Page - 157 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 We recall that for everypositive integer q> 0 thevector spaceCq(B,V) is a leftmoduleofB. The leftactionof s∈B isdefinedby (s · f)(ξ)= s · f(ξ)− f(s ·ξ). Definition15. Acochain f ∈Cq(B,V) is calleda left invariant cochain if s · f=0 ∀s∈B ∀s. Astraightforwardconsequenceof thisdefinition is that a left invariant cochain is a cocycleof bothC∗KV andC∗τ.Thevector subspaceof left invariantq-cochainsofB isdenotedbyHqe(B,V). It is easy tosee that Zqτ(B,V)∩ZqKV(B,V)=Hqe(B,V), Zqτ(A,V)∩ZqKV(A,V)=Hqe(A,V). Definition 16. AKVcochain of degree qwhose coboundary is left invariant invariant is called a residual KVcocycles. (1) Thevector subspaceof residualKVcocycles ofdegreeq isdenotedbyZqKVres. (2) ThevectorsubspaceofresidualcoboundariesofdegreeqisdefinedbyBqKVres=H q e(B,V)+dKV(Cq−1KV (B,V)). TheresidualKVcohomologyspaceofdegreeq is thequotientvector space. (3) HqKVres(B,V)= ZqKVres BqKVres . (4) By replacing the KV complex by the total KV complex one defines the vector space of residual total cocyclesZqτres and the space of residual total coboundariesB q τres. Thereforeweget the residual totalKV cohomologyspace Hqτres(A,V)= Z q τres Bqτ,res The definitions above lead to the cohomological exact sequences which is similar to those constructedbyEilenbergmachinery [35].Wearegoingtopayaspecialattentionto twocohomology exact sequences. (1)Atoneside theoperatordKV yieldsacanonical linearmap HqKVres(B,V)→Hq+1e (B,V). (2)AtanothersideeveryKVcocycle isaresidualcocycleandeveryKVcoboundary isaresidual coboundaryaswell. Thenonehasacanonical linearmap HqKV(B,V)→HqKVres(A,V). Thosecanonical linearmappingsyield the followingexact sequences →Hq−1KVres(B,V)→Hqe(B,V)→HqKV(B,V)→HqKVres(B,V)→ →Hq−1τres(B,V)→Hqe(B,V)→Hqτ(B,V)→Hqτres(B,V)→ 157