\(p\)-Borel principal ideals (Q1358719)

Let \(K\) be a field and \(R=K[x_1, \dots, x_n]\) the polynomial ring over \(K\) in \(n\) variables. Let \(u\in R\) be a monomial, and write \(x_i^l|u\) to express that \(x^l_i\) divides \(u\) but \(x_i^{l+1}\) does not. Now fix a prime number \(p\). For non-negative integers \(k\) and \(l\) with \(p\)-adic expansion \(k=\sum_ik_ip^i\) and \(l=\sum_i l_ip^i\), set \(k\leq_pl\) if \(k_i\leq l_i\) for all \(i\). Using this notation, a monomial ideal \(I\) is said to be \(p\)-Borel if and only if it satisfies the following condition: If \(u\) is a monomial in \(I\) and \(x^l_j|u\), then \((x_i/x_j)^k u\in I\) for all \(i<j\), and all \(k\leq_pl\). Let \(u\in R\) be a monomial; then \(\langle u\rangle\) denotes the smallest \(p\)-Borel ideal which contains \(u\). The ideal \(\langle u\rangle\) is called \(p\)-Borel principal with Borel generator \(u\). In his thesis [``Nonstandard Borel-fixed ideals'' (Brandeis Univ. 1994)], \textit{K. Pardue} conjectured a formula for the regularity of a \(p\)-Borel principal ideal, and proves his conjecture in the case that at most two variables (in successive order) divide \(u\). As one of our main results in this paper we show that Pardue's formula is indeed a lower bound for the regularity of a \(p\)-Borel principal ideal. We prove this by exhibiting certain Koszul cycles which we discover when we compute the Koszul homology of a \(p\)-Borel principal Cohen-Macaulay ideal. It is noted by Pardue that a \(p\)-Borel principal ideal \(\langle u\rangle\) is Cohen-Macaulay if and only if the Borel generator is of the form \(u=x^a_i\). For such \(p\)-Borel ideals we give the explicit minimal free resolution of \(p\)-Borel principal Cohen-Macaulay ideals.