Families of Auslander-Reiten components for simply connected differential graded algebras. (English) Zbl 1244.16010
Let \(\Bbbk\) be a field, \(X\) a simply connected topological space and \(C^*(X,\Bbbk)\) the corresponding singular cochain differential graded algebra. By P. Jørgensen [Comment. Math. Helv. 79, No. 1, 160-182 (2004; Zbl 1053.55010)], \(X\) has Poincaré duality if and only if the compact derived categories of both left and right DG modules over \(C^*(X,\Bbbk)\) have Auslander-Reiten triangles. In the latter case one can define the Auslander-Reiten quiver for \(X\) and it was shown by Jørgensen that all components are of the form \(\mathbb ZA_\infty\). The present paper addresses the questions about the number of these components and their suitable parameterization.
The main result is as follows: Theorem. Let \(A\) be a simply connected Gorenstein differential graded algebra of finite type. Then the Auslander-Reiten quiver of \(\mathbf D^c(A)\) has finitely many components if and only if \(\dim_\Bbbk\mathbf H^*A=2\). In the latter case the number of components equals \(\sup\{i\mid\mathbf H^iA\neq 0\}-1\). Moreover, if \(\dim_\Bbbk\mathbf H^iA\geq 2\) for some \(i\), then there is an \(n\)-parameter family of Auslander-Reiten components for each \(n\in\mathbb N\), in fact, there are objects, each lying in different components, that can be parameterized by \(\mathbb P^1(\Bbbk)^n\).
The main result is as follows: Theorem. Let \(A\) be a simply connected Gorenstein differential graded algebra of finite type. Then the Auslander-Reiten quiver of \(\mathbf D^c(A)\) has finitely many components if and only if \(\dim_\Bbbk\mathbf H^*A=2\). In the latter case the number of components equals \(\sup\{i\mid\mathbf H^iA\neq 0\}-1\). Moreover, if \(\dim_\Bbbk\mathbf H^iA\geq 2\) for some \(i\), then there is an \(n\)-parameter family of Auslander-Reiten components for each \(n\in\mathbb N\), in fact, there are objects, each lying in different components, that can be parameterized by \(\mathbb P^1(\Bbbk)^n\).
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 16E45 | Differential graded algebras and applications (associative algebraic aspects) |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 16E35 | Derived categories and associative algebras |
| 55P62 | Rational homotopy theory |
Keywords:
Auslander-Reiten quivers; Auslander-Reiten triangles; components; differential graded algebras; topological spaces; derived categoriesCitations:
Zbl 1053.55010References:
| [1] | Adams J.F., Hilton P.J.: On the chain algebra of a loop space. Comment. Math. Helv. 30, 305–330 (1956) · Zbl 0071.16403 · doi:10.1007/BF02564350 |
| [2] | Auslander, M.: Functors and morphisms determined by objects. In: Representation theory of algebras (Proc. Conf., Temple University, Philadelphia, PA, 1976). Lecture Notes in Pure Appl. Mathematics, vol. 37, pp. 1–244. Dekker, New York (1978) |
| [3] | Avramov L.L., Foxby H.-B.: Locally Gorenstein homomorphisms. Am. J. Math. 114(5), 1007–1047 (1992) · Zbl 0769.13007 · doi:10.2307/2374888 |
| [4] | Avramov, L.L., Foxby, H.-B., Halperin, S.: Resolutions for dg modules. Preprint (2008, version from 7 March 2008) |
| [5] | Crawley-Boevey W.W.: On tame algebras and bocses. Proc. Lond. Math. Soc. (3) 56(3), 451–483 (1988) · Zbl 0661.16026 · doi:10.1112/plms/s3-56.3.451 |
| [6] | Crawley-Boevey, W.W.: Matrix problems and Drozd’s theorem. Topics in algebra, Part 1 (Warsaw, 1988), vol. 26, pp. 199–222. Banach Center Publ. PWN, Warsaw (1990) |
| [7] | Drozd, J.A.: Tame and wild matrix problems. Representations and quadratic forms (Russian). Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, vol. 154, pp. 39–74 (1979) |
| [8] | Dwyer W.G., Greenlees J.P.C., Iyengar S.: Duality in algebra and topology. Adv. Math. 200(2), 357–402 (2006) · Zbl 1155.55302 · doi:10.1016/j.aim.2005.11.004 |
| [9] | Félix Y., Halperin S., Thomas J.-C.: Gorenstein spaces. Adv. Math. 71(1), 92–112 (1988) · Zbl 0659.57011 · doi:10.1016/0001-8708(88)90067-9 |
| [10] | Félix Y., Halperin S., Thomas J.-C.: Rational homotopy theory Graduate. Texts in Mathematics, vol. 205. Springer, New York (2001) |
| [11] | Frankild A., Iyengar S., Jørgensen P.: Dualizing differential graded modules and Gorenstein differential graded algebras. J. Lond. Math. Soc. 68(2), 288–306 (2003) · Zbl 1064.16009 · doi:10.1112/S0024610703004496 |
| [12] | Frankild, A., Jørgensen, P.: Homological identities for differential graded algebras II. Part of PhD thesis A. Frankild, (2002) · Zbl 1041.16005 |
| [13] | Frankild A., Jørgensen P.: Gorenstein differential graded algebras. Isr. J. Math. 135, 327–353 (2003) · Zbl 1067.13013 · doi:10.1007/BF02776063 |
| [14] | Freyd, P.: Stable homotopy. In: Proceedings of conference on categorical algebra (La Jolla, CA, 1965), pp. 121–172. Springer, New York (1966) |
| [15] | Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. In: London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988) · Zbl 0635.16017 |
| [16] | Happel D., Keller B., Reiten I.: Bounded derived categories and repetitive algebras. J. Algebra 319(4), 1611–1635 (2008) · Zbl 1187.16016 · doi:10.1016/j.jalgebra.2007.10.001 |
| [17] | Jørgensen P.: Auslander–Reiten theory over topological spaces. Comment. Math. Helv. 79(1), 160–182 (2004) · Zbl 1053.55010 · doi:10.1007/s00014-001-0795-4 |
| [18] | Jørgensen, P.: Amplitude inequalities for differential graded modules (2006). arXiv:math/0601416v1 [math.RA] · Zbl 1096.18005 |
| [19] | Jørgensen P.: The Auslander–Reiten quiver of a Poincaré duality space. Algebr. Represent. Theory 9(4), 323–336 (2006) · Zbl 1123.55005 · doi:10.1007/s10468-006-9007-4 |
| [20] | Jørgensen, P.: Calabi-Yau categories and Poincaré duality spaces. In: Trends in Representation Theory of Algebras and Related Topics. (Toruń, 2007). Congress Reports Series, vol. 1, pp. 399–432. European Mathematical Society, Zürich (2008) |
| [21] | Keller B.: Deriving DG categories. Ann. Sci. École Norm. Sup. (4) 27(1), 63–102 (1994) · Zbl 0799.18007 |
| [22] | Reiten, I., Van den Bergh, M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Am. Math. Soc. 2(15), 295–366 (2002, electronic) · Zbl 0991.18009 |
| [23] | Schmidt, K.: Auslander–Reiten theory for simply connected differential graded algebras. PhD thesis, University of Paderborn (2007). arXiv:0801.0651v1[math.RT] · Zbl 1303.16015 |
| [24] | Spaltenstein N.: Resolutions of unbounded complexes. Compos. Math. 65(2), 121–154 (1988) · Zbl 0636.18006 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
