Lemma 108.32.1. There exists an epimorphism of local rings of dimension $0$ which is not a surjection.

## 108.32 An epimorphism of zero-dimensional rings which is not surjective

In [Lazard-deux] and [Autour] one can find the following example. Let $k$ be a field. Consider the ring homomorphism

which maps $x_ i$ to $x_ i$ and $z_ i$ to $x_ iy_ i$. Note that $y_ i^{4^{i + 1}}$ is zero in the right hand side but that $y_1$ is not zero (details omitted). This map is not surjective: we can think of the above as a map of $\mathbf{Z}$-graded algebras by setting $\deg (x_ i) = -1$, $\deg (z_ i) = 0$, and $\deg (y_ i) = 1$ and then it is clear that $y_1$ is not in the image. Finally, the map is an epimorphism because

hence the tensor product of the target over the source is isomorphic to the target.

**Proof.**
See discussion above.
$\square$

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