Pruefer modules in filtration categories of semibricks
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918384550019072 |
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| author | Lukas, Frank |
| author_facet | Lukas, Frank |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13142 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pruefer modules in filtration categories of semibricks Lukas, Frank Representation Theory Rings and Algebras 16G70, 16D90, 16G10 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. |
| title | Pruefer modules in filtration categories of semibricks |
| topic | Representation Theory Rings and Algebras 16G70, 16D90, 16G10 |
| url | https://arxiv.org/abs/2402.13142 |