Rocky Mountain Journal of Mathematics

Local homology, Koszul homology and Serre classes

Kamran Divaani-Aazar, Hossein Faridian, and Massoud Tousi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Given a Serre class $\mathcal {S}$ of modules, we compare the containment of Koszul homology, Ext modules, Tor modules, local homology and local cohomology in $\mathcal {S}$ up to a given bound $s \geq 0$. As applications, we give a full characterization of Noetherian local homology modules. Furthermore, we establish a comprehensive vanishing result which readily leads to formerly known descriptions of the numerical invariants' width and depth in terms of Koszul homology, local homology and local cohomology. In addition, we recover a few renowned vanishing criteria scattered throughout the literature.

Article information

Rocky Mountain J. Math., Volume 48, Number 6 (2018), 1841-1869.

First available in Project Euclid: 24 November 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13C05: Structure, classification theorems 13D07: Homological functors on modules (Tor, Ext, etc.) 13D45: Local cohomology [See also 14B15]

Koszul homology module local cohomology module local homology module Serre class


Divaani-Aazar, Kamran; Faridian, Hossein; Tousi, Massoud. Local homology, Koszul homology and Serre classes. Rocky Mountain J. Math. 48 (2018), no. 6, 1841--1869. doi:10.1216/RMJ-2018-48-6-1841.

Export citation


  • M. Aghapournahr and L. Melkersson, A natural map in local cohomology, Ark. Matem. 48 (2010).
  • ––––, Cofiniteness and coassociated primes of local cohomology modules, Math. Scand. 105 (2009), 161–170.
  • ––––, Local cohomology and Serre subcategories, J. Algebra 320 (2008), 1275–1287.
  • M. Asgharzadeh and M. Tousi, A unified approach to local cohomology modules using Serre classes, Canadian Math. Bull. 53 (2010), 577–586.
  • K. Bahmanpour and M. Aghapournahr, A note on cofinite modules, Comm. Algebra 44 (2016), 3683–3691.
  • K. Bahmanpour, I. Khalili and R. Naghipour, Cofiniteness of torsion functors of cofinite modules, Colloq. Math. 136 (2014), 221–230.
  • L.W. Christesen, Gorenstein dimensions, Lect. Notes Math. 1747 (2000).
  • N.T. Cuong and T.T. Nam, A local homology theory for linearly compact modules, J. Algebra 319 (2008), 4712–4737.
  • ––––, The I-adic completion and local homology for Artinian modules, Math. Proc. Cambr. Philos. Soc. 131 (2001), 61–72.
  • D. Delfino and T. Marley, Cofinite modules and local cohomology, J. Pure Appl. Alg. 121 (1997), 45–52.
  • K. Divaani-Aazar, H. Faridian and M. Tousi, Local homology, finiteness of Tor modules and cofiniteness, J. Alg. Appl. 16 (2017), 1750240.
  • A. Frankild, Vanishing of local homology, Math. Z. 244 (2003), 615–630.
  • J.P.C. Greenlees and J.P. May, Derived functors of $I$-adic completion and local homology, J. Algebra 149 (1992), 438–453.
  • M. Hatamkhani and K. Divaani-Aazar, On the vanishing of local homology modules, Glasgow Math. J. 55 (2013), 457–464.
  • G. Huneke and J. Koh, Cofiniteness and vanishing of local cohomology modules, Math. Proc. Cambr. Philos. Soc. 110 (1991), 421–429.
  • T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscr. Math. 104 (2001), 519–525.
  • F.M.A. Mashhad and K. Divaani-Aazar, Local homology and Gorenstein flat modules, J. Alg. Appl. 11 (2012), 1250022.
  • E. Matlis, The higher properties of $R$-sequences, J. Algebra 50 (1978), 77–112.
  • ––––, The Koszul complex and duality, Comm. Algebra 1 (1974), 87–144.
  • L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra 285 (2005), 649–668.
  • ––––, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambr. Philos. Soc. 107 (1990), 267–271.
  • A.S. Richardson, Co-localization, co-support and local homology, Rocky Mountain J. Math. 36 (2006), 1679–1703.
  • J.J. Rotman, An introduction to homological algebra, Universitext, Springer, New York, 2009.
  • S. Sather-Wagstaff and R. Wicklein, Support and adic finiteness for complexes, Comm. Alg. 45 (2017), 2569–2592.
  • P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), 161–180.
  • A.M. Simon, Some homological properties of complete modules, Math. Proc. Cambr. Philos. Soc. 108 (1990), 231–246.
  • ––––, Adic-completion and some dual homological results, Publ. Mat. Cambr. 36 (1992), 965–979.
  • M. Stokes, Some dual homological results for modules over commutative rings, J. Pure Appl. Alg. 65 (1990), 153–162.
  • L.A. Tarrío, A.J. López and J. Lipman, Local homology and cohomology on schemes, Ann. Sci. Ecole Norm. 30 (1997), 1–39.
  • W. Vasconcelos, Devisor theory in module categories, North-Holland, Amsterdam, 1974.
  • S. Yassemi, Coassociated primes, Comm. Algebra 23 (1995), 1473–1498.
  • K.-I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimension one, Nagoya Math. J. 147 (1997), 179–191.
  • H. Zöschinger, Der Krullsche Durchschnittssatz für kleine Untermoduln, Arch. Math. 62 (1994), 292–299.