zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] Atiyah, M.F.; MacDonald, I.G., Introduction to commutative algebra, (1969), Addison-Wesley Reading, Mass · Zbl 0175.03601 [2] Fröberg, R., An inequality for hubert series of graded algebras, () [3] Hermann, G., Die frage der endlich vielen schritte in der theorie der polynomideale, Math. ann., 95, 736-788, (1926) · JFM 52.0127.01 [4] Iarrobino, A., Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. amer. math. soc., 285, 337-378, (1984) · Zbl 0548.13009 [5] Iarrobino, A., Punctual Hilbert schemes, () · Zbl 0267.14005 [6] Macaulay, F.S., Some properties of enumeration in the theory of modular systems, (), 531-555 · JFM 53.0104.01 [7] Stanley, R.P., Hilbert functions of graded algebras, Adv. in math., 28, 57-83, (1978) · Zbl 0384.13012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.