Volume 8, Issue 2 (Vol. 8,No. 2, 2022)                   mmr 2022, 8(2): 65-75 | Back to browse issues page

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Bandari S. Study and characterization of some classes of polymatroidal ideals. mmr 2022; 8 (2) :65-75
URL: http://mmr.khu.ac.ir/article-1-3133-en.html
Buein Zahra Technical University , somayeh.bandari@yahoo.com
Abstract:   (772 Views)
Introduction
Throughout this paper, we consider monomial ideals of the polynomial ring  over a filed. We try to give some properties of the polymatroidal ideals, which are the special class of monomial ideals. Herzog and Takayama constructed explicit resolutions for all ideals with linear quotients which admit regular decomposition functions. They also shaw that this class contains all matroidal ideals. We generalize their result to the polymatroidal ideals. Therefore, we can give an explicit linear resolution for any polymatroidal ideal. We also characterize generic polymatroidal ideals. The author and Jafari [1] characterized generalized Cohen-Macaulay polymatroidal ideals. Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.
Material and methods
A monomial ideal  is said to be polymatroidal, if it is single degree and for any two elements  such that  there exists an index  with  such that. In the case that the polymatroidal ideal is squarefree, it is called matroidal.
We know that the powers of a polymatroidal ideal are again polymatroidal and polymatroidal ideals have linear quotients. Therefore all powers of polymatroidal ideal have linear resolutions.
Let  has linear quotients with the order  of elements of. We can associate a unique decomposition function, that is a function  which maps a monomial  to, if  is the smallest index such that , where . The decomposition function  is called regular, if  for all  and
We show that any polymatroidal ideal has a regular decomposition function. Therefore we can give an explicit linear resolution for any polymatroidal ideal. By an example, we show that our result can not be extended to the weakly polymatroidal ideals even if they are generated in a single degree.
Recall that, a monomial ideal  is called generic if two distinct minimal generators  and  have the same positive degree in some variable , there is a third generator  which  and , where  is the least common multiple of  and .
In the next result, we characterize generic polymatroidal ideals.
A monomial ideal  is called generalized Cohen-Macaulay, whenever  is equidimensional and monomial localization  is Cohen-Macaulay for all monomial prime ideals, where  is unique homogenous maximal ideal of .
Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.
Results and discussion
For the first result, we show that any polymatroidal ideal has a regular decomposition function. So we have an explicit linear resolution of any polymatrodal ideal.
In the next, we show that ifis a fully supported polymatroidal ideal generated in degree. Then  is generic if and only if  is either a complete intersection or.
Finally, we prove that if   is a fully supported monomial ideal in  and generated in degree. Then is a generalized Cohen-Macaulay polymatroidal ideal for all  if and only if where  and  for some integers  and one of the following statements holds true:
1.  is a principal ideal.
2.  is a Veronese ideal.
3. is equidimensional and  for all .
4.  is an unmixed matroidal ideal of degree 2.
Conclusion
The following conclusions were drawn from this research:
• Any polymatroidal ideal has a regular decomposition function.
• characterization of generic ideals.
• characterization of monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.