The Stacks project

Example 17.10.9. Let $X$ be countably many copies $L_1, L_2, L_3, \ldots $ of the real line all glued together at $0$; a fundamental system of neighbourhoods of $0$ being the collection $\{ U_ n\} _{n \in \mathbf{N}}$, with $U_ n \cap L_ i = (-1/n, 1/n)$. Let $\mathcal{O}_ X$ be the sheaf of continuous real valued functions. Let $f : \mathbf{R} \to \mathbf{R}$ be a continuous function which is identically zero on $(-1, 1)$ and identically $1$ on $(-\infty , -2) \cup (2, \infty )$. Denote $f_ n$ the continuous function on $X$ which is equal to $x \mapsto f(nx)$ on each $L_ j = \mathbf{R}$. Let $1_{L_ j}$ be the characteristic function of $L_ j$. We consider the map

\[ \bigoplus \nolimits _{j \in \mathbf{N}} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{j, i \in \mathbf{N}} \mathcal{O}_ X, \quad e_ j \longmapsto \sum \nolimits _{i \in \mathbf{N}} f_ j 1_{L_ i} e_{ij} \]

with obvious notation. This makes sense because this sum is locally finite as $f_ j$ is zero in a neighbourhood of $0$. Over $U_ n$ the image of $e_ j$, for $j > 2n$ is not a finite linear combination $\sum g_{ij} e_{ij}$ with $g_{ij}$ continuous. Thus there is no neighbourhood of $0 \in X$ such that the displayed map is given by a “matrix” as in the proof of Lemma 17.10.8 above.

Note that $\bigoplus \nolimits _{j \in \mathbf{N}} \mathcal{O}_ X$ is the sheaf associated to the free module with basis $e_ j$ and similarly for the other direct sum. Thus we see that a morphism of sheaves associated to modules in general even locally on $X$ does not come from a morphism of modules. Similarly there should be an example of a ringed space $X$ and a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ such that $\mathcal{F}$ is not locally of the form $\mathcal{F}_ M$. (Please email if you find one.) Moreover, there should be examples of locally compact spaces $X$ and maps $\mathcal{F}_ M \to \mathcal{F}_ N$ which also do not locally come from maps of modules (the proof of Lemma 17.10.8 shows this cannot happen if $N$ is free).

Comments (2)

Comment #4318 by Minh-Tien Tran on

Since is a continuous function and is defined to be zero on , it is identically zero on as well. The same goes for . So I believe this is the definition of you have thought of. It is not very important but can lead to unnecessary confusion.

Comment #4476 by on

The values at are never used. So it doesn't matter what the value is at those points. The reason for picking would be that it is a fixed open neighbourhood of in .

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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01BL. Beware of the difference between the letter 'O' and the digit '0'.