Characterization of the kernel and cokernel of the natural homomorphism between a module and its double dual.

Question

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Suppose $$ G \overset{\varphi}{\rightarrow} F \to M \to 0$$ is exact where $F,G$ are finite free modules. Suppose $D(M)=\operatorname{coker}\varphi^{*}$. Then $\ker h=\operatorname{Ext}^1(D(M),R)$ and $\operatorname{coker}h=\operatorname{Ext}^2(D(M,R))$, where $M \overset{h}{\rightarrow} M^{**}$ is the natural homomorphism between $M$ and its double dual. (Bruns and Herzog, Exercise 1.4.21.)

Answer 1

Dualizing $G \overset{\varphi}\to F \overset{\pi}\to M \to 0$ we get $0\to M^*\overset{\pi^*}\to F^* \overset{\varphi^*}\to G^* \to D(M) \to 0$. Now take $K\to H\overset{p}\to M^*\to 0$ a finite presentation of $M^*$. Thus we get a part of a free resolution of $D(M)$, that is, $$K\to H\to F^* \overset{\varphi^*}\to G^* \to D(M) \to 0,$$ and therefore we can compute $\operatorname{Ext}_R^i(D(M),R)$ for $i=1,2$ using it.

Dualizing $K\to H\overset{p}\to M^*\to 0$ we obtain $0\to M^{**}\overset{p^*}\to H^*\to K^*$. Now we get a sequence $G \overset{\varphi}\to F\to H^*\to K^*$, where the second left arrow is $F\overset{\pi}\to M\overset{h}\to M^{**}\overset{p^*}\to H^*$.

At this moment I think you have enough data to finish the proof.