Geodesic
Bounds on Betti Numbers
Canadian journal of mathematics, Tome 34 (1982) no. 3, pp. 589-592

Voir la notice de l'article provenant de la source Cambridge University Press

The Betti numbers βn (k) of the residue class field k = R/m of a commutative local ring (R, m) have been studied for about 20 years, primarily as the coefficients of the Poincaré series of E . Several authors have obtained results about the growth of the sequence {βn (k)}.For example, Gulliksen [3] showed that when R is non-regular, the sequence is non-decreasing. More recently, Avramov [1] studied asymptotic properties of {βn (k)} and found that under certain conditions the growth is exponential, i.e., there is a natural number p such that for all n, βpn (k) ≧ 2n .In this paper, we examine the sequence {βn (M)} for arbitrary finitely generated non-free modules M over any commutative local artin ring R. We establish the following bounds: 1 2 3 where l(X) is the length of X. 
Ramras, Mark. Bounds on Betti Numbers. Canadian journal of mathematics, Tome 34 (1982) no. 3, pp. 589-592. doi: 10.4153/CJM-1982-040-0
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