Motivation for gerbes on a site

One of the main motivations for considering gerbes on a site is to consider the following naive question: if the Cech cohomology group ${\displaystyle H^{1}({\mathcal {U}},G)}$ for a suitable covering ${\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}}$ of a space ${\displaystyle X}$ gives the isomorphism classes of principal ${\displaystyle G}$-bundles over ${\displaystyle X}$, what does the iterated cohomology functor ${\displaystyle H^{1}(-,H^{1}(-,G))}$ represent? Meaning, we are gluing together the groups ${\displaystyle H^{1}(U_{i},G)}$ via some one cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group ${\displaystyle H^{2}({\mathcal {U}},G)}$. It is expected this intuition should hold for higher gerbes.

From two-term complexes of coherent sheaves

Every two-term complex of coherent sheaves

${\displaystyle {\mathcal {E}}^{\bullet }=[{\mathcal {E}}^{-1}\xrightarrow {d} {\mathcal {E}}^{0}]}$

on a scheme ${\displaystyle X\in {\text{Sch}}}$ has a canonical sheaf of groupoids associated to it, where on an open subset ${\displaystyle U\subseteq X}$ there is a two-term complex of ${\displaystyle X(U)}$-modules

${\displaystyle {\mathcal {E}}^{-1}(U)\xrightarrow {d} {\mathcal {E}}^{0}(U)}$

giving a groupoid. It has objects given by elements ${\displaystyle x\in {\mathcal {E}}^{0}(U)}$ and a morphism ${\displaystyle x\to x'}$ is given by an element ${\displaystyle y\in {\mathcal {E}}^{-1}(U)}$ such that

${\displaystyle dy+x=x'}$

In order for this stack to be a gerbe, the cohomology sheaf ${\displaystyle {\mathcal {H}}^{0}({\mathcal {E}})}$ must always have a section. This hypothesis implies the category constructed above always has objects. Note this can be applied to the situation of comodules over Hopf-algebroids to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded Hopf-algebroid is used). In addition, two-term spectra from the stabilization of the derived category of comodules of Hopf-algebroids ${\displaystyle (A,\Gamma )}$ with ${\displaystyle \Gamma }$ flat over ${\displaystyle A}$ give additional models of gerbes which are non-strict.

Moduli stack of stable bundles on a curve

Consider a smooth projective curve ${\displaystyle C}$ over ${\displaystyle k}$ of genus ${\displaystyle g>1}$. Let ${\displaystyle {\mathcal {M}}_{r,d}^{s}}$ be the moduli stack of stable vector bundles on ${\displaystyle C}$ of rank ${\displaystyle r}$ and degree ${\displaystyle d}$. It has a coarse moduli space ${\displaystyle M_{r,d}^{s}}$, which is a quasiprojective variety. These two moduli problems parametrize the same objects, but the stacky version remembers automorphisms of vector bundles. For any stable vector bundle ${\displaystyle E}$ the automorphism group ${\displaystyle Aut(E)}$ consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to ${\displaystyle \mathbb {G} _{m}}$. It turns out that the map ${\displaystyle {\mathcal {M}}_{r,d}^{s}\to M_{r,d}^{s}}$ is indeed a ${\displaystyle \mathbb {G} _{m}}$-gerbe in the sense above.^[9] It is a trivial gerbe if and only if ${\displaystyle r}$ and ${\displaystyle d}$ are coprime.

Root stacks

Another class of gerbes can be found using the construction of root stacks. Informally, the ${\displaystyle r}$-th root stack of a line bundle ${\displaystyle L\to S}$ over a scheme is a space representing the ${\displaystyle r}$-th root of ${\displaystyle L}$ and is denoted

${\displaystyle {\sqrt[{r}]{L/S}}.\,}$^{[10]pg 52}

The ${\displaystyle r}$-th root stack of ${\displaystyle L}$ has the property

${\displaystyle \bigotimes ^{r}{\sqrt[{r}]{L/S}}\cong L}$

as gerbes. It is constructed as the stack

${\displaystyle {\sqrt[{r}]{L/S}}:(\operatorname {Sch} /S)^{op}\to \operatorname {Grpd} }$

sending an ${\displaystyle S}$-scheme ${\displaystyle T\to S}$ to the category whose objects are line bundles of the form

${\displaystyle \left\{(M\to T,\alpha _{M}):\alpha _{M}:M^{\otimes r}\xrightarrow {\sim } L\times _{S}T\right\}}$

and morphisms are commutative diagrams compatible with the isomorphisms ${\displaystyle \alpha _{M}}$. This gerbe is banded by the algebraic group of roots of unity ${\displaystyle \mu _{r}}$, where on a cover ${\displaystyle T\to S}$ it acts on a point ${\displaystyle (M\to T,\alpha _{M})}$ by cyclically permuting the factors of ${\displaystyle M}$ in ${\displaystyle M^{\otimes r}}$. Geometrically, these stacks are formed as the fiber product of stacks

${\displaystyle {\begin{matrix}X\times _{B\mathbb {G} _{m}}B\mathbb {G} _{m}&\to &B\mathbb {G} _{m}\\\downarrow &&\downarrow \\X&\to &B\mathbb {G} _{m}\end{matrix}}}$

where the vertical map of ${\displaystyle B\mathbb {G} _{m}\to B\mathbb {G} _{m}}$ comes from the Kummer sequence

${\displaystyle 1\xrightarrow {} \mu _{r}\xrightarrow {} \mathbb {G} _{m}\xrightarrow {(\cdot )^{r}} \mathbb {G} _{m}\xrightarrow {} 1}$

This is because ${\displaystyle B\mathbb {G} _{m}}$ is the moduli space of line bundles, so the line bundle ${\displaystyle L\to S}$ corresponds to an object of the category ${\displaystyle B\mathbb {G} _{m}(S)}$ (considered as a point of the moduli space).

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 algebras
• Deformations of infinitesimal thickenings
• Twisted forms of projective varieties
• Fiber functors for motives