Dual module
≈ 2 min read
Names in other languages
No names in other languages are held for this entry yet.
Article bodies are English and are not machine-translated. A name in another language is not a translation of the article.
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.
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.
See also
Dual sheaf of a sheaf of modules
References
Text derived from https://en.wikipedia.org/w/index.php?oldid=1294003972 under CC BY-SA 4.0; modifications have been made.