Let $A=\mathbb{C}[x_1,\ldots,x_n]$ be a polynomial algebra over the complex numbers. I am interested in injective modules over $A$.

Since $A$ is projective over itself, the $\mathbb{C}$-dual module $A^\ast=\mathbb{C}[[x_1,\cdots,x_n]]$ is known to be injective and the generatings $x_i$ of $A$ act on $A^\ast$ by partial derivatives $\partial/\partial x^i$. As the ring $A$ is Noetherian, $A^\ast$ splits in the direct sum of indecomposable injective modules. My question is about the structure of this decomposition.

As I can see, for any $y\in \mathbb{C}^n$ the module $A^\ast$ has the indecomposible $A$-submodules $$M_y=\{p(x)e^{yx}| p(x)\in A\}\,,$$ where $yx$ stands for the inner product of vectors in $\mathbb{C}^n$. Furthermore, it seems that $M_y$ is isomorphic to the injective hull $E_A(\mathbb{C})$ of the trivial $A$-module $\mathbb{C}$ on which $p(x)\in A$ acts by multiplication by $p(y)\in \mathbb{C}$.

Is it true that

$$ A^\ast\simeq \bigoplus_{y\in \mathbb{C}^n} M_y \quad? $$ This would be rather strange, as any finite sum of functions of the form $p(x)e^{yx}$ for some $p$'s and $y$'s can't produce a divergent power series from $A^\ast$.

What is a good reference on the subject?


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