Is the injective hull $E(R / \mathfrak{m})$ a reflexive module if $(R, \mathfrak{m})$ is Artinian?

Question

For the definition of the reflexive modules, I refer to the Stacks project, tag 0AUY.

If $(R,{\mathfrak m})$ is a commutative Artinian ring, is the injective hull $E(R/\mathfrak{m})$ reflexive?

Answer 1

To expand on Mohan’s comment, if $(R,\mathfrak{m},k)$ is an Artinian local ring, then $E_R(k)$ is reflexive if and only if $R$ is Gorenstein. Indeed, every module is reflexive over an Artinian Gorenstein ring. On the other hand, suppose $E_R(k)$ is reflexive. Write $(-)^*:=\operatorname{Hom}_R(-,R)$ and $(-)^{\vee}:=\operatorname{Hom}_R(-,E_R(k))$.

Note that given any $R$-module $M$, we have $$M^*=\operatorname{Hom}_R(M,R) \cong \operatorname{Hom}_R(M,\operatorname{Hom}_R(E_R(k),E_R(k))) \cong (M \otimes_R E_R(k))^{\vee} \,.$$

Using this remark, we have $$E_R(k) \cong E_R(k)^{**} \cong (E_R(k)^* \otimes_R E_R(k))^{\vee} \,.$$

Applying, $(-)^{\vee}$ to both sides reveals that $E_R(k)^* \otimes_R E_R(k) \cong R$. This forces $E_R(k)$ to be cyclic, i.e., for $R$ to have type $1$. Thus $R$ is Gorenstein.