On the embedded associated primes of monomial ideals

Official URL: http://dx.doi.org/10.1216/rmj.2022.52.275

Let I be a square-free monomial ideal in a polynomial ring R = K[x(1),...,x(n)] over a field K, m = (x(1),...,x(n)) be the graded maximal ideal of R, and {u(1),...,u(beta 1(I))} be a maximal independent set of minimal generators of I such that m\x(i) is not an element of Ass(R/(I\x(i))(t)) for all x(i) vertical bar Pi(beta 1(I))(i=1) u(i) and some positive integer t, where I\x(i) denotes the deletion of I at x(i) and beta(1)(I) denotes the maximum cardinality of an independent set in I. We prove that if m is an element of Ass(R/I-t), then t >= beta(1) (I) + 1. As an application, we verify that under certain conditions, every unmixed Konig ideal is normally torsion-free, and so has the strong persistence property. In addition, we show that every square-free transversal polymatroidal ideal is normally torsion-free. Next, we state some results on the corner elements of monomial ideals. In particular, we prove that if I is a monomial ideal in a polynomial ring R = K[x(1),...,x(n)] over a field K and z is an I(t )corner element for some positive integer t such that m\x(i) is not an element of Ass(I\x(i))(t) for some 1 <= i <= n, then x(i) divides z.