Filtrations and local syzygies of multiplier ideals II

Let aa be a non-zero ideal sheaf on a smooth affine variety XX of dimension dd and let cc be a positive rational number. Let xx be a closed point of XX and let mxmx be the maximal ideal sheaf at xx. In [Robert Lazarsfeld, Kyungyong Lee, Local syzygies of multiplier ideals, Invent. Math. 167 (2007) 409–418] the authors studied the local syzygies of the multiplier ideal J(ac)J(ac). Motivated by their result, the asymptotic behavior of the local syzygies of the multiplier ideal J(mxk⋅ac) at xx for k≥d−2k≥d−2 was studied in [Seunghun Lee, Filtrations and local syzygies of multiplier ideals, J. Algebra (2007) 629–639]. In this note, we study the local syzygies of J(mxk⋅ac) at xx for 1≤k≤d−31≤k≤d−3. As a by-product we give a different proof of the main theorem in the former reference cited above.