Study and characterization of some classes of polymatroidal ideals

Author
Somayeh Bandari
Buein Zahra Technical University
Abstract
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:

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
1. Bandari S., Jafari R., "On certain equidimensional polymatroidal ideals", Manuscripta Math., 149(1-2) (2016) 223-233.

2. Conca A., Herzog J., "Castelnuovo–Mumford regularity of products of ideals", Collect. Math. 54(2) (2003) 137–152.

3. Herzog J., Hibi T., "Monomial Ideals", Graduate Texts in Mathematics, 260,

Springer-Verlag, London, (2011).

4. Herzog J., Hibi T., "Cohen-Macaulay polymatroidal ideals", European J. Combin., 27(4) (2006) 513-517.

5. Herzog J., Rauf A., Vladoiu M., "The stable set of associated prime ideals of a polymatroidal ideal", J. Algebraic Combin., 37(2) (2013) 289-312.

6. Herzog J., Takayama Y., "Resolutions by mapping cones", Homology, Homotopy Appl., 4(2) (2002) pp. 277-294.

7. Herzog, J., Vladoiu, M. "Monomial ideals with primary components given by powers

of monomial prime ideals", Electron. J. Comb., 21(1) (2014), # P1.69.

8. Mohammadi F., Moradi S., "Weakly polymatroidal ideals with applications to vertex cover ideals", Osaka J. Math., 47 (2010) 627-636.