The Betti-numbers of a graded ideal I in a polynomial ring and the Betti-numbers of its generic initial ideal Gin(I) are compared. In characteristic zero it is shown that if these Betti-numbers coincide in some homological degree, then they coincide in all higher homological degrees. We also compare the Betti-numbers of componentwise linear ideals which are contained in each other and have the same Hilbert polynomial.
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Jürgen Herzog, Aldo Conca, Takayuki Hibi, Rigid resolutions and big Betti numbers. Comment. Math. Helv. 79 (2004), no. 4, pp. 826–839DOI 10.1007/S00014-004-0812-2