Homological dimensions of complexes of R-modules

Subject Areas : Applied Mathematics

N. Tayarzadeh ¹ , Esmaiel Hosseini ² , Sh. Niknejad ³

1 - Department of Mathematics, Islamic Azad University, Gachsaran branch, Gachsaran, Iran.
2 - Department of Mathematics, Islamic Azad University, Gachsaran branch, Gachsaran, Iran.
3 - Department of Mathematics, Islamic Azad University, Gachsaran Branch, Gachsaran, Iran.

Received: 2013-02-03 Accepted : 2013-08-22 Published : 2015-05-01

Keywords:

Abstract :

Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.

References:

[1] M. Ansari and E. Hosseini, The behavior of homological
dimensions, Mathematics Scientic Journal Vol.7, No. 1, (2011),
1-10.
[2] S. E. Dickson, A torsion theory for abelian categories, Trans.
Amer. Math. Soc. 121, (1966), 223-235.
[3] E. Enochs, J. Garca Rozas, Flat covers of complexes, J. Algebra,
210,(1998), 86-102.
[4] P. Eklof, J. Trlifaj, How to make Ext vanish, Bull. London
Math. Soc, 33, no. 1 (2001), 44-51.
[5] R. Gobel, S. Shelah, Cotorsion theories and spliters, Trans.
Amer. Math. Soc. 352, No. 11, (2000), 5357-5379.
102

[6] L. Salce, Cotorsion theories for abelian groups, Symposia

Mathematica, Vol. XXIII (Conf. Abelian Groups and their
Relationship to the Theory of Modules, INDAM, Rome, 1977),
Academic Press, London, 1979, pp.11-32. 11.
[7] D. Simson, A remark on projective dimension of at modules,
Math. Ann. 209 (1974), 181-182.