process

374 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan5.3.1 Decomposition of local cochains. As was noted in 5.2.1, for n; q D 0;1;:::there is a direct sum decompositionC k .Mat.A/ q / loc DMC I .Mat.A/ q / loc : (5.9)I 2Œq kC1In what follows we will interpret a multi-index I D .i 0 ;:::;i n / 2 Œq kC1 as a mapI Wf0;:::;kg!Œq. For I as above let s.I/ DjIm.I /j 1. The map I factorsuniquely into the compositionf0;:::;kg I 0 ! Œs.I / m.I / ! Œqwhere the second map is a morphism in (i.e. is order preserving). Then, the isomorphismsm.I / A I 0 .i/I 0 .j / Š A I.i/I.j/ induce the isomorphismm.I / C I 0 .Mat.A/ s.I/ / loc ! C I .Mat.A/ q / loc :Therefore, the decomposition (5.9) may be rewritten as follows:C k .Mat.A/ q / loc D M Me C I .Mat.A/ p / loc (5.10)ewhere the summation is over injective (monotone) maps e W Œs.e/ ! Œq and surjectivemaps I Wf0;:::;kg!Œs.e/. Note that, for e, I as above, there is an isomorphisme C I .Mat.A/ p / loc Š C eBI .e ] Mat.A/ s.e/ / loc :5.3.2 Filtrations. Let F C k .Mat.A/ q / loc denote the decreasing filtration defined byF s C k .Mat.A/ q / loc DM Me C I .Mat.A/ p / locIeWs.e/sNote that F s C k .Mat.A/ q / loc D 0 for s>kand Gr s C k .Mat.A/ q / loc D 0 for s0.I