Axiomatic closure operations, phantom extensions, and solidity

Abstract

In this article, we generalize a previously defined set of axioms for a closure operation that induces balanced big Cohen–Macaulay modules. While the original axioms were only defined in terms of finitely generated modules, these new ones will apply to all modules over a local domain. The new axioms will lead to a notion of phantom extensions for general modules, and we will prove that all modules that are phantom extensions can be modified into balanced big Cohen–Macaulay modules and are also solid modules. As a corollary, if R has characteristic p>0 and is F-finite, then all solid algebras are phantom extensions. If R also has a big test element (e.g., if R is complete), then solid algebras can be modified into balanced big Cohen–Macaulay modules. (Hochster and Huneke have previously demonstrated that there exist solid algebras that cannot be modified into balanced big Cohen–Macaulay algebras.) We also point out that tight closure over local domains in characteristic p generally satisfies the new axioms and that the existence of a big Cohen–Macaulay module induces a closure operation satisfying the new axioms.