The subadditivity conjecture for plus test ideals

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Let XX be a regular scheme. Subadditivity conjecture. For divisors Δ1\Delta_1 and Δ2\Delta_2 on XX, one has

τ+(OX,Δ1+Δ2)⊆τ+(OX,Δ1)⋅τ+(OX,Δ2).\tau_+(\mathcal{O}_X,\Delta_1+\Delta_2)\subseteq\tau_+(\mathcal{O}_X,\Delta_1)\cdot\tau_+(\mathcal{O}_X,\Delta_2).

This is the mixed-characteristic analogue of subadditivity for multiplier ideals and test ideals. The source says that the local case is proved for a variant of perturbed test ideals, while the stated global assertion remains open.

References

Primary source

Christopher Hacon, Alicia Lamarche and Karl Schwede, “Global generation of test ideals in mixed characteristic and applications”, arXiv:2106.14329 (2022).

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