Definition: An invertible module over a commutative ring R is a finitely generated projective R-module of rank 1. Equivalently, it is an R-module M such that there exists another R-module N with the property that the tensor product M ⊗_R N is isomorphic to R as an R-module.
Overview: Invertible modules are fundamental objects in commutative algebra and algebraic geometry. They play a key role in the study of line bundles in geometry and are closely related to the Picard group of a ring or scheme. The set of isomorphism classes of invertible modules over a ring R forms an abelian group under the tensor product operation; this group is known as the Picard group of R, denoted Pic(R). Invertible modules generalize the notion of invertible ideals in integral domains.
Etymology/Origin: The term "invertible" stems from the algebraic property that such a module has a tensor inverse. This mirrors the concept of multiplicative inverses in rings: just as a unit in a ring has a multiplicative inverse, an invertible module has a "tensor inverse." The concept emerged naturally in the mid-20th century with the development of sheaf theory and modern algebraic geometry, notably in the works of Jean-Pierre Serre and Alexander Grothendieck.
Characteristics:
- An invertible module is projective of rank 1, meaning it is locally free of rank 1.
- For a commutative ring R, a module M is invertible if and only if it is finitely generated, projective, and for every prime ideal 𝔭 of R, the localization M_𝔭 is a free R_𝔭-module of rank 1.
- The tensor product of two invertible modules is again invertible.
- Every invertible module over a local ring is free of rank 1.
- In the context of Dedekind domains, the invertible ideals (i.e., nonzero fractional ideals) correspond precisely to invertible modules, and the Picard group is equivalent to the ideal class group.
Related Topics:
- Picard group
- Projective module
- Locally free sheaf
- Line bundle
- Fractional ideal
- Ring theory
- Scheme theory
- Commutative algebra
- Sheaf cohomology
Source: Standard references include "Commutative Algebra" by Bourbaki, "Algebraic Geometry" by Robin Hartshorne, and "Éléments de géométrie algébrique" by Grothendieck.