process

Duality for topological abelian group stacks and T -duality 299neighbourhood U of a such that ..q k Œq/s/ jU D 0 and .ps/ jU D 0 for all q 2 Z mult .For i D 4, instead of ..q k Œq/s/ jU D 0 we must find P v 2 M23 v (depending on Uand s) (see 4.5.10 for notation) such that P v s jU D 0. For i D 1 we must show thats jU D 0.We perform the argument in detail for i D 2; 3. It is a refinement of the proof ofLemma 4.50. The cases i D 1 and i D 4 are very similar.Let s be represented by a cycle Os 2 Ccont i .G; Map.A; Z//. Note that Os W Gi A ! Zis locally constant. The sets ¹Os 1 .h/ j h 2 Zº form an open covering of G i A. SinceG and therefore G i is compact and A 2 S is compactly generated by assumption onS, the compactly generated topology (this is the topology we use here) on the productG i A coincides with the product topology ([Ste67, Theorem 4.3]). This allows thefollowing construction.The set of subsets¹Os 1 .h/ j h 2 Zºforms an open covering of G i A. Using the compactness of the subset G i ¹aºG i A we choose a finite set h 1 ;:::;h r 2 Z such that G i ¹aº S riD1 Os 1 .h i /.There exists a neighbourhood U A of a such that G i U S riD1 Os 1 .h i /.We now use that G is profinite by Lemma 4.58. Since Os.G i U/is a finite set (asubset of ¹h 1 ;:::;h r º) there exists a finite quotient group G ! xG such that Os jG i U hasa factorization Ns W xG i U ! Z. In our case xG Š .Z=pZ/ r for a suitable r 2 N. Notethat Ns is a cycle in Ccont i . xG;Map.U; Z//. Now by Lemma 4.56 we know that .q k Œq/Nsand pNs are the boundaries of some Nt; Nt 1 2 Ccont i 1.xG;Map.U; Z//. Pre-composing Nt; Nt 1with the projection G i U ! xG i U we get Ot; Ot 1 2 Ccont i .G; Map.U; Z// whoseboundaries are .q k Œq/Os and pOs, respectively. This shows that .q k Œqs/ jU D 0and .ps/ jU D 0.4.5.14 We consider a compact group G and form the double complex Hom ShAb S.U ;I /defined in 4.5.7. Taking the cohomology first in the U and then in the I -direction weget a second spectral sequence .Frp;q ;d r / with second pageF p;q2Š Ext p Sh Ab S .H q U ; Z/:In the following we calculate the term F 1;22and show the vanishing of several otherentries.Lemma 4.60. The left lower corner of F p;q2has the following form.3 02 0 Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/1 0 Ext 1 Sh Ab S .G; Z/ Ext2 Sh Ab S .G; Z/ Ext3 Sh Ab S .G; Z/0 Z 0 0 0 0 00 1 2 3 4 5