Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 24.25.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $T$ and for each $t \in T$ an injective map $\mathcal{N}_ t \to \mathcal{N}'_ t$ of graded $\mathcal{A}$-modules such that an object $\mathcal{I}$ of $\textit{Mod}(\mathcal{A})$ is injective if and only if for every solid diagram

\[ \xymatrix{ \mathcal{N}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{N}'_ t \ar@{..>}[ru] } \]

a dotted arrow exists in $\textit{Mod}(\mathcal{A})$ making the diagram commute.

Proof. This is true in any Grothendieck abelian category, see Injectives, Lemma 19.11.6. By Lemma 24.11.1 the category $\textit{Mod}(\mathcal{A})$ is a Grothendieck abelian category. $\square$


Comments (0)


Your email address will not be published.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.