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Der Residuenkomplex in der lokalen algebraischen und analytischen Geometrie. (The residue complex in local algebraic and analytic geometry). (German) Zbl 0582.32010
The author constructs a complex of differential module, denoted by \(D_{\Omega}(A/P)\) over \((\Omega_{A/P},d)\), the algebra of exterior differential forms of A over P where A is a flat algebra over a local Noetherian and commutative ring P; here one assumes that either A and P are analytic algebras or A is finitely generated. \(D_{\Omega}(A/P)\) is called the residue complex.
Detailed construction of the residue complex is carried out in this paper by using the Cousin complex which was introduced previously by the author [Math. Z. 182, 389-402 (1983; Zbl 0489.13006)]. The Cousin complex is denoted by \(C^.(A/P).\)
In the special case where P is a field of any characteristic, it is shown that \(D_{\Omega}(A)\) is an injective \(\Omega_ A\)-module (notice that one drops the label P). Furthermore a result of Grothendieck duality type is established, namely a duality between \(M\otimes C^ r(A)\) and \(Hom_{\Omega_ A}(M,\omega_ A)\) where \(\omega_ A\) denotes the zero cohomology of the complex \(D_{\Omega}(A)\), \(r=\dim A\) and M is any \((\Omega_ A,d)\)-module.
Reviewer: Vo Van Tan

32B05 Analytic algebras and generalizations, preparation theorems
32A27 Residues for several complex variables
13D25 Complexes (MSC2000)
13F25 Formal power series rings
13N05 Modules of differentials
Full Text: DOI EuDML
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