Гомологиялық алгебра мен топологиядағы гербелер

In mathematics, a gerbe (/dʒ//ɜr//b/; ʒɛʁb) 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 fibre bundles 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.

A gerbe on a topological space is a stack of groupoids over that is locally non empty (each point has an open neighbourhood over which the section category of the gerbe is not empty) and transitive (for any two objects and of for any open set , there is an open covering of such that the restrictions of and to each are connected by at least one morphism). A canonical example is the gerbe of principal bundles with a fixed structure group : the section category over an open set is the category of principal bundles on 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 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.

One of the main motivations for considering gerbes on a site is to consider the following naive question: if the Cech cohomology group for a suitable covering of a space gives the isomorphism classes of principal bundles over , what does the iterated cohomology functor represent? Meaning, we are gluing together the groups via some one cocycle. Gerbes are a technical response for this question: they give geometric representations of elements in the higher cohomology group It is expected this intuition should hold for higher gerbes.

One of the main theorems concerning gerbes is their cohomological classification whenever they have automorphism groups given by a fixed sheaf of abelian groups , that all gerbes representing torsion cohomology classes in are represented by a bundle of finite dimensional algebras for a fixed complex vector space In addition, the non torsion classes are represented as infinite dimensional principal bundles of the projective group of unitary operators on a fixed infinite dimensional separable Hilbert space Note this is well defined because all separable Hilbert spaces are isomorphic to the space of square summable sequences
The homotopy theoretic interpretation of gerbes comes from looking at the homotopy fiber squareanalogous to how a line bundle comes from the homotopy fiber squarewhere , giving as the group of isomorphism classes of line bundles on .

There are natural examples of Gerbes that arise from studying the algebra of compactly supported complex valued functions on a paracompact space pg 3. Given a cover of there is the Cech groupoid defined aswith source and target maps given by the inclusionsand the space of composable arrows is justThen a degree 2 cohomology class is just a mapWe can then form a non commutative C* algebra , which is associated to the set of compact supported complex valued functions of the spaceIt has a non commutative product given bywhere the cohomology class twists the multiplication of the standard algebra product.

Let be a variety over an algebraically closed field , an algebraic group, for example Recall that a G torsor over is an algebraic space with an action of and a map , such that locally on (in étale topology or fppf topology) is a direct product A G gerbe over M may be defined in a similar way. It is an Artin stack with a map , such that locally on M (in étale or fppf topology) is a direct product Here denotes the classifying stack of , i. e. a quotient of a point by a trivial 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 and are the same, but in each point is equipped with a stabilizer group isomorphic to .

Every two term complex of coherent sheaveson a scheme has a canonical sheaf of groupoids associated to it, where on an open subset there is a two term complex of modulesgiving a groupoid. It has objects given by elements and a morphism is given by an element such thatIn order for this stack to be a gerbe, the cohomology sheaf 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 with flat over give additional models of gerbes that are non strict.

Consider a smooth projective curve over of genus Let be the moduli stack of stable vector bundles on of rank and degree It has a coarse moduli space , 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 the automorphism group consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to It turns out that the map is indeed a gerbe in the sense above. It is a trivial gerbe if and only if and are coprime.

Another class of gerbes can be found using the construction of root stacks. Informally, the th root stack of a line bundle over a scheme is a space representing the th root of and is denotedpg 52 The th root stack of has the propertyas gerbes. It is constructed as the stacksending an scheme to the category whose objects are line bundles of the formand morphisms are commutative diagrams compatible with the isomorphisms This gerbe is banded by the algebraic group of roots of unity , where on a cover it acts on a point by cyclically permuting the factors of in Geometrically, these stacks are formed as the fiber product of stackswhere the vertical map of comes from the Kummer sequenceThis is because is the moduli space of line bundles, so the line bundle corresponds to an object of the category (considered as a point of the moduli space).

There is another related construction of root stacks with sections. Given the data above, let be a section. Then the th root stack of the pair is defined as the lax 2 functorsending an scheme to the category whose objects are line bundles of the formand 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. Given an affine scheme , all line bundles are trivial, hence and any section is equivalent to taking an element Then, the stack is given by the stack quotientwithIf then this gives an infinitesimal extension of .

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 version of abelian gerbes belonging more to the hierarchy starting with principal bundles 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.