Abstract
Let R be a commutative Noetherian ring that is a smooth \(\mathbb {Z}\)-algebra. For each ideal \(\mathfrak {a}\) of R and integer k, we prove that the local cohomology module \(H^{k}_{\mathfrak {a}}(R)\) has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.
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Notes
Added in proof: In the formal power series case one cannot use [8, Theorem 2.4] for a proof of the finiteness of the prime ideals not containing u because the ring is not finitely generated over a field. A proof of this remains the same as in [10, pp. 5880 (from line −5)–5882]; but our proof in the formal power series case of the finiteness of the primes containing u is much simpler than in [10].
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B.B. was supported by NSF grants DMS 1160914 and DMS 1128155, M.B. by DFG grants SFB/TRR45 and his Heisenberg Professorship, G.L. by NSF grant DMS 1161783, A.K.S. by NSF grant DMS 1162585, and W.Z. by NSF grant DMS 1068946. A.K.S. thanks Uli Walther for several valuable discussions. The authors are grateful to the American Institute of Mathematics (AIM) for supporting their collaboration. All authors were also supported by NSF grant 0932078000 while in residence at MSRI.
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Bhatt, B., Blickle, M., Lyubeznik, G. et al. Local cohomology modules of a smooth \(\mathbb{Z}\)-algebra have finitely many associated primes. Invent. math. 197, 509–519 (2014). https://doi.org/10.1007/s00222-013-0490-z
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DOI: https://doi.org/10.1007/s00222-013-0490-z