Dual module

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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 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 is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring k, then the Cartier dual is the Spec of the dual k-module of A.

References[edit]

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