Editing Algebraic stack (section)

==== Explanation of technical conditions ====

===== Using the fppf topology =====
First of all, the fppf-topology is used because it behaves well with respect to [[Descent theory|descent]]. For example, if there are schemes <math>X,Y \in \operatorname{Ob}(\mathrm{Sch}/S)</math> and <math>X \to Y</math>can be refined to an fppf-cover of <math>Y</math>, if <math>X</math> is flat, locally finite type, or locally of finite presentation, then <math>Y</math> has this property.<ref>{{Cite web|title=Lemma 35.11.8 (06NB)—The Stacks project|url=https://stacks.math.columbia.edu/tag/06NB|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> this kind of idea can be extended further by considering properties local either on the target or the source of a morphism <math>f:X\to Y</math>. For a cover <math>\{X_i \to X\}_{i \in I}</math> we say a property <math>\mathcal{P}</math> is '''local on the source''' if<blockquote><math>f:X\to Y</math> has <math>\mathcal{P}</math> if and only if each <math>X_i \to Y</math> has <math>\mathcal{P}</math>.</blockquote>There is an analogous notion on the target called '''local on the target'''. This means given a cover <math>\{Y_i \to Y \}_{i \in I}</math><blockquote><math>f:X\to Y</math> has <math>\mathcal{P}</math> if and only if each <math>X\times_YY_i \to Y_i</math> has <math>\mathcal{P}</math>.</blockquote>For the fppf topology, having an immersion is local on the target.<ref>{{Cite web|title=Section 35.21 (02YL): Properties of morphisms local in the fppf topology on the target—The Stacks project|url=https://stacks.math.columbia.edu/tag/02YL|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> In addition to the previous properties local on the source for the fppf topology, <math>f</math> being universally open is also local on the source.<ref>{{Cite web|title=Section 35.25 (036M): Properties of morphisms local in the fppf topology on the source—The Stacks project|url=https://stacks.math.columbia.edu/tag/036M|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> Also, being locally Noetherian and Jacobson are local on the source and target for the fppf topology.<ref>{{Cite web|title=Section 35.13 (034B): Properties of schemes local in the fppf topology—The Stacks project|url=https://stacks.math.columbia.edu/tag/034B|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> This does not hold in the fpqc topology, making it not as "nice" in terms of technical properties. Even though this is true, using algebraic stacks over the fpqc topology still has its use, such as in [[chromatic homotopy theory]]. This is because the [[Moduli stack of formal group laws]] <math>\mathcal{M}_{fg}</math> is an fpqc-algebraic stack<ref>{{Cite web|last=Goerss|first=Paul|title=Quasi-coherent sheaves on the Moduli Stack of Formal Groups|url=https://sites.math.northwestern.edu/~pgoerss/papers/modfg.pdf|url-status=live|archive-url=https://web.archive.org/web/20200829022756/https://sites.math.northwestern.edu/~pgoerss/papers/modfg.pdf|archive-date=29 August 2020}}</ref><sup>pg 40</sup>.

===== Representable diagonal =====
By definition, a 1-morphism <math>f:\mathcal{X} \to \mathcal{Y}</math> of categories fibered in groupoids is '''representable by algebraic spaces'''<ref>{{Cite web|title=Section 92.9 (04SX): Morphisms representable by algebraic spaces—The Stacks project|url=https://stacks.math.columbia.edu/tag/04SX|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> if for any fppf morphism <math>U \to S</math> of schemes and any 1-morphism <math>y: (Sch/U)_{fppf} \to \mathcal{Y}</math>, the associated category fibered in groupoids<blockquote><math>(Sch/U)_{fppf}\times_{\mathcal{Y}} \mathcal{X}</math></blockquote>is '''representable as an algebraic space''',<ref>{{Cite web|title=Section 92.7 (04SU): Split categories fibred in groupoids—The Stacks project|url=https://stacks.math.columbia.edu/tag/04SU|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref><ref>{{Cite web|title=Section 92.8 (02ZV): Categories fibred in groupoids representable by algebraic spaces—The Stacks project|url=https://stacks.math.columbia.edu/tag/02ZV|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> meaning there exists an algebraic space<blockquote><math>F:(Sch/S)^{op}_{fppf} \to Sets</math></blockquote>such that the associated fibered category <math>\mathcal{S}_F \to (Sch/S)_{fppf}</math><ref><math>Sets \to Cat</math> is the embedding sending a set <math>S</math> to the category of objects <math>S</math> and only identity morphisms. Then, the Grothendieck construction can be applied to give a category fibered in groupoids</ref> is equivalent to <math>(Sch/U)_{fppf}\times_{\mathcal{Y}} \mathcal{X}</math>. There are a number of equivalent conditions for representability of the diagonal<ref>{{Cite web|title=Lemma 92.10.11 (045G)—The Stacks project|url=https://stacks.math.columbia.edu/tag/045G|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> which help give intuition for this technical condition, but one of main motivations is the following: for a scheme <math>U</math> and objects <math>x, y \in \operatorname{Ob}(\mathcal{X}_U)</math> the sheaf <math>\operatorname{Isom}(x,y)</math> is representable as an algebraic space. In particular, the stabilizer group for any point on the stack <math>x : \operatorname{Spec}(k) \to \mathcal{X}_{\operatorname{Spec}(k)}</math> is representable as an algebraic space.

Another important equivalence of having a representable diagonal is the technical condition that the intersection of any two algebraic spaces in an algebraic stack is an algebraic space. Reformulated using fiber products<blockquote><math>\begin{matrix}
Y \times_{\mathcal{X}}Z & \to & Y \\
\downarrow & & \downarrow \\
Z & \to & \mathcal{X}
\end{matrix}</math></blockquote>the representability of the diagonal is equivalent to <math>Y \to \mathcal{X}</math> being representable for an algebraic space <math>Y</math>. This is because given morphisms <math>Y \to \mathcal{X}, Z \to \mathcal{X}</math> from algebraic spaces, they extend to maps <math>\mathcal{X}\times\mathcal{X}</math> from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on <math>(F/S)_{fppf}</math> as an algebraic space.<ref>{{Cite web|title=Section 78.5 (046I): Bootstrapping the diagonal—The Stacks project|url=https://stacks.math.columbia.edu/tag/046I|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref>

Note that an analogous condition of representability of the diagonal holds for some formulations of [[higher stacks]]<ref>{{cite arXiv|last=Simpson|first=Carlos|date=1996-09-17|title=Algebraic (geometric) ''n''-stacks|eprint=alg-geom/9609014}}</ref> where the fiber product is an <math>(n-1)</math>-stack for an <math>n</math>-stack <math>\mathcal{X}</math>.