The Stacks project

110.3 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 110.2. Let $K$ be a field and set

\[ V_ i = \bigoplus \nolimits _{s \in S_ i} K \]

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

\[ 0 \to (W_ i) \to (V_ i) \to (K) \to 0 \]

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 $.

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