Dual module explained

In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of left (respectively right) R-module homomorphisms from M to R with the pointwise right (respectively left) module structure.^{[1] [2]} The dual module is typically denoted M^∗ or .

If the base ring R is a field, then a dual module is a dual vector space.

Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective.

Example: If

G = \operatorname{Spec}(A) is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring R, then the Cartier dual G ^{D} is the Spec of the dual R-module of A.

Notes and References

1. Book: Nicolas Bourbaki

   . Algebra I. Nicolas Bourbaki. Nicolas Bourbaki. 1974. Springer. 9783540193739.

2. Book: Serge Lang

   . Algebra. Serge Lang. Serge Lang. 2002. Springer. 978-0387953854.