Lemma 109.10.1. Let $R$ be a ring and let $I \subset R$ be a finitely generated ideal. The category of $I$-adically complete $R$-modules has kernels and cokernels but is not abelian in general.

## 109.10 The category of complete modules is not abelian

Let $R$ be a ring and let $I \subset R$ be a finitely generated ideal. Consider the category $\mathcal{A}$ of $I$-adically complete $R$-modules, see Algebra, Definition 10.96.2. Let $\varphi : M \to N$ be a morphism of $\mathcal{A}$. The cokernel of $\varphi $ in $\mathcal{A}$ is the completion $(\mathop{\mathrm{Coker}}(\varphi ))^\wedge $ of the usual cokernel (as $I$ is finitely generated this completion is complete, see Algebra, Lemma 10.96.3). Let $K = \mathop{\mathrm{Ker}}(\varphi )$. We claim that $K$ is complete and hence is the kernel of $\varphi $ in $\mathcal{A}$. Namely, let $K^\wedge $ be the completion. As $M$ is complete we obtain a factorization

Since $\varphi $ is continuous for the $I$-adic topology, $K \to K^\wedge $ has dense image, and $K = \mathop{\mathrm{Ker}}(\varphi )$ we conclude that $K^\wedge $ maps into $K$. Thus $K^\wedge = K \oplus C$ and $K$ is a direct summand of a complete module, hence complete.

We will give an example that shows that $\mathop{\mathrm{Im}}\not= \mathop{\mathrm{Coim}}$ in general. We take $R = \mathbf{Z}_ p = \mathop{\mathrm{lim}}\nolimits _ n \mathbf{Z}/p^ n\mathbf{Z}$ to be the ring of $p$-adic integers and we take $I = (p)$. Consider the map

where the left hand side is the $p$-adic completion of the direct sum. Hence an element of the left hand side is a vector $(x_1, x_2, x_3, \ldots )$ with $x_ i \in \mathbf{Z}_ p$ with $p$-adic valuation $v_ p(x_ i) \to \infty $ as $i \to \infty $. This maps to $(x_1, px_2, p^2x_3, \ldots )$. Hence we see that $(1, p, p^2, \ldots )$ is in the closure of the image but not in the image. By our description of kernels and cokernels above it is clear that $\mathop{\mathrm{Im}}\not= \mathop{\mathrm{Coim}}$ for this map.

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

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)