# Torsion classes of finite type and spectra

### Grigory Garkusha

University of Wales, Swansea, UK### Mike Prest

University of Manchester, UK

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## 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*,**O***R*) → (Spec(Mod *R*),**O**Mod_R_)

and

(Proj *A*,**O**Proj A) → (Spec(QGr *A*),**O**QGr *A*),

where (Spec(Mod *R*),**O**Mod_R_) and (Spec(QGr *A*),**O**QGr *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 **D**per(*R*) and the torsion classes of finite type in Mod *R* is established.