Equivalence of the HC→HH map with Connes’ map

Determine whether, for each integer n≥0 and each differential graded algebra (A, φ), the maps HC^n(A^0)→HH^n(A^0) arising from the morphisms of spectral sequences between the cyclic-cohomology spectral sequences and the mixed filtration spectral sequence on the cyclic tricomplex EC(A) coincide with Connes’ canonical morphism I: HC^n(A^0)→HH^n(A^0) in the SBI long exact sequence relating cyclic cohomology and Hochschild cohomology of the associative algebra A^0 (the degree-zero part of A).

Background

From morphisms between the spectral sequences constructed via different filtrations of the cyclic tricomplex EC(A), the authors deduce a corollary that provides natural maps between various cohomology groups, including a map HC^n(A^0)→HH^n(A^0). Here A^0 denotes the degree-zero subalgebra of the differential graded algebra A.

Connes’ SBI long exact sequence is a classical tool connecting cyclic cohomology and Hochschild cohomology of associative (non-graded) algebras, featuring the canonical map I: HC^n→HH^n. The authors explicitly note uncertainty about whether their spectral-sequence-induced maps from HC^n(A^0) to HH^n(A^0) agree with Connes’ I map, prompting the need to compare these constructions and establish equivalence or identify differences.