Lemma 19.4.1. Let $X$ be a topological space. The category of abelian sheaves on $X$ has enough injectives. In fact it has functorial injective embeddings.

## 19.4 Abelian sheaves on a space

**Proof.**
For an abelian group $A$ we denote $j : A \to J(A)$ the functorial injective embedding constructed in More on Algebra, Section 15.55. Let $\mathcal{F}$ be an abelian sheaf on $X$. By Sheaves, Example 6.7.5 the assignment

is an abelian sheaf. There is a canonical map $\mathcal{F} \to \mathcal{I}$ given by mapping $s \in \mathcal{F}(U)$ to $\prod _{x \in U} j(s_ x)$ where $s_ x \in \mathcal{F}_ x$ denotes the germ of $s$ at $x$. This map is injective, see Sheaves, Lemma 6.11.1 for example.

It remains to prove the following: Given a rule $x \mapsto I_ x$ which assigns to each point $x \in X$ an injective abelian group the sheaf $\mathcal{I} : U \mapsto \prod _{x \in U} I_ x$ is injective. Note that

is the product of the skyscraper sheaves $i_{x, *}I_ x$ (see Sheaves, Section 6.27 for notation.) We have

see Sheaves, Lemma 6.27.3. Hence it is clear that each $i_{x, *}I_ x$ is injective. Hence the injectivity of $\mathcal{I}$ follows from Homology, Lemma 12.27.3. $\square$

## 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Your email address will not be published.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)