In 2000, Ein, Lazarsfeld, and Smith found a uniform bound for the size of symbolic powers of ideals in a regular ring. Their original proof, in characteristic zero, uses multiplier ideals, though the same proof works in positive characteristic using test ideals instead. The main reason for the regularity assumption is so that these test ideals satisfy the subadditivity property: the test ideal of a product is contained in the product of test ideals.
In this talk, we’ll discuss a new subaddivity formula that works for non-regular rings. This formula will use the formalism of Cartier algebras. Along the way, we’ll see some constructions in the toric case and applications to bounding symbolic powers of ideals.