Skip to main content
SpringerLink
Log in

Compact DG modules and Gorenstein DG algebras

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

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 the 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 Gorensteiness.

For a finite connected DG algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D blf (A) and D blf (A op) instead, when A is a regular DG algebra.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Iversen B. Amplitude inequalities for complexes. Ann Sci École Norm Sup, 10: 547–558 (1977)

    MATH  MathSciNet  Google Scholar 

  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

    Google Scholar 

  4. He J W, Wu Q S. Koszul differential graded algebras and BGG correspondence. J Algebra, 320: 2934–2962 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Félix Y, Halperin S, Thomas J C. Gorenstein spaces. Adv Math, 71: 92–112 (1988)

    Article  MATH  Google Scholar 

  6. Avramov L L, Foxby H V. Locally Gorenstein homomorphisms. Amer J Math Soc, 114: 1007–1047 (1992)

    MATH  MathSciNet  Google Scholar 

  7. Dwyer W, Greenlees J P C, Iyengar S. Duality in algebra and topology. Adv Math, 200: 357–402 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Frankild A J, Jøgensen P. Gorenstein differential graded algebras. Israel J Math, 135: 327–354 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. J/ℷensen P. Auslander-Reiten theory over topological spaces. Comment Math Helv, 79: 160–182 (2004)

    Article  MathSciNet  Google Scholar 

  11. Frankild A J, Jøgensen P. Homological properties of cochain differential graded algebras. J Algebra, 320: 3311–3326 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mao X F, Wu Q S. Homological invariants for connected DG algebra. Comm Algebra, 36: 3050–3072 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Keller B. Deriving DG categories. Ann Sci É cole Norm Sup, 27: 6–102 (1994)

    Google Scholar 

  14. Yekutieli A, Zhang J J. Rigid complexes via DG algebras. Trans Amer Math Soc, 360: 3211–3248 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Krause H. Auslander-Reiten theory via Brown representability. K-theory, 20: 331–344 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Krause H. Auslander-Reiten triangles and a theorem of Zimmermann. Bull London Math Soc, 37: 361–372 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jøgensen P. Non-commutative graded homological identities. J London Math Soc, 57: 336–350 (1998)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ringel C M. Tame algebras and integral quadratic forms. In: Lecture Notes in Math, Vol 1099. New York: Springer-Verlag, 1984

    MATH  Google Scholar 

  24. Bökstedt M, Neeman A. Homotopy limits in triangulated categories. Compositio Math, 86: 209–234 (1993)

    MATH  MathSciNet  Google Scholar 

  25. Kock J. Frobenius algebras and 2D topological quantum Field theories. In: Math Soc Stud Texts, Vol. 59. London: Cambridge University Press, 2003

    Google Scholar 

  26. Frankild A J, Jøgensen P. Homological identities for differential graded algebras. J Algebra, 265: 114–136 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    XueFeng Mao & QuanShui Wu

Authors
  1. XueFeng Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. QuanShui Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to XueFeng Mao.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mao, X., Wu, Q. Compact DG modules and Gorenstein DG algebras. Sci. China Ser. A-Math. 52, 648–676 (2009). https://doi.org/10.1007/s11425-008-0175-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0175-z

Keywords

MSC(2000)

Search

Navigation