Abstract
Let C be a triangulated category with a proper class E of triangles. We prove that there exists an Avramov–Martsinkovsky type exact sequence in C, which connects E-cohomology, E-Tate cohomology and E-Gorenstein cohomology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asadollahi, J., Salarian, Sh.: Cohomology theories based on Gorenstein injective modules. Trans. Amer. Math. Soc., 358, 2183–2203 (2006)
Asadollahi, J., Salarian, Sh.: Gorenstein objects in triangulated categories. J. Algebra, 281, 264–286 (2004)
Asadollahi, J., Salarian, Sh.: Tate cohomology and Gorensteinness for triangulated categories. J. Algebra, 299, 480–502 (2006)
Auslander, M., Bridger, M.: Stable Module Theory, Mem. Amer. Math. Soc., vol.94, American Mathematical Society, Providence, R.I., 1969
Avramov, L. L., Martsinkovsky, A.: Absolute, relative and Tate cohomology of modules of finite Gorenstein dimensions. Proc. London Math. Soc., 85, 393–440 (2002)
Beligiannis, A.: Relative homological algebra and purity in triangulated categories. J. Algebra, 227, 268–361 (2000)
Christensen, L. W.: Gorenstein Dimensions, Lecture Notes in Mathematics, vol. 1747, Springer-Verlag, Berlin, 2000
Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules. Math. Z., 220, 611–633 (1995)
Enochs, E. E., Jenda, O. M. G.: Relative homological algebra, De Gruyter Expositions in Mathematics no. 30, Walter De Gruyter, New York, 2000
Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra, 189, 167–193 (2004)
Neeman, A.: Triangulated Categories, Ann. of Math. Stud., vol. 148, Princeton Univ. Press, Princeton, 2001
Puppe, D.: On the structure of stable homotopy theory. In Colloquim on Algebraic Topology, Aarhus Universitet Matematisk Institut, 1962, 65–71
Quillen, D.: Algebraic K-Theory I. Proc. Conf., Battelle Memorial Inst., Seattle,WA, 1972. In: Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, 85–147
Ren, W., Liu, Z. K.: Balance of Tate cohomology in triangulated categories. Appl. Categ. Structures, 23 6, 819–828 (2015)
Ren, W., Liu, Z. K.: Gorenstein homological dimensions for triangulated categories. J. Algebra, 410, 258–276 (2014)
Veliche, O.: Gorenstein projective dimension for complexes. Trans. Amer. Math. Soc., 358, 1257–1283 (2006)
Verdier, J. L.: Catégories Dérivées, Etat 0. Lecture Notes in Math., vol. 569, Springer-Verlag, 1977, 262–311
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (Grant Nos. 11401476, 11361052, 11261050)
Rights and permissions
About this article
Cite this article
Ren, W., Zhao, R.Y. & Liu, Z.K. Cohomology theories in triangulated categories. Acta. Math. Sin.-English Ser. 32, 1377–1390 (2016). https://doi.org/10.1007/s10114-016-5280-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-016-5280-2