It is a trivial gerbe if and only if and are coprime. Root stacks 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). Root stacks with sections 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 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 . 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 Differential geometry and -gerbes: Jean-Luc Brylinski's approach 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 version of abelian gerbes belonging more to