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 ) (\mathcal{A},\mathcal{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 ) (\mathcal{A},\mathcal{T}) , and R G R_{\mathcal{G}} 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 [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R G R_{\mathcal{G}} has projective dimension at most one as an 𝑅-module. Moreover, we show that 𝒜 is covering if and only if both the localisation R G R_{\mathcal{G}} and the quotient rings R / J R/J are perfect rings for every J ∈ G J\in\mathcal{G} . Rings satisfying the latter two conditions are called 𝒢-almost perfect.

$\mathcal{P}_1$-covers over commutative rings

“…In a forthcoming paper [BLG20] we consider 1-tilting cotorsion pairs (A, B) over commutative rings R and characterise the rings over which A is covering using a purely algebraic approach.…”