Gysin maps and cycle classes for Hodge cohomology

If f:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps {Mathematical expression} for the Hodge cohomology groups, where d-dim Y-dim X. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map {Mathematical expression} where X is quasi-projective over a field, S is the singular locus, and CH ⁱ(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective variety X over C with {Mathematical expression}, where n=dim X. One also recovers examples of Nori of affine n-dimensional varieties which support indecomposable vector bundles of rank n.