Bricks and τ-tilting theory under base field extensions

2025/08/01 by Erlend D. Børve, Børve, Erlend D., Eric J. Hanson +3 · 1 citation
Mathematics · #12F10 #16G10 #16G60 #18E40 #52A20 #55P20 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2508.01040

openalex publication_date 2025/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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⊗_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 \mathfrakW(Λ) of Λ to the τ-cluster morphism category \mathfrakW(Λ_K) of Λ_K. In particular, this establishes a faithful functor from \mathfrakW(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.

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Cited by

• Scattering diagrams for Artin algebras

Related

• τ-tilting theory
• A construction of support τ-tilting modules over τ-tilting finite algebras
• τ-tilting theory -- An introduction
• Semibricks
• A brick version of a theorem of Auslander