process

388 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganProof. It is easy to see that Id˝cotr is a morphism of graded Lie algebras, which satisfies.r p˝IdCId˝rcan /ı.Id˝cotr/ D .Id˝cotr/ır can and ıı.Id˝cotr/ D .Id˝cotr/ıı.Since the domain of .Id ˝ cotr/ is the normalized complex, in view of Lemma 7.13 wealso have rF p ı .Id ˝ cotr/ D .Id ˝ cotr/ ı !. This implies that .Id ˝ cotr/ is amorphism of DGLA.To see that this map is a quasiisomorphism, introduce filtration on N p U byF i N p U D iN p U and consider the complexes xC .J Np U/Œ1 and C .Mat.A// p ˝J Np U/Œ1 equipped with the trivial filtration. The map (7.15) is a morphism of filteredcomplexes with respect to the induced filtrations on the source and the target. Thedifferentials induced on the associated graded complexes are ı (or, more precisely,Id ˝ ı) and the induced map of the associated graded objects is Id ˝ cotr which is aquasi-isomorphism. Therefore, the map (7.15) is a quasiisomorphism as claimed.The map (7.15) therefore induces a morphism Id ˝cotr W G DR .J/ ! ! H for everyW Œn ! . These morphisms are clearly compatible with the cosimplicial structureand hence induce a quasiisomorphism of cosimplicial DGLAsId ˝ cotr W G DR .J/ ! ! Hwhere the differential in the right hand side is given by (7.12).We summarize our consideration in the following:Theorem 7.18. For a any choice of 2 Isom 0 .A 01 ˝J N1 U; J.A 01 //, r2C .A 01 /and F 0 as in (7.6) the composition ˆ;r;F WD ı exp. F / ı .Id ˝ cotr/is a quasiisomorphism of cosimplicial DGLAs.ˆ;r;F W G DR .J/ ! ! G DR .J.A// (7.16)Let W V ! U be a refinement of the cover U and let .V; A / be the induced descentdatum. We will denote the corresponding cosimplicial DGLAs by G DR .J/ ! .U/ andG DR .J/ ! .V/ respectively. Then the map NW N V ! N U induces a morphism ofcosimplicial DGLAs.N/ W G DR .J.A// ! G DR .J.A // (7.17)and.N/ W G DR .J/ ! .U/ ! G DR .J/ ! .V/: (7.18)Notice also that the choice that the choice of data , r, F 0 on N U induces thecorresponding data .N/ , .N/ r, .N/ F 0 on N V. This data allows one toconstruct using the equation (7.16) the mapˆ.N/ ;.N/ r;.N/ F W G DR .J/ ! ! G DR .J.A //: (7.19)The following proposition is an easy consequence of the description of the map ˆ,and we leave the proof to the reader.