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On Flasque Sheaves and Flasque Modules

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Abstract

We show that each sheaf of modules admits a flasque hull, such that homomorphisms into flasque sheaves factor over the flasque hull. On the other hand, we give examples of modules over non-noetherian rings that do not inject into flasque modules. This reveals the impossibility to extend the proof of Serre’s vanishing result for affine schemes with flasque quasicoherent resolutions to the non-noetherian setting. However, we outline how hypercoverings can be used for a reduction to the noetherian case.

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Acknowledgements

I wish to thank the referee and the editor for useful suggestions, and for pointing out Quillen’s small object argument in [11] and Kempf’s proof for Serre Vanishing in [13].

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  1. Mathematisches Institut, Heinrich-Heine-Universität, 40204, Düsseldorf, Germany

    Stefan Schröer

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  1. Stefan Schröer
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Correspondence to Stefan Schröer.

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Communicated by Amnon Neeman.

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Schröer, S. On Flasque Sheaves and Flasque Modules. Appl Categor Struct 26, 1113–1122 (2018). https://doi.org/10.1007/s10485-018-9520-8

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  • DOI: https://doi.org/10.1007/s10485-018-9520-8

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