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

TL;DR

This paper explores the local and colocal properties of modules and complexes over commutative rings, introducing the concept of antilocality and analyzing its implications in homological algebra and cotorsion theories.

Contribution

It introduces the notion of antilocality in modules and complexes, providing new insights into how global properties are controlled locally in commutative algebra.

Findings

Antilocality applies to injectivity, contraadjustedness, and cotorsion properties.

Certain classes like flat contraadjusted modules are proven to be antilocal.

The paper establishes conditions under which classes are local or colocal in cotorsion theories.

Abstract

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles.