
Mathematics 2006
Torsion classes of finite type and spectraAbstract: Given a commutative ring R (respectively a positively graded commutative ring $A=\ps_{j\geq 0}A_j$ which is finitely generated as an A_0algebra), 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\subset Spec R (respectively Y\subset Proj A) of the form Y=\cup_{i\in\Omega}Y_i, with Spec R\Y_i (respectively Proj A\Y_i) quasicompact and open for all i\in\Omega, 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 perf(R) and the torsion classes of finite type in Mod R is established.
