Gerbe

 ( Homological Algebra ) 

In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.

"Gerbe" is a French (and archaic English) word that literally means "wheat sheaf."

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Definitions

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Gerbes on a topological space A gerbe on a topological space $S$

(2026). 9783540749561, Springer. ISBN 9783540749561

is a stack $\backslash mathcal\{X\}$ of over $S$ that is locally non-empty (each point $p\; \backslash in\; S$ has an open neighbourhood $U\_p$ over which the section category $\backslash mathcal\{X\}(U\_p)$ of the gerbe is not empty) and transitive (for any two objects $a$ and $b$ of $\backslash mathcal\{X\}(U)$ for any open set $U$, there is an open covering $\backslash mathcal\{U\}\; =\; \backslash \{U\_i\; \backslash \}\_\{i\; \backslash in\; I\}$ of $U$ such that the restrictions of $a$ and $b$ to each $U\_i$ are connected by at least one morphism).

A canonical example is the gerbe $BH$ of principal bundles with a fixed structure group $H$: the section category over an open set $U$ is the category of principal $H$-bundles on $U$ with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle $X\; \backslash times\; H\; \backslash to\; X$ shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.

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Gerbes on a site The most general definition of gerbes are defined over a site. Given a site $\backslash mathcal\{C\}$ a $\backslash mathcal\{C\}$-gerbe $G$

Giraud, J. (Jean) (1971). 9783540053071, Springer-Verlag. ISBN 9783540053071

is a category fibered in groupoids $G\; \backslash to\; \backslash mathcal\{C\}$ such that

1. There exists a refinement $\backslash mathcal\{C\}\text{'}$ of $\backslash mathcal\{C\}$ such that for every object $S\; \backslash in\; \backslash text\{Ob\}(\backslash mathcal\{C\}\text{'})$ the associated fibered category $G\_S$ is non-empty
2. For every $S\; \backslash in\; \backslash text\{Ob\}(\backslash mathcal\{C\})$ any two objects in the fibered category $G\_S$ are locally isomorphic

Note that for a site $\backslash mathcal\{C\}$ with a final object $e$, a category fibered in groupoids $G\; \backslash to\; \backslash mathcal\{C\}$ admits a local section if and only if $\backslash text\{Ob\}(G\_e)\; \backslash neq\; \backslash varnothing$, in which case $G\; \backslash to\; \backslash mathcal\{C\}$ automatically satisfies the first axiom.

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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 $H^1(\backslash mathcal\{U\},G)$ for a suitable covering $\backslash mathcal\{U\}\; =\; \backslash \{U\_i\backslash \}\_\{i\; \backslash in\; I\}$ of a space $X$ gives the isomorphism classes of principal $G$-bundles over $X$, what does the iterated cohomology functor $H^1(-,H^1(-,G))$ represent? Meaning, we are gluing together the groups $H^1(U\_i,G)$ via some 1-cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group $H^2(\backslash mathcal\{U\},G)$. It is expected this intuition should hold for .

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Cohomological classification One of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups $\backslash underline\{L\}$, called a band. For a gerbe $\backslash mathcal\{X\}$ on a site $\backslash mathcal\{C\}$, an object $U\; \backslash in\; \backslash text\{Ob\}(\backslash mathcal\{C\})$, and an object $x\; \backslash in\; \backslash text\{Ob\}(\backslash mathcal\{X\}(U))$, the automorphism group of a gerbe is defined as the automorphism group $L\; =\; \backslash underline\{\backslash text\{Aut\}\}\_\{\backslash mathcal\{X\}(U)\}(x)$. Notice this is well defined whenever the automorphism group is always the same. Given a covering $\backslash mathcal\{U\}\; =\; \backslash \{U\_i\; \backslash to\; X\; \backslash \}\_\{i\; \backslash in\; I\}$, there is an associated class

$c(\backslash underline\{L\})\; \backslash in\; H^3(X,\backslash underline\{L\})$

representing the isomorphism class of the gerbe $\backslash mathcal\{X\}$ banded by $L$.

For example, in topology, many examples of gerbes can be constructed by considering gerbes banded by the group $U(1)$. As the classifying space $B(U(1))\; =\; K(\backslash mathbb\{Z\},2)$ is the second Eilenberg–Maclane space for the integers, a bundle gerbe banded by $U(1)$ on a topological space $X$ is constructed from a homotopy class of maps in

$X,\; =\; X,K(\backslash mathbb\{Z\},3)$,

which is exactly the third singular homology group $H^3(X,\backslash mathbb\{Z\})$. It has been found that all gerbes representing torsion cohomology classes in $H^3(X,\backslash mathbb\{Z\})$ are represented by a bundle of finite dimensional algebras $\backslash text\{End\}(V)$ for a fixed complex vector space $V$. In addition, the non-torsion classes are represented as infinite-dimensional principal bundles $PU(\backslash mathcal\{H\})$ of the projective group of unitary operators on a fixed infinite dimensional Separable space Hilbert space $\backslash mathcal\{H\}$. Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square-summable sequences $\backslash ell^2$.

The homotopy-theoretic interpretation of gerbes comes from looking at the homotopy fiber square

analogous to how a line bundle comes from the homotopy fiber square

where $BU(1)\; \backslash simeq\; K(\backslash mathbb\{Z\},2)$, giving $H^2(S,\backslash mathbb\{Z\})$ as the group of isomorphism classes of line bundles on $S$.

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Examples

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C*-algebras

There are natural examples of Gerbes that arise from studying the algebra of compactly supported complex valued functions on a paracompact space $X$^{pg 3}. Given a cover $\backslash mathcal\{U\}\; =\; \backslash \{U\_i\backslash \}$ of $X$ there is the Cech groupoid defined as

$\backslash mathcal\{G\}\; =\; \backslash left\backslash \{\; \backslash coprod\_\{i,j\}U\_\{ij\}\; \backslash rightrightarrows\; \backslash coprod\; U\_i\; \backslash right\backslash \}$

with source and target maps given by the inclusions

$\backslash begin\{align\}\; s:\; U\_\{ij\}\; \backslash hookrightarrow\; U\_j\; \backslash \backslash \; t:\; U\_\{ij\}\; \backslash hookrightarrow\; U\_i\; \backslash end\{align\}$

and the space of composable arrows is just

$\backslash coprod\_\{i,j,k\}U\_\{ijk\}$

Then a degree 2 cohomology class $\backslash sigma\; \backslash in\; H^2(X;U(1))$ is just a map

$\backslash sigma:\; \backslash coprod\; U\_\{ijk\}\; \backslash to\; U(1)$

We can then form a non-commutative C*-algebra $C\_c(\backslash mathcal\{G\}(\backslash sigma))$, which is associated to the set of compact supported complex valued functions of the space

$\backslash mathcal\{G\}\_1\; =\; \backslash coprod\_\{i,j\}U\_\{ij\}$

It has a non-commutative product given by

$a*\; b(x,i,k)\; :=\; \backslash sum\_j\; a(x,i,j)b(x,j,k)\backslash sigma(x,i,j,k)$

where the cohomology class $\backslash sigma$ twists the multiplication of the standard $C^*$-algebra product.

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Algebraic geometry

Let $M$ be a variety over an algebraically closed field $k$, $G$ an algebraic group, for example $\backslash mathbb\{G\}\_m$. Recall that a G-torsor over $M$ is an algebraic space $P$ with an action of $G$ and a map $\backslash pi:P\backslash to\; M$, such that locally on $M$ (in étale topology or fppf topology) $\backslash pi$ is a direct product $\backslash pi|\_U:G\backslash times\; U\backslash to\; U$. A G -gerbe over M may be defined in a similar way. It is an Artin stack $\backslash mathcal\{M\}$ with a map $\backslash pi\backslash colon\backslash mathcal\{M\}\; \backslash to\; M$, such that locally on M (in étale or fppf topology) $\backslash pi$ is a direct product $\backslash pi|\_U\backslash colon\; \backslash mathrm\{B\}G\; \backslash times\; U\; \backslash to\; U$. Here $BG$ denotes the classifying stack of $G$, i.e. a quotient $$ of a point by a trivial $G$-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 spaces of $\backslash mathcal\{M\}$ and $M$ are the same, but in $\backslash mathcal\{M\}$ each point is equipped with a stabilizer group isomorphic to $G$.

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From two-term complexes of coherent sheaves

Every two-term complex of coherent sheaves

$\backslash mathcal\{E\}^\backslash bullet\; =\; \backslash mathcal\{E\}^\{-1\}$

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

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

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

$dy\; +\; x\; =\; x\text{'}$

In order for this stack to be a gerbe, the cohomology sheaf $\backslash mathcal\{H\}^0(\backslash mathcal\{E\})$ 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 $(A,\backslash Gamma)$ with $\backslash Gamma$ flat over $A$ give additional models of gerbes that are non-strict.

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Moduli stack of stable bundles on a curve

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

(2026). 9780817649333 ISBN 9780817649333

It is a trivial gerbe if and only if $r$ and $d$ are coprime.

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Root stacks

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

$\backslash sqrtr\{L/S\}.\backslash ,$^{pg 52}

The $r$-th root stack of $L$ has the property

$\backslash bigotimes^r\backslash sqrt\{r\}\{L/S\}\; \backslash cong\; L$

as gerbes. It is constructed as the stack

$\backslash sqrtr\{L/S\}:\; (\backslash operatorname\{Sch\}/S)^\{op\}\; \backslash to\; \backslash operatorname\{Grpd\}$

sending an $S$-scheme $T\; \backslash to\; S$ to the category whose objects are line bundles of the form

$\backslash left\backslash \{\; (M\; \backslash to\; T,\backslash alpha\_M)\; :\; \backslash alpha\_M:\; M^\{\backslash otimes\; r\}\; \backslash xrightarrow\{\backslash sim\}\; L\backslash times\_ST\; \backslash right\backslash \}$

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

where the vertical map of $B\backslash mathbb\{G\}\_m\; \backslash to\; B\backslash mathbb\{G\}\_m$ comes from the Kummer sequence

$1\; \backslash xrightarrow\{\}\; \backslash mu\_r\; \backslash xrightarrow\{\}\; \backslash mathbb\{G\}\_m\; \backslash xrightarrow\{\; (\backslash cdot)^r\}\; \backslash mathbb\{G\}\_m\; \backslash xrightarrow\{\}\; 1$

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

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Root stacks with sections

There is another related construction of root stacks with sections. Given the data above, let $s:\; S\; \backslash to\; L$ be a section. Then the $r$-th root stack of the pair $(L\backslash to\; S,s)$ is defined as the lax 2-functor

$\backslash sqrtr\{(L,s)/S\}:\; (\backslash operatorname\{Sch\}/S)^\{op\}\; \backslash to\; \backslash operatorname\{Grpd\}$

sending an $S$-scheme $T\; \backslash to\; S$ to the category whose objects are line bundles of the form

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. Locally, we may assume $S\; =\; \backslash text\{Spec\}(A)$ and the line bundle $L$ is trivial, hence any section $s$ is equivalent to taking an element $s\; \backslash in\; A$. Then, the stack is given by the stack quotient

$\backslash sqrtr\{(L,s)/S\}\; =\; \backslash text\{Spec\}(B)/\backslash mu\_r$

with

$B\; =\; \backslash frac\{Ax\}\{x^r\; -\; s\}$

If $s\; =\; 0$ then this gives an infinitesimal extension of $\backslash text\{Spec\}(A)/\backslash mu\_r$.

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Examples throughout algebraic geometry

These and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookkeeping tools:

• Deformations of infinitesimal thickenings
• Twisted forms of projective varieties
• for motives

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Differential geometry

• $H^3(X,\backslash mathbb\{Z\})$ and $\backslash mathcal\{O\}\_X^*$-gerbes: Jean-Luc Brylinski's approach

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History

Gerbes first appeared in the context of algebraic geometry. They were subsequently developed in a more traditional geometric framework by Brylinski . One can think of gerbes as being a natural step in a hierarchy of mathematical objects providing geometric realizations of integral cohomology classes.

A more specialised notion of gerbe was introduced by Murray and called bundle gerbes. Essentially they are a smooth function version of abelian gerbes belonging more to the hierarchy starting with than sheaves. Bundle gerbes have been used in gauge theory and also string theory. Current work by others is developing a theory of non-abelian bundle gerbes.

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See also

• Twisted sheaf
• Azumaya algebra
• Twisted K-theory
• Algebraic stack
• Bundle gerbe
• String group

• .
• .

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External links

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Introductory articles

• - Stuart Johnson
• , Ernesto Lupercio, Bernado Uribe.
• What is a Gerbe?, by Nigel Hitchin in Notices of the AMS
• , Michael Murray.

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Gerbes in topology

• , Zhi-Ming Luo

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Twisted K-theory

• Twisted Bundles and Twisted K-Theory - Karoubi

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Applications in string theory

• - contains examples of gerbes in appendix using the Brauer group
• - Very down-to earth introduction with applications to Mirror symmetry
• - Gives techniques for describing groups such as the String group as a gerbe

Categories: Homological Algebra, Sheaf Theory

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