Free module

Free modules are the free objects in $𝑅 𝖬 𝗈 𝖽$ : modules formed by freely applying module operations to a given set $𝑆$ . #m/def/module Departing from classical tradition, we call a module equipped with an isomorphism from a free module an $𝑆$ -framed module, while a module for which there merely exists such an isomorphism is a $𝑆$ -framable module. Assuming the Axiom of Choice, any module over a division ring is framable: See Assuming choice, every vector space has a basis.

Notation

In these notes, we have two conventions for the free module over $𝑅$ generated by a set $𝑆$ . The first is

𝑅 (𝑆) = s p a n {1 𝑠 : 𝑠 \in 𝑆} \leq 𝑅 𝑆

where we think of elements as maps of finite support $𝑆 \to 𝑅$ , and we identify $𝑠 \in 𝑆$ with $1 𝑠 : 𝑡 \mapsto [𝑠 = 𝑡]$ . The second is

𝑅 \cdot {𝑥 𝑠} 𝑠 \in 𝑆 = s p a n 𝑅 {𝑥 𝑠 : 𝑠 \in 𝑆}

which allows for the explicit naming of the basis to be used.

Being framable is equivalent to the existence of a basis, where by a basis $B$ for an $𝑅$ -module $𝑉$ , we mean an $𝑅$ -spanning set $B$ such that each $𝑣 \in 𝑉$ is given by a unique $𝑅$ -linear combination of $B$ elements.

Universal property

Let $𝑅$ be a ring and $𝑆$ be a set. The free module is a pair consisting of an $𝑅$ -module $𝑅 (𝑆)$ and a function $𝜄 : 𝑆 \to 𝑅 (𝑆)$ such that given any $𝑅$ -module $𝑀$ and function $𝑓 : 𝑆 \to 𝑀$ there exists a unique module homomorphism $¯ 𝑓 : 𝑅 (𝑆) \to 𝑀$ such that the following diagram commutes

This has a unique extension to a functor such that

𝜄 : 1 \Rightarrow 𝑅 (-) : 𝖲 𝖾 𝗍 \to 𝖲 𝖾 𝗍

becomes a natural transformation.

Monoidal functor

When $𝑅 = K$ is a commutative ring this forms a monoidal functor with respect to the cartesian structure on $𝖲 𝖾 𝗍$ and the tensor product on $K 𝖬 𝗈 𝖽$ . #m/thm/cat If $𝟙 = {∙}$ in $𝖲 𝖾 𝗍$ , then we let

𝛾 : K \to K {∙} 1 \mapsto 1 ∙ 𝜇 𝑋, 𝑌 : K (𝑋) \otimes K (𝑌) \to K (𝑋 \times 𝑌) 1 𝑥 \otimes 1 𝑦 \mapsto 1 (𝑥, 𝑦)

Construction as maps

Let $𝑆$ be a set and $𝑅$ be a ring. The free module $𝑅 (𝑆)$ is the set of maps of finite support $𝑆 \to 𝑅$ with addition and scaling induced by those of $𝑅$ , #m/def/module
i.e. for all $𝑠 \in 𝑆$

(𝛼 𝑎 + 𝛽 𝑏) (𝑠) = 𝛼 𝑎 (𝑠) + 𝛽 𝑏 (𝑠)

where we identify $𝑠 \in 𝑆$ with $1 𝑠 : 𝑡 \mapsto [𝑡 = 𝑠]$ invoking an Iverson bracket.

Proof of universal property

Clearly $𝑅 (𝑆)$ as constructed is an $𝑅$ -module with basis ${1 𝑠} 𝑠 \in 𝑆$ Now let $𝑀$ be an $𝑅$ -module and $𝑓 : 𝑆 \to 𝑀$ be a function. For a module homomorphism $¯ 𝑓 : 𝑅 (𝑆) \to 𝑀$ to make the diagram commute, we require that $¯ 𝑓 (1 𝑠) = 𝑓 (𝑠)$ for all $𝑠 \in 𝑆$ , which fully specifies $¯ 𝑓$ so that for $𝑎 \in 𝑅 (𝑆)$

¯ 𝑓 (𝑎) = \sum 𝑠 \in 𝑆 𝑓 (𝑠) 𝑎 (𝑠)

as required.

Further results

$𝑅 (𝑆)$ carries the additional structure of an $𝑅$ -comonoid, namely the Free $K$ -comonoid

#state/tidy | #lang/en | #SemBr