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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

##### MSC:
 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
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##### References:
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