The Stacks project

Proof. Consider the category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ introduced in Remark 106.6.1. The proposition states that there exists an object of $\mathcal{C}$ lying over $W$. Namely, if we have such an object $(W, W', a, i, y', \alpha )$ then $W = \mathcal{X} \times _{\mathcal{X}'} W'$. Hence $W \to W'$ is a thickening of algebraic spaces so $W'$ is affine by More on Morphisms of Spaces, Lemma 76.9.5 and More on Morphisms, Lemma 37.2.3.

Lemma 106.6.7 tells us $\mathcal{C}$ is a gerbe over $W_{spaces, {\acute{e}tale}}$. This means we can étale locally find a solution and these local solutions are étale locally isomorphic; this part does not require the assumption that the thickening is first order. By Lemma 106.6.8 the automorphism sheaves of objects of our gerbe are abelian and fit together to form a quasi-coherent module $\mathcal{G}$ on $W_{spaces, {\acute{e}tale}}$. We will verify conditions (1) and (2) of Cohomology on Sites, Lemma 21.11.1 to conclude the existence of an object of $\mathcal{C}$ lying over $W$. Condition (1) is true: the étale coverings $\{ W_ i \to W\} $ with each $W_ i$ affine are cofinal in the collection of all coverings. For such a covering $W_ i$ and $W_ i \times _ W W_ j$ are affine and $H^1(W_ i, \mathcal{G})$ and $H^1(W_ i \times _ W W_ j, \mathcal{G})$ are zero: the cohomology of a quasi-coherent module over an affine algebraic space is zero for example by Cohomology of Spaces, Proposition 69.7.2. Finally, condition (2) is that $H^2(W, \mathcal{G}) = 0$ for our quasi-coherent sheaf $\mathcal{G}$ which again follows from Cohomology of Spaces, Proposition 69.7.2. This finishes the proof. $\square$