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Torsion classes of finite type and spectraGrigory Garkusha and Mike Prest1 IntroductionNon-commutative geometry comes in various flavours. One is based on abelian and triangulatedcategories, the latter being replacements of classical schemes. This is basedon classical results of Gabriel and later extensions, in particular byThomason. Precisely,Gabriel [6] proved that any noetherian scheme X can be reconstructed uniquely up toisomorphism from the abelian category, Qcoh X, of quasi-coherent sheaves over X.This reconstruction result has been generalized to quasi-compact schemes by Rosenbergin [15]. Based on Thomason’s classification theorem, Balmer [3] reconstructs anoetherian scheme X from the triangulated category of perfect complexes D per .X/.This result has been generalized to quasi-compact, quasi-separated schemes by Buan–Krause–Solberg [5].In this paper we reconstruct affine and projective schemes from appropriate abeliancategories. Our approach, similar to that used in [8], [9], is different from Rosenberg’s[15] and less abstract. Moreover, some results of the paper are of independentinterest.Let Mod R (respectively QGr A) denote the category of R-modules (respectivelygraded A-modules modulo torsion modules) with R (respectively A D L n0 A n)acommutative ring (respectively a commutative graded ring). We first demonstrate thefollowing result (cf. [8], [9]).Theorem (Classification). Let R (respectively A) be a commutative ring (respectivelycommutative graded ring which is finitely generated as an A 0 -algebra). Then the mapsV 7! S DfM 2 Mod R j supp R .M / V g;S 7! V D [ M 2Ssupp R .M /andV 7! S DfM 2 QGr A j supp A .M / V g;S 7! V D [ M 2Ssupp A .M /induce bijections between1. the set of all subsets V Spec R (respectively V Proj A) of the form V DSi2 Y i with Spec R n Y i (respectively Proj A n Y i ) quasi-compact and open forall i 2 , This paper was written during the visit of the first author to the University of Manchester supported bythe MODNET Research Training Network in Model Theory. He would like to thank the University for thekind hospitality.