Homological dimension of categories of modules Let $A$ be a Noetherian ring. We have two categories:
(a) category of $A$-modules
(b) category of finite type $A$-modules.
Do their homological dimensions agree? The homological dimension of an abelian category $C$ is defined as $\inf\{m \ | \ \text{Ext}^{m+1}_C(-,-)=0\}$.
 A: Yes, both homological dimensions agree.
First, by Baer's criterion for injectivity, for any $A$-module $M$ you have $$\text{injdim}_A(M)=\max\{n\ |\ \text{Ext}^n_A(A/I,M)\neq 0\text{ for some }I\},$$ so you may restrict to finitely generated modules in the first argument when defining the homological dimension. For finitely generated modules $M$ (over a Noetherian ring), however, $\text{Ext}(M,-)$ commutes with directed colimits, so it also suffices to consider finitely generated modules in the second argument.
Previous The question has changed completely, the rest is the answer to the question of whether $\text{Ext}$ commutes with directed limits in the first and second argument.
No, this is not possible in either argument.
Concerning a direct limit in the second argument, it is not even true that $$\text{Hom}(M,\text{colim}_I N_i)\cong\text{colim}_I(M,N_i).$$ As an example, consider ${\mathbb Q} = \text{colim}({\mathbb Z}\xrightarrow{\cdot 2} {\mathbb Z}\xrightarrow{\cdot 3} \ \ldots)$: you have $\text{Hom}({\mathbb Q}, {\mathbb Z})=0$ but $\text{Hom}({\mathbb Q},{\mathbb Q})\neq 0$. The modules $M$ for which $\text{Hom}(M,-)$ commutes with directed limits are the finitely presented ones.
Concerning a direct limit in the first argument, you have the desired property for $\text{Hom}$, but not for $\text{Ext}$, since inverse limits are not exact. Concretely, any non-projective but flat module gives an example, since any flat module is the colimit of a diagram of free modules; again, ${\mathbb Q}$ as a ${\mathbb Z}$-module is an example.
