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Traces in strict Frobenius algebras and strict complete intersections. (English) Zbl 0626.13009
Let R denote a commutative ring and S a commutative R-algebra, i.e., S/R. For a filtration \(F=\{F_ i\}_{i\in {\mathbb{Z}}}\) of the R-algebra S (the \(F_ i\subset S\) are R-submodules) consider the corresponding graded ring \(gr_ FS\). The filtered algebra (S/R,F) is called a strict Gorenstein (Frobenius, complete intersection) algebra, if \(gr_ FS/R\) is a Gorenstein (Frobenius, complete intersection) algebra. If S is a graded algebra and there is a homogeneous map \(\sigma: S\to R,\) S/R has a homogeneous trace. A subtle investigation of the duality isomorphism via a homogeneous trace for a strict Gorenstein (Frobenius, complete intersection) algebra yields to generalizations of the following classical theorems about zero-dimensional subschemes of \({\mathbb{P}}^ n:\)
1. Bezout’s theorem. 2. The Cayley-Bacharach theorem. 3. A formula of Jacobi related to the residue theorem on \({\mathbb{P}}^ n\). \(4.\quad A\quad result\) of Humbert on the intersection of curves in \({\mathbb{P}}^ 2.\)
Here the affine coordinate rings of the corresponding schemes are algebras of this type over the ground field. As a technical tool deformation theory is used in order to prove a statement - known in the “graded case” - in general.
Reviewer: P.Schenzel

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M10 Complete intersections
16W50 Graded rings and modules (associative rings and algebras)
13D10 Deformations and infinitesimal methods in commutative ring theory
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