A new subadditivity formula for test ideals

Abstract

We exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to [DEL00] and [HY03]. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras [Bli13]. We also show that our subadditivity containment is sharper than ones shown previously in [Tak06] and [Eis10]. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas from [Tak08] and [Eis10] to show that the adjoint ideal J X pA, Zq reduces mod p to Takagi's adjoint test ideal, even when the ambient space is singular, provided that A is regular at the generic point of X. One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting.

The uniform symbolic topology property for diagonally F-regular algebras

“…In [Smo18], the second named author introduced the Cartier algebra consisting of p −elinear maps compatible with the diagonal closed embedding ∆ 2 : R ⊗ R − → R. Here we generalize this construction to higher diagonals and verify these have the required basic properties, including an analogous subadditivity formula.…”

“…Proof. The second statement is obvious, whereas the former is a consequence of Kunz's theorem [Kun69] just as in [Smo18,§7]. Indeed, if R is smooth over k, then R ⊗n is smooth and therefore regular for all n. Thus Kunz's theorem tells us that F e * R ⊗n is a projective R ⊗n -module, which implies that D (n) (R) = C R for all n. Similarly, if R is a localization of S, where S is a smooth k algebra, then R ⊗n is a localization of S ⊗n and so R ⊗n is still regular.…”

“…e a so we are done. The proof for the subadditivity formula (c) is similar to the one in [Smo18], though here we do not assume that k is perfect. As (F e * R) ⊗n canonically surjects onto F e * (R ⊗n ), we have C R ⊗n ,e ⊂ Hom R ⊗n (F e * R) ⊗n , R ⊗n for all e > 0.…”

“…holds for our choice of t. Following [Smo18], we will construct t using the so-called diagonal Cartier algebras. Namely, we set t = τ D (n) ; p (hn) ,…”

“…

Let k be a field of positive characteristic. Building on the work of [Smo18], we define a new class of k-algebras, called diagonally F -regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth k-algebras.

…”