The Stacks project

Example 6.4.5. Let $X$ be a topological space. For each $x \in X$ suppose given an abelian group $M_ x$. For $U \subset X$ open we set

\[ \mathcal{F}(U) = \bigoplus \nolimits _{x \in U} M_ x. \]

We denote a typical element in this abelian group by $\sum _{i = 1}^ n m_{x_ i}$, where $x_ i \in U$ and $m_{x_ i} \in M_{x_ i}$. (Of course we may always choose our representation such that $x_1, \ldots , x_ n$ are pairwise distinct.) We define for $V \subset U \subset X$ open a restriction mapping $\mathcal{F}(U) \to \mathcal{F}(V)$ by mapping an element $s = \sum _{i = 1}^ n m_{x_ i}$ to the element $s|_ V = \sum _{x_ i \in V} m_{x_ i}$. We leave it to the reader to verify that this is a presheaf of abelian groups.

Comments (0)

There are also:

  • 4 comment(s) on Section 6.4: Abelian presheaves

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Your email address will not be published.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 006L. Beware of the difference between the letter 'O' and the digit '0'.