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
{\mathcal{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
{\mathcal{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
{\mathcal{G}}
, we show that if
𝒜
{\mathcal{A}}
is covering, then
𝒢
{\mathcal{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
{\mathcal{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
{\mathcal{G}}
-almost perfect.
Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups
