Lemma 70.9.6. Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let \mathcal{A} be a finite quasi-coherent \mathcal{O}_ X-algebra. Then \mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i is a directed colimit of finite and finitely presented quasi-coherent \mathcal{O}_ X-algebras with surjective transition maps.
Proof. By Lemma 70.9.3 there exists a finitely presented \mathcal{O}_ X-module \mathcal{F} and a surjection \mathcal{F} \to \mathcal{A}. Using the algebra structure we obtain a surjection
Denote \mathcal{J} the kernel. Write \mathcal{J} = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_ i as a filtered colimit of finite type \mathcal{O}_ X-submodules \mathcal{E}_ i (Lemma 70.9.2). Set
where (\mathcal{E}_ i) indicates the ideal sheaf generated by the image of \mathcal{E}_ i \to \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}). Then each \mathcal{A}_ i is a finitely presented \mathcal{O}_ X-algebra, the transition maps are surjective, and \mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i. To finish the proof we still have to show that \mathcal{A}_ i is a finite \mathcal{O}_ X-algebra for i sufficiently large. To do this we choose an étale surjective map U \to X where U is an affine scheme. Take generators f_1, \ldots , f_ m \in \Gamma (U, \mathcal{F}). As \mathcal{A}(U) is a finite \mathcal{O}_ X(U)-algebra we see that for each j there exists a monic polynomial P_ j \in \mathcal{O}(U)[T] such that P_ j(f_ j) is zero in \mathcal{A}(U). Since \mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i by construction, we have P_ j(f_ j) = 0 in \mathcal{A}_ i(U) for all sufficiently large i. For such i the algebras \mathcal{A}_ i are finite. \square
Comments (0)