Closedness of local cohomology support

Must a local-cohomology module over a normal domain have finitely many minimal primes? The counterexample target is H²_J for a codimension-one prime J in a five-dimensional finitely generated flat normal domain over Z or a field, with a corresponding six-dimensional normal local example.

Why it matters and what counts as progress

Why it matters: It identifies whether normality suffices for this finiteness property; characteristic two requires its own construction.

What counts as progress: Normal-domain examples with infinitely many minimal primes in the stated local cohomology, or finiteness theorems identifying additional necessary hypotheses.

How the problem developed

1990 · Finiteness conjecture [1]
Craig Huneke Conjectured that local cohomology modules have only finitely many associated primes.
2002 · Counterexample [1]
Mordechai Katzman Constructed a local k-algebra whose local cohomology module has infinitely many associated primes, settling Huneke’s finiteness conjecture for local rings.
30 Jul 2026 · AI-assisted construction [2]
Bhargav Bhatt and Linquan Ma Constructed noetherian rings with local cohomology modules whose support is not closed, equivalently with infinitely many minimal primes, answering a question of Huneke and Lyubeznik.

AI contribution

Local cohomology modules with nonclosed support

Bhargav Bhatt, Linquan Ma

Bhatt and Ma construct a five-dimensional finitely generated flat normal domain over Z or any field with a codimension-one prime J for which H²_J has infinitely many minimal primes. The characteristic-two case uses a separate construction; a cone gives a six-dimensional normal local example.

AI involvementAI-centered

ImportanceNotable

Technical verificationReported

Human understandingFull