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Multiplier ideals of sums via cellular resolutions

2008; International Press of Boston; Volume: 15; Issue: 2 Linguagem: Inglês

10.4310/mrl.2008.v15.n2.a13

1945-001X

Shin-Yao Jow, Ezra Miller,

Algebraic structures and combinatorial models

Fix nonzero ideal sheaves a 1 , . . ., ar and b on a normal Q-Gorenstein complex variety X.For any positive real numbers α and β, we construct a resolution of the multiplier ideal J ((a 1 + • • • + ar) α b β ) by sheaves that are direct sums of multiplier ideals J (aThe resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex.The CW-complex is naturally expressed as a triangulation ∆ of the simplex of nonnegative real vectors λ ∈ R r with P r i=1 λ i = α.The acyclicity of our resolution reduces to that of a cellular free resolution, supported on ∆, of a related monomial ideal.Our resolution implies the multiplier ideal sum formulageneralizing Takagi's formula for two summands [Tak05], and recovering Howald's multiplier ideal formula for monomial ideals [How01] as a special case.Our resolution also yields a new exactness proof for the Skoda complex [Laz04, Section 9.6.C].