Page - 156 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Weuse thesedata forconstructingtwocochaincomplexes. Theyaredenotedby (C∗KV,dKV)and by (C∗τ,dτ) respectively. Theunderlyinggradedvectorspacesaredefinedby CKV= J(V)⊕∑ q>0 Cq(B,V), Cτ=V⊕∑ q>0 Cq(B,V). Theircoboundaryoperatorsaredefinedby (dKVv)(s)=−s ·v, (dτw)(s)=−sw, dKV(f)= q ∑ 1 (−1)jdj(f) if q>0, dτ(f)= q+1 ∑ 1 (−1)jdj(f) if q>0. Thesimplicial formula (5a)yields the identities d2KV=0, d2τ=0. Thecohomologyspace HKV(B,V)=∑ q HqKV(B,V) is calledtheV-valuedKVcohomologyofB. Thecohomologyspace Hτ(B,V)=∑ q Hqτ(B,V) is calledtheV-valuedtotalKVcohomologyofB. ThealgebraA isa two-sidedidealof theKValgebraB.Mutatismutandisourconstructiongives thecohomologyspacesHKV(A,V)andHτ(A,V). TheyarecalledtheV-valuedKVcohomologyand theV-valuedtotalKVcohomologyofA. Comments. Thoughthe spectral sequencesarenot thepurposeof thispaperwerecall that thepair (A⊂B)gives rise toa spectral sequencesEijr [32–34]. The termE ij 0 isnothingother thanHKV(A,V) [29]. Inotherwordsonehas HqKV(A,V)= ∑ 0≤j≤q Ej,q−j0 . 3.2.7. ResidualCohomology Beforepursuingweintroduce thenotionof residualcohomology. Itwillbeused in thesectionbe devotedthehomological statisticalmodels. ThemachinerywearegoingtointroduceissimilartothemachineryofEilenberg[35]. Inparticular we introduce the residual cohomology. Our construction leads to an exact cohomology sequence which links theresidualcohomologywith theequivariantcohomology.Werestrict theattention to the categoryof leftmodulesofKValgebroids.Wekeepourpreviousnotation. 156