The Stacks project

Lemma 67.6.4. Let $S$ be a scheme. Let $X$ be a quasi-compact reasonable algebraic space. Then there exists a directed system of quasi-compact and quasi-separated algebraic spaces $X_ i$ such that $X = \mathop{\mathrm{colim}}\nolimits _ i X_ i$ (colimit in the category of sheaves). Moreover we can arrange it such that

  1. for every quasi-compact scheme $T$ over $S$ we have $\mathop{\mathrm{colim}}\nolimits X_ i(T) = X(T)$,

  2. the transition morphisms $X_ i \to X_{i'}$ of the system and the coprojections $X_ i \to X$ are surjective and étale, and

  3. if $X$ is a scheme, then the algebraic spaces $X_ i$ are schemes and the transition morphisms $X_ i \to X_{i'}$ and the coprojections $X_ i \to X$ are local isomorphisms.

Proof. We sketch the proof. By Properties of Spaces, Lemma 65.6.3 we have $X = U/R$ with $U$ affine. In this case, reasonable means $U \to X$ is universally bounded. Hence there exists an integer $N$ such that the “fibres” of $U \to X$ have degree at most $N$, see Definition 67.3.1. Denote $s, t : R \to U$ and $c : R \times _{s, U, t} R \to R$ the groupoid structural maps.

Claim: for every quasi-compact open $A \subset R$ there exists an open $R' \subset R$ such that

  1. $A \subset R'$,

  2. $R'$ is quasi-compact, and

  3. $(U, R', s|_{R'}, t|_{R'}, c|_{R' \times _{s, U, t} R'})$ is a groupoid scheme.

Note that $e : U \to R$ is open as it is a section of the étale morphism $s : R \to U$, see Étale Morphisms, Proposition 41.6.1. Moreover $U$ is affine hence quasi-compact. Hence we may replace $A$ by $A \cup e(U) \subset R$, and assume that $A$ contains $e(U)$. Next, we define inductively $A^1 = A$, and

\[ A^ n = c(A^{n - 1} \times _{s, U, t} A) \subset R \]

for $n \geq 2$. Arguing inductively, we see that $A^ n$ is quasi-compact for all $n \geq 2$, as the image of the quasi-compact fibre product $A^{n - 1} \times _{s, U, t} A$. If $k$ is an algebraically closed field over $S$, and we consider $k$-points then

\[ A^ n(k) = \left\{ (u, u') \in U(k) : \begin{matrix} \text{there exist } u = u_1, u_2, \ldots , u_ n \in U(k)\text{ with} \\ (u_ i , u_{i + 1}) \in A \text{ for all }i = 1, \ldots , n - 1. \end{matrix} \right\} \]

But as the fibres of $U(k) \to X(k)$ have size at most $N$ we see that if $n > N$ then we get a repeat in the sequence above, and we can shorten it proving $A^ N = A^ n$ for all $n \geq N$. This implies that $R' = A^ N$ gives a groupoid scheme $(U, R', s|_{R'}, t|_{R'}, c|_{R' \times _{s, U, t} R'})$, proving the claim above.

Consider the map of sheaves on $(\mathit{Sch}/S)_{fppf}$

\[ \mathop{\mathrm{colim}}\nolimits _{R' \subset R} U/R' \longrightarrow U/R \]

where $R' \subset R$ runs over the quasi-compact open subschemes of $R$ which give étale equivalence relations as above. Each of the quotients $U/R'$ is an algebraic space (see Spaces, Theorem 64.10.5). Since $R'$ is quasi-compact, and $U$ affine the morphism $R' \to U \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} U$ is quasi-compact, and hence $U/R'$ is quasi-separated. Finally, if $T$ is a quasi-compact scheme, then

\[ \mathop{\mathrm{colim}}\nolimits _{R' \subset R} U(T)/R'(T) \longrightarrow U(T)/R(T) \]

is a bijection, since every morphism from $T$ into $R$ ends up in one of the open subrelations $R'$ by the claim above. This clearly implies that the colimit of the sheaves $U/R'$ is $U/R$. In other words the algebraic space $X = U/R$ is the colimit of the quasi-separated algebraic spaces $U/R'$.

Properties (1) and (2) follow from the discussion above. If $X$ is a scheme, then if we choose $U$ to be a finite disjoint union of affine opens of $X$ we will obtain (3). Details omitted. $\square$

Comments (2)

Comment #6119 by comment_bot on

It would be helpful to make the statement more detailed by explicitly mentioning there that:

  1. The maps are étale.
  2. If is a scheme, then one may choose all the to also be schemes and the maps to be local isomorphisms.
  3. For every quasi-compact -scheme (I suppose that in the proof was meant to be an -scheme---this should be mentioned), the map is bijective.

All of these aspects are actually useful in practice, see Remark 4.6 in Bhatt's article "Algebraization and Tannaka duality."

Comment #6201 by on

Yes, thanks! Strangely I never realized this argument says something even in the case of schemes until Bhargav pointed it out to me (of course he came up with the argument independently). The change is here.

There are also:

  • 2 comment(s) on Section 67.6: Reasonable and decent algebraic spaces

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