The Stacks project

Proof. The only point here is that $f$ induces a morphism of reductions $X_{red} \to Y_{red}$ by Lemma 87.12.1. Hence this lemma is immediate from the corresponding lemma for morphisms of algebraic spaces (Properties of Spaces, Lemma 66.18.8). $\square$

Proof. This will follow immediately from the case of algebraic spaces (Properties of Spaces, Lemma 66.18.11) if we can show that the induced morphism $X_{red} \to Y_{red}$ is étale. Observe that $X \times _ Y Y_{red}$ is an algebraic space, étale over the reduced algebraic space $Y_{red}$, and hence reduced itself (by our definition of reduced algebraic spaces in Properties of Spaces, Section 66.7. Hence $X_{red} = X \times _ Y Y_{red}$ as desired. $\square$

Proof. Denote $\mathcal{C}$ the category introduced in the lemma. Observe that for $\varphi : U \to X$ in $\mathcal{C}$ the morphism $\varphi $ is representable (by schemes) and affine, see Lemma 87.19.7. Recall that $X_{affine, {\acute{e}tale}} = X_{red, affine, {\acute{e}tale}}$. Hence we can define a functor

because $U \times _ X X_{red}$ is an affine scheme.

To finish the proof we will construct a quasi-inverse. Namely, write $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 87.9.1. For each $\lambda $ we have $X_{red} \subset X_\lambda $ is a thickening. Thus for every $\lambda $ we have an equivalence

for example by More on Algebra, Lemma 15.11.2. Hence if $U_{red} \to X_{red}$ is an étale morphism with $U_{red}$ affine, then we obtain a system of étale morphisms $U_\lambda \to X_\lambda $ of affine schemes compatible with the transition morphisms in the system defining $X$. Hence we can take

as our affine formal algebraic space over $X$. The construction gives that $U \times _ X X_\lambda = U_\lambda $. This shows that $U \to X$ is representable and étale. We omit the verification that the constructions are mutually inverse to each other. $\square$

Proof. Combine Lemmas 87.34.4 and 87.19.13. $\square$

Proof. Denote $\mathcal{C}$ the category introduced in the lemma. Recall that $X_{spaces, {\acute{e}tale}} = X_{red, spaces, {\acute{e}tale}}$. Hence we can define a functor

because $U \times _ X X_{red}$ is an algebraic space étale over $X_{red}$.

To finish the proof we will construct a quasi-inverse. Choose an object $\psi : V \to X_{red}$ of $X_{red, spaces, {\acute{e}tale}}$. Consider the functor $U_{V, \psi } : (\mathit{Sch}/S)_{fppf} \to \textit{Sets}$ given by

We claim that the transformation $U_{V, \psi } \to X$, $(a, b) \mapsto a$ defines an object of the category $\mathcal{C}$. First, let's prove that $U_{V, \psi }$ is a formal algebraic space. Observe that $U_{V, \psi }$ is a sheaf for the fppf topology (some details omitted). Next, suppose that $X_ i \to X$ is an étale covering by affine formal algebraic spaces as in Definition 87.11.1. Set $V_ i = V \times _{X_{red}} X_{i, red}$ and denote $\psi _ i : V_ i \to X_{i, red}$ the projection. Then we have

by a formal argument because $X_{i, red} = X_ i \times _ X X_{red}$ (as $X_ i \to X$ is representable by algebraic spaces and étale). Hence it suffices to show that $U_{V_ i, \psi _ i}$ is an affine formal algebraic space, because then we will have a covering $U_{V_ i, \psi _ i} \to U_{V, \psi }$ as in Definition 87.11.1. On the other hand, we have seen in the proof of Lemma 87.34.3 that $\psi _ i : V_ i \to X_ i$ is the base change of a representable and étale morphism $U_ i \to X_ i$ of affine formal algebraic spaces. Then it is not hard to see that $U_ i = U_{V_ i, \psi _ i}$ as desired.

We omit the verification that $U_{V, \psi } \to X$ is representable by algebraic spaces and étale. Thus we obtain our functor $(V, \psi ) \mapsto (U_{V, \psi } \to X)$ in the other direction. We omit the verification that the constructions are mutually inverse to each other. $\square$

Proof. This follows by combining Lemmas 87.34.6 and 87.18.3. $\square$