process

364 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe sheaf A p iis endowed with the associative R-algebra structure induced by thaton A. We denote by A p ii the Ap i˝R .A p i /op -module A p i, with the module structuregiven by the left and right multiplication.The identities pr 2 01 ı pr1 001 D pr0 00 ı pr1 0 ,pr2 12 ı pr1 001 D pr2 02 ı pr1 001D Id implythat the pull-back of A 012 to N 1 U by pr 1 001gives the pairing.pr 1 001 / .A 012 /W A 1 0 ˝R A 01 ! A 01 : (4.3)The associativity condition implies that the pairing (4.3) endows A 01 with a structureof a A 1 0 -module. Similarly, the pull-back of A 012 to N 1 U by pr 1 011 endows A 01 witha structure of a .A 1 1 /op -module. Together, the two module structures define a structureof a A 1 0 ˝R .A 1 1 /op -module on A 01 .The map (4.1) factors through the map.pr 2 01 / A 01 ˝A2 1.pr 2 12 / A 01 ! .pr 2 02 / A 01 : (4.4)Definition 4.6. A unit for a convolution datum A is a morphism of R-modulessuch that the compositions1W R ! AA 011˝Id! A10 ˝R A 01.pr 1 001 / .A 012 /! A 01andA 01Id˝1! A 01 ˝R A 1 1.pr 1 011 / .A 012 /! A 01are equal to the respective identity morphisms.4.2.2 Descent dataDefinition 4.7. A descent datum on X is an R-linear convolution datum .U; A/ on Xwith a unit which satisfies the following additional conditions:1. A 01 is locally free of rank one as a A 1 0 -module and as a .A1 1 /op -module;2. the map (4.4) is an isomorphism.4.2.3 1-morphisms. Suppose given convolution data .U; A/ and .U; B/ as in Definition4.5.Definition 4.8. A 1-morphism of convolution data W .U; A/ ! .U; B/