2011/10/13 by Kustin, Andrew R., Vraciu, Adela
#13D02 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1110.2822
Let A=\pmb k[x1,...,xn]/(x1d,...,xnd), where \pmb k is an infinite field. If \pmb k has characteristic zero, then Stanley proved that A has the Weak Lefschetz Property (WLP). Henceforth, \pmb k has positive characteristic p. If n=3, then Brenner and Kaid have identified all d, as a function of p, for which A has the WLP. In the present paper, the analogous project is carried out for 4≤ n. If 4≤ n and p=2, then A has the WLP if and only if d=1. If n=4 and p is odd, then we prove that A has the WLP if and only if d=kq+r for integers k,q,d with 1≤ k≤ \fracp-12, r∈\fracq-12,\fracq+12, and q=pe for some non-negative integer e. If 5≤ n, then we prove that A has the WLP if and only if \lfloor\fracn(d-1)+32\rfloor≤ p. We first interpret the WLP for the ring \pmb k[x1, ..., xn]/(x1d, ..., xnd) in terms of the degrees of the non-Koszul relations on the elements x1d, ..., xn-1d, (x1+ ... +xn-1)d in the polynomial ring \pmb k[x1, ..., xn-1]. We then exhibit a sufficient condition for \pmb k[x1, ..., xn]/(x1d, ..., xnd) to have the WLP. This condition is expressed in terms of the non-vanishing in \pmb k of determinants of various Toeplitz matrices of binomial coefficients. Frobenius techniques are used to produce relations of low degree on x1d, ..., xn-1d, (x1+ ... +xn-1)d. From this we obtain a necessary condition for A to have the WLP. We prove that the necessary condition is sufficient by showing that the relevant determinants are non-zero in \pmb k.