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Invertible module

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated May 4, 2024

In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible sheaves in algebraic geometry.

Formally, a finitely generated module M over a ring R is said to be invertible if it is locally a free module of rank 1. In other words, M_{P}\cong R_{P} for all primes P of R. Now, if M is an invertible R-module, then its dual M* = Hom(M,R) is its inverse with respect to the tensor product, i.e. M\otimes _{R}M^{*}\cong R.

The theory of invertible modules is closely related to the theory of codimension one varieties including the theory of divisors.

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This article is adapted from the Wikipedia article “Invertible module”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Picard group

In mathematics, the Picard group of a ringed space X, denoted by Pic, is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.