Editing Algebraic stack (section)

== Structure sheaf ==
The structure sheaf of an algebraic stack is an object pulled back from a universal structure sheaf <math>\mathcal{O}</math> on the site <math>(Sch/S)_{fppf}</math>.<ref>{{Cite web|title=Section 94.3 (06TI): Presheaves—The Stacks project|url=https://stacks.math.columbia.edu/tag/06TI|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref> This '''universal structure sheaf'''<ref>{{Cite web|title=Section 94.6 (06TU): The structure sheaf—The Stacks project|url=https://stacks.math.columbia.edu/tag/06TU|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref> is defined as<blockquote><math>\mathcal{O}:(Sch/S)_{fppf}^{op} \to Rings, \text{ where } U/X \mapsto \Gamma(U,\mathcal{O}_U)</math></blockquote>and the associated structure sheaf on a category fibered in groupoids<blockquote><math>p:\mathcal{X} \to (Sch/S)_{fppf}</math></blockquote>is defined as<blockquote><math>\mathcal{O}_\mathcal{X} := p^{-1}\mathcal{O}</math></blockquote>where <math>p^{-1}</math> comes from the map of Grothendieck topologies. In particular, this means is <math>x \in \text{Ob}(\mathcal{X})</math> lies over <math>U</math>, so <math>p(x) = U</math>, then <math>\mathcal{O}_\mathcal{X}(x)=\Gamma(U,\mathcal{O}_U)</math>. As a sanity check, it's worth comparing this to a category fibered in groupoids coming from an <math>S</math>-scheme <math>X</math> for various topologies.<ref>{{Cite web|title=Section 94.8 (076N): Representable categories—The Stacks project|url=https://stacks.math.columbia.edu/tag/076N|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref> For example, if <blockquote><math>(\mathcal{X}_{Zar},\mathcal{O}_\mathcal{X}) = ((Sch/X)_{Zar}, \mathcal{O}_X)</math></blockquote>is a category fibered in groupoids over <math>(Sch/S)_{fppf}</math>, the structure sheaf for an open subscheme <math>U \to X</math> gives<blockquote><math>\mathcal{O}_\mathcal{X}(U) = \mathcal{O}_X(U) = \Gamma(U,\mathcal{O}_X)</math></blockquote>so this definition recovers the classic structure sheaf on a scheme. Moreover, for a [[quotient stack]] <math>\mathcal{X} = [X/G]</math>, the structure sheaf this just gives the <math>G</math>-invariant sections<blockquote><math>\mathcal{O}_{\mathcal{X}}(U) = \Gamma(U,u^*\mathcal{O}_X)^{G}</math></blockquote>for <math>u:U\to X</math> in <math>(Sch/S)_{fppf}</math>.<ref>{{Cite web|title=Lemma 94.13.2 (076S)—The Stacks project|url=https://stacks.math.columbia.edu/tag/076S|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref><ref>{{Cite web|title=Section 76.12 (0440): Quasi-coherent sheaves on groupoids—The Stacks project|url=https://stacks.math.columbia.edu/tag/0440|access-date=2020-10-01|website=stacks.math.columbia.edu}}</ref>