Closure operations induced via resolutions of singularities in characteristic zero
Neil Epstein, Peter M. McDonald, Rebecca R. G., Karl Schwede
TL;DR
This work defines a Koszul–Hironaka (KH) closure in characteristic zero by pushing forward a resolution’s structure to a Cohen–Macaulay complex, producing a closure $J^{\mathrm{KH}}$ that is independent of choices, idempotent, and persistent, and that commutes with localization/completion. KH closure satisfies strong colon-capturing and a KH–Briançon–Skoda property, and it detects rational singularities (rationality ⇔ all ideals KH-closed). It is strictly tighter than characteristic-zero tight closure and relates to canonical alteration closure via inclusion $J^{\mathrm{KH}}\subseteq J^{\mathrm{calt}}\subseteq J^{\mathrm{cl}_{\Gamma(\omega_Y)}}$, with parameter ideals yielding equal closures and multiplier-ideal-type test ideals emerging in the Cohen–Macaulay case. The paper also develops alternate, alteration-based closures, and connects these zero-characteristic notions to positive-characteristic theories through reduction mod $p$, showing that KH closures align with parameter test ideals in favorable situations. Computational aspects are showcased via a Macaulay2 implementation, and several counterexamples highlight KH closure’s distinct behavior for products and powers, inviting further exploration of its algebraic and geometric implications.
Abstract
Using the fact that the structure sheaf of a resolution of singularities, or regular alteration, pushes forward to a Cohen-Macaulay complex in characteristic zero with a differential graded algebra structure, we introduce a tight-closure-like operation on ideals in characteristic zero using the Koszul complex, which we call KH closure (Koszul-Hironaka). We prove it satisfies various strong colon capturing properties and a version of the Briançon-Skoda theorem, and it behaves well under finite extensions. It detects rational singularities and is tighter than characteristic zero tight closure. 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 closure of the product, as one might expect.