## 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 ⊂ 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^{!} O_{S}, as in [RD]. We start with I-C. Huang's theory of pseudofunctors on modules with 0-dimensional support, which provides a graded sheaf ⊕_{q} K^{q}_{X/S}. We then use smooth formal embeddings to obtain the coboundary operator δ: K^{q}_{X/S} → K^{q+1}_{X/S}. We exhibit a canonical isomorphism between the complex (K^{.}_{X/S},δ) and the residue complex of [RD]. When π is equidimensional of dimension n and generically smooth we show that H^{-n}K^{.}_{X/S} is canonically isomorphic to to the sheaf of regular differentials of Kunz-Waldi [KW]. 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}.

Original language | English |
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Pages (from-to) | 863-896 |

Number of pages | 34 |

Journal | Canadian Journal of Mathematics |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - 1 Jan 1998 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics (all)