POLYMATROIDAL IDEALS AND LINEAR RESOLUTION

Let S=K[x_۱,\ldots,x_n] be a polynomial ring over a field K andI\subset S be a monomial ideal with a linearresolution. Let\frak{m}=(x_۱,\ldots,x_n) be the unique homogeneous maximal ideal and I\frak{m} be apolymatroidal ideal. We prove that if either I\frak{m} is polymatroidal with strongexchange property, or I is a monomial ideal in at most ۴variables, then I is polymatroidal. We also show that the firsthomological shift ideal of polymatroidal ideal is againpolymatroidal.

Keywords:

polymatroidal ideals , monomial localization , linear quotients , linear resolution , homological shift ideal