Duality between injective envelopes and flat covers
Ville Puuska
TL;DR
This paper explores a duality between injective envelopes and flat covers over commutative Noetherian rings, revealing that a morphism is an injective envelope if and only if its Matlis dual is a flat cover, but the converse does not always hold.
Contribution
It establishes a specific duality between injective envelopes and flat covers and clarifies the limitations of this duality when roles are swapped.
Findings
A morphism is an injective envelope iff its Matlis dual is a flat cover.
Swapping injective envelopes and flat covers breaks the duality in general.
The duality provides new insights into module theory over Noetherian rings.
Abstract
We establish a duality between injective envelopes and flat covers over a commutative Noetherian ring. One case of this duality states that a morphism is an injective envelope, if and only if its Matlis dual is a flat cover. We also show that if we swap injective envelopes and flat covers in this duality, neither implication is true in general.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models