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Betti numbers, Castelnuovo Mumford regularity, and generalisations of Macaulay’s theorem. (English) Zbl 0908.13007
Let \(I(H)\) be the set of homogeneous ideals in \(k[x_0, \dots, x_n]\) with Hilbert function \(H\). Then the set of sequences of Betti numbers, \(\beta (I(H))= \{(\beta_0 (I), \dots, \beta_l(I))\); \(I\in I(H)\}\), is finite. The set \(\beta (I(H))\) is partially ordered by \((a_0, \dots,a_l) \leq(b_0, \dots, b_l)\) if \(a_i\leq b_i\) for all \(i\). Bigatti and Hulett have independently proved that \(\beta (I(H))\) has a unique maximal element which corresponds to the lexicographical monomial ideal in \(I(H)\).
In this paper the subsets \(I(H,m)\) and \(IS(H,m)\) of \(I(H)\), consisting of homogeneous (saturated homogeneous, resp.) ideals with Castelnuovo-Mumford regularity at most \(m\) are considered. It is shown that, in characteristic 0, \(\beta (I(H,m))\) and \(\beta (IS(H,m))\) contain maximal elements and these are calculated.

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D02 Syzygies, resolutions, complexes and commutative rings
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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