Localization of Module Homomorphism is Exact

Theorem

Let $A$ be a commutative ring with unity.

Let $S \subseteq A$ be a multiplicatively closed subset.

Let $S^{-1}A$ be the localization of $A$ at $S$.

Let:

$M_1 \stackrel {f_1} \longrightarrow M_2 \stackrel {f_2} \longrightarrow M_3$

be an exact sequence of $A$-modules.

For $i=1,2,3$, let $\struct { S^{-1}M_i, \iota_i}$ be the localization of $M_i$ at $S$.

For $i=1,2$, let:

$S^{-1}f_i : S^{-1}M_i \to S^{-1}M_{i+1}$

be the unique $S^{-1}A$-homomorphism such that:

$\iota_{i+1} \circ f_i = \paren {S^{-1}f_i} \circ \iota_i$

Then:

$S^{-1}M_1 \stackrel {S^{-1}f_1} \longrightarrow S^{-1}M_2 \stackrel {S^{-1}f_2} \longrightarrow S^{-1}M_3$

is an exact sequence of $S^{-1}A$-modules.

Proof

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Also known as

The operation $S^{-1}$ is called exact.

Sources

• 1969: M.F. Atiyah and I.G. MacDonald: Introduction to Commutative Algebra: Chapter $3$: Rings and Modules of Fractions