On the Algebraic de Rham Homology of Quasi-projective Schemes

Abstract

Let Y be a nonempty, connected, quasi-projective scheme over a field k of characteristic 0, embedded in Pnk. We show that the object En,n1 on the E1 page of the Hodge-de Rham spectral sequence abutting to the algebraic de Rham homology of Y is a one dimensional k-space when Y is complete of dimension at least one, an infinite dimensional k-space when Y is zero dimensional, and zero when Y is not complete. Further, we show that when Y is of dimension at least one, En,n1 is a stable object, and when Y is of dimension zero, En,n2 is a stable object. In the latter case, the spectral sequence degenerates on the E2 page. Together these results allow us to compute the zero-th algebraic de Rham homology of Y .