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Zbl 0786.13004
Avramov, Luchezar L.; Buchweitz, Ragnar-Olaf
Lower bounds of Betti numbers. (English)
[J] Compos. Math. 86, No.2, 147-158 (1993). ISSN 0010-437X; ISSN 1570-5846/e

An old conjecture on Betti numbers, namely $b\sb i\sp R(M)=\dim\sb K \text {Tor}\sb i\sp R(M,K) \ge {d \choose i}$ (for a local or graded $d$- dimensional noetherian ring $R$ and an $R$-module $M$ of finite length and finite projective dimension), implies that $\beta\sp R(M)=\sum b\sb i\sp R(M) \ge 2\sp d$.\par It is proved that $\log\sb 2\beta\sp R (M) \ge (d-d(M)+p\sp m-1)$ $(\log\sb 2p) /p\sp m (p\sp m-1)$ holds for a positively graded algebra $R$ over a field $K$, which is finitely generated by elements of degree 1 and $M$ is a nonzero f. g. graded $R$-module of finite projective dimension, $p$ is a prime, $d(M)$ is a Krull dimension of $M$, $m$ is a natural number depending on $p$ and $M$. This implies $\beta\sp R(M) \ge 2\sp d$ for graded modules $M= \bigoplus M\sb i$ of finite length and such that $\sum(-1)\sp i \dim\sb KM\sb i \ne 0$. Moreover it is proved that for an equicharacteristic local noetherian ring $R$ of Krull dimension $d \ge 5$ and $R$-module $M$ of finite length and finite projective dimension $\beta\sp R(M) \ge 3/2(d-1)\sp 2+8$.
[S.Balcerzyk (Toruń)]

MSC 2000:: *13D03 (Co)homology of commutative rings and algebras
13D40 Hilbert-Samuel functions and Poincare series

Keywords: Hilbert series; multiplicity; graded algebra; Krull dimension

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