J an 2 00 7 Adjoints of ideals

We characterize ideals whose adjoints are determined by their Rees valuations. We generalize the notion of a regular system of parameters, and prove that for ideals generated by monomials in such elements, the integral closure and adjoints are generated by monomials. We prove that the adjoints of such ideals and of all ideals in twodimensional regular local rings are determined by their Rees valuations. We prove special cases of subadditivity of adjoints. Adjoint ideals and multiplier ideals have recently emerged as a fundamental tool in commutative algebra and algebraic geometry. In characteristic 0 they may be defined using resolution of singularities. In all characteristics, even mixed, Lipman gave the following definition: Definition 0.1: Let R be a regular domain, I an ideal in R. The adjoint adj I of I is defined as follows: adj I = ⋂ v {r ∈ R | v(r) ≥ v(I)− v(JRv/R)}, where the intersection varies over all valuations v on the field of fractions K of R that are non-negative on R and for which the corresponding valuation ring Rv is a localization of a finitely generated R-algebra. The symbol JRv/R denotes the Jacobian ideal of Rv over R. By the assumption on v, each valuation in the definition of adj I is Noetherian. Many valuations v have the same valuation ring Rv; any two such valuations are positive real multiples of each other, and are called equivalent. In the definition of adj I above, one need only use one v from each equivalence class. In the sequel, we will always choose normalized valuations, that is, the integer-valued valuation v such that for all r ∈ R, v(r) equals that non-negative integer n which satisfies that rRv equals the nth power of the maximal ideal of Rv. Lipman proved that for any ideal I in R and any x ∈ R, adj(xI) = x adj(I). In particular, adj(xR) = (x). A crucial and very powerful property is the subadditivity of adjoints: adj(IJ) ⊆ adj(I) adj(J). This was proved in characteristic zero by Demailly, Ein and Lazarsfeld [2], and is unknown in general. We prove it for generalized monomial ideals in Section 4, and for ideals in two-dimensional regular domains in Section 5. The case of subadditivity of adjoints for ordinary monomial ideals can be deduced from Howald’s work [3] using toric resolutions, and the two-dimensional case has been proved by Takagi and Watanabe [17] using multiplier ideals. The case for generalized monomial ideals proved here is new. ∗ Partially supported by the National Science Foundation 2000 Mathematics Subject Classification 13A18, 13A30, 13B22, 13H05.