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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.01040 |
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| _version_ | 1866912515939631104 |
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| author | Børve, Erlend D. Hanson, Eric J. Kaipel, Maximilian |
| author_facet | Børve, Erlend D. Hanson, Eric J. Kaipel, Maximilian |
| contents | Let $K:k$ be a field extension and let $Λ$ be a finite-dimensional $k$-algebra. We investigate the relationship between $Λ$ and $Λ_K = Λ\otimes_k K$ with particular emphasis on various aspects of $τ$-tilting theory and bricks. We show that many types of objects for $Λ$ lift injectively to the same type of object for $Λ_K$, and many common constructions in $τ$-tilting theory commute with the process of extending the base field. One of our main applications is the construction of a faithful functor from the $τ$-cluster morphism category $\mathfrak{W}(Λ)$ of $Λ$ to the $τ$-cluster morphism category $\mathfrak{W}(Λ_K)$ of $Λ_K$. In particular, this establishes a faithful functor from $\mathfrak{W}(Λ)$ to a group whenever $k$ is of characteristic zero which has many important consequences. In the appendix, E. J. Hanson shows the analogous result whenever $k$ is a finite field. Moreover, we give some nontrivial examples to illustrate the behaviour of $τ$-tilting finiteness under base field extension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01040 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bricks and $τ$-tilting theory under base field extensions Børve, Erlend D. Hanson, Eric J. Kaipel, Maximilian Representation Theory 12F10, 16G10, 16G60, 18E40, 52A20, 55P20 Let $K:k$ be a field extension and let $Λ$ be a finite-dimensional $k$-algebra. We investigate the relationship between $Λ$ and $Λ_K = Λ\otimes_k K$ with particular emphasis on various aspects of $τ$-tilting theory and bricks. We show that many types of objects for $Λ$ lift injectively to the same type of object for $Λ_K$, and many common constructions in $τ$-tilting theory commute with the process of extending the base field. One of our main applications is the construction of a faithful functor from the $τ$-cluster morphism category $\mathfrak{W}(Λ)$ of $Λ$ to the $τ$-cluster morphism category $\mathfrak{W}(Λ_K)$ of $Λ_K$. In particular, this establishes a faithful functor from $\mathfrak{W}(Λ)$ to a group whenever $k$ is of characteristic zero which has many important consequences. In the appendix, E. J. Hanson shows the analogous result whenever $k$ is a finite field. Moreover, we give some nontrivial examples to illustrate the behaviour of $τ$-tilting finiteness under base field extension. |
| title | Bricks and $τ$-tilting theory under base field extensions |
| topic | Representation Theory 12F10, 16G10, 16G60, 18E40, 52A20, 55P20 |
| url | https://arxiv.org/abs/2508.01040 |