We say an object $X$ in category $C=R$-Mod is of finite type if for any functor $F: I \rightarrow C$ with $I$ a directed poset, the natural map $$\underrightarrow{\lim} Hom_{\mathcal{C}}(X,F(i))\to Hom_{\mathcal{C}}(X,\underrightarrow{\lim}F)$$ is injective. I want to show that this definition coincides with the usual definition of finitely generated module.
In this answer, Hanno has proved that a finite type object in R-Mod is a finitely generated module. But I still don't know how to prove that a finitely generated module satisfies the above condition. If anyone can help me, thanks a lot.
Best Answer
For an elementary solution, you need an elementary formulation of filtered colimits of $R$-modules. Namely, the filered colimit of $R$-modules is the $R$-module structure on the union of the filtered colimit of the underlying sets, which is the set-theoretic union of the $R$-modules.
Then a homomorphism from a finitely generated $R$-module $X$ to $F(i)$ consisets of choice of finitely many elements of $F(i)$ (the images of the generators) satisfying the defining relations. The filtered colimit $\varinjlim Hom_{\mathcal C}(X,F(i))$ the union of all such choices inside the set-theoretic union of the $F(i)$. The $R$-module structure on the union is such that if the relations hold for a choice of elements in one of the $F(i)$, then they hold in the resulting $R$-module structure. In particular, each choice of elements of $F(i)$ gets sent to a unique choice of elements in the union.
A purely categorical answer proceeds as follows (and justifies at the end the filtered colimit interpretation above).
Consider a finite diagram $J\colon D\to C$ with colimit $X$.
Because contravariant representable functors send colimits to limits, the natural map $\varinjlim_i Hom_{\mathcal{C}}(\mathrm{colim}_dJ(d),F(i))\to Hom_{\mathcal{C}}(\mathrm{colim}_dJ(d),\varinjlim_i F(i))$ is the same as the natural map $\varinjlim_i\lim_d Hom_{\mathcal{C}}(J(d),F(i))\to\lim_d Hom_{\mathcal{C}}(J(d),\varinjlim_i F(i))$.
Because filtered colimits commute with finite limits in Set, the above is the same as the natural map $\lim_d \varinjlim_i Hom_{\mathcal{C}}(J(d),F(i))\to\lim_d Hom_{\mathcal{C}}(J(d),\varinjlim_i F(i))$, which is the (finite) limit of the natural maps $\varinjlim_i Hom_{\mathcal{C}}(J(d),F(i))\to Hom_{\mathcal{C}}(J(d),\varinjlim_i F(i))$.
If each of those is a monomorphism, resp. isomorphism, then so is the limit. In particular, finitely generated, resp. finitely presented, objects are stable under finite colimits.
Moreover, an epimorphism $X\twoheadrightarrow Y$ induces morphisms $Hom_{\mathcal C}(Y,F(i))\hookrightarrow Hom_{\mathcal C}(X,F(i))$ whose pullback along itself consist of identities, i.e. monomorphisms, whence $\varinjlim_i Hom_{\mathcal C}(Y,F(i))\to\varinjlim_i Hom_{\mathcal C}(X,F(i))$ also has pullback along itself consisting of identities, i.e. is also a monomorphism. We thus have two equal composites $\varinjlim_iHom_{\mathcal C}(Y, F(i))\hookrightarrow\varinjlim_iHom_{\mathcal C}(X, F(i))\to Hom_{\mathcal C}(X,\varinjlim_iF(i))$ and $\varinjlim_iHom_{\mathcal C}(Y, F(i))\to Hom_{\mathcal C}(Y,\varinjlim_iF(i))\hookrightarrow \varinjlim_iHom_{\mathcal C}(X, F(i))$.
If the natural map $\varinjlim_iHom_{\mathcal C}(X, F(i))\to Hom_{\mathcal C}(X,\varinjlim_iF(i))$ is a monomorphism, then so are the composites, and hence so is $\varinjlim_iHom_{\mathcal C}(Y, F(i))\to Hom_{\mathcal C}(Y,\varinjlim_iF(i))$. In particular, if $X$ is finitely generated and $X\twoheadrightarrow Y$ is an epimorphism, then $Y$ is finitely generated. In particular, any epimorphism whose domain is a finite colimit of finitely generated objects has codomain a finitely generated object.
Now by definition a finitely generated $R$-module is the codomain of an epimorphism whose domain is a finite coproduct of copies of $R$, whence it suffices to show that $R$ is finitely generated in the categorical sense. Since $Hom_{\mathcal R-mod}(R,-)$ is the forgetful functor from $R$-modules to sets, this is equivalent to showing the forgetful functor preserves filtered colimits. The fact that $R$-modules are specified by morphisms out of finite products satisfying certain equations implies the forgetful functor preserves sifted colimits, i.e. ones commuting with finite products, so in particular it preserves filtered colimits, as desired.