A Zariski-local notion of F-total acyclicity for complexes of sheaves1

Lars Winther Christensen, Sergio Estrada, Alina Iacob

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We study a notion of total acyclicity for complexes of flat sheaves over a scheme. It is Zariski-local—i.e. it can be verified on any open affine covering of the scheme—and for sheaves over a quasi-compact semi-separated scheme it agrees with the categorical notion. In particular, it agrees, in their setting, with the notion studied by Murfet and Salarian for sheaves over a noetherian semi-separated scheme. As part of the study we recover, and in several cases extend the validity of, recent results on existence of covers and precovers in categories of sheaves. One consequence is the existence of an adjoint to the inclusion of these totally acyclic complexes into the homotopy category of complexes of flat sheaves.

Original languageEnglish
Pages (from-to)197-214
Number of pages18
JournalQuaestiones Mathematicae
Issue number2
StatePublished - Apr 3 2017


  • F-total acyclicity
  • Gorenstein flat precover
  • ascent-descent property


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