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Bhatt–Iyengar–Ma Tor-Vanishing Problem

About 7 years old · traced to

Let kk be an algebraically closed field of characteristic 00, and let (R,m,k)(R,\mathfrak m,k) be a complete Noetherian local domain.

Let R+R^+ denote the absolute integral closure of RR, i.e. the integral closure of RR in an algebraic closure of Frac⁡(R)\operatorname{Frac}(R).

Question. If

Tor⁡iR(k,R+)=0\operatorname{Tor}_i^R(k,R^+)=0

for some i≥1i\geq 1, must RR be regular?

This is a special case of a question of Bhatt, Iyengar, and Ma.

References

References

B. Bhatt, S. B. Iyengar, and L. Ma, Regular Rings and Perfect(oid) Algebras, Comm. Algebra 47 (2019), 2367–2383. See also M. Asgharzadeh and S. Patankar, Remarks on some Homological Problems regarding Infinite Integral Extensions, arXiv:2608.09473, Question ((Q_1)).

commutative-algebra
Posted by Shravan Patankar Posted

Progress summary

Refreshed
Claimed solved

A recent paper says the answer is yes, but its hypotheses and proof have not yet been independently checked against the full question.

Bhatt, Iyengar, and Ma posed the question in 2020: whether vanishing of Tor⁡iR(k,R+)\operatorname{Tor}_i^R(k,R^+) for some i≥1i\geq 1 forces regularity in characteristic zero.

Known results

  • Positive characteristic: vanishing of Tor⁡iR(k,R+)\operatorname{Tor}_i^R(k,R^+) implies regularity for excellent local domains (Bhatt–Iyengar–Ma, 2020).
  • Mixed characteristic: the same conclusion holds when dim⁡R≤3\dim R\leq 3 (Bhatt–Iyengar–Ma, 2020).
  • Characteristic zero: the conclusion was proved for two-dimensional finitely generated graded rings and for complete local domains with dim⁡R≥4\dim R\geq 4 (2022).

August 2026 claimed affirmative result

A paper released in August 2026 repeats the full characteristic-zero question and displays the claim that vanishing for some i>0i>0 implies regularity. However, the surrounding discussion uses narrower NBIM hypotheses, so the claim’s scope is unclear and it is not yet confirmation of the stated problem’s solution. A June 2026 paper gives further characteristic-zero results but likewise does not visibly resolve every complete local domain.

Current status (as of August 2026): A full affirmative answer is claimed in an August paper, but its applicability to the exact stated hypotheses remains unverified; the problem should not yet be treated as settled.

Sources

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