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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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