Study and characterization of some classes of polymatroidal ideals

Bandari, Somayeh

Somayeh Bandari

Buein Zahra Technical University , somayeh.bandari@yahoo.com

Abstract: (711 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:

is a principal ideal.
is a Veronese ideal.
is equidimensional and for all .
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.

Keywords: Polymatroidal ideals, Regular decomposition function, Generic ideals, generalized Cohen-Macaulay ideals, Linear quotients.

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