Editing Gerbe (section)

===Algebraic geometry ===
Let <math>M</math> be a [[algebraic variety|variety]] over an [[algebraically closed field]] <math>k</math>, <math>G</math> an [[algebraic group]], for example <math>\mathbb{G}_m</math>. Recall that a [[Torsor (algebraic geometry)|''G''-torsor]] over <math>M</math> is an [[algebraic space]] <math>P</math> with an action of <math>G</math> and a map <math>\pi:P\to M</math>, such that locally on <math>M</math> (in [[étale topology]] or [[fppf topology]]) <math>\pi</math> is a direct product <math>\pi|_U:G\times U\to U</math>. A '''''G''-gerbe over ''M''''' may be defined in a similar way. It is an [[Artin stack]] <math>\mathcal{M}</math> with a map <math>\pi\colon\mathcal{M} \to M</math>, such that locally on ''M'' (in étale or fppf topology) <math>\pi</math> is a direct product <math>\pi|_U\colon \mathrm{B}G \times U \to U</math>.<ref>{{cite journal| first1=Dan | last1 = Edidin | first2 = Brendan | last2 = Hassett|first3 = Andrew | last3 = Kresch | first4 = Angelo | last4 = Vistoli | title = Brauer groups and quotient stacks | journal = [[American Journal of Mathematics]] | year = 2001 | volume = 123 | issue = 4 | pages = 761–777 | arxiv=math/9905049 | doi=10.1353/ajm.2001.0024| s2cid = 16541492 }}</ref> Here <math>BG</math> denotes the [[classifying stack]] of <math>G</math>, i.e. a quotient <math>[ * / G ]</math> of a point by a trivial <math>G</math>-action. There is no need to impose the compatibility with the group structure in that case since it is covered by the definition of a stack. The underlying [[topological space]]s of <math>\mathcal{M}</math> and <math>M</math> are the same, but in <math>\mathcal{M}</math> each point is equipped with a stabilizer group isomorphic to <math>G</math>.

==== From two-term complexes of coherent sheaves ====
Every two-term complex of coherent sheaves<blockquote><math>\mathcal{E}^\bullet = [\mathcal{E}^{-1} \xrightarrow{d} \mathcal{E}^0]</math></blockquote>on a scheme <math>X \in \text{Sch}</math> has a canonical sheaf of groupoids associated to it, where on an open subset <math>U \subseteq X</math> there is a two-term complex of <math>X(U)</math>-modules<blockquote><math>\mathcal{E}^{-1}(U) \xrightarrow{d} \mathcal{E}^0(U)</math></blockquote>giving a groupoid. It has objects given by elements <math>x \in \mathcal{E}^0(U)</math> and a morphism <math>x \to x'</math> is given by an element <math>y \in \mathcal{E}^{-1}(U)</math> such that<blockquote><math>dy + x = x' </math></blockquote>In order for this stack to be a gerbe, the cohomology sheaf <math>\mathcal{H}^0(\mathcal{E})</math> must always have a section. This hypothesis implies the category constructed above always has objects. Note this can be applied to the situation of [[Comodule over a Hopf algebroid|comodules over Hopf-algebroids]] to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded [[Hopf algebroid|Hopf-algebroid]] is used). In addition, two-term spectra from the stabilization of the [[derived category]] of comodules of Hopf-algebroids <math>(A,\Gamma)</math> with <math>\Gamma</math> flat over <math>A</math> give additional models of gerbes that are [[Abelian 2-group|non-strict]].

==== Moduli stack of stable bundles on a curve ====
Consider a smooth [[projective variety|projective]] [[algebraic curve|curve]] <math>C</math> over <math>k</math> of genus <math>g > 1</math>. Let <math>\mathcal{M}^s_{r, d}</math> be the [[moduli space|moduli stack]] of [[stable vector bundle]]s on <math>C</math> of rank <math>r</math> and degree <math>d</math>. It has a [[Moduli space#Coarse moduli spaces|coarse moduli space]] <math>M^s_{r, d}</math>, which is a [[quasiprojective variety]]. These two moduli problems parametrize the same objects, but the stacky version remembers [[automorphism]]s of vector bundles. For any stable vector bundle <math>E</math> the automorphism group <math>Aut(E)</math> consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to <math>\mathbb{G}_m</math>. It turns out that the map <math>\mathcal{M}^s_{r, d} \to M^{s}_{r, d}</math> is indeed a <math>\mathbb{G}_m</math>-gerbe in the sense above.<ref>{{cite journal|last1=Hoffman|first1=Norbert|year=2010|title=Moduli stacks of vector bundles on curves and the King-Schofield rationality proof|journal=Cohomological and Geometric Approaches to Rationality Problems|series=Progress in Mathematics |volume=282 |pages=133–148|doi=10.1007/978-0-8176-4934-0_5|arxiv=math/0511660|isbn=978-0-8176-4933-3|s2cid=5467668}}</ref> It is a trivial gerbe if and only if <math>r</math> and <math>d</math> are [[coprime]].

==== Root stacks ====
Another class of gerbes can be found using the construction of root stacks. Informally, the <math>r</math>-th root stack of a line bundle <math>L \to S</math> over a [[Scheme (mathematics)|scheme]] is a space representing the <math>r</math>-th root of <math>L</math> and is denoted<blockquote><math>\sqrt[r]{L/S}.\,</math><ref name=":0">{{cite arXiv|last1=Abramovich|first1=Dan|last2=Graber|first2=Tom|last3=Vistoli|first3=Angelo|date=2008-04-13|title=Gromov-Witten theory of Deligne-Mumford stacks|eprint=math/0603151}}</ref><sup>pg 52</sup> </blockquote>The <math>r</math>-th root stack of <math>L</math> has the property<blockquote><math>\bigotimes^r\sqrt[{r}]{L/S} \cong L</math></blockquote>as gerbes. It is constructed as the stack<blockquote><math>\sqrt[r]{L/S}: (\operatorname{Sch}/S)^{op} \to \operatorname{Grpd}</math></blockquote>sending an <math>S</math>-scheme <math>T \to S</math> to the category whose objects are line bundles of the form<blockquote><math>\left\{
(M \to T,\alpha_M) : \alpha_M: M^{\otimes r} \xrightarrow{\sim} L\times_ST
\right\}</math></blockquote>and morphisms are commutative diagrams compatible with the isomorphisms <math>\alpha_M</math>. This gerbe is banded by the [[algebraic group]] of roots of unity <math>\mu_r</math>, where on a cover <math>T \to S</math> it acts on a point <math>(M\to T,\alpha_M)</math> by cyclically permuting the factors of <math>M</math> in <math>M^{\otimes r}</math>. Geometrically, these stacks are formed as the fiber product of stacks<blockquote><math>\begin{matrix}
X\times_{B\mathbb{G}_m} B\mathbb{G}_m & \to & B\mathbb{G}_m \\
\downarrow & & \downarrow \\
X & \to & B\mathbb{G}_m
\end{matrix}</math></blockquote>where the vertical map of <math>B\mathbb{G}_m \to B\mathbb{G}_m</math> comes from the [[Kummer sequence]]<blockquote><math>1 \xrightarrow{} \mu_r \xrightarrow{} \mathbb{G}_m \xrightarrow{ (\cdot)^r} \mathbb{G}_m \xrightarrow{} 1</math></blockquote>This is because <math>B\mathbb{G}_m</math> is the moduli space of line bundles, so the line bundle <math>L \to S</math> corresponds to an object of the category <math>B\mathbb{G}_m(S)</math> (considered as a point of the moduli space).

===== Root stacks with sections =====
There is another related construction of root stacks with sections. Given the data above, let <math>s: S \to L</math> be a section. Then the <math>r</math>-th root stack of the pair <math>(L\to S,s)</math> is defined as the lax 2-functor<ref name=":0" /><ref name=":1">{{cite journal|last=Cadman|first=Charles|year=2007|title=Using stacks to impose tangency conditions on curves|url=https://www.charlescadman.com/pdf/stacks.pdf|journal=Amer. J. Math.|volume=129|issue=2|pages=405–427|arxiv=math/0312349|doi=10.1353/ajm.2007.0007|s2cid=10323243}}</ref><blockquote><math>\sqrt[r]{(L,s)/S}: (\operatorname{Sch}/S)^{op} \to \operatorname{Grpd}</math></blockquote>sending an <math>S</math>-scheme <math>T \to S</math> to the category whose objects are line bundles of the form<blockquote><math>\left\{
(M \to T,\alpha_M, t) :
\begin{align}
&\alpha_M: M^{\otimes r} \xrightarrow{\sim} L\times_ST \\
& t \in \Gamma(T,M) \\
&\alpha_M(t^{\otimes r}) = s
\end{align}
\right\}</math></blockquote>and morphisms are given similarly. These stacks can be constructed very explicitly, and are well understood for affine schemes. In fact, these form the affine models for root stacks with sections.<ref name=":1" />{{rp|4}} Locally, we may assume <math>S = \text{Spec}(A)</math> and the line bundle <math>L</math> is trivial, hence any section <math>s</math> is equivalent to taking an element <math>s \in A</math>. Then, the stack is given by the stack quotient<blockquote><math>\sqrt[r]{(L,s)/S} = [\text{Spec}(B)/\mu_r]</math><ref name=":1" />{{rp|9}}</blockquote>with<blockquote><math>B = \frac{A[x]}{x^r - s}</math></blockquote>If <math>s = 0</math> then this gives an infinitesimal extension of <math>[\text{Spec}(A)/\mu_r]</math>.

==== Examples throughout algebraic geometry ====
These and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookkeeping tools:
* [[Azumaya algebra]]s
* Deformations of infinitesimal thickenings
* Twisted forms of projective varieties
* [[Fiber functor]]s for [[motive (algebraic geometry)|motive]]s