process

322 U. Bunke, T. Schick, M. Spitzweck, and A. Thom5.2.3 The final step in the verification of (47) follows from a consideration of thediagramD.B/Œ2 ıuıY 0 .H/ RD.B/Œ2 RD.ˇ/Y 0 .H/H D.A/H 0 D.A/RD.K A /RD.Y.K//D.A/D.A/RD.˛/ RD.A/D 0 .Y.K//showing the marked equality in.D.P// D Y 0 .H/ Š D D.Y.K// D D..P //:It remains to show thatD W Ext 2 Sh Ab S .B; A/ ! Ext2 Sh Ab S .D.A/; D.B//is an isomorphism.To this end we look at the following commutative web of mapsR 2 Hom ShAb S.D.A/; D.B//DR 2 Hom ShAb S.B; A/Š R 2 Hom ShAb S.B; D.D.A///ŠR 2 Hom ShAb S.D.A/; RD.B//uD 0Š vŠ z R 2 Hom ShAb S.D.A/ ˝L B;T/ R 2 Hom ShAb S.B; RD.D.A//RDŠR 2 Hom ShAb S.RD.A/; RD.B//Š R 2 Hom ShAb S.RD.A/ ˝L B;T/ŠR 2 Hom ShAb S.B; RD.RD.A///.The horizontal isomorphisms in the two lower rows are given by the derived adjointnessof the tensor product and the internal homomorphisms. The horizontal isomorphismin the first row is induced by the isomorphism A ! D.D.A//. The maps u and v areisomorphisms since we assume that D.A/ is admissible, compare the correspondingargument in the proof of Lemma 5.14. The map z is induced by the canonical mapA ! RD.RD.A//.