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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:
[1] Elzein, F.: Complexe dualisant et applications. Thèse de doctorat d’état
[2] Grothendieck, A.: Local cohomology. In: Lecture Notes in Mathematics, Vol. 41. Berlin, Heidelberg, New York: Springer 1967 · Zbl 0185.49202
[3] Hartshorne, R.: Residues and duality. In: Lecture in Mathematics, Vol. 20. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0212.26101
[4] Herzog, J., Kunz, E.: Der kanonische Modul eines Cohen-Macaulay-Rings. In: Lecture Notes in Mathematics, Vol. 238. Berlin, Heidelberg, New York: Springer 1971 · Zbl 0231.13009
[5] Kersken, M.: Reguläre Differentialformen. Dissertation Osnabrück (1981)
[6] Kersken, M.: Cousinkomplexe und Nennersysteme. Math. Z.182, 389-402 (1983) · Zbl 0502.13008 · doi:10.1007/BF01179758
[7] Kunz, E.: Residuen von Differentialformen auf Cohen-Macaulay-Varietäten. Math. Z.152 (1977) · Zbl 0342.14022
[8] Ramis, P., Ruget, G.: Complexe dualisant et theorèmes de dualité en géométrie analytique complexe. Publ. Math.32, 77-91 (1967) · Zbl 0206.25006
[9] Scheja, G.: Differentialmoduln lokaler analytischer Algebren. Schriftenreihe Math. Inst. Fribourg2 (1970) · Zbl 0199.36003
[10] Scheja, G., Storch, U.: Differentielle Eigenschaften der Lokalisierungen analytischer Algebren. Math. Ann.197, 137-170 (1972) · Zbl 0229.14002 · doi:10.1007/BF01419591
[11] Scheja, G., Storch, U.: Lokale Verzweigungstheorie. Schriftenreihe Math. Inst. Fribourg5 (1973/74)
[12] Scheja, G., Storch, U.: Über Spurfunktionen bei vollständigen Durchschnitten. J. reine angewandte Math.278/279 (1975) · Zbl 0316.13003
[13] Scheja, G., Storch, U.: Residuen bei vollständigen Durchschnitten. Math. Nachrichten91, 157-170 (1979) · Zbl 0455.13002 · doi:10.1002/mana.19790910113
[14] Serre, J.P.: Algebre locale ? Multiplicites. In: Lecture Notes in Mathematics, Vol. 11. Berlin, Heidelberg, New York: Springer 1965
[15] Sharp, R.Y.: The Cousin complex for a module over a commutative noetherian ring. Math. Z.112, 340-356 (1969) · Zbl 0182.06103 · doi:10.1007/BF01110229
[16] Sharp, R.Y.: Gorenstein modules. Math. Z.115, 117-139 (1970) · Zbl 0189.03904 · doi:10.1007/BF01109819
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