Componentwise linear ideals. (English) Zbl 0930.13018
A componentwise linear ideal is a graded ideal \(I\) of a polynomial ring such that, for each degree \(q\), the ideal generated by all homogeneous polynomials of degree \(q\) belonging to \(I\) has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal \(I_\Delta\) arising from a simplicial complex \(\Delta\) is componentwise linear if and only if the Alexander dual of \(\Delta\) is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.
Reviewer: Takayuki Hibi (Osaka)
MSC:
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
Keywords:
linear resolution of the Stanley-Reisner ideal; componentwise linear ideal; polynomial ring; simplicial complex; Alexander dualReferences:
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