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Thin algebras of embedding dimension three. (English) Zbl 0588.13013
Let \(R=k[X_ 1,...,X_ g]/(f_ 1,...,f_ r)\) where the \(f_ i's\) are homogeneous and suppose that g, r, \(d_ 1,...,d_ r\) are specified in advance. This article addresses the question which Hilbert series \(H_ R(t)=\sum_{i}(\dim_ kR_ i)t^ i\) can occur. A lower bound is the coefficientwise inequality \(H_ R(t)\geq | (1-t)^{- g}\prod^{r}_{i=1}(1-t^{d_ i})|\) where absolute value symbols denote the initial non-negative segment of a power series. It is known that the lower bound can be obtained in case \(g=2\) or in case \(r\leq g+1\) and char k\(=0\). This paper settles the case \(g=3\) affirmatively.
Reviewer: R.Fröberg

MSC:
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13C13 Other special types of modules and ideals in commutative rings
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