
## 2. Other and Related Problems

1. #### Problem 2.05.

[de Fernex] Generalize the Hara-Yoshida Theorem on restriction of (Hacon-de Fernex) multiplier ideals to test ideals to the non-$\Q$-Gorenstein case. Can this be done in the numerically Gorenstein setting?
• #### Problem 2.1.

[Takagi] Investigate the subadditivity of (Hacon-de Fernex) multiplier ideals in the non-$\Q$-Gorenstein case.
1. Remark. This is closely related to problem generalizedHaraYoshida.
• #### Problem 2.15.

[Ishii] Define a “test ideal" that relates to the Mather discrepancy multiplier ideal, in the non-$\mathbb{Q}$-Gorenstein case.
• #### Problem 2.2.

[Patakfalvi] Investigate lifting sections (for cohomology) using test ideals.
• #### Problem 2.25.

[Mustaţă] Given ideals $I_1, \ldots, I_n$ of a regular, $F$-finite ring $R$, consider the map sending an $r$-tuple of nonnegative real numbers $(\lambda_1, \ldots, \lambda_r)$ to $\tau\left( {I_1}^{\lambda_1} \cdot \ldots \cdot{I_n}^{\lambda_n}\right)$, the mixed test ideal. If we bound all the $\lambda_i$ by some fixed $M$, does there exist a rational polyhedral decomposition such that the function described is constant on the interior of each region?
• #### Problem 2.3.

[Hochster] Realize effective computations of test ideals.
• ### Direct summand conjecture

#### Problem 2.35.

[Hochster] Prove/disprove the Direct Summand Conjecture.
• #### Problem 2.4.

[Hochster] Extend tight closure to mixed characteristic.
• #### Problem 2.45.

[Schwede] Suppose that $R$ is a local, equidimensional, $F$-pure ring of dimension $d$ with embedding dimension $n$. Is the multiplicity of $R$ at most $\binom{n}{d}$?
Answered positively by Huneke-Watanabe at the AIM workshop.
• #### Problem 2.5.

[Lyubeznik] Given a smooth, projective variety $X$ over a number field, does there exist a dense set of primes for which the action of Frobenius on the coherent cohomology $H^i(X_p, \mathcal{O}_{X_p})$ is not nilpotent?
• #### Problem 2.55.

[Singh] Do determinantal rings have finite $F$-representation type?
The 2 $\times$ 2 minors of a generic matrix has been done by Smith and Van den Bergh; try 3 $\times$ 3 minors and higher.
• #### Problem 2.6.

[Brenner] For $p \geq 11$, does $\mathbb{F}_p[x,y,z]/(x^2+y^3+z^7)$ have finite $F$-representation type?
• #### Problem 2.65.

[Schwede] Suppose that $R$ is a local ring and $f \in R$ is a regular element such that $R/(f)$ is $F$-injective. Does this imply that $R$ is $F$-injective?
This is known in the Cohen-Macaulay case. An analogous result is known to hold for Du Bois singularities in characteristic zero, in generality.
• #### Problem 2.7.

[Blickle] Given a log pair $(X ,\Delta)$, we know that there exists a finite map $\phi: Y \to X$ such that $\operatorname{im}\left(\phi_{*} \mathcal{O}_Y \left( K_Y - \phi^*\left( K_X + \Delta \right)\right) \overset{\text{trace}}{\longrightarrow} \mathcal{O}_X \right) = \tau(X, \Delta)$. Is this statement true if further decorated by $\mathfrak{a}^t$? What about for alterations?
• #### Problem 2.75.

[Mustaţă] Identify a possible positive characteristic analog of minimal log discrepancy.
• #### Problem 2.8.

[Schwede/Patakfalvi] Investigate possible Bertini theorems for $F$-singularities.
• #### Problem 2.85.

[Joshi] Investigate the applications of the topics of the conference (e.g., test ideals) to projective geometry.

Cite this as: AimPL: Test ideals and multiplier ideals, available at http://aimpl.org/testandmultiplierideals.