Dual module

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

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 k, then the Cartier dual $G^{D}$ is the Spec of the dual k-module of A.

References

^ N. Bourbaki (1974). Algebra I. Springer. ISBN 9783540193739.
^ S. Lang (2002). Algebra. Springer. ISBN 978-0387953854.

This algebra-related article is a stub. You can help Wikipedia by expanding it.

[1] N. Bourbaki (1974). Algebra I. Springer. ISBN 9783540193739.

[2] S. Lang (2002). Algebra. Springer. ISBN 978-0387953854.

[1]

[2]