Filtrations and local syzygies of multiplier ideals II

Abstract Let a be a non-zero ideal sheaf on a smooth affine variety X of dimension d and let c be a positive rational number. Let x be a closed point of X and let m x be the maximal ideal sheaf at x . 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 ( a c ) . Motivated by their result, the asymptotic behavior of the local syzygies of the multiplier ideal J ( m x k ⋅ a c ) at x for k ≥ 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 ( m x k ⋅ a c ) at x for 1 ≤ k ≤ d − 3 . As a by-product we give a different proof of the main theorem in the former reference cited above.