Editing Gerbe (section)

===== 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>.