process

370 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe identities pr 2 01 ı pr1 010 D Id, pr2 12 ı pr1 010 D and pr2 02 ı pr1 010 D pr0 00 ı pr1 0imply that the pull-back of A 012 by pr 1 010gives the pairing.pr 1 010 / .A 012 /W A 01 ˝R A 10 ! A 1 00 : (5.1)which is a morphism of A 1 0 ˝K .A 1 0 /op -modules. Similarly, we have the pairing.pr 1 101 / .A 012 /W A 10 ˝R A 01 ! A 1 11 : (5.2)The pairings (4.2), (4.3), (5.1) and (5.2), A ij ˝R A jk ! A ik are morphisms ofA 1 i ˝R .A 1 k /op -modules which, as a consequence of associativity, factor through mapsA ij A ˝A1 jk ! A ik (5.3)jinduced by A ij k D .pr 1 ij k / .A 012 /; here i;j;k D 0; 1. Define now for every p 0the sheaves A p ij , 0 i;j p, onN pU by A p ij D .prp ij / A 01 . Define also A p ij k D.pr p ij k / .A 012 /. We immediately obtain for every p the morphismsA p ij k W Ap ij ˝A p jA p jk ! Ap ik : (5.4)5.1.2 Matrix algebras. Let Mat.A/ 0 D A; thus, Mat.A/ 0 is a sheaf of algebras onN 0 U.Forp D 1;2;:::let Mat.A/ p denote the sheaf on N p U defined bypMMat.A/ p D A p ij :i;jD0The maps (5.4) define the pairingMat.A/ p ˝ Mat.A/ p ! Mat.A/ pwhich endows the sheaf Mat.A/ p with a structure of an associative algebra by virtueof the associativity condition. The unit section 1 is given by 1 D P piD0 1 ii, where 1 iiis the image of the unit section of A p ii .5.1.3 Combinatorial restriction. The algebras Mat.A/ p , p D 0;1;:::, do not forma cosimplicial sheaf of algebras on N U in the usual sense. They are, however, relatedby combinatorial restriction which we describe presently.For a morphism f W Œp ! Œq in define a sheaf on N q U bypMf ] Mat.A/ q D A q f.i/f.j/ :i;jD0Note that f ] Mat.A/ q inherits a structure of an algebra.Recall from Section 2.3.1 that the morphism f induces the map f W N q U ! N p Uand that f denotes the pull-back along f . We will also use f to denote the canonicalisomorphism of algebrasf W f Mat.A/ p ! f ] Mat.A/ q (5.5)induced by the isomorphisms f A p ij Š Aq f.i/f.j/ .