Closure operations induced via resolutions of singularities in characteristic zero

Abstract

Using the fact that the structure sheaf of a resolution of singularities, or regular alteration, pushes forward to a Cohen-Macaulay complex in equal characteristic zero with a differential graded algebra structure, we introduce a tight-closure-like operation on ideals in equal characteristic zero using the Koszul complex, which we call KH (Koszul-Hironaka). We prove it satisfies various strong colon capturing properties, a substantial case of the Brian\c{c}on-Skoda theorem, and it behaves well under finite extensions. It detects rational singularities and is tighter than tight closure in equal characteristic zero. Furthermore, its formation commutes with localization and it can be computed effectively. On the other hand, the product of the KH closures of ideals is not always contained in the KH of the product, as one might expect. We also explore a related closure operation (canonical alteration closure), induced by canonical modules of regular alterations, which detects KLT-type singularities in equal characteristic zero and which is closely related to tight closure in characteristic $p > 0$. For parameter ideals we show both these closure operations coincide and reduce modulo $p \gg 0$ to tight closure. Finally, we explore an intermediate operation (Hironaka pre-closure) which which satisfies numerous desired properties, but for which we have not been able to prove idempotence.