Induced module

Let $𝐴$ be a $𝕂$ -monoid, $𝐵 \leq 𝐴$ be a $𝕂$ -submonoid, and $𝑉$ be a $𝐵$ -module. The $𝐴$ -module induced by the $𝐵$ -module $𝑉$ is a canonical way of extending $𝑉$ to accomodate a representation of $𝐴$ , 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 $𝐴$ be $𝕂$ -ring, $𝐵 \leq 𝐴$ be a $𝕂$ -subring, and $𝑉$ be a $𝐵$ -module. The $𝐴$ -module induced by the $𝐵$ -module $𝑉$ is a pair consisting of an $𝐴$ -module $I n d 𝐴 𝐵 𝑉 = 𝐴 \otimes 𝐵 𝑉$ and a $𝐵$ -Module homomorphism $𝜄 : 𝑉 \to I n d 𝐴 𝐵 𝑉$ such that given any $𝐴$ -module $𝑊$ a $𝐵$ -module homomorphism $𝑓 : 𝑉 \to 𝑊$ factorizes uniquely through $𝜄$ #m/def/ralg

such that $¯ 𝑓 : I n d 𝐴 𝐵 𝑉 \to 𝑊$ is an $𝐴$ -module homomorphism. This admits a unique extension to a functor $I n d 𝐴 𝐵 : 𝐵 𝖬 𝗈 𝖽 \to 𝐴 𝖬 𝗈 𝖽$ such that $𝜄 : 1 \Rightarrow I n d 𝐴 𝐵 : 𝐵 𝖬 𝗈 𝖽 \to 𝐵 𝖬 𝗈 𝖽$ becomes a natural transformation.

Construction

Let $𝐴 \otimes 𝕂 𝑉$ be the $𝕂$ -tensor product with the bilinear map $(\otimes) : 𝐴 \times 𝑉 \to 𝐴 \otimes 𝕂 𝑉$ . Let $𝐾$ denote the vector subspace generated by any elements of the form

𝑎 𝑏 \otimes 𝑣 - 𝑎 \otimes 𝑏 \cdot 𝑣

for any $𝑎 \in 𝐴$ , $𝑏 \in 𝐵$ , and $𝑣 \in 𝑉$ . We construct the induced module as the quotient vector space

𝐴 \otimes 𝐵 𝑉 = 𝐴 \otimes 𝕂 𝑉 𝐾

with its natural projection $𝜋 : 𝐴 \otimes 𝕂 𝑉 ↠ 𝐴 \otimes 𝐵 𝑉$ . The map

(\otimes 𝐵) = 𝜋 \circ (\otimes 𝕂)

defines a representation of $𝐴$ , and the inclusion is given by

𝜄 : 𝑉 ↪ 𝐴 \otimes 𝐵 𝑉 𝑣 \mapsto 1 \otimes 𝐵 𝑣

Proof of the universal property

Let $𝑊$ be an $𝐴$ -module and $𝑓 : 𝑉 \to 𝑊$ be a $𝐵$ -module homomorphism. Then for the above diagram to commute, we require that $¯ 𝑓 (𝜄 (𝑣)) = ¯ 𝑓 (1 \otimes 𝐵 𝑣) = 𝑓 (𝑣)$ for $𝑣 \in 𝑉$ . For $¯ 𝑓$ to be an $𝐴$ -module homomorphism, it follows $¯ 𝑓 (𝑎 \otimes 𝐵 𝑣) = 𝑎 \cdot 𝑓 (𝑣)$ for $𝑎 \in 𝐴$ and $𝑣 \in 𝑉$ . Since elements of this form span $𝐴 \otimes 𝐵 𝑉$ , this fully defines $¯ 𝑓$ , hence it is unique.

Graded structure

Let $𝐴$ be a $𝔄$ -graded $𝕂$ -monoid, $𝐵 \leq 𝐴$ be unital graded subalgebra, and $𝑉$ be a graded $𝐵$ -module. Then $I n d 𝐴 𝐵 𝑉$ has a natural graded structure, where for any $𝑎 \in 𝐴 𝛼$ and $𝑣 \in 𝑉 𝛽$ , $d e g (𝑎 \otimes 𝐵 𝑣) = 𝛼 + 𝛽$ .

Induced module

Universal property

Construction

Graded structure

See also

Footnotes