Editing Gerbe (section)

==== 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>.