Subadditivity Formulas for Test Ideals

Abstract

Given a ring $R$ of positive characteristic, an ideal $\mathfrak{a}$ in $R$, and a positive number $t$, one can construct what’s called the test ideal of this data, denoted $\tau(\mathfrak a^t)$. This notion was introduced by Hara and Yoshida in 2003, based on work in tight closure by Hochster and Huneke, and it measures the singularities of $R$ and $\mathfrak a$. Hara and Yoshida also showed that on regular rings, test ideals obey a subadditivity formula, namely $\tau(\mathfrak a^s \mathfrak b^t) \subseteq \tau(\mathfrak a^s) \tau(\mathfrak b^t)$, and Takagi generalized this formula to the affine case. This formula has a number of important applications, such as bounding the growth of symbolic powers of ideals.

In this talk, I will discuss progress towards an improved subaddivity formula for non-regular rings using the formalism of Cartier algebras. Along the way, I’ll show some constructions for the toric case.