Relative homological algebra in the category of quasi-coherent sheaves

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Abstract

In this paper, we prove the existence of a flat cover and of a cotorsion envelope for any quasi-coherent sheaf over a scheme (X, OX). Indeed we prove something more general. We define what it is understood by the category of quasi-coherent R-modules, where R is a representation by rings of a quiver Q, and we prove the existence of a flat cover and a cotorsion envelope for quasi-coherent R-modules. Then we use the fact that the category of quasi-coherent sheaves on (X, OX is equivalent to the category of quasi-coherent R-modules for some Q and R to get our result. © 2004 Elsevier Inc. All rights reserved.

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Enochs, E., & Estrada, S. (2005). Relative homological algebra in the category of quasi-coherent sheaves. Advances in Mathematics, 194(2), 284–295. https://doi.org/10.1016/j.aim.2004.06.007

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