Let $A$ be a noetherian integral domain and $\mathfrak{p}\subset A$ a prime ideal with residue field $k(\mathfrak{p}):= A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$.

I've seen in many places the symbol $E(k(\mathfrak{p}))$ denoting the injective hull of this field and i've seen a non-constructive proof for the existence of injective hulls in the general case. However i've seen very few few actual injective hulls (among them the prufer groups $\mathbb{Q}_p/\mathbb{Z}_{p}$ as hulls of the finite fields $\mathbb{F}_p$).

Does the following hold in the general (for a noetherian integral domain): $$E(k(\mathfrak{p})) \cong Frac(\widehat A_\mathfrak{p})/ \widehat A_\mathfrak{p}$$

Edit: This is apparently wrong. Thouh the main question still stands:

What is an explicit description of the injective hull of the residue field?

Commutative Ring Theoryfor the local ring of affine space at the origin. $\endgroup$Residues and Duality(especially section 5, Proposition 6.1, and Theorem 9.1(vi)). In a nutshell, teach yourself local duality. $\endgroup$3more comments