The Stacks project

Email of Matthew Emerton dated April 27, 2016.

Proposition 104.6.9 (Emerton). Let $\mathcal{X} \subset \mathcal{X}'$ be a first order thickening of algebraic stacks. Let $W$ be an affine scheme and let $W \to \mathcal{X}$ be a smooth morphism. Then there exists a cartesian diagram

\[ \xymatrix{ W \ar[d] \ar[r] & W' \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X}' } \]

with $W' \to \mathcal{X}'$ smooth and $W'$ affine.

Proof. Consider the category $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ introduced in Remark 104.6.1. 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 74.9.5 and More on Morphisms, Lemma 37.2.3.

Lemma 104.6.7 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 104.6.8 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 67.7.2. 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 67.7.2. This finishes the proof. $\square$

Comments (2)

Comment #5826 by alexis bouthier on

Maybe one needs to add that is also an affine scheme.

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