zbMATH — the first resource for mathematics

Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. (English) Zbl 1115.16005
While Auslander-Reiten theory is two-dimensional in the sense that almost split sequences can be regarded as projective resolutions of simple functors of length two, the present article exhibits a class of subcategories of classical representation categories with generalized almost split sequences of length \(n\).
Let \(R\) be a complete regular local ring of dimension \(d\), and let \(\Lambda\) be a Cohen-Macaulay \(R\)-order, i.e., an \(R\)-algebra which is finitely generated and free over \(R\). Assume that \(\Lambda\) represents an isolated singularity in the sense of Auslander. Within the category \(\Lambda\text{CM}\) of maximal Cohen-Macaulay modules over \(\Lambda\), the author considers functorially finite full subcategories \(\mathcal C\) which are maximally \(l\)-orthogonal for some \(l>0\). This means that \(\mathcal C\) consists of all objects \(X\) in \(\Lambda\text{CM}\) which satisfy \(\text{Ext}^i(X,Y)=0\) for all \(Y\) in \(\mathcal C\) and \(0<i\leq l\), and similarly, \(\mathcal C\) consists of all \(X\) in \(\Lambda\text{CM}\) with \(\text{Ext}^i(Y,X)=0\) for all \(Y\) in \(\mathcal C\) and \(0<i\leq l\). It turns out that such categories \(\mathcal C\) admit generalized almost split sequences of length \(2+l\).
Among other things, the author proves a higher Auslander-Reiten formula, he shows that long almost split sequences are found in the socle of higher Ext-groups, he generalizes Auslander’s description of the Auslander-Reiten translate (in dimension \(d\)), and characterizes the higher Auslander-Reiten translate by means of derived functors.
For representation-finite Gorenstein orders \(\Lambda\) of arbitrary dimension \(d\), he gives a complete classification of maximal 1-orthogonal subcategories of \(\Lambda\text{CM}\). This classification is related to configurations of bijectives in dimension 0 (Riedtmann) and 1 (Wiedemann). The author contributes to the ubiquity of Catalan numbers by showing that in case \(\mathbb{A}_n\), maximal 1-orthogonal subcategories can be enumerated by triangulations of polygons. In the \(\mathbb{B}\)-, \(\mathbb{C}\)-, and \(\mathbb{D}\)-cases, he gives a similar enumeration.
As an interesting example, he exhibits a maximal \((d-2)\)-orthogonal subcategory of the category \(\Lambda\text{CM}\), where the isolated singularity \(\Lambda\) is the ring of invariants of a \(d\)-dimensional power series ring under a finite group without pseudo-reflections. Last, but not least, he shows that the unpleasant fact that for \(n>1\), the group \(\text{Ext}^n(X,Y)\) does not describe \(n\)-extensions up to isomorphism, vanishes if they are restricted to a maximal \((n-1)\)-orthogonal subcategory.

16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16G30 Representations of orders, lattices, algebras over commutative rings
Full Text: DOI arXiv
[1] Artin, M.; Schelter, W.F., Graded algebras of global dimension 3, Adv. math., 66, 2, 171-216, (1987) · Zbl 0633.16001
[2] Artin, M.; Verdier, J.-L., Reflexive modules over rational double points, Math. ann., 270, 1, 79-82, (1985) · Zbl 0553.14001
[3] Auslander, M., Coherent functors, (), 189-231 · Zbl 0192.10902
[4] Auslander, M., Functors and morphisms determined by objects, (), 1-244
[5] Auslander, M., Isolated singularities and existence of almost split sequences, (), 194-242
[6] Auslander, M., Rational singularities and almost split sequences, Trans. amer. math. soc., 293, 2, 511-531, (1986) · Zbl 0594.20030
[7] Auslander, M.; Bridger, M., Stable module theory, Mem. amer. math. soc., 94, (1969), Amer. Math. Soc., Providence, RI · Zbl 0204.36402
[8] Auslander, M.; Buchweitz, R., The homological theory of maximal Cohen-Macaulay approximations, Colloque en l’honneur de pierre Samuel, Orsay, 1987, Mem. soc. math. fr. (N.S.), 38, 5-37, (1989) · Zbl 0697.13005
[9] Auslander, M.; Reiten, I., Stable equivalence of dualizing R-varieties, Adv. math., 12, 306-366, (1974) · Zbl 0285.16027
[10] Auslander, M.; Reiten, I., Almost split sequences in dimension two, Adv. math., 66, 1, 88-118, (1987) · Zbl 0625.13013
[11] Auslander, M.; Smalo, S.O., Almost split sequences in subcategories, J. algebra, 69, 2, 426-454, (1981) · Zbl 0457.16017
[12] Auslander, M.; Reiten, I.; Smalo, S.O., Representation theory of Artin algebras, Cambridge stud. adv. math., vol. 36, (1995), Cambridge Univ. Press Cambridge · Zbl 0834.16001
[13] Buan, A.; Marsh, R.; Reineke, M.; Reiten, I.; Todorov, G., Tilting theory and cluster combinatorics, Adv. math., 204, 2, 572-618, (2006) · Zbl 1127.16011
[14] Caldero, P.; Chapoton, F.; Schiffler, R., Quivers with relations arising from clusters (\(A_n\) case), Trans. amer. math. soc., 358, 3, 1347-1364, (2006) · Zbl 1137.16020
[15] Cartan, H.; Eilenberg, S., Homological algebra, (1956), Princeton Univ. Press Princeton, NJ · Zbl 0075.24305
[16] Curtis, C.W.; Reiner, I., Methods of representation theory, vol. I. with applications to finite groups and orders, A wiley-interscience publication, (1990), John Wiley & Sons New York
[17] Drozd, Y.A.; Kiričenko, V.V.; Roĭter, A.V., Hereditary and bass orders, Izv. akad. nauk SSSR ser. mat., 31, 1415-1436, (1967), (in Russian)
[18] Esnault, H., Reflexive modules on quotient surface singularities, J. reine angew. math., 362, 63-71, (1985) · Zbl 0553.14016
[19] Evans, E.G.; Griffith, P., Syzygies, London math. soc. lecture note ser., vol. 106, (1985), Cambridge Univ. Press Cambridge · Zbl 0569.13005
[20] Fossum, R.M.; Griffith, P.A.; Reiten, I., Trivial extensions of abelian categories, () · Zbl 0255.16014
[21] Gabriel, P.; Roiter, A.V., Representations of finite-dimensional algebras, (1997), Springer Berlin
[22] Geigle, W.; Lenzing, H., A class of weighted projective curves arising in representation theory of finite-dimensional algebras, (), 265-297
[23] Goto, S.; Nishida, K., Finite modules of finite injective dimension over a Noetherian algebra, J. London math. soc. (2), 63, 2, 319-335, (2001) · Zbl 1047.13003
[24] Goto, S.; Nishida, K., Towards a theory of bass numbers with application to Gorenstein algebras, Colloq. math., 91, 2, 191-253, (2002) · Zbl 1067.16014
[25] Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London math. soc. lecture note ser., vol. 119, (1988), Cambridge Univ. Press Cambridge · Zbl 0635.16017
[26] Happel, D.; Preiser, U.; Ringel, C.M., Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to Dtr-periodic modules, (), 280-294 · Zbl 0446.16032
[27] Hilton, P.J.; Stammbach, U., A course in homological algebra, Grad. texts in math., vol. 4, (1997), Springer New York · Zbl 0863.18001
[28] Iyama, O., τ-categories I: ladders, Algebr. represent. theory, 8, 3, 297-321, (2005) · Zbl 1093.16015
[29] Iyama, O., τ-categories II: Nakayama pairs and rejective subcategories, Algebr. represent. theory, 8, 4, 449-477, (2005) · Zbl 1124.16017
[30] Iyama, O., τ-categories III: Auslander orders and Auslander-Reiten quivers, Algebr. represent. theory, 8, 5, 601-619, (2005) · Zbl 1091.16012
[31] Iyama, O., Auslander correspondence, Adv. math., 210, 1, 51-82, (2007), (this issue) · Zbl 1115.16006
[32] McKay, J., Graphs, singularities, and finite groups, (), 183-186
[33] Matsumura, H., Commutative ring theory, Cambridge stud. adv. math., vol. 8, (1989), Cambridge Univ. Press Cambridge
[34] Riedtmann, C., Representation-finite self-injective algebras of class \(A_n\), (), 449-520
[35] Riedtmann, C., Algebren, darstellungskocher, überlagerungen und zurück, Comment. math. helv., 55, 2, 199-224, (1980), (in German) · Zbl 0444.16018
[36] Ringel, C.M., Tame algebras and integral quadratic forms, Lecture notes in math., vol. 1099, (1984), Springer Berlin · Zbl 0546.16013
[37] Reiten, I.; Van den Bergh, M., Two-dimensional tame and maximal orders of finite representation type, Mem. amer. math. soc., 80, (1989) · Zbl 0677.16002
[38] Roggenkamp, K.W.; Schmidt, J.W., Almost split sequences for integral group rings and orders, Comm. algebra, 4, 10, 893-917, (1976) · Zbl 0361.16007
[39] Simson, D., Linear representations of partially ordered sets and vector space categories, Algebra logic appl., vol. 4, (1992), Gordon and Breach Science Publishers Montreux · Zbl 0818.16009
[40] Stanley, R.P., Enumerative combinatorics, vol. 2, Cambridge stud. adv. math., vol. 62, (1999), Cambridge Univ. Press Cambridge · Zbl 0928.05001
[41] Van den Bergh, M., Non-commutative crepant resolutions, (), 749-770 · Zbl 1082.14005
[42] Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings, London math. soc. lecture note ser., vol. 146, (1990), Cambridge Univ. Press Cambridge · Zbl 0745.13003
[43] Wiedemann, A., Classification of the Auslander-Reiten quivers of local Gorenstein orders and a characterization of the simple curve singularities, J. pure appl. algebra, 41, 2-3, 305-329, (1986) · Zbl 0597.16009
[44] Wiedemann, A., Die Auslander-Reiten kocher der gitterendlichen gorensteinordnungen, Bayreuth. math. schr., 23, 1-134, (1987) · Zbl 0698.16003
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.