Projective and reflexive modules

Question

Is there an easy way to prove that if $M$ is finitely generated projective $R$-module, then $M$ is a reflexive module using just the definition of the natural map?

The natural map is $\theta: M \rightarrow M^{**}$ is defined by $\theta(m)=\widehat{m},$ where given $f \in M^{*},$ we have $\widehat{m}(f)=f(m).$

Using just the natural map and the dual basis for projective modules, it is easy to see that $M$ is torsionless (or that $\theta$ is injective), but I'm struggling to see that $\theta$ is surjective.

I'm using this page's notation, and they also give another path to prove what I'm asking.

Answer 1

Hint:

$M$ is a direct summand of a finitely generated free $R$-module $F\simeq R^n$ for some $n$. As the $\operatorname{Hom}$ functor commutes with direct sums, you can suppose $M$ is free, and ultimately that it is (isomorphic to) $R$.

Now, any linear form on $R$ is multiplication $m_\lambda$ by some $\lambda\in R$. A linear form $u\colon \operatorname{Hom}(R,R)\to R $ satisfies $$u(m_\lambda)=u(\lambda\operatorname{Id})=\lambda u(\operatorname{Id})=m_\lambda(u(\operatorname{Id}))$$ so that $u=\theta( u(\operatorname{Id}))$.

Answer 2

Let $M\oplus N=R^k$; then applying $\operatorname{Hom}_R(-,R)$ to the split exact sequence $0\to M\to R^k\to N\to0$ preserves split exactness. Applying it again you get the commutative diagram with split exact rows: $$\require{AMScd} \begin{CD} 0 @>>> M @>>> R^k @>>> N @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> M^{**} @>>> (R^k)^{**} @>>> N^{**} @>>> 0 \end{CD} $$ The middle vertical arrow is easily seen to be an isomorphism. By diagram chasing, $M\to M^{**}$ is injective and by symmetry also $N\to N^{**}$ is injective. Hence, by diagram chasing, $M\to M^{**}$ is surjective.