A generalization of tight closure and multiplier ideals

Abstract:

We introduce a new variant of tight closure associated to any fixed ideal $\mathfrak {a}$, which we call $\mathfrak {a}$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau (\mathfrak {a})$ of all $\mathfrak {a}$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $\tau (\mathfrak {a})$ and the multiplier ideal associated to $\mathfrak {a}$ (or, the adjoint of $\mathfrak {a}$ in Lipman’s sense) in normal $\mathbb {Q}$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau (\mathfrak {a})$ similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau (\mathfrak {a})$ and the F-rationality of Rees algebras.