standard duality on modules over algebras
Let be a field and let be an associative unital algebra. Throughout we will assume that all -modules over are unital. If is a right -module, then the space of all linear mappings
can be equipped with a left -module structure as follows: for any and put
Note that action direction need to be reversed, because
Analogously takes left -modules to right -modules. Also this action is compatible with functoriality of , which means that it takes -homomorphisms to -homomorphisms. In particular we obtain a (contravariant) functor from category of left (right) -modules to category of right (left) -modules. Obviously does not change the dimension of spaces, so we have well defined functors
Proof. Let be a finite dimensional -module. We need to define a natural isomorphism between and . Indeed, define
We will show that each is an isomorphism.
is an epimorphism. Indeed, let be a linear mapping. We need to show, that there is such that
for any . Since is finite dimensional, then let be a -basis of . Of course is a basis of , where is given by if and otherwise. Define
We leave it as a simple exercise, that .
What remains is to prove, that is natural. Consider an -homomorphism . We need to show that the following diagram commutes:
Indeed, if , then let . We have that
and evaluating this at we have
In particular we obtain that
which means that
which completes the proof.
|Title||standard duality on modules over algebras|
|Date of creation||2013-12-11 15:25:39|
|Last modified on||2013-12-11 15:25:39|
|Last modified by||joking (16130)|