- Volume 53 Issue 2
Recently a construction of local cohomology functors for compactly generated triangulated categories admitting small coproducts is introduced and studied by Benson, Iyengar, Krause, Asadollahi and their coauthors. Following their idea, we introduce the depth of objects in such triangulated categories and get that when (R, m) is a graded-commutative Noetherian local ring, the depth of every cohomologically bounded and cohomologically finite object is not larger than its dimension.
- J. Asadollahi, S. Salarian, and R. Sazeedeh, On the local cohomology and support for triangulated categories, Kyoto J. Math. 51 (2011), no. 4, 811-829. https://doi.org/10.1215/21562261-1424866
- L. L. Avramov and S. B. Iyengar, Constructing modules with prescribed cohomological support, Illinois J. Math. 51 (2007), no. 1, 1-20.
- D. J. Benson, S. Iyengar, and H. Krause, Local cohomology and support for triangulated categories, Ann. Sci. Ec. Norm. Super. 41 (2008), no. 4, 573-619.
- D. J. Benson, S. Iyengar, and H. Krause, Colocalizing subcategories and cosupport, J. Reine Angew. Math. 673 (2012), 161-207.
- M. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geo-metric applications, Cambridge Studies in advanced Mathematics No. 60, Cambridge University Press, Cambridge, 1998.
- W. Burns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in advanced Math-ematics 39, Cambridge University Press, Cambridge, 1998.
- H.-B. Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra 15 (1979), no. 2, 149-172. https://doi.org/10.1016/0022-4049(79)90030-6
- H.-B. Foxby and S. Iyengar, Depth and amplitude for unbounded complexes, Commuta-tive algebra (Grenoble/Lyon, 2001), 119-137, Contemp. Math., 331, Amer. Math. Soc., Providence, RI, 2003 https://doi.org/10.1090/conm/331/05906
- R. Hartshorne, Residues and duality, Lecture Notes Math. 20, Springer-Verlag, New York, 1966.
- H. Krause, Localization for triangulated categories. Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge University Press, Cambridge, 2010.
- J. Lipman, Lectures on local cohomology and duality, Local cohomology and its applications (Guanajuato, 1999), 39-89, Lecture Notes in Pure and Appl. Math., 226, Dekker, New York, 2002.
- A. Neeman, Triangulated categories, Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001.
Supported by : National Natural Science Foundation of China