Bhatt–Iyengar–Ma Tor-Vanishing Problem

Let $k$ be an algebraically closed field of characteristic $0$ , and let $(R,\mathfrak m,k)$ be a complete Noetherian local domain.

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

Question. If

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

for some $i\geq 1$ , must $R$ be regular?

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

Progress summary

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 $\operatorname{Tor}_i^R(k,R^+)$ for some $i\geq 1$ forces regularity in characteristic zero.

Known results

Positive characteristic: vanishing of $\operatorname{Tor}_i^R(k,R^+)$ implies regularity for excellent local domains (Bhatt–Iyengar–Ma, 2020).
Mixed characteristic: the same conclusion holds when $\dim R\leq 3$ (Bhatt–Iyengar–Ma, 2020).
Characteristic zero: the conclusion was proved for two-dimensional finitely generated graded rings and for complete local domains with $\dim R\geq 4$ (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 $i>0$ 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

Sources & referencesView supporting material

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 1 week ago