The Stacks project

95.11 The stack in groupoids of finitely generated quasi-coherent sheaves

Let $p : \mathcal{QC}\! \mathit{oh}_{fg} \to (\mathit{Sch}/S)_{fppf}$ be the stack introduced in Section 95.5 (using the abuse of notation introduced there). We can turn this into a stack in groupoids $p' : \mathcal{QC}\! \mathit{oh}_{fg}' \to (\mathit{Sch}/S)_{fppf}$ by the procedure of Categories, Lemma 4.35.3, see Stacks, Lemma 8.5.3. In this particular case this simply means $\mathcal{QC}\! \mathit{oh}_{fg}'$ has the same objects as $\mathcal{QC}\! \mathit{oh}_{fg}$ but the morphisms are pairs $(f, g) : (U, \mathcal{F}) \to (U', \mathcal{F}')$ where $g$ is an isomorphism $g : f^*\mathcal{F}' \to \mathcal{F}$.