×

zbMATH — the first resource for mathematics

\(n\)-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:
18E99 Categorical algebra
18E10 Abelian categories, Grothendieck categories
18E30 Derived categories, triangulated categories (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Amiot, C., Iyama, O., Reiten, I.: Stable categories of Cohen-Macaulay modules and cluster categories. arXiv:1104.3658 (2011) · Zbl 1450.18003
[2] Artin, M; Zhang, J, Noncommutative projective schemes, Adv. Math., 109, 228-287, (1994) · Zbl 0833.14002
[3] Auslander, M.: Coherent functors. In: Proceedings of Conference on Categorical Algebra (La Jolla, CA, 1965), pp. 189-231. Springer, New York (1966)
[4] Auslander, M.: Representation Dimension of Artin Algebras. Lecture Notes. Queen Mary College, London (1971) · Zbl 1208.16015
[5] Auslander, M; Reiten, I, Stable equivalence of dualizing r-varieties, Adv. Math., 12, 306-366, (1974) · Zbl 0285.16027
[6] Auslander, M; Reiten, I, Applications of contravariantly finite subcategories, Adv. Math., 86, 111-152, (1991) · Zbl 0774.16006
[7] Auslander, M; Smalø, SO, Almost split sequences in subcategories, J. Algebra, 69, 426-454, (1981) · Zbl 0457.16017
[8] Auslander, M., Unger, L.: Isolated singularities and existence of almost split sequences. In: Dlab, V., Gabriel, P., Michler, G. (eds.) Representation Theory II Groups and Orders, Number 1178 in Lecture Notes in Mathematics, pp. 194-242. Springer, Berlin (1986) · Zbl 1021.16017
[9] Barot, M; Kussin, D; Lenzing, H, The cluster category of a canonical algebra, Trans. Am. Math. Soc., 362, 4313-4330, (2010) · Zbl 1209.18010
[10] Beilinson, A, Coherent sheaves on \(\text{ P }^d\) and problems in linear algebra, Funktsional. Anal. i Prilozhen, 12, 68-69, (1978) · Zbl 0402.14006
[11] Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, pp. 5-171. Soc. Math. France, Paris (1982) · Zbl 0118.26104
[12] Bergh, PA; Thaule, M, The axioms for \(n\)-angulated categories, Algebr. Geom. Topol., 13, 2405-2428, (2013) · Zbl 1272.18008
[13] Bergh, PA; Thaule, M, The Grothendieck group of an \(n\)-angulated category, J. Pure Appl. Algebra, 218, 354-366, (2014) · Zbl 1291.18015
[14] Bergh, P.A., Jasso, G., Thaule, M.: Higher \(n\)-angulations from local rings. To appear in J. Lond. Math. Soc. (2). arXiv:1311.2089 [math] (2013) · Zbl 1371.18009
[15] Bondal, AI; Kapranov, MM, Framed triangulated categories, Math. Sb., 181, 669-683, (1990) · Zbl 0719.18005
[16] Buan, AB; Marsh, R; Reineke, M; Reiten, I; Todorov, G, Tilting theory and cluster combinatorics, Adv. Math., 204, 572-618, (2006) · Zbl 1127.16011
[17] Buchweitz, R.-O.: Maximal Cohen-Macaulay Modules and Tate-Cohomology Over Gorenstein Rings. University of Hannover (1986). https://tspace.library.utoronto.ca/handle/1807/16682 · Zbl 0628.18003
[18] Bühler, T, Exact categories, Expo. Math., 28, 1-69, (2010) · Zbl 1192.18007
[19] Fomin, S; Zelevinsky, A, Cluster algebras i: foundations, J. Am. Math. Soc., 15, 497-529, (2001) · Zbl 1021.16017
[20] Frerick, L., Sieg, D.: Exact categories in functional analysis (2010). https://www.math.uni-trier.de/abteilung/analysis/HomAlg.pdf · Zbl 1127.16011
[21] Geiß, C; Keller, B; Oppermann, S, \(n\)-angulated categories, J. Reine Angew. Math., 675, 101-120, (2013) · Zbl 1271.18013
[22] Geiß, C; Leclerc, B; Schröer, J, Auslander algebras and initial seeds for cluster algebras, J. Lond. Math. Soc., 75, 718-740, (2007) · Zbl 1135.17007
[23] Geiß, C., Leclerc, B., Schröer, J.: Preprojective algebras and cluster algebras. In: Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pp. 253-283. Eur. Math. Soc., Zürich (2008) · Zbl 1291.18015
[24] Grothendieck, A, Sur quelques points d’algèbre homologique, i, Tohoku Math. J. (2), 9, 119-221, (1957) · Zbl 0118.26104
[25] Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Number 119 in London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1988) · Zbl 1271.18013
[26] Herschend, M; Iyama, O, Selfinjective quivers with potential and 2-representation-finite algebras, Compos. Math., 147, 1885-1920, (2011) · Zbl 1260.16016
[27] Herschend, M., Iyama, O., Minamoto, H., Oppermann, S.: Representation theory of Geigle-Lenzing complete intersections. arXiv:1409.0668 (2014) · Zbl 0118.26104
[28] Herschend, M; Iyama, O; Oppermann, S, \(n\)-representation infinite algebras, Adv. Math., 252, 292-342, (2014) · Zbl 1339.16020
[29] Iyama, O, Auslander correspondence, Adv. Math., 210, 51-82, (2007) · Zbl 1115.16006
[30] Iyama, O, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math., 210, 22-50, (2007) · Zbl 1115.16005
[31] Iyama, O, Cluster tilting for higher Auslander algebras, Adv. Math., 226, 1-61, (2011) · Zbl 1233.16014
[32] Iyama, O; Oppermann, S, \(n\)-representation-finite algebras and \(n\)-APR tilting, Trans. Am. Math. Soc., 363, 6575-6614, (2011) · Zbl 1264.16015
[33] Iyama, O; Oppermann, S, Stable categories of higher preprojective algebras, Adv. Math., 244, 23-68, (2013) · Zbl 1338.16018
[34] Iyama, O; Yoshino, Y, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math., 172, 117-168, (2008) · Zbl 1140.18007
[35] Jasso, G.: \(τ ^2\)-Stable tilting complexes over weighted projective lines. arXiv:1402.6036 (2014) · Zbl 1453.14051
[36] Keller, B, Chain complexes and stable categories, Manuscripta Math., 67, 379-417, (1990) · Zbl 0753.18005
[37] Keller, B.: On differential graded categories. In: International Congress of Mathematicians. Vol. II, pp. 151-190. Eur. Math. Soc., Zürich (2006) · Zbl 1140.18008
[38] Keller, B; Reiten, I, Acyclic Calabi-Yau categories, Compos. Math., 144, 1332-1348, (2008) · Zbl 1171.18008
[39] Keller, B; Vossieck, D, Sous LES catégories dérivées, C. R. Acad. Sci. Paris Sér. I Math., 305, 22-228, (1987) · Zbl 0628.18003
[40] Minamoto, H, Ampleness of two-sided tilting complexes, Int. Math. Res. Not., 2012, 67-101, (2012) · Zbl 1237.14008
[41] Neeman, A, The derived category of an exact category, J. Algebra, 135, 388-394, (1990) · Zbl 0753.18004
[42] Quillen, D.: Higher algebraic \(k\)-theory. i. In Algebraic \(K\)-theory, I: Higher \(K\)-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), number 341 in Lecture Notes in Math., pp. 85-147. Springer, Berlin (1973) · Zbl 0292.18004
[43] Ringel, CM, The self-injective cluster-tilted algebras, Arch. Math., 91, 218-225, (2008) · Zbl 1208.16015
[44] Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque, (239):xii + 253 pp. With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis (1996) · Zbl 1115.16005
[45] Weibel, C.A.: An Introduction to Homological Algebra, Volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)
[46] Yoshino, Y.: Cohen-Macaulay Modules over Cohen-Macaulay Rings. Cambridge University Press, Cambridge (1990) · Zbl 0745.13003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.