Lesson plan
Dual module
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. 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.
Article summary
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. 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.
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Discussion questions
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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