Editing Algebraic stack (section)

===== 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>.