Dual module
In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of module homomorphisms from M to R with the pointwise right (respectively left) module structure.[1][2] The dual module is typically denoted M∗ or HomR(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 [math]\displaystyle{ G = \operatorname{Spec}(A) }[/math] is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring k, then the Cartier dual [math]\displaystyle{ G^D }[/math] is the Spec of the dual k-module of A.
References
- ↑ Nicolas Bourbaki (1974). Algebra I. Springer. ISBN 9783540193739.
- ↑ Serge Lang (2002). Algebra. Springer. ISBN 978-0387953854.
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