Skip to main content
SpringerLink
Log in

Totally Acyclic Approximations

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Let \(Q \rightarrow R\) be a surjective homomorphism of Noetherian rings such that Q is Gorenstein and R as a Q-bimodule admits a finite resolution by modules which are projective on both sides. We define an adjoint pair of functors between the homotopy category of totally acyclic R-complexes and that of Q-complexes. This adjoint pair is analogous to the classical adjoint pair of functors between the module categories of R and Q. As a consequence, we obtain a precise notion of approximations of totally acyclic R-complexes by totally acyclic Q-complexes.

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

Access this article

Log in via an institution

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. Auslander, M., Bridger, M.: Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence (1969)

  2. Auslander, M., Buchweitz, R.-O.: The homological theory of maximal Cohen–Macaulay approximations, (French summary) Colloque en l’honneur de Pierre Samuel (Orsay, 1987), Mém. Soc. Math. France (N.S.) No. 38, pp. 5–37 (1989)

  3. Auslander, M., Reiten, I.: Homologically finite subcategories. In: Representations of Algebras and Related Topics (Kyoto, 1990), London Mathematical Society Lecture Note Series, 168, pp. 1–42. Cambridge University Press, Cambridge (1992)

  4. Auslander, M., Smalø, S.O.: Preprojective modules over Artin algebras. J. Algebra 66(1), 61–122 (1980)

    Article  MathSciNet  Google Scholar 

  5. Avramov, L.L., Martsinkovsky, A.: Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension. Proc. Lond. Math. Soc. (3) 85(2), 393–440 (2002)

    Article  MathSciNet  Google Scholar 

  6. Bergh, P.A., Jorgensen, D.A.: A generalized Dade’s lemma for local rings. Algebr. Represent. Theory 21(6), 1369–1380 (2018)

    Article  MathSciNet  Google Scholar 

  7. Bergh, P.A., Jorgensen, D.A.: Complete intersections and equivalences with categories of matrix factorizations. Homol. Homotopy Appl. 18(2), 377–390 (2016)

    Article  MathSciNet  Google Scholar 

  8. Bergh, P.A., Jorgensen, D.A., Oppermann, S.: The Gorenstein defect category. Q. J. Math. 66(2), 459–471 (2015)

    Article  MathSciNet  Google Scholar 

  9. Buchweitz, R.-O.: Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings, 155 pp (1987). https://tspace.library.utoronto.ca/handle/1807/16682

  10. Christensen, L.W.: Gorenstein Dimensions. Lecture Notes in Mathematics, vol. 1747. Springer, Berlin (2000)

  11. Christensen, L.W., Foxby, H.B., Holm, H.: Derived category methods in commutative algebra (in preparation)

  12. Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton (1956)

    MATH  Google Scholar 

  13. Dalezios, G., Estrada, S., Holm, H.: Quillen equivalences for stable categories. J. Algebra 501, 130–149 (2018)

    Article  MathSciNet  Google Scholar 

  14. Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260(1), 35–64 (1980)

    Article  MathSciNet  Google Scholar 

  15. Eisenbud, D., Peeva, I.: Minimal Free Resolutions Over complete Intersections. Lecture Notes in Mathematics, vol. 2152. Springer, Cham (2016)

    Book  Google Scholar 

  16. Enochs, E.E.: Injective and flat covers, envelopes and resolvents. Israel J. Math. 39(3), 189–209 (1981)

    Article  MathSciNet  Google Scholar 

  17. Holm, T., Jørgensen, P.: Triangulated categories: definitions, properties, and examples. In: Triangulated Categories. London Mathematical Society Lecture Note Series, 375, pp. 1–51. Cambridge University Press, Cambridge (2010)

  18. Jorgensen, D.A., Şega, L.: Asymmetric complete resolutions and vanishing of ext over Gorenstein rings Internat. Math. Res. Notices 56, 3459–3477 (2005)

    Article  Google Scholar 

  19. Krause, H.: Approximations and adjoints in homotopy categories. Math. Ann. 353(3), 765–781 (2012)

    Article  MathSciNet  Google Scholar 

  20. Krause, H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)

    Article  MathSciNet  Google Scholar 

  21. Lindokken, S.: Private communication

  22. Neeman, A.: Triangulated categories. Annals of Mathematics Studies, 148. Princeton University Press, Princeton (2001)

  23. Neeman, A.: Some adjoints in homotopy categories. Ann. Math. (2) 171(3), 2143–2155 (2010)

    Article  MathSciNet  Google Scholar 

  24. Oppermann, S., Psaroudakis, C., Stai, T.: Change of rings and singularity categories. Adv. Math. 350, 190–241 (2019)

    Article  MathSciNet  Google Scholar 

  25. Orlov, D.: Triangulated categories of singularities and D-branes in Landau–Ginzburg models (in Russian) Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 240–262; translation in Proc. Steklov Inst. Math. 246 (2004), no. 3, 227–248

  26. Steele, N.: Support and rank varieties of totally acyclic complexes. arXiv:1603.02731

  27. Veliche, O.: Gorenstein projective dimension for complexes. Trans. Am. Math. Soc. 358(3), 1257–1283 (2006)

    Article  MathSciNet  Google Scholar 

  28. Zheng, Y., Huang, Z.: Triangulated equivalences involving Gorenstein projective modules. Can. Math. Bull. 60(4), 879–890 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referee for multiple useful comments.

Author information

Authors and Affiliations

  1. Institutt for matematiske fag, NTNU, 7491, Trondheim, Norway

    Petter A. Bergh

  2. Department of mathematics, University of Texas at Arlington, Arlington, TX, 76019, USA

    David A. Jorgensen

  3. Department of Mathematics and Statistics, Wake Forest University, Winston-Salem, NC, 27109, USA

    W. Frank Moore

Authors
  1. Petter A. Bergh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David A. Jorgensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. Frank Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David A. Jorgensen.

Additional information

Communicated by Henning Krause.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bergh, P.A., Jorgensen, D.A. & Moore, W.F. Totally Acyclic Approximations. Appl Categor Struct 29, 729–745 (2021). https://doi.org/10.1007/s10485-021-09633-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-021-09633-1

Keywords

Mathematics Subject Classification

Search

Navigation