Smooth Formal Embeddings and the Residue Complex

Abstract

Let π: X -> S be a finite type morphism of noetherian schemes. A smooth formal embedding of X (over S) is a bijective closed immersion X -> \frak{X}, where \frak{X} is a noetherian formal scheme, formally smooth over S. An example of such an embedding is the formal completion \frak{X} = Y_{/X} where X \subset Y is an algebraic embedding. Smooth formal embeddings can be used to calculate algebraic De Rham (co)homology. Our main application is an explicit construction of the Grothendieck residue complex when S is a regular scheme. By definition the residue complex is the Cousin complex of π^{!} \cal{O}_{S}. We start with Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf \cal{K}^{.}_{X/S}. We then use smooth formal embeddings to obtain the coboundary operator on \cal{K}^{.}_{X / S}. We exhibit a canonical isomorphism between the complex (\cal{K}^{.}_{X/S}, δ) and the residue complex of Grothendieck. When πis equidimensional of dimension n and generically smooth we show that H^{-n} \cal{K}^{.}_{X/S} is canonically isomorphic to the sheaf of regular differentials of Kunz-Waldi. Another issue we discuss is Grothendieck Duality on a noetherian formal scheme \frak{X}. Our results on duality are used in the construction of \cal{K}^{.}_{X/S}.