Auslander-Reiten theory, derived categories, and higher dimensional homological algebra

Abstract

Auslander-Reiten theory plays an important role in the study of abelian and triangulated categories (in classic homological algebra) and in their higher analogues (in the more recent higher homological algebra). The classic setup studies module categories of the form mod and their bounded derived categories Db(mod ), where is a nite dimensional algebra over a eld k and mod is the category of nitely generated (right) -modules. If gldim B 1, Br uning proved there is a bijection between the wide subcategories of the abelian category mod and those of the triangulated category Db(mod ). When T is a suitable triangulated category, J rgensen described Auslander-Reiten triangles in the extension closed subcategories of T . If X b mod is a precovering extension closed subcategory, Kleiner proved that any indecomposable not Ext-projective X > X appears as the end term of an Auslander-Reiten sequence in X and he further described the case when End (X) modulo the morphisms factoring through a projective is a division ring. Letting d be a positive integer, we study higher homological algebra and higher Auslander- Reiten theory. Geiss, Keller and Oppermann generalised triangulated categories to (d+2)- angulated categories and Jasso likewise generalised abelian categories to d-abelian categories. Note that the case d = 1 recovers classic homological algebra. Assuming there is a d-cluster tilting subcategory F b mod , consider F = add{ idF S i > Z} b Db(mod ): Then F is d-abelian and plays the role of a higher mod having for higher derived category the (d + 2)-angulated category F. With this in mind, we generalise Br uning, J rgensen and Kleiner's results for higher values of d. We also use higher Auslander-Reiten theory to generalise results on Grothendieck groups of a suitable triangulated category T . We present \higher cluster tilting" versions of results by Xiao and Zhu and by Palu and a \higher angulated" version of Palu's result. Our results express K0(T ) as a quotient of the split Grothendieck group of higher-cluster tilting subcategories of T . We prove analogues of results by Kleiner on subcategories of mod in the corresponding setup of subcategories of a suitable triangulated category T with a precovering extension closed subcategory C. In particular, we introduce indecomposable Ext-projective objects C in C, show that such a C appears in what we call a left-weak Auslander-Reiten triangle in C and prove how these objects are related to the concept of Iyama and Yoshino's mutation. Contents