Smooth formal embeddings and the residue complex

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23 Scopus citations


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, 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 KqX/S. We then use smooth formal embeddings to obtain the coboundary operator δ: KqX/S → Kq+1X/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-nK.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 languageEnglish
Pages (from-to)863-896
Number of pages34
JournalCanadian Journal of Mathematics
Issue number4
StatePublished - 1 Jan 1998
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics (all)


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