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 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 W(Λ) of Λ the τ-cluster morphism category W(Λ_K) of Λ_K. In particular, this establishes a faithful functor from W(Λ) to a group whenever K is of characteristic zero which has many important consequences. In the appendix, the analogous result is shown to hold whenever K is a finite field. Moreover, we give some nontrivial examples to illustrate the behaviour of τ-tilting finiteness under base field extension.

We characterise those basic and connected Nakayama algebras Λ for which the mutation of τ-exceptional sequences respects the braid group relations. We show that this is the case if and only if Λ is hereditary or all indecomposable projective Λ-modules have length at least |Λ|.

A mutation operation for τ-exceptional sequences of modules over any finite-dimensional algebra was recently introduced, generalising the mutation for exceptional sequences of modules over hereditary algebras. We interpret this mutation in terms of TF-ordered τ-rigid modules, which are in bijection with τ-exceptional sequences. As an application we show that the mutation is transitive for Nakayama algebras, by providing an explicit combinatorial description of mutation over this class of algebras.

We take a novel lattice-theoretic approach to the τ-cluster morphism category T(A) of a finite-dimensional algebra A and define the category via the lattice of torsion classes torsA. Using the lattice congruence induced by an ideal I of A we establish a functor F_I: T(A) → T(A/I). If torsA is finite, F_I is a regular epimorphism in the category of small categories and we characterise when F_I if full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the τ-cluster morphism category up to equivalence.

In this paper, we introduce the notion of an admissible partition of a simplicial polyhedral fan and define the category of a partitioned fan as a generalisation of the τ-cluster morphism category of a finite-dimensional algebra. This establishes a complete lattice of categories around the τ-cluster morphism category, which is closely tied to the fan structure. We prove that the classifying spaces of these categories are cube complexes, which reduces the process of determining if they are K(π,1) spaces to three sufficient conditions. We characterise when these conditions are satisfied for fans in ℝ² and prove that the first one, the existence of a certain faithful functor, is satisfied for hyperplane arrangements whose normal vectors lie in the positive orthant. As a consequence we obtain a new infinite class of algebras for which the τ-cluster morphism category admits a faithful functor and for which the cube complexes are K(π,1) spaces. In the final section we also offer a new algebraic proof of the relationship between an algebra and its g-vector fan.

The wall-and-chamber structure is a geometric invariant that can be associated to any algebra. In these notes we give the definition of this object and we explain its relationship with torsion classes and τ-tilting theory.