Injective Cogenerators in the Category of Modules over a Noetherian Ring Let $R$ be a Noetherian ring and let $\mathcal{A}$ be an injective $R$-module. The injectivity of $\mathcal{A}$ is equivalent to the exactness of the functor $Hom_R(-,\mathcal{A})$, i.e.
whenever we have an exact sequence of $R$-modules $M \rightarrow N \rightarrow P, \, \, \, (1)$, then the sequence $Hom_R(P,\mathcal{A}) \rightarrow Hom_R(N,\mathcal{A}) \rightarrow Hom_R(M,\mathcal{A}), \, \, \, (2)$ is also exact. Now assume that $\mathcal{A}$ is also a cogenerator in the category of $R$-modules, i.e. $Hom_R(M,\mathcal{A})=0 \Rightarrow M=0$. How can we prove that exactness of $(2)$ implies exactness of $(1)$?
 A: Let $\mathcal{C}$ be an abelian category, and suppose $A$ is an injective cogenerator in $\mathcal{C}$. For any morphism $f : X \to Y$ in $\mathcal{C}$, consider the following exact sequence in $\mathcal{C}$:
$$0 \longrightarrow W \longrightarrow X \longrightarrow Y \longrightarrow Z \longrightarrow 0$$
Since $A$ is injective, we get an exact sequence of abelian groups:
$$0 \longrightarrow \mathcal{C}(Z, A) \longrightarrow \mathcal{C}(Y, A) \longrightarrow \mathcal{C}(X, A) \longrightarrow \mathcal{C}(W, A) \longrightarrow 0$$
So, if $\mathcal{C}(f, A) : \mathcal{C}(Y, A) \to \mathcal{C}(X, A)$ is an isomorphism in $\textbf{Ab}$, then $\mathcal{C}(Z, A)$ and $\mathcal{C}(W, A)$ must be trivial, and this implies $Z$ and $W$ are zero in $\mathcal{C}$ because $A$ is a cogenerator. Thus, $\mathcal{C}(f, A) : \mathcal{C}(Y, A) \to \mathcal{C}(X, A)$ is an isomorphism if and only if $f : X \to Y$ is an isomorphism in $\mathcal{C}$; in other words, $\mathcal{C}(-, A) : \mathcal{C}^\textrm{op} \to \textbf{Ab}$ is a conservative functor.
Now, it is a general fact that conservative functors reflect all limits and colimits that they preserve, so this implies $\mathcal{C}(-, A)$ reflects exact sequences. For example, suppose we have a chain complex in $\mathcal{C}$
$$X \longrightarrow Y \longrightarrow Z \longrightarrow 0$$
whose image in $\textbf{Ab}$ is exact:
$$0 \longrightarrow \mathcal{C}(Z, A) \longrightarrow \mathcal{C}(Y, A) \longrightarrow \mathcal{C}(X, A)$$
Now compute the cokernel in $\mathcal{C}$ so that we get an exact sequence
$$X \longrightarrow Y \longrightarrow C \longrightarrow 0$$
and the universal property of $C$ ensures there is a chain morphism from this complex to the first one; but because the image of the first complex is already exact, the image of the morphism $C \to Z$ must be an isomorphism in $\textbf{Ab}$, so $C \to Z$ itself must be an isomorphism in $\mathcal{C}$, and therefore the first complex must be exact in $\mathcal{C}$.
