Torsion classes of finite type and spectra

Abstract

Given a commutative ring R (respectively a positively graded commutative ring A = ⊕j ≥ 0 Aj which is finitely generated as an A_0-algebra), a bijection between the torsion classes of finite type in Mod R (respectively tensor torsion classes of finite type in QGr A) and the set of all subsets Y ⊆ spec R (respectively Y ⊆ Proj A) of the form Y = ∪i ∈ Ω_Yi, with Spec R \Yi (respectively Proj A \Yi) quasi-compact and open for all i ∈ Ω, is established. Using these bijections, there are constructed isomorphisms of ringed spaces

(Spec R,OR) → (Spec(Mod R),OMod_R_)

and

(Proj A,OProj A) → (Spec(QGr A),OQGr A),

where (Spec(Mod R),OMod_R_) and (Spec(QGr A),OQGr A) are ringed spaces associated to the lattices _L_tor(Mod R) and _L_tor(QGr A) of torsion classes of finite type. Also, a bijective correspondence between the thick subcategories of perfect complexes Dper(R) and the torsion classes of finite type in Mod R is established.