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Periodic cyclic homology of algebras over the complex numbers was invented by Connes [Co] as a noncommutative generalization of de Rham cohomology. It is defined as the homology of the periodic cyclic bicomplex. This is a Z=2Z-graded complex of di¤erential forms over the given algebra with grading given by the parity of forms. The degree of algebraic di¤erential forms yields a descending filtration, called the Hodge filtration, of the periodic cyclic bicomplex by subcomplexes. The corresponding filtration on (co)homology is known as the dimension filtration [Co]. This filtration yields an important additional structure on periodic cyclic (co)homology. In particular, it provides the link between the periodic and ordinary cyclic groups [Co], [FT]. The associated spectral sequence gives rise to Connes’ exact sequence [Co] relating cyclic and Hochschild homology. In general it is di‰cult to determine the Hodge filtration on the periodic cyclic (co)homology of a given algebra.

Excision and the Hodge Filtration in periodic cyclic homology: the case of splitting and invertible extensions