Smooth Formal Embeddings and the Residue Complex

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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 -> X, where X is a noetherian formal scheme, formally smooth over S. An example of such an embedding is the formal completion X = Y/ₓ where X ⊂ 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! Oₛ. We start with Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf K.ₓ/ₛ. We then use smooth formal embeddings to obtain the coboundary operator on K.ₓ / ₛ. We exhibit a canonical isomorphism between the complex (K.ₓ/ₛ, δ) and the residue complex of Grothendieck. When πis equidimensional of dimension n and generically smooth we show that H⁻ⁿ 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 X. Our results on duality are used in the construction of K.ₓ/ₛ.