Proposition 9.2.0.1 (063A)

Proposition 9.2.0.1. Let $R$ be an associative ring and let $M$ be a left $R$-module. Then $M$ is finitely presented if and only if the functor

\[ \{ \textnormal{$R$-modules} \} \rightarrow \operatorname{Set}\quad \quad N \mapsto \operatorname{Hom}_{R}(M,N) \]

preserves small filtered colimits.

Proof. Assume first that $R$ is finitely presented: that is, there exists an exact sequence of left $R$-modules

\[ R^{m} \xrightarrow {f} R^{n} \rightarrow M \rightarrow 0. \]

For any filtered diagram of $R$-modules $\{ N_ i \} $ having colimit $N$, we have a commutative diagram of exact sequences

\[ \xymatrix@R =50pt@C=40pt{ 0 \ar [r] & \varinjlim _{i} \operatorname{Hom}_{R}(M, N_ i) \ar [r] \ar [d] & \varinjlim _{i} \operatorname{Hom}_{R}( R^{n}, N_ i ) \ar [d]^{\sim } \ar [r] & \varinjlim _{i} \operatorname{Hom}_{R}( R^{m}, N_ i) \ar [d]^{\sim } \\ 0 \ar [r] & \operatorname{Hom}_{R}(M, N) \ar [r] & \operatorname{Hom}_{R}( R^{n}, N) \ar [r] & \operatorname{Hom}_{R}( R^{m}, N). } \]

Since the middle and right vertical maps are bijective, the left vertical map is also bijective.

We now prove the converse. Let $\{ M_{\alpha } \} $ be the collection of all finitely generated submodules of $M$, partially ordered by inclusion. Then $M$ is the colimit of the filtered diagram $\{ M_{\alpha } \} $. If the functor $\operatorname{Hom}_{R}(M, \bullet )$ preserves small filtered colimits, then the tautological map

\[ \varinjlim _{\alpha } \operatorname{Hom}_{R}(M, M_{\alpha } ) \rightarrow \operatorname{Hom}_{R}(M, M) \]

is a bijection. In particular, the identity map $\operatorname{id}_{M}: M \rightarrow M$ factors through a finitely generated submodule of $M$, so $M$ is generated by a finite collection of elements $x_1, x_2, \cdots , x_ n$. Let $R^{n}$ be the free $R$-module on generators $\widetilde{x}_1, \cdots , \widetilde{x}_{n}$, so there is surjective $R$-module homomorphism $f: R^{n} \twoheadrightarrow M$ satisfying $f( \widetilde{x}_ i ) = x_ i$ for $1 \leq i \leq n$. Let $K$ be the kernel of $f$, and let $\{ K_{\beta } \} $ be the collection of all finitely generated submodules of $K$, partially ordered by inclusion. If the functor $\operatorname{Hom}_{R}(M, \bullet )$ preserves small filtered colimits, then the tautological map

\[ \varinjlim _{\beta } \operatorname{Hom}_{R}( M, R^{n} / K_{\beta }) \rightarrow \operatorname{Hom}_{R}(M, R^{N}/K ) = \operatorname{Hom}_{R}(M,M) \]

is bijective. In particular, the identity map $\operatorname{id}_{M}$ factors as a composition

\[ M \xrightarrow {g} R^{n} / K' \twoheadrightarrow R^{n} / K \simeq M, \]

where $K'$ is some finitely generated submodule of $K$. Enlarging $K'$ if necessary, we may assume that $g$ carries each $x_ i$ to the image of $\widetilde{x}_ i$ in $R^{n} / K'$: that is, the composite map $g \circ f$ coincides with the tautological map from $R^{n}$ to $R^{n}/K'$. We then have $K = \ker (f) \subseteq \ker (g \circ f) = K'$, so that $K = K'$ is finitely generated and $M = R^{n} / K$ is finitely presented, as desired. $\square$