Relationship between Homology of a Simplicial Semimodule and Homology of its Module Completion

摘要

Let K(Λ) denote the ring completion of a semiring Λ and let S be a simplicial Λ -semimodule, H_n(S) the n-th homology Λ-semimodule of S introduced in our earlier paper, K(S) the K(Λ) -module completion of S, H_n(K(S)) the n-th homology K(Λ) -module of K(S) and k_S : S → K(S) the canonical simplicial map. We prove (1) that the induced map H_n(k_S): H_n(S) → H_n(K(S)) is an injective Λ-homomorphism for all n; (2) that if S satisfies the Kan condition and the Λ-semimodule of path components of S is a K(Λ)-module, then H_n(S) is a K(Λ)-module and the induced map H_n(k_S): H_n(S) → H_n(K(S)) is a K(Λ)-isomorphism for all n.