Smooth Formal Embeddings and the Residue Complex

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dc.creatorYekutieli, Amnon
dc.date1995-10-05
dc.date1998-02-16
dc.date.accessioned2026-07-07T08:58:01Z
dc.date.available2026-07-07T08:58:01Z
dc.descriptionLet 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}.
dc.description33 pages, AMSLaTeX, final version (some corrections, section on D-modules omitted), to appear in Canadian Math. J
dc.identifierhttps://arxiv.org/abs/alg-geom/9510007
dc.identifierhttp://arxiv.org/abs/alg-geom/9510007
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/147163
dc.subjectAlgebraic Geometry
dc.subject14B20 (Primary) 14F10, 14B15, 14F20 (Secondary)
dc.titleSmooth Formal Embeddings and the Residue Complex
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