## 109.3 A zero limit

Let $(S_ i)_{i \in I}$ be a directed inverse system of nonempty sets with surjective transition maps and with $\mathop{\mathrm{lim}}\nolimits S_ i = \emptyset $, see Section 109.2. Let $K$ be a field and set

Then the transition maps $V_ i \to V_ j$ are surjective for $i \geq j$. However, $\mathop{\mathrm{lim}}\nolimits V_ i = 0$. Namely, if $v = (v_ i)$ is an element of the limit, then the support of $v_ i$ would be a finite subset $T_ i \subset S_ i$ with $\mathop{\mathrm{lim}}\nolimits T_ i \not= \emptyset $, see Categories, Lemma 4.21.7.

For each $i$ consider the unique $K$-linear map $V_ i \to K$ which sends each basis vector $s \in S_ i$ to $1$. Let $W_ i \subset V_ i$ be the kernel. Then

is a nonsplit short exact sequence of inverse systems of vector spaces over the directed set $I$. Hence $W_ i$ is a directed system of $K$-vector spaces with surjective transition maps, vanishing limit, and nonvanishing $R^1\mathop{\mathrm{lim}}\nolimits $.

## 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Your email address will not be published.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)