满足$\mathcal{X}_1\subseteq \mathcal{Y}_2$，我们证明$(\rm Smd\langle \mathcal{X}_1,\mathcal{X}_2 \rangle,\mathcal{Y}_1\cap \mathcal{Y}_2)$也是一个（hereditary）完全余内射对，其中$\rm Smd\langle \mathcal{X}_1,\mathcal{X}_2 \rangle$是扩展$\mathcal{X}_1$和$\mathcal{X}_2$的直接和因子类。作为应用，我们构造了完全余内射对，例如$(^{\perp}\mathcal{GI}^{\leqslant n},\mathcal{GI}^{\leqslant n})$，其中$\mathcal{GI}^{\leqslant n}$是Gorenstein内射维数至多$n$的模类。我们还刻画了精确复形内射模的左正交类以及具有有限Gorenstein射影、Gorenstein平坦和PGF维数的模类。
英文摘要：
Given two (hereditary) complete cotorsion pairs $(\mathcal{X}_1,\mathcal{Y}_1)$ and $(\mathcal{X}_2,\mathcal{Y}_2)$ in an exact category with $\mathcal{X}_1\subseteq \mathcal{Y}_2$, we prove that $\left({\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle,\mathcal{Y}_1\cap \mathcal{Y}_2\right)$ is also a (hereditary) complete cotorsion pair, where ${\rm Smd}\langle \mathcal{X}_1,\mathcal{X}_2 \rangle$ is the class of direct summands of extension of $\mathcal{X}_1$ and $\mathcal{X}_2$. As an application, we construct complete cotorsion pairs, such as $(^\perp\mathcal{GI}^{\leqslant n},\mathcal{GI}^{\leqslant n})$, where $\mathcal{GI}^{\leqslant n}$ is the class of modules of Gorenstein injective dimension at most $n$. And we also characterize the left orthogonal class of exact complexes of injective modules and the classes of modules with finite Gorenstein projective, Gorenstein flat, and PGF dimensions.
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