Vol. 287, No. 1, 2017

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ISSN: 0030-8730
On locally coherent hearts

Manuel Saorín

Vol. 287 (2017), No. 1, 199–221
Abstract

Let G be a locally coherent Grothendieck category. We show that, under particular conditions, if a t-structure τ in the unbounded derived category D(G) restricts to the bounded derived category Db( fp(G)) of its category of finitely presented (i.e, coherent) objects, then its heart τ is a locally coherent Grothendieck category on which τ Db( fp(G)) is the class of finitely presented objects. Those particular conditions are always satisfied when G is arbitrary and τ is the Happel–Reiten–Smalø t-structure in D(G) associated to a torsion pair in  fp(G) or when G =  Qcoh(X) is the category of quasicoherent sheaves on a noetherian affine scheme X and τ is any compactly generated t-structure in D(X) := D( Qcoh(X)) which restricts to Db(X) := Db( coh(X)). In particular, the heart of any t-structure in Db(X) is the category of finitely presented objects of a locally coherent Grothendieck category.

Keywords
locally coherent Grothendieck category, triangulated category, derived category, t-structure, heart of a t-structure
Mathematical Subject Classification 2010
Primary: 18E15, 18E30, 13DXX, 14AXX, 16EXX
Milestones
Received: 19 May 2016
Revised: 20 September 2016
Accepted: 1 October 2016
Published: 6 February 2017
Authors
Manuel Saorín
Departamento de Matemáticas
Universidad de Murcia
Aptdo. 4021
30100 Murcia
Spain