Let L:k be a field extension and let A be a finite-dimensional k-algebra. The extension of scalars of A along L:k is the L-algebra $A^L$, obtained by tensoring A and L over k. In the early 1980s, Jensen and Lenzing showed that extension of scalars preserves many module-theoretic and homological properties, particularly when L:k is MacLane separable. In particular, representation-finiteness is preserved in this case. However, if A is tau-tilting finte, i.e. it admits only a finite number of support tau-tilting modules up to isomorphism, this need not be true for $A^L$. We explore some examples and counter-examples of when tau-tilting finiteness is preserved. Along the way, we explain how tau-tilting theory and related notions lift under extension of scalars.\\ The talk is based on joint work in progress with Max Kaipel (Cologne).

It is well known that for Auslander algebras, the category of all (finitely generated) projective modules is an abelian category and this property of abelianness characterizes Auslander algebras by Tachikawa's theorem in 1974. Let n be a positive integer. In this talk, by using torsion theoretic methods, we show that n-Auslander algebras can be characterized by the abelianness of the category of modules with projective dimension less than n and a certain additional property, extending the classical Auslander-Tachikawa theorem. By Auslander-Iyama correspondence a categorical characterization of the class of Artin algebras having n-cluster tilting modules is obtained. Since higher Auslander algebras are a special case of higher Auslander-Gorenstein algebras, the results are given in the general setting as extending previous results of Kong. Moreover, as an application of some results, we give categorical descriptions for the semisimplicity and selfinjectivity of an Artin algebra. Higher Auslander-Gorenstein algebras are also studied from the viewpoint of cotorsion pairs and, as application, we show that they satisfy in two nice equivalences.