For a scheme $X$, I have a reference - https://stacks.math.columbia.edu/tag/077P - that says there are enough injectives in the category $\text{QCoh}(X)$. I am looking for a reference that says the analogous statement when I replace $X$ with a stack $\mathcal{X}$. I am happy with any reasonable restrictions like locally noetherian etc. I was only able to find https://stacks.math.columbia.edu/tag/01DL which proves it in the category of abelian sheaves.

## 2 Answers

In the paper

A functorial formalism for quasi-coherent sheaves on a geometric stack, *Expo. Math.* **33** (2015) 452–501,

in Corollary 5.10 it is shown that $\mathrm{Qco}(X)$ is a Grothendieck category for an algebraic stack $X$ with affine diagonal, i.e. a *geometric* stack. In this paper this is proved by showing that the category of comodules over a Hopf algebroid is Grothendieck. It is a classical fact that a Grothendieck Abelian category (i.e. an Abelian category satisfying AB5 with a family of generators) has enough injectives.

The paper is also accesible at https://arxiv.org/abs/1304.2520 .

Proposition 4.2 in Ethan Pribble's 2004 Ph.D. thesis is the result you asked for, under the hypothesis that $X$ is an algebraic stack with affine diagonal: Pribble shows that $Mod_{qcoh}(O_X)$ then has enough injectives. I suspect that isn't the earliest place this result has appeared in the literature, but it's the earliest I remember.