Graded Betti numbers of componentwise linear ideals

Details Graded Betti numbers of componentwise linear ideals Graded Betti numbers of componentwise linear ideals

2011-11-02

arxiv.org/abs/1111.0442v1

Mathematics

Commutative Algebra

21 pages

Scientific paper

In this paper, we study the Betti tables of homogeneous ideals in a polynomial ring. Especially, we concentrate ourselves on componentwise linear ideals, so that, exploiting the deformation to the generic initial ideal, we can reduce the problem to study the Betti tables of strongly stable ideals. We obtain a complete numerical characterization of the graded Betti numbers of ideals with linear resolution, giving two different proofs. We provide a necessary (in general not sufficient) numerical condition for a table being the Betti table of a componentwise linear ideal. Such a condition leads to a characterization of the Betti tables of componentwise linear ideals in three variables. Furthermore we identify the Betti tables of Gotzmann ideals. Eventually, provided the characteristic of the base field is $0$, we succeed to characterize the possible extremal Betti numbers (values as well as positions) of any homogeneous ideal.

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