Lemma 76.56.6. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space with the resolution property. Then $X$ has affine diagonal over $\mathbf{Z}$ (as in Properties of Spaces, Definition 66.3.1).
Proof. We could prove this as in the case of schemes, but instead we will deduce the lemma from the case of schemes. First, we may and do assume $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Next, we choose a scheme $Y$ and a surjective integral morphism $f : Y \to X$, see Decent Spaces, Lemma 68.9.2. Then $f$ is affine, hence $Y$ has the resolution property by Derived Categories of Spaces, Lemma 75.28.3. Hence by the case of schemes, the scheme $Y$ has affine diagonal, see Derived Categories of Schemes, Lemma 36.36.10. Next, we consider the commutative diagram
\[ \xymatrix{ Y \ar[d] \ar[rr]_{\Delta _ Y} & & Y \times _{\mathbf{Z}} Y \ar[d] \\ X \ar[rr]^{\Delta _ X} & & X \times _{\mathbf{Z}} X } \]
Observe that the right vertical arrow is integral, in particular affine. Let $W \to X \times _{\mathbf{Z}} X$ be a morphism with $W$ affine. Then we see that
\[ Y \times _{X \times _{\mathbf{Z}} X} W = Y \times _{\Delta _ Y, Y \times _{\mathbf{Z}} Y} (Y \times _{\mathbf{Z}} Y) \times _{X \times _{\mathbf{Z}} X} W \]
is affine. On the other hand, $Y \to X$ is integral and surjective hence
\[ Y \times _{X \times _{\mathbf{Z}} X} W \longrightarrow X \times _{X \times _{\mathbf{Z}} X} W \]
is integral surjective as the base change of $Y \to X$ to $W$. We conclude that the target of this arrow is affine by Limits of Spaces, Proposition 70.15.2. It follows that $\Delta _ X$ is affine as desired. $\square$
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