×

Compact DG modules and Gorenstein DG algebras. (English) Zbl 1197.16011

Summary: When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.
When \(A\) is a regular DG algebra such that \(H(A)\) is a Koszul graded algebra, \(H(A)\) is proved to have finite global dimension. And we give an example to illustrate that the global dimension of \(H(A)\) may be infinite, if the condition that \(H(A)\) is Koszul is weakened to the condition that \(A\) is a Koszul DG algebra. For a general regular DG algebra \(A\), we give some equivalent conditions for the Gorensteinness.
For a finite connected DG algebra \(A\), we prove that \(D^c(A)\) and \(D^c(A^{\text{op}})\) admit Auslander-Reiten triangles if and only if \(A\) and \(A^{\text{op}}\) are Gorenstein DG algebras. When \(A\) is a non-trivial regular DG algebra such that \(H(A)\) is locally finite, \(D^c(A)\) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in \(D^b_{\text{lf}}(A)\) and \(D^b_{\text{lf}}(A^{\text{op}})\) instead, when \(A\) is a regular DG algebra.

MSC:

16E45 Differential graded algebras and applications (associative algebraic aspects)
16E10 Homological dimension in associative algebras
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16S37 Quadratic and Koszul algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Iversen B. Amplitude inequalities for complexes. Ann Sci École Norm Sup, 10: 547–558 (1977) · Zbl 0433.13003
[2] Jøgensen P. Amplitude inequalities for differential graded modules. arXiv: math. RA/0601416 v1
[3] Félix Y, Halperin S, Thomas J C. Rational Homotopy Theory. In: Grad Texts in Math, 205. Berlin: Springer, 2000 · Zbl 0961.55002
[4] He J W, Wu Q S. Koszul differential graded algebras and BGG correspondence. J Algebra, 320: 2934–2962 (2008) · Zbl 1193.16012 · doi:10.1016/j.jalgebra.2008.06.021
[5] Félix Y, Halperin S, Thomas J C. Gorenstein spaces. Adv Math, 71: 92–112 (1988) · Zbl 0659.57011 · doi:10.1016/0001-8708(88)90067-9
[6] Avramov L L, Foxby H V. Locally Gorenstein homomorphisms. Amer J Math Soc, 114: 1007–1047 (1992) · Zbl 0769.13007
[7] Dwyer W, Greenlees J P C, Iyengar S. Duality in algebra and topology. Adv Math, 200: 357–402 (2006) · Zbl 1155.55302 · doi:10.1016/j.aim.2005.11.004
[8] Frankild A J, Jøgensen P. Gorenstein differential graded algebras. Israel J Math, 135: 327–354 (2003) · Zbl 1067.13013 · doi:10.1007/BF02776063
[9] Frankild A J, Iyengar S, Jøgensen P. Dualizing differential graded modules and Gorenstein differential graded algebras. J London Math Soc, 68: 288–306 (2003) · Zbl 1064.16009 · doi:10.1112/S0024610703004496
[10] J/ensen P. Auslander-Reiten theory over topological spaces. Comment Math Helv, 79: 160–182 (2004) · Zbl 1053.55010 · doi:10.1007/s00014-001-0795-4
[11] Frankild A J, Jøgensen P. Homological properties of cochain differential graded algebras. J Algebra, 320: 3311–3326 (2008) · Zbl 1192.16009 · doi:10.1016/j.jalgebra.2008.08.001
[12] Mao X F, Wu Q S. Homological invariants for connected DG algebra. Comm Algebra, 36: 3050–3072 (2008) · Zbl 1163.16007 · doi:10.1080/00927870802110870
[13] Keller B. Deriving DG categories. Ann Sci É cole Norm Sup, 27: 6–102 (1994) · Zbl 0799.18007
[14] Yekutieli A, Zhang J J. Rigid complexes via DG algebras. Trans Amer Math Soc, 360: 3211–3248 (2008) · Zbl 1165.18009 · doi:10.1090/S0002-9947-08-04465-6
[15] Krause H. Auslander-Reiten theory via Brown representability. K-theory, 20: 331–344 (2000) · Zbl 0970.18012 · doi:10.1023/A:1026571214620
[16] Krause H. Auslander-Reiten triangles and a theorem of Zimmermann. Bull London Math Soc, 37: 361–372 (2005) · Zbl 1070.18006 · doi:10.1112/S0024609304004011
[17] Neeman A. The connection between the K-theory localization theorem of Thomason. Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann Sci É cole Norm Sup, 25: 547–566 (1992) · Zbl 0868.19001
[18] Kriz I, May J P. Operads, Algebras, Modules and Motives. Astérisque, 233, 1995
[19] He J W, Lu D M. Higher Koszul algebras and A-infinity algebras. J Algebra, 293: 335–362 (2005) · Zbl 1143.16027 · doi:10.1016/j.jalgebra.2005.05.025
[20] Jøgensen P. Non-commutative graded homological identities. J London Math Soc, 57: 336–350 (1998) · doi:10.1112/S0024610798006164
[21] Apassov D. Homological dimensions over differential graded rings. In Complexes and Differential Graded Modules. Dissertation for the Doctoral Degree, Lund University, 1999, 25–39
[22] Happel D. On the derived category of a finite-dimensional algebra. Comment Math Helv, 62: 339–389 (1987) · Zbl 0626.16008 · doi:10.1007/BF02564452
[23] Ringel C M. Tame algebras and integral quadratic forms. In: Lecture Notes in Math, Vol 1099. New York: Springer-Verlag, 1984 · Zbl 0546.16013
[24] Bökstedt M, Neeman A. Homotopy limits in triangulated categories. Compositio Math, 86: 209–234 (1993) · Zbl 0802.18008
[25] Kock J. Frobenius algebras and 2D topological quantum Field theories. In: Math Soc Stud Texts, Vol. 59. London: Cambridge University Press, 2003 · Zbl 1046.57001
[26] Frankild A J, Jøgensen P. Homological identities for differential graded algebras. J Algebra, 265: 114–136 (2003) · Zbl 1041.16005 · doi:10.1016/S0021-8693(03)00025-5
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.