process

Duality for topological abelian group stacks and T -duality 3336.3.3 In this paragraph we identify the extension groups in the sequence (58) with sheafcohomology. For a site S with final object and X; Y 2 Sh Ab S we have a local-globalspectral sequence with second termE p;q2Š R p .I Ext q Sh Ab S.X; Y //which converges to Ext pCqSh Ab S.X; Y /.We apply this first to the sheaves Z jB ; T jB 2 Sh Ab S=B. The final object of the siteS=B is idW B ! B. WehaveExt q Sh Ab S=B .Z jB ; T jB / Š 0 for q 1 so that this spectralsequence degenerates at the second page and givesExt p Sh Ab S=B .Z jB ; T jB / Š H p .BI Hom ShAb S=B .Z jB ; T jB //Š H p .BI T jB / Š H p .BI T/ Š H pC1 .BI Z/:Since the group T n is admissible by Theorem 4.30, and by Corollary 3.14 we haveExt q Sh Ab S=B .T n jB ; T jB / Š 0 for q D 1; 2,wegetExt p Sh Ab S=B .T n jB ; T jB / Š H p .BI Hom ShAb S=B .T n jB ; T jB //Š H p .BI Hom ShAb S .T n ; T/ jB /Š H p .BI Z n jB /Š H p .BI Z n /for p D 1; 2. Therefore the sequence (58) has the formH 1 .BI Z n / ˛! H 3 .BI Z/ ! Q EOc! H 2 .BI Z n / ˇ! H 4 .BI Z/: (59)In this picture the maps ˛; ˇ are both given by the cup-product with the Chern classc.E/ 2 H 2 .BI Z n /, i.e. ˛..x i // D P x i [ c i .E/.6.3.4 Recall from 6.1.3 the decreasing filtration .F k H .EI Z// k0 and the spectralsequence associated to the decomposition of functors .EI :::/ D .B;:::/ıp W Sh Ab S=E ! Ab. This spectral sequence converges to GrH .EI Z/. Its secondpage is given by E i;j2Š H i .BI R j p Z/. The edge sequence for F 2 H 3 .EI Z/ has theformker.d 0;22W E 0;22! E 2;12 / d 3! 1;1coker.d 1;12W E 1;12! E 3;02 /2;1! F 2 H 3 .EI Z/ ! .E 2;1 0;22=im.d2 W E 0;22! E 2;1 d2//2! E 4;0We now make this explicit. Since the fibre of p is an n-torus, R 0 p Z Š Z,R 1 p Z Š Z n , R 2 p Z Š ƒ 2 Z n . The differential d 2 can be expressed in terms of theChern class c.E/. We get the following web of exact sequences, where K, A, B aredefined as the appropriate kernels and cokernels.2 :