Given a ring R, two classes A and B of R-modules are said to form a cotorsion pair (A, beta) in Mod R if A = Ker Ext(R)(1)(-, beta) and beta = KerExt(R)(1)(A, -). We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of R. We show that Findim R < infinity if and only if the following dimensions are finite for some cotorsion pair (A, beta) in Mod R: the relative projective dimension of A with respect to itself, and the A-resolution dimension of the category P of all R-modules of finite projective dimension. Moreover, we obtain an analogous result for findim R, and we characterize when Findim R = findim R.
Homological dimensions in cotorsion pairs
ANGELERI, LIDIA;
2009-01-01
Abstract
Given a ring R, two classes A and B of R-modules are said to form a cotorsion pair (A, beta) in Mod R if A = Ker Ext(R)(1)(-, beta) and beta = KerExt(R)(1)(A, -). We investigate relative homological dimensions in cotorsion pairs. This can be applied to study the big and the little finitistic dimension of R. We show that Findim R < infinity if and only if the following dimensions are finite for some cotorsion pair (A, beta) in Mod R: the relative projective dimension of A with respect to itself, and the A-resolution dimension of the category P of all R-modules of finite projective dimension. Moreover, we obtain an analogous result for findim R, and we characterize when Findim R = findim R.
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