A Counterexample to Bhatt-Lurie's Cohomological Dimension Conjecture

Abstract: We exhibit a counterexample to a conjecture of Bhatt--Lurie on the cohomological dimension of the Hodge--Tate locus for regular local rings. The example arises from a non-excellent discrete valuation ring constructed by Datta--Smith. We also explain how the same mechanism yields broader families of counterexamples, while the expected bound is recovered under an excellence hypothesis.

• The paper presents a counterexample to Bhatt-Lurie's cohomological dimension conjecture by constructing non-excellent DVRs where the Hodge-Tate cohomology exceeds expected bounds.
• It employs detailed computations of the cotangent complex and the Hodge-Tate complex to reveal nonzero obstruction groups in a key degree.
• The findings emphasize that excellence, ensuring finiteness and regularity, is critical for adhering to cohomological limits in prismatic and related p-adic cohomology theories.

Counterexamples to Bhatt-Lurie's Cohomological Dimension Conjecture

Background and Formulation of the Conjecture

Bhatt-Lurie’s cohomological dimension conjecture emerges from the framework of prismatic cohomology, which unifies $p$-adic cohomology theories and geometrizes the Hodge-Tate locus via the stack $\mathrm{WCart}_X$. Formally, the conjecture [BL22b, Conjecture 10.1] postulates that for a $p$-complete, noetherian, regular local ring $R$ with perfect residue field, the Hodge-Tate locus $\mathrm{WCart}_{X}^{HT}$ satisfies

$$\mathrm{cd}(\mathrm{WCart}_{\mathrm{Spf}(R)}^{HT}) \leq \dim(R)$$

for the cohomological dimension (cd) with respect to the structure sheaf. This expected bound is motivated by the formal smoothness heuristics of regular rings over "the field with one element" and is closely related to structural properties of crystalline and prismatic theories.

Construction and Properties of the Counterexample

Guo Li constructs a counterexample to the conjecture using a non-excellent discrete valuation ring (DVR) $V_p$ of characteristic $p > 0$, originally due to Datta-Smith [DS18]. This ring is defined as $V_p := k[t] \cap K$ where $k = \mathbb{F}_p$, $\mathrm{WCart}_X$0, and $\mathrm{WCart}_X$1, $\mathrm{WCart}_X$2 for some transcendental power series $\mathrm{WCart}_X$3 that is algebraically independent over $\mathrm{WCart}_X$4. Key properties of $\mathrm{WCart}_X$5:

• It is a Noetherian, regular, local ring of Krull dimension $\mathrm{WCart}_X$6 with perfect residue field $\mathrm{WCart}_X$7.
• $\mathrm{WCart}_X$8 is $\mathrm{WCart}_X$9-complete but not excellent, nor $p$0-finite.
• $p$1 is not smooth over $p$2 and is not complete in its maximal ideal topology.

Crucially, the absence of excellence in $p$3 (and its generalizations) lies at the heart of the counterexample, as excellence was previously implicit in the scope of the conjecture due to its near ubiquity among Noetherian rings in algebraic geometry.

Computation of the Obstructing Cohomology

The central computation demonstrates that

$p$4

This is achieved through a careful analysis of the cotangent complex and the Hodge-Tate complex $p$5, along with its conjugate filtration. The ring $p$6 is shown to be Cartier smooth over $p$7, allowing the identification of the graded pieces of the Hodge-Tate complex with exterior powers of Kähler differentials.

By leveraging the filtered colimit representation of $p$8 via smooth $p$9-algebras and the explicit structure of $R$0, the paper establishes that the relevant Hodge-Tate cohomology group is nonzero due to the transcendence degree of the fraction field. This implies the cohomological dimension of the Hodge-Tate locus exceeds $R$1, directly contradicting the conjecture’s bound.

Generalizations and Flexibility

The method of construction is not isolated: by varying the choice of the transcendental power series, one obtains uncountably many non-excellent DVRs yielding analogous counterexamples. Furthermore, similar constructions can be performed in higher Krull dimensions, as cleverly outlined in the formation

$R$2

which gives regular local rings of dimension $R$3 with the failure manifest in $R$4. The obstruction arises in each case from the failure of excellence and the infinite rank of the module of differentials in the absence of $R$5-finiteness.

Resolution Under Excellence

Notably, the conjecture is shown to be valid whenever $R$6 is assumed to be excellent (or $R$7-finite). Using results of Bhatt-Mathew [BM23] and earlier foundational work, the cohomological bound is restored under such hypotheses. The $R$8-finiteness plays a pivotal role in controlling the cotangent complex and ensures finiteness properties necessary for the expected bound. This identifies excellence not merely as a technical convenience, but as an essential structural property for the validity of the conjecture.

Implications and Future Directions

This work delineates the boundaries of the Bhatt-Lurie conjecture, illustrating that excellence is indispensable for cohomological control in prismatic formal geometry. The explicit constructions highlight that non-excellent, non-$R$9-finite regular local rings—rare in classical algebraic geometry but essential in valuation-theoretic contexts—display wild behaviors invisible in the excellent case. The techniques also establish a template for constructing further counterexamples to cohomological bounds in related geometric settings.

For future developments, this result suggests attention to excellence must remain explicit in formulating conjectures in prismatic, derived, or stack-theoretic settings, especially when considering "large" or "transcendental" base rings. It also motivates the potential study of prismatic and mixed-characteristic phenomena in the non-excellent, non-noetherian regime.

Conclusion

Guo Li’s analysis provides a definitive counterexample to the Bhatt-Lurie cohomological dimension conjecture in the absence of excellence, leveraging the structure of non-excellent DVRs to demonstrate the necessity of finiteness and regularity assumptions. The results not only clarify the scope of the conjecture but also illuminate subtle behaviors in the geometrization of $\mathrm{WCart}_{X}^{HT}$0-adic cohomology, enriching the landscape for both foundational and applied investigations in algebraic geometry and arithmetic geometry (2606.05260).