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Universal property of an inverse limit of modules

Statement

Let $M_{1} \overset{φ_{2}}{\leftarrow} M_{2} \overset{φ_{3}}{\leftarrow} M_{3} \overset{}{\leftarrow} \dots$ be an inverse system of $R$ -modules. For every $R$ -module $N$ , giving an $R$ -linear map $f ⁣ : N \to \underset{\leftarrow}{l i m} M_{n}$ is equivalent to giving a family of $R$ -linear maps $f_{n} ⁣ : N \to M_{n}$ such that $φ_{n} \circ f_{n} = f_{n - 1} (n \geq 2) .$

Equivalently, the projections $π_{n} ⁣ : \underset{\leftarrow}{l i m} M_{n} \to M_{n}$ form a terminal compatible cone.

Facts & Assumptions

Given: An inverse system $(M_{n}, φ_{n})$ of $R$ -modules and an $R$ -module $N$ .

[L1]

The inverse limit is the compatible-element submodule of the product $\underset{\leftarrow}{l i m} M_{n} = {(x_{n}) \in \prod M_{n} : φ_{n} (x_{n}) = x_{n - 1} for n \geq 2}$ with projections to the coordinates (Inverse systems and inverse limits of modules).

Proof

technique · direct

1.1L1given

If $f ⁣ : N \to \underset{\leftarrow}{l i m} M_{n}$ is $R$ -linear, define $f_{n} : = π_{n} \circ f ⁣ : N \to M_{n} .$ For each $x \in N$ , the element $f (x)$ lies in the compatible submodule from [L1], so its coordinates satisfy $φ_{n} (f_{n} (x)) = f_{n - 1} (x)$ for every $n \geq 2$ . Thus the family $(f_{n})$ is compatible.

1.2L1givenconstruct

Conversely, let $(f_{n})$ be a compatible family and define $f (x) : = (f_{n} (x))_{n \geq 1} \in \prod M_{n} .$ Compatibility says $φ_{n} (f_{n} (x)) = f_{n - 1} (x)$ for every $n \geq 2$ , so $f (x)$ actually lies in $\underset{\leftarrow}{l i m} M_{n}$ . Since products and coordinate maps are $R$ -linear, $f$ is $R$ -linear.

2.1step 1.1step 1.2

The two constructions are inverse to each other: starting from $f$ and then taking coordinates recovers each $f_{n}$ , while starting from $(f_{n})$ and then forming $f$ gives the unique map whose $n$ th coordinate is $f_{n}$ .

3.1step 2.1∎

Therefore maps $N \to \underset{\leftarrow}{l i m} M_{n}$ are in bijection with compatible families $(f_{n})$ , which is exactly the terminal-cone universal property.

Depends on

Inverse systems and inverse limits of modules

Used by

Equivalent adic filtrations have canonically isomorphic completions Example
Inverse limits preserve kernels Theorem
Kernel and universal property of adic completion Theorem

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Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §22.5 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, Proposition 9.5 (standard reference, not scraped)
The Stacks Project, Section 10.87 (standard reference, not scraped)