Support and injective resolutions of complexes over commutative rings
Xiao-Wu Chen, Srikanth B. Iyengar
TL;DR
This paper explores the limitations of using minimal semi-injective resolutions to determine the support of complexes over commutative noetherian rings, highlighting that localizations may not preserve semi-injectivity.
Contribution
It provides explicit examples demonstrating that support cannot always be inferred from minimal semi-injective resolutions and that localizations of semi-injective complexes may lose semi-injectivity.
Findings
Support of complexes may not be read from minimal semi-injective resolutions.
Localization of semi-injective complexes may not remain semi-injective.
Examples illustrate the failure of certain expected properties in homological algebra.
Abstract
Examples are given to show that the support of a complex of modules over a commutative noetherian ring may not be read off the minimal semi-injective resolution of the complex. The same examples also show that a localization of a semi-injective complex need not be semi-injective.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Homotopy and Cohomology in Algebraic Topology