Arithmetical rank of squarefree monomial ideals of small arithmetic degree. (English) Zbl 1244.13016
The authors explore the arithmical rank of certain classes of squarefree monomial ideals. The starting point of their article is the following chain of inequalities for a squarefree monomial ideal \(I \in k[\underline{x}]=:R\):
\[
\text{ht} I \leq \text{pd}_R R/I \leq \text{arank} I \leq \mu (I)
\]
where arank denotes the arithmetical rank of \(I\), i.e. the minimal number of elements of \(I\) needed to generate an ideal with the same radical as \(I\), and \(\mu(I)\) the minimal number of generators of \(I\). They then identify and study three classes of ideals for which equality in the middle holds.
These three classes are: 1) \(\mu(I) \leq \text{pd}_R R/I +1\).
In this case, they not only prove the equality, but proceed further to study this class establishing a classification.
2) \(\mathrm{arithdeg} I = \mathrm{reg} I\).
Here the number of prime components equals the Castelnuovo-Mumford regularity of the ideal. Such ideals arise as the Alexander dual of squarefree monomial ideals for which the minimal number of generators coincides with the projective dimension.
3) \(\mathrm{arithdeg} I = \mathrm{indeg} I + 1\).
This case covers the Alexander dual of ideals from class 1.
These three classes are: 1) \(\mu(I) \leq \text{pd}_R R/I +1\).
In this case, they not only prove the equality, but proceed further to study this class establishing a classification.
2) \(\mathrm{arithdeg} I = \mathrm{reg} I\).
Here the number of prime components equals the Castelnuovo-Mumford regularity of the ideal. Such ideals arise as the Alexander dual of squarefree monomial ideals for which the minimal number of generators coincides with the projective dimension.
3) \(\mathrm{arithdeg} I = \mathrm{indeg} I + 1\).
This case covers the Alexander dual of ideals from class 1.
Reviewer: Anne Frühbis-Krüger (Hannover)
MSC:
| 13F55 | Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes |
Keywords:
arithmetical rank; almost complete intersection; Alexander duality; regularity; arithmetic degree; initial degreeReferences:
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