The Stacks project

Theorem 19.14.3. Let $\mathcal{A}$ be a Grothendieck abelian category. Then there exists a (noncommutative) ring $R$ and functors $G : \mathcal{A} \to \text{Mod}_ R$ and $F : \text{Mod}_ R \to \mathcal{A}$ such that

  1. $F$ is the left adjoint to $G$,

  2. $G$ is fully faithful, and

  3. $F$ is exact.

Moreover, the functors are the ones constructed above.

Proof. We first prove $G$ is fully faithful, or equivalently that $F \circ G \to \text{id}$ is an isomorphism, see Categories, Lemma 4.24.4. First, given an object $A$ the map $F(G(A)) \to A$ is surjective, because every map of $U \to A$ factors through $F(G(A))$ by construction. On the other hand, the map $F(G(A)) \to A$ is the adjoint of the map $\text{id} : G(A) \to G(A)$ and hence injective by Lemma 19.14.2.

The functor $F$ is right exact as it is a left adjoint. Since $\text{Mod}_ R$ has enough projectives, to show that $F$ is exact, it is enough to show that the first left derived functor $L_1F$ is zero. To prove $L_1F(M) = 0$ for some $R$-module $M$ choose an exact sequence $0 \to K \to P \to M \to 0$ of $R$-modules with $P$ free. It suffices to show $F(K) \to F(P)$ is injective. Now we can write this sequence as a filtered colimit of sequences $0 \to K_ i \to P_ i \to M_ i \to 0$ with $P_ i$ a finite free $R$-module: just write $P$ in this manner and set $K_ i = K \cap P_ i$ and $M_ i = \mathop{\mathrm{Im}}(P_ i \to M)$. Because $F$ is a left adjoint it commutes with colimits and because $\mathcal{A}$ is a Grothendieck abelian category, we find that $F(K) \to F(P)$ is injective if each $F(K_ i) \to F(P_ i)$ is injective. Thus it suffices to check $F(K) \to F(P)$ is injective when $K \subset P = R^{\oplus n}$. Thus $F(K) \to U^{\oplus n}$ is injective by an application of Lemma 19.14.2. $\square$


Comments (0)

There are also:

  • 3 comment(s) on Section 19.14: The Gabriel-Popescu theorem

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Your email address will not be published.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 0F5U. Beware of the difference between the letter 'O' and the digit '0'.