process

320 U. Bunke, T. Schick, M. Spitzweck, and A. ThomNote that RD descends to a functor between derived categories RD W D b .Sh Ab S/ op !D C .Sh Ab S/. We now consider the following web of maps:Hom D C .Sh Ab S/.B; AŒ2/RDHom D C .Sh Ab S/.RD.AŒ 2/; RD.B//D 0DD.A/!RD.A/Hom D C .Sh Ab S/.D.A/; RD.B/Œ2/uD.B/!RD.B/Hom D C .Sh Ab S/.D.A/; D.B/Œ2/.Since B is admissible the map D.B/ ! RD.B/ is an isomorphism in cohomologyin degree 0; 1; 2 (because D.B/ is concentrated in degree 0 and H k .RD.B// DR k Hom ShAb S.B; T/ D Ext k Sh Ab S .B; T/). Since D.A/ is acyclic in non-zero degree,the map u is an isomorphism. For this, observe that D.B/ ! RD.B/ can be replacedup to quasi-isomorphism by an embedding 0 ! RD.B/ B ! RD.B/ ! Q ! 0such that the quotient is zero in degrees 0; 1; 2. The statement then follows from thelong exact Ext-sequence for Hom ShAb S.D.A/; /, because Ext i Sh Ab S .D.A/; B/ D 0 fori D 0; 1; 2. Therefore the diagram defines the mapD WD u 1 ı D 0 W Hom D C .Sh Ab S/.B; AŒ2/ ! Hom D C .Sh Ab S/.D.A/; D.B/Œ2/:5.2.2 We now show the relation (47). By Lemma 2.13 it suffices to show (47) forP 2 PIC.S/ of the form P D ch.K/ for complexesAs in 2.5.9 we considerK W 0 ! A ! X ! Y ! B ! 0; K W 0 ! X ! Y ! 0:K A W 0 ! X ! Y ! B ! 0with B in degree 0. Then by Definition (12) of the map and the Yoneda map Y , theelement .ch.K// 2 Hom D C .Sh Ab S/.B; AŒ2/ is represented by the composition (see(10))Y.K/W B ! ˇ ˛K A AŒ2:Since RD preserves quasi-isomorphisms we get RD.˛ 1/ D RD.˛/ 1 . It followsthatRD.Y.K// W RD.AŒ2/ RD.˛/ ! 1 RD.K A / RD.ˇ/ ! RD.B/:We read off thatD 0 .Y.K//W D.A/ ! RD.A/ RD.˛/ 1 ! RD.K A /Œ2 RD.ˇ/ ! RD.B/Œ2: