Ideals containing the squares of the variables. (English) Zbl 1141.13016
Let \(S=K[x_1,\dots,x_n]\) be a polynomial ring over a field \(K\), let \(I\) be a squarefree ideal and let \(P=(x_1^2,\ldots,x_n^2)\). It is well known that there exists a squarefree lex-segment ideal \(L\) such that \(L+P\) has the same Hilbert function as \(I+P\). The ideal \(L+P\) is called lex-plus-squares. In this paper the authors study the graded Betti numbers of \(I+L\). First they provide a relation between the Betti numbers of \(I\) and those of \(I+P\). Then they prove that the Betti numbers of \(L+P\) are greater than or equal to those of \(I+P\). This gives a positive answer to the lex-plus-powers conjecture for this class of ideals.
Reviewer: Emanuela De Negri (Genova)
MSC:
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
Software:
LexIdealsReferences:
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