Mixed characteristic analogue of Hochster–Huneke Lemma 3.6

Develop a mixed characteristic analogue of Lemma 3.6 from Hochster and Huneke’s comparison of symbolic and ordinary powers, establishing an appropriate Jacobian-ideal-based or related criterion that uniformly multiplies symbolic or Frobenius symbolic powers into ordinary or bracket powers for all ideals in mixed characteristic rings, thereby enabling extensions of Uniform Symbolic Topology results to BCM-singularities in mixed characteristic.

Background

The main results in the paper for prime characteristic rely on properties of Jacobian ideals described in Lemma 3.6 of Hochster–Huneke (HHComparison), which furnish uniform multipliers ensuring containments between symbolic/Frobenius symbolic and ordinary/bracket powers. These techniques are essential for establishing the Uniform Symbolic Topology Property for strongly F-regular (or BCM) singularities.

To extend these methods to mixed characteristic settings and to BCM-singularities, a suitable analogue of HHComparison Lemma 3.6 is needed. The authors note that such an analogue is currently unavailable, and its establishment would address a core obstacle in generalizing their approach beyond positive characteristic.