Generalized Local Homology Modules of Complexes

Author
Abstract
The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing these modules by using Gorenstein flat resolutions. We also find a lower bound for vanishing of generalized local homology modules over a commutative Noetherian ring and we give an upper bound for vanishing of these modules over a commutative Noetherian ring possessing a dualizing complex.
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J. Bartijn, Flatnes, completions, regular sequences un m´enage`a trois, Thesis, Utrecht (1985). L.W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin (2000). L.W. Christensen and H. Holm, Ascent properties of Auslander categories, Canad. J. Math. , 61(1) (2009) 76-108. L.W. Christensen, A. Frankild and H. Holm, On Gorenstein projective, injective and flat dimensions—a functorial description with applications, J. Algebra, 302(1) (2006) 231-279. E.E. Enoch and O.M.G. Jenda, Relative homological algebra, de Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin (2000). H-B. Foxby, Hyperhomological algebra & commutative rings, in preparation. A. Frankild, Vanishing of local homology, Math. Z., 244 (2003) 615-630. J.P.C. Greenlees and J.P. May, Derived functors of the I-adic completion and local homology, J. Algebra, 149 (1992) 438-453. H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004) 167-193. J. Lipman, A.J. L´opez and L.A. Tarrio, Local homology and cohomology on schemes, Ann. Sci. Ecole Norm. Sup. , 30(4) (1997) 1-39. E. Matlis, The Koszul complex and duality, Comm. Algebra, 1(2) (1974) 87-144. F. Mohammadi Aghjeh Mashhad and K. Divaani-Aazar, local homology and Gorenstein flat modules, Journal of Algebra and its Applications, 11(2) (2012) 12500221-12500228. T.T. Nam, Generalized local homology modules for artinian modules, Algebra Colloquium, 19(1) (2012) 1205-1212. T.T. Nam, Left derived functors of the generalized I-adic completion and generalized local homology, comm. In Algebra, 38 (2010) 440-453. A-M. Simon, Some homological properties of complete modules, Math. Proc. Cambridge Philos. Soc., 108 (2) (1990) 231-246. S. Yassemi, Generalized section functors, J. Pure Appl. Algebra, 95(1) (1994) 103-119.