process

Deformations of gerbes on smooth manifolds 363The associated stack A C is canonically equivalent to the stack of locally free A op -modules of rank one. The canonical morphism A C ! A C sends the unique (locallydefined) object of A C to the free module of rank one.1-morphisms and 2-morphisms of R-algebroid stacks are those of stacks in R-linearcategories. We denote the 2-category of R-algebroid stacks by AlgStack R .X/.4.2 Descent data4.2.1 Convolution dataDefinition 4.5. An R-linear convolution datum is a triple .U; A 01 ; A 012 / consistingof• a cover U 2 Cov.X/,• a sheaf A 01 of R-modules A 01 on N 1 U,• a morphismA 012 W .pr 2 01 / A 01 ˝R .pr 2 12 / A 01 ! .pr 2 02 / A 01 (4.1)of R-modulessubject to the associativity condition expressed by the commutativity of the diagram.pr 3 01 / A 01 ˝R .pr 3 12 / A 01 ˝R .pr 3 23 / A 01.pr 3 012 / .A 012/˝Id .pr302/ A 01 ˝R .pr 3 23 / A 01Id˝.pr 3 123 / .A 012.pr 3 023 / .A 012/.pr 3 01 / A 01 ˝R .pr 3 .pr 313 / 013A / .A 012/01 .pr303/ A 01 .For a convolution datum .U; A 01 ; A 012 / we denote by A the pair .A 01 ; A 012 / andabbreviate the convolution datum by .U; A/.For a convolution datum .U; A/ let• A WD .pr 0 00 / A 01 ; A is a sheaf of R-modules on N 0 U,• A p iWD .pr p i / A; thus for every p we get sheaves A p i , 0 i p on N pU.The identities pr 0 01 ı pr0 000 D pr0 12 ı pr0 000 D pr 02 ı pr0 000 D pr0 00imply that thepull-back of A 012 to N 0 U by pr 0 000gives the pairing.pr 0 000 / .A 012 /W A ˝R A ! A: (4.2)The associativity condition implies that the pairing (4.2) endows A with a structure ofa sheaf of associative R-algebras on N 0 U.