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What is the injective hull of $\mathbb{C}(x,y)/\mathbb{C}[x,y]$ as a $\mathbb{C}[x,y]$-module? Is it isomorphic to any familiar module?

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Let me try and give a general answer - though this may not be very helpful if you're looking for an explicit description of the injective envelope.

If $R$ is a Noetherian domain with field of fractions $Q$, there is an exact sequence

$$0 \to R \to Q \to Q/R \to 0$$

Now $E(R) \cong Q$ (as $Q$ is torsionfree, divisible, and an essential extension of $R$). This implies that $E(Q/R)$ is the second term in the minimal injective resolution,

$$0 \to R \to I^0 = Q \to I^1 = E(Q/R) \to I^2 \to \ldots $$

Now the $i^\text{th}$ module $I^i$ is a direct sum of indecomposable injectives $E(R/p)$, where a prime $p \in \operatorname{Spec}R$ appears with multiplicity $\mu_i(p, R)$, the $i^\text{th}$ Bass number of $R$ with respect to $p$. If $R$ is regular (which $\mathbb{C}[x,y]$ is), hence Gorenstein, then $\mu_i(p, R) = \delta_{i, \text{ht} p}$ (Kronecker delta). Thus,

$$E(Q/R) = I^1 = \bigoplus_{\text{ht}p=1} E(R/p)$$

Understanding this module would thus involve knowing all height $1$ primes in $\mathbb{C}[x,y]$, i.e. all irreducible polynomials in $2$ variables over $\mathbb{C}$, which is not a particularly easy task.

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