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Local, colocal, and antilocal properties of modules and complexes over commutative rings

Leonid Positselski
Dec 2022
摘要
This paper is a commutative algebra introduction to the homological theory ofquasi-coherent sheaves and contraherent cosheaves over quasi-compactsemi-separated schemes. Antilocality is an alternative way in which globalproperties are locally controlled in a finite affine open covering. Forexample, injectivity of modules over non-Noetherian commutative rings is notpreserved by localizations, while homotopy injectivity of complexes of modulesis not preserved by localizations even for Noetherian rings. The latter alsoapplies to the contraadjustedness and cotorsion properties. All the mentionedproperties of modules or complexes over commutative rings are actuallyantilocal. They are also colocal, if one presumes contraadjustedness.Generally, if the left class in a (hereditary complete) cotorsion theory formodules or complexes of modules over commutative rings is local and preservedby direct images with respect to affine open immersions, then the right classis antilocal. If the right class in a cotorsion theory for contraadjustedmodules or complexes of contraadjusted modules is colocal and preserved by suchdirect images, then the left class is antilocal. As further examples, the classof flat contraadjusted modules is antilocal, and so is the class of acycliccomplexes of contraadjusted modules.
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