[[Module theory MOC]]
# Induced module
Let $A$ be a [[K-monoid|$\mathbb K$-ring]], $B \leq A$ be a [[Unital subalgebra|$\mathbb K$-subring]], and $V$ be a $B$-[[Module over a unital associative algebra|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]].[^1988]
We have the adjunction

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![[induced-restricted adjunction.svg#invert|250]]

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

## Universal property

Let $A$ be [[K-monoid|$\mathbb K$-ring]], $B \leq A$ be a [[Unital subalgebra|$\mathbb K$-subring]], and $V$ be a $B$-[[Module over a unital associative algebra|module]]. The $A$-module **induced** by the $B$-module $V$ is a pair consisting of an $A$-module $\Ind_{B}^A V = A \otimes_{B} V$ and a $B$-[[Module homomorphism]] $\iota : V \to \Ind^A_{B} V$
such that given any $A$-module $W$ a $B$-module homomorphism $f : V \to W$
factorizes uniquely through $\iota$ #m/def/falg 

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such that $\bar{f} : \Ind^A_{B} V \to W$ is an $A$-module homomorphism.
This admits a unique extension to a [[functor]] $\Ind^A_{B} : \lMod B \to \lMod A$ such that $\iota : 1 \Rightarrow \Ind^A_{B} : \lMod B \to \lMod B$ becomes a [[natural transformation]].

  [^1988]: 1988\. [[Sources/@frenkelVertexOperatorAlgebras1988|Vertex operator algebras and the Monster]], §1.5, p. 11

## Construction

Let $A \otimes_{\mathbb{K}} V$ be the $\mathbb{K}$-[[Tensor product of vector spaces|tensor product]] with the bilinear map $(\otimes): A \times V \to A \otimes_{\mathbb{K}} V$.
Let $K$ denote the [[vector subspace]] generated by any elements of the form
$$
\begin{align*}
ab \otimes v - a \otimes b \cdot v
\end{align*}
$$
for any $a \in A$, $b \in B$, and $v \in V$.
We construct the induced module as the [[quotient vector space]]
$$
\begin{align*}
A \otimes_{B} V = \frac{A \otimes_{\mathbb{K}} V}{K}
\end{align*}
$$
with its natural projection $\pi : A \otimes_{\mathbb{K}} V \twoheadrightarrow A \otimes_{B} V$.
The map
$$
\begin{align*}
(\otimes_{B}) = \pi \circ (\otimes_{\mathbb{K}})
\end{align*}
$$
defines a representation of $A$,
and the inclusion is given by
$$
\begin{align*}
\iota : V &\hookrightarrow A \otimes_{B} V \\
v &\mapsto 1 \otimes_{B} v
\end{align*}
$$

> [!check]- 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 $\bar f( \iota(v)) = \bar f(1 \otimes_B v) = f(v)$ for $v \in V$.
> For $\bar f$ to be an $A$-module homomorphism, it follows $\bar f(a \otimes_B v) = a \cdot f(v)$ for $a \in A$ and $v \in V$.
> Since elements of this form span $A \otimes_B V$, this fully defines $\bar f$, hence it is unique. <span class="QED"/>

## Graded structure

Let $A$ be a $\mathfrak{A}$-[[Graded algebra|graded]] [[K-monoid]], $B \leq A$ be [[Unital subalgebra|unital]] [[graded subalgebra]],
and $V$ be a [[Graded module|graded]] $B$-[[Module over a unital associative algebra|module]].
Then $\Ind^A_{B} V$ has a natural graded structure, where for any $a \in A_{\alpha}$ and $v \in V_{\beta}$, $\deg(a \otimes_{B} v) = \alpha + \beta$.

## See also

- [[Complexification]]

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