Décompositions de l'homologie cyclique des algèbres différentielles graduées commutatives. (Decompositions of cyclic homology of commutative differential graded algebras)
Let (A,d) be a differential graded commutative algebra over a field K of characteristic 0. Let HH(A,d) resp. HC(A,d) be the Hochschild resp. cyclic homology of (A,d). The author proves that there exist natural decompositions \[ HH_ q(A,d)=HH_ q^{(0)}(A,d)\oplus...\oplus HH_ q^{(q)}(A,d), \] \[ HC_ q(A,d)=HC_ q^{(0)}(A,d)\oplus...\oplus HC_ q^{(q)}(A,d) \] with \(HH_ q^{(0)}(A,d)=H_ 0(A,d)\) and \(HH_ q^{(q)}(A,d)=HH_ q^{(q)}(H_ 0(A,d))\), similarly for HC. A map f: (A,d)\(\to (B,d)\) inducing an isomorphism \(H_*(A,d)\to H_*(B,d)\) induces isomorphisms on the groups \(HH_ q^{(p)}\) and \(HC_ q^{(p)}.\) The decomposition above coincides with the decomposition described by \textit{D. Burghelea} and \textit{M. Vigué-Poirrier} [Lect. Notes Math. 1318, 51-72 (1988; Zbl 0666.13007)], which was in terms of minimal models. In certain cases, the groups \(HH_ q^{(p)}\) and \(HC_ q^{(p)}\) vanish, if \(p>p(q)\). Namely if A is of finite type and generated by r elements, then one may pose \(p(q)=[(q+r)/2],\) if \(A=K[x_ 1,...,x_ r]/I,\) where I is generated by a regular sequence, one can take \(p(q)=[(n+1)/2]\).
scientific article; zbMATH DE number 4208352