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Subadditivity Formulas for Test Ideals


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.

AMS Special Session on Commutative Algebra
Pullman, WA
Daniel Smolkin
Postdoctoral Researcher