non-commutative product given bywhere the cohomology class twists the multiplication of the standard -algebra product. Algebraic geometry 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 . From two-term complexes of coherent sheaves 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 which are non-strict. Moduli stack of stable bundles on a curve 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.