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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.21592 |
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| _version_ | 1866910674382225408 |
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| author | Paquette, Charles Prasad, Deepanshu Wehlau, David |
| author_facet | Paquette, Charles Prasad, Deepanshu Wehlau, David |
| contents | For an algebraically closed field $\mathbb{K}$, we consider a Galois $G$-covering $\mathcal{B} \to \mathcal{A}$ between locally bounded $\mathbb{K}$-categories given by bound quivers, where $G$ is torsion-free and acts freely on the objects of $\mathcal{B}$. We define the notion of $(G,τ_{\mathcal{B}})$-rigid subcategory and of support $(G,τ_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. These are the analogues of the similar concepts in the context of a finite-dimensional algebra, where we additionally require that the subcategory be $G$-equivariant. When $\mathcal{A}$ is a finite-dimensional algebra, we show that the corresponding push-down functor $\mathcal{F}_λ: \mathcal{B}$-$\rm mod$ $\to \mathcal{A}$-$\rm mod$ sends $(G,τ_{\mathcal{B}})$-rigid subcategories (respectively support $(G,τ_{\mathcal{B}})$-tilting pairs) to $τ_{\mathcal{A}}$-rigid modules (respectively support $τ_{\mathcal{A}}$-tilting pairs). We further show that there is a notion of mutation for support $(G,τ_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. Mutations of support $τ_\mathcal{A}$-tilting pairs and of support $(G,τ_\mathcal{B})$-tilting pairs commute with the push-down functor. We derive some consequences of this, and in particular, we derive a $τ$-tilting analogue of the result of P. Gabriel that locally representation-finiteness is preserved under coverings. Finally, we prove that when the Galois group $G$ is finitely generated free, any rigid $\mathcal{A}$-module (and in particular $τ_\mathcal{A}$-rigid $\mathcal{A}$-modules) lies in the essential image of the push-down functor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21592 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galois Coverings, $τ$-Rigidity and Mutations Paquette, Charles Prasad, Deepanshu Wehlau, David Representation Theory For an algebraically closed field $\mathbb{K}$, we consider a Galois $G$-covering $\mathcal{B} \to \mathcal{A}$ between locally bounded $\mathbb{K}$-categories given by bound quivers, where $G$ is torsion-free and acts freely on the objects of $\mathcal{B}$. We define the notion of $(G,τ_{\mathcal{B}})$-rigid subcategory and of support $(G,τ_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. These are the analogues of the similar concepts in the context of a finite-dimensional algebra, where we additionally require that the subcategory be $G$-equivariant. When $\mathcal{A}$ is a finite-dimensional algebra, we show that the corresponding push-down functor $\mathcal{F}_λ: \mathcal{B}$-$\rm mod$ $\to \mathcal{A}$-$\rm mod$ sends $(G,τ_{\mathcal{B}})$-rigid subcategories (respectively support $(G,τ_{\mathcal{B}})$-tilting pairs) to $τ_{\mathcal{A}}$-rigid modules (respectively support $τ_{\mathcal{A}}$-tilting pairs). We further show that there is a notion of mutation for support $(G,τ_{\mathcal{B}})$-tilting pairs over $\mathcal{B}$-$\rm mod$. Mutations of support $τ_\mathcal{A}$-tilting pairs and of support $(G,τ_\mathcal{B})$-tilting pairs commute with the push-down functor. We derive some consequences of this, and in particular, we derive a $τ$-tilting analogue of the result of P. Gabriel that locally representation-finiteness is preserved under coverings. Finally, we prove that when the Galois group $G$ is finitely generated free, any rigid $\mathcal{A}$-module (and in particular $τ_\mathcal{A}$-rigid $\mathcal{A}$-modules) lies in the essential image of the push-down functor. |
| title | Galois Coverings, $τ$-Rigidity and Mutations |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2410.21592 |