In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the dual module of a left (respectively right)
module ''M'' over a
ring ''R'' is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
left (respectively right) ''R''-module homomorphisms from ''M'' to ''R'' with the
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...
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
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
. A
torsionless module is one for which the canonical homomorphism is
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
.
Example: If
is a finite commutative
group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups hav ...
represented by a
Hopf algebra ''A'' over a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
''R'', then the
Cartier dual is the Spec of the dual ''R''-module of ''A''.
See also
*
Dual sheaf of a sheaf of modules
References
Module theory
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