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$$n$$-abelian and $$n$$-exact categories. (English) Zbl 1356.18005
The main motivation of this interesting paper is to introduce the notion of $$n$$-abelian category as a generalization of the classical notion of abelian category defined by A. Grothendieck [Tohoku Math. J. (2) 9, 119–221 (1957; Zbl 0118.26104)] to axiomatize the properties of the category of modules over a ring and of the category of sheaves over a scheme. A brief description of the contents of this article is the following: In Section 2 the author introduce the basic concepts behind the definitions of $$n$$-abelian and $$n$$-exact categories: $$n$$-cokernels, $$n$$-kernels, $$n$$-exact sequences, and $$n$$-pushout and $$n$$-pullback diagrams. The central point of Section 3 is the notion of $$n$$-abelian category. In this section the author define these kind of categories as categories inhabited by certain exact sequences with $$n+2$$ terms, called $$n$$-exact sequences. Also, section 3 contains the proofs of basic properties of $$n$$-abelian categories, including the existence of $$n$$-pushout (resp. $$n$$-pullback) diagrams. Moreover, in this third section we can find a characterization of semisimple categories in terms of $$n$$-abelian categories, the notion of projective object in $$n$$-abelian categories and their basic properties. Finally, in this section the author proved that $$n$$-cluster-tilting subcategories of abelian categories are $$n$$-abelian, and, if we assume that there exists enough projectives, a partial converse was given.
Using a parallel way to T. Bühler’s exposition of the basics of the theory of exact categories given in [Expo. Math. 28, No. 1, 1–69 (2010; Zbl 1192.18007)], the notion of $$n$$-exact category is introduced in Section 4. The author prove that the class of $$n$$-exact categories contains that of $$n$$-abelian categories, and also proves that if $${\mathcal M}$$ is an $$n$$-cluster-tilting subcategory of an exact category, then $${\mathcal M}$$ is an $$n$$-exact category (for the definition of $$n$$-cluster-tilting subcategory see the papers of O. Iyama [Adv. Math. 210, No. 1, 51–82 (2007; Zbl 1115.16006); Adv. Math. 210, No. 1, 22–50 (2007; Zbl 1115.16005)], and the one of A. B. Buan et al. [Adv. Math. 204, No. 2, 572–618 (2006; Zbl 1127.16011)].
In Section 5 the author define the notion of Frobenius $$n$$-exact category proving that their stable categories admits a structure of $$(n+2)$$-angulated category; this property permits to introduce the notion of algebraic $$(n+2)$$-angulated categories. Also, in this section we can find a method to construct Frobenius $$n$$-exact categories from certain $$n$$-cluster-tilting subcategories of Frobenius exact categories. As was proved in the final part of this section, this construction is closely related with the standard construction of $$(n+2)$$-angulated categories given by C. Geiss et al. [J. Reine Angew. Math. 675, 101–120 (2013; Zbl 1271.18013)]. Finally, working with finite dimensional algebras over an algebraically closed field, in Section 6 the author provides several examples to illustrate the main results of this paper.

##### MSC:
 1.8e+100 Categorical algebra 1.8e+11 Abelian categories, Grothendieck categories 1.8e+31 Derived categories, triangulated categories (MSC2010)
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