Covering classes and $1$-tilting cotorsion pairs over commutative rings

Abstract

We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair (A, T ) provides for covers, that is when the class A is a covering class. We use Hrbek's bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if G is the Gabriel topology associated to the 1-tilting cotorsion pair (A, T ), and RG is the ring of quotients with respect to G, we show that if A is covering then G is a perfect localisation (in Stenström's sense [Ste75]) and the localisation RG has projective dimension at most one. Moreover, we show that A is covering if and only if both the localisation RG and the quotient rings R/J are perfect rings for every J ∈ G. Rings satisfying the latter two conditions are called G-almost perfect.

Flat commutative ring epimorphisms of almost Krull dimension zero

“…Let G denote the Gabriel filter of ideals in R related to u. We show that pd R U ≤ 1 whenever the quotient ring R/I is semilocal of Krull dimension zero for every ideal I ∈ G. The argument is based on the notion of I-contramodule R-modules for an ideal I in a commutative ring R. This result of ours has already found its uses in the work of Bazzoni and Le Gros on envelopes and covers in the tilting cotorsion pairs related to 1-tilting modules over commutative rings; see [4,Remark 8.6 and Theorem 8.7] and [5,Remark 8.2 and Theorem 8.17].…”

“…We will say that the ring R is F-h-local [9, Section IV.3], [5,Section 7] if every ideal I ∈ F is contained only in finitely many maximal ideals of R and every prime ideal of R belonging to F is contained in a unique maximal ideal. In other words, R is F-h-local if and only if for every I ∈ F the quotient ring R/I is semilocal and every prime ideal of R/I is contained in a unique maximal ideal.…”

“…Let R be a commutative ring with a filter of ideals F. We will say that R is F-h-nil [6, Section 6], [5,Section 7] if every ideal I ∈ F is contained only in finitely many maximal ideals of R and all the prime ideals of R belonging to F are maximal. In other words, R is F-h-nil if and only if for every I ∈ F the quotient ring R/I is semilocal of Krull dimension zero.…”