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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
URL: http://mmr.khu.ac.ir/article-1-3133-en.html
Buein Zahra Technical University , somayeh.bandari@yahoo.com
Abstract: (1289 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 if
is 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:
The following conclusions were drawn from this research:
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 if
is 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:
is a principal ideal.- is a Veronese ideal.
is equidimensional and 
for all 
.- is an unmixed matroidal ideal of degree 2.
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.
Keywords: Polymatroidal ideals, Regular decomposition function, Generic ideals, generalized Cohen-Macaulay ideals, Linear quotients.
Type of Study: Original Manuscript |
Subject:
Mat
Received: 2020/09/29 | Revised: 2022/11/16 | Accepted: 2021/03/7 | Published: 2022/05/21 | ePublished: 2022/05/21
Received: 2020/09/29 | Revised: 2022/11/16 | Accepted: 2021/03/7 | Published: 2022/05/21 | ePublished: 2022/05/21
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