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Reduction exponent and degree bound for the defining equations of graded rings. (English) Zbl 0641.13016
Generalizing results of Hong, Cho, and Achilles and Schenzel the author gives upper degree bounds for the defining equations of certain graded rings $$R=\oplus^{\infty}_{i=0}R_ i\quad in$$ terms of the reduction exponent r(I) of a minimal reduction I of $$R^+$$ and the multiplicity e(R). (It seems to be tacitly understood that R is generated by $$R_ 1$$ as an $$R_ 0$$-algebra.)
The first main result: Let R be a graded Buchsbaum (resp. Cohen-Macaulay) ring with $$R_ 0$$ 0-dimensional local; then the degrees of the defining equations of R can be bounded above by $$r(I)+1$$, in particular by $$e(R)+1$$ (resp. e(R)).
The second main result concerns the associated graded ring $$G_{{\mathfrak a}}(A)$$ of a Buchsbaum (resp. Cohen-Macaulay) local ring (A,$${\mathfrak m}):$$ Let $$d=\dim (A)\geq 1$$ and $$depth(G_{{\mathfrak a}}(A))\geq d-1$$, and let $${\mathfrak b}$$ be a minimal reduction of $${\mathfrak a}$$; then (i) $$r({\mathfrak b})$$ is independent of $${\mathfrak b}$$, (ii) the degrees of the defining equations of $$G_{{\mathfrak a}}(A)$$ can be bounded above by r($${\mathfrak b})+1$$, and (iii) $$r({\mathfrak b})\leq e(A)t^{d-1}$$ (resp. $$e(A)t^{d-1}- 1)$$, t being the last integer with $${\mathfrak m}^ t\subset {\mathfrak a}$$. The author derives several auxiliary results about filter-regular sequences and reduction exponents which are of independent interest. Furthermore it is pointed out that an argument in a paper by Y. Cho [Proc. Am. Soc. 89, 569-573 (1983; Zbl 0558.13014)] is incorrect.
Reviewer: W.Bruns

##### MSC:
 13H15 Multiplicity theory and related topics 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 16W50 Graded rings and modules (associative rings and algebras) 13C05 Structure, classification theorems for modules and ideals in commutative rings
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