Editing Gerbe (section)

==== From two-term complexes of coherent sheaves ====
Every two-term complex of coherent sheaves<blockquote><math>\mathcal{E}^\bullet = [\mathcal{E}^{-1} \xrightarrow{d} \mathcal{E}^0]</math></blockquote>on a scheme <math>X \in \text{Sch}</math> has a canonical sheaf of groupoids associated to it, where on an open subset <math>U \subseteq X</math> there is a two-term complex of <math>X(U)</math>-modules<blockquote><math>\mathcal{E}^{-1}(U) \xrightarrow{d} \mathcal{E}^0(U)</math></blockquote>giving a groupoid. It has objects given by elements <math>x \in \mathcal{E}^0(U)</math> and a morphism <math>x \to x'</math> is given by an element <math>y \in \mathcal{E}^{-1}(U)</math> such that<blockquote><math>dy + x = x' </math></blockquote>In order for this stack to be a gerbe, the cohomology sheaf <math>\mathcal{H}^0(\mathcal{E})</math> must always have a section. This hypothesis implies the category constructed above always has objects. Note this can be applied to the situation of [[Comodule over a Hopf algebroid|comodules over Hopf-algebroids]] to construct algebraic models of gerbes over affine or projective stacks (projectivity if a graded [[Hopf algebroid|Hopf-algebroid]] is used). In addition, two-term spectra from the stabilization of the [[derived category]] of comodules of Hopf-algebroids <math>(A,\Gamma)</math> with <math>\Gamma</math> flat over <math>A</math> give additional models of gerbes that are [[Abelian 2-group|non-strict]].