过滤范畴中普雷弗模块
Pruefer modules in filtration categories of semibricks
AI总结：
本文研究了在半砖类X的过滤范畴中，通过构造X^⊥-包络来定义普雷弗模块，并证明了该范畴中的投射对象可分解为这些普雷弗模块的直接和。
摘要：
设R为具有单位元的环，X是模块范畴Mod R中的一个半砖类，即由成对正交的有限生成模块组成的类，其极限环为除环。我们研究由所有具有因子在X中的模块组成的满子范畴Filt(X)。我们证明Filt(X)是Mod R的一个宽子范畴。对于Ext正交类[ X^⊥ = {M ∈ Mod R | Ext^1_R(X,M)=0 对所有X ∈ X} ]，我们为每个模块Y构造一个X^⊥-包络Y_{X}(∞)作为迭代通用短正合列的极限。假设每个X ∈ X具有射射性至多为一且Hom_R(X,R)=0对所有X ∈ X。则正则模块的包络R_{X}(∞)同构于Schofield意义下的X正交局部化R_X。模块在X中的X^⊥包络被称为普雷弗模块，因为它们与经典普雷弗群和 tame 标准代数上的普雷弗模块有许多共同性质。我们证明Filt(X)中的每个投射对象都是此类普雷弗模块的直接和。
英文摘要：
Let $R$ be a ring with unity and $\mathcal{X}$ a semibrick in the module category $\mathrm{Mod}\,R$, that is, a class of pairwise orthogonal finitely presented modules whose endomorphism rings are division rings. We study the full subcategory $\mathrm{Filt}(\mathcal{X})$ consisting of all modules admitting a filtration with factors in $\mathcal{X}$. We show that $\mathrm{Filt}(\mathcal{X})$ is a wide subcategory of $\mathrm{Mod}\,R$. For the Ext-orthogonal class \[ \mathcal{X}^{\perp} = \{M \in \mathrm{Mod}\,R \mid \operatorname{Ext}^1_R(X,M)=0 \text{ for all } X \in \mathcal{X}\} \] we construct, for every module $Y$, an $\mathcal{X}^{\perp}$-envelope $Y_{\mathcal{X}}(\infty)$ as a direct limit of iterated universal short exact sequences. Assume that every $X \in \mathcal{X}$ has projective dimension at most one and that $\operatorname{Hom}_R(X,R)=0$ for all $X \in \mathcal{X}$. Then the envelope $R_{\mathcal{X}}(\infty)$ of the regular module is isomorphic to the universal localization $R_{\mathcal{X}}$ of $R$ at $\mathcal{X}$ in the sense of Schofield. The $\mathcal{X}^{\perp}$-envelopes of modules in $\mathcal{X}$ are called Prüfer modules since they share many properties with classical Prüfer groups and with Prüfer modules over tame hereditary algebras. We prove that every injective object in $\mathrm{Filt}(\mathcal{X})$ is a direct sum of such Prüfer modules.
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