Hellus's associated-prime conjecture for Matlis duals of local cohomology

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Let (R,m)(R,\mathfrak{m}) be a Noetherian regular local ring, let II be a non-zero ideal of RR, and let E:=ER(R/m)E:=E_R(R/\mathfrak{m}) be an injective hull of the residue field. For any RR-module MM, write D(M):=Hom⁡R(M,E)D(M):=\operatorname{Hom}_R(M,E), and write H⁡Ii(R)\operatorname{H}^i_I(R) for the ii-th local cohomology module supported in II.

Hellus's conjecture. If H⁡Ii(R)≠0\operatorname{H}^i_I(R)\neq 0, then

0∈Ass⁡R(D(H⁡Ii(R))).0\in \operatorname{Ass}_R\bigl(D(\operatorname{H}^i_I(R))\bigr).

This extends Hellus's equivalent domain condition from ideals generated by a specified number of elements to arbitrary non-zero ideals in regular local rings. The supplied text says that the corresponding associated-prime conjecture is known in many cases, but gives no resolution of this proposed generalization.

References

Primary source

Gennady Lyubeznik and Tuğba Yıldırım, “On the matlis duals of local cohomology modules”, arXiv:1707.00501 (2017).

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