Induced module

Let be a -monoid, be a -submonoid, and be a -module. The -module induced by the -module is a canonical way of extending to accomodate a representation of , as formalized by the Universal property.¹ We have the adjunction

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with the Restricted module and more generally we can consider Change of ring along a ring homomorphism.

Universal property

Let be -ring, be a -subring, and be a -module. The -module induced by the -module is a pair consisting of an -module and a -Module homomorphism such that given any -module a -module homomorphism factorizes uniquely through #m/def/ralg

such that is an -module homomorphism. This admits a unique extension to a functor such that becomes a natural transformation.

Construction

Let be the -tensor product with the bilinear map . Let denote the vector subspace generated by any elements of the form

for any , , and . We construct the induced module as the quotient vector space

with its natural projection . The map

defines a representation of , and the inclusion is given by

Proof of the universal property

Let be an -module and be a -module homomorphism. Then for the above diagram to commute, we require that for . For to be an -module homomorphism, it follows for and . Since elements of this form span , this fully defines , hence it is unique.

Graded structure

Let be a -graded -monoid, be unital graded subalgebra, and be a graded -module. Then has a natural graded structure, where for any and , .