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QUILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS AND THE RESOLUTION PROPERTY

Part of: Abelian categories Homological algebra (Co)homology theory

Published online by Cambridge University Press:  09 March 2020

SERGIO ESTRADA  and
ALEXANDER SLÁVIK
SERGIO ESTRADA*
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100Murcia, Spain
ALEXANDER SLÁVIK
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 186 75 Prague 8, Czech Republic e-mail: Slavik.Alexander@seznam.cz
*
e-mail: sestrada@um.es
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Abstract

We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to ‘mock’ the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.

Keywords

Resolution propertyvery flat sheafmodel categoryQuillen equivalencehomotopy categoryrestricted flat Mittag-Leffler

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 18E10: Exact categories, abelian categories 18E30: Derived categories, triangulated categories 18G55: Homotopical algebra
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 3 , June 2021 , pp. 302 - 320
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Henderson

The first author is supported by the grant MTM2016-77445-P and FEDER funds and the grant 19880/GERM/15 by the Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia.

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