A three-dimensional singularity that no finite module can make Cohen–Macaulay

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Family 195 · Algebra
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

A singular ring may fail to behave like a smooth coordinate ring, yet a module over it can sometimes restore enough regularity for homological arguments. The small Cohen–Macaulay module question asks whether every complete local domain admits such a finite algebraic substitute.

Family 195 claims a counterexample already in dimension three over the complex numbers. Its ring is complete, normal, and has no zero divisors, but every nonzero finitely generated module has depth less than three.

The strength lies in “every finite module.” Earlier obstructions for finite extension algebras or graded modules leave room for a module with no multiplication or compatible grading. The manuscript’s construction is designed to close those possibilities at its chosen singularity.

2. Background and essential definitions

Depth measures successive regularity

Let $(R,\mathfrak m)$ be a Noetherian local ring and $M$ a nonzero finite module. An element $x\in\mathfrak m$ is regular on $M$ if multiplication by $x$ is injective. A sequence $x_1,\ldots,x_r$ is regular on $M$ if each element remains regular after quotienting by the preceding ones, with the final quotient nonzero.

The depth of $M$ is the longest possible length of such a sequence. It cannot exceed the ring’s dimension. A finite module is maximal Cohen–Macaulay if

$$\operatorname{depth}_R M=\dim R.$$

“Small Cohen–Macaulay module” means the same thing here, with “small” emphasizing finite generation. Over the regular ring $\mathbb C[[x,y,z]]$, the ring itself works: multiplication successively by $x,y,z$ has the required injectivity. A singular ring may fail this test on itself while still possessing another module that passes it.

A domain has no nonzero zero divisors. A normal domain is integrally closed in its fraction field. Completeness means that compatible information modulo all powers of $\mathfrak m$ comes from an element of the ring. These conditions make the counterexample a well-controlled local algebra, rather than a pathology caused by dropping standard finiteness assumptions.

Modules, algebras, and infinite substitutes

A finite extension algebra $B$ is also a finite $R$-module, but a module need not have multiplication. Ruling out Cohen–Macaulay extension algebras therefore does not rule out all small Cohen–Macaulay modules.

A big Cohen–Macaulay module may require infinitely many generators. In the balanced formulation, every system of parameters acts regularly and the quotient by $\mathfrak m$ is nonzero. Big objects can exist even when no small one does. The claimed counterexample concerns finite generation, not the general availability of big Cohen–Macaulay methods.

A cone built from a surface

Let $X$ be a smooth projective complex surface and $H$ an ample globally generated divisor. “Globally generated” means that its line bundle has sections with no common zero; “ample” supplies sufficiently many sections in large powers to embed the surface projectively.

The section ring is

$$T(X,H)=\bigoplus_{j\ge0}H^0(X,\mathcal O_X(jH)).$$

Its geometry is a three-dimensional cone over the surface. Localizing at the positive-degree ideal isolates the vertex, and completing gives the local ring $R(X,H)$. The construction will choose the surface so that its numerical geometry obstructs every hypothetical small module at this vertex.

3. Development history and previous partial results

Hochster formulated the finite-module existence problem in early work on Cohen–Macaulay modules and homological conjectures. His later account records both the original existence expectation and a subsequent expectation of counterexamples.

Dimensions at most two have positive answers for complete local domains: finite normalization is Cohen–Macaulay. Dimension three is therefore the first possible dimension for the claimed domain counterexample.

There are also significant positive results in characteristic $p$. Work of Hartshorne, Peskine–Szpiro, and Hochster gives graded small Cohen–Macaulay modules for three-dimensional finitely generated nonnegatively graded domains over perfect fields of positive characteristic. These hypotheses do not include the characteristic-zero cone used here.

Allowing infinite generation changes the landscape. Hochster established big Cohen–Macaulay modules in equal characteristic. Hochster and Huneke showed that absolute integral closures of excellent positive-characteristic domains provide big Cohen–Macaulay algebras. Perfectoid methods later supplied the mixed-characteristic existence theorem. None of these statements promises a finite module.

Nonexistence results for small algebras were known earlier. In characteristic zero, trace makes a normal domain a direct summand of each finite extension domain, so a Cohen–Macaulay extension would force the original ring to be Cohen–Macaulay. Bhatt constructed positive-characteristic examples without finite Cohen–Macaulay extensions. Arbitrary finite modules evade the multiplication and trace framework.

The immediate geometric predecessors involve vector bundles on surfaces. Bogomolov’s inequality and the Hodge index theorem give numerical restrictions on Ulrich bundles. Beauville described such a route under a rank-one divisor-class assumption; Anghel developed a surface inequality without that restriction and used complete-quadrangle covers.

The manuscript carefully records two September 2026 comparisons. Anghel’s 14 September preprint excludes arithmetically Cohen–Macaulay bundles and graded maximal Cohen–Macaulay modules for relevant section rings, using a sharper numerical threshold. Liang Chen’s 21 September preprint proposes a completed-local counterexample under a proportionality condition between the canonical divisor and the polarization. Thus the existential completed-module claim overlaps with that prior proposal. The present manuscript claims a different obstruction without that proportionality hypothesis, and uses an exponent-seven cover rather than the exponent-six example discussed there.

4. What the new manuscripts claim

The manuscript, dated 23 September 2026, claims a three-dimensional complete Noetherian normal local domain $R$ containing $\mathbb C$, with residue field $\mathbb C$, such that

$$\operatorname{depth}_R M<3 \quad\text{for every nonzero finitely generated }R\text{-module }M.$$

It imposes no grading, algebra structure, or special rank on the module being excluded.

The principal intermediate claim is a geometric obstruction. Write $K_X$ for the canonical divisor of a smooth projective surface, $K_X^2$ for its self-intersection, and $c_2(X)$ for the second Chern number, equal here to the topological Euler characteristic. If $H$ is ample and globally generated and the completed cone admits a small Cohen–Macaulay module, then

$$K_X^2-2c_2(X)\le6H^2.$$

These are numerical invariants of the chosen surface and polarization. The manuscript constructs a pair with

$$H^2=1715,\qquad K_X^2-2c_2(X)=13377>10290=6H^2.$$

The claimed obstruction therefore excludes every small Cohen–Macaulay module over its completed cone.

A corollary also excludes such modules over the uncompleted vertex localization of the same section ring. This is a three-dimensional local domain essentially of finite type over $\mathbb C$. If it had a suitable finite module, faithfully flat completion would preserve its nonzeroness and regular sequence, producing the forbidden completed module.

5. Why the advance matters

Provisional importance: 5/5. If correct, this gives a negative answer to the domain form of a central finite-module existence question, in the first possible dimension.

The theorem separates the finite and infinite worlds sharply. Big Cohen–Macaulay methods can remain available while no finite module reproduces the required depth. It also explains why approaches to other conjectures cannot universally assume a small Cohen–Macaulay or Ulrich module over each source ring.

The nearby multiplicity problems illustrate this point. Family 194 uses complexes and normalized lengths to formulate its claimed general comparison without making such finite-module existence a prerequisite. A failure of one ideal auxiliary object need not invalidate every numerical method that it once motivated.

6. The main idea

Carry an arbitrary module onto a surface

Choose a finite map $f:X\to\mathbb P^2$ defined by sections of $H$. The completed cone is finite over the regular ring

$$A=\mathbb C[[x_0,x_1,x_2]].$$

A hypothetical depth-three finite module over the cone is free as an $A$-module, by the regular local homological theory. Away from the vertex it determines a vector bundle.

The manuscript replaces the cone vertex by the exceptional surface $X$ on a regular model. It extends the actual bundle to a coherent sheaf on that model. This preserves information from an arbitrary nongraded module; it does not assume that the module descends to a graded or Ulrich bundle on $X$.

Balance slopes without changing the punctured bundle

A torsion-free sheaf $B$ on the surface has normalized slope

$$\mu(B)=\frac{c_1(B)\cdot H}{\operatorname{rank}(B)H^2}.$$

Here $c_1(B)$ records its first Chern class. Slope compares its average twisting in the direction of the polarization. A Harder–Narasimhan filtration separates a sheaf into semistable pieces with decreasing slopes; semistability means that smaller-rank subsheaves cannot have a greater slope.

Elementary modifications supported on the exceptional surface leave the bundle outside that surface unchanged. The manuscript uses them to put the slopes of the restriction’s semistable pieces into an interval of length at most one.

Freeness over $A$ yields one Chern-character inequality after pushing this restriction to $\mathbb P^2$. Bogomolov’s inequality, the Hodge index theorem, and the bounded slope interval yield an inequality in the opposite direction. Together, they give

$$0\le \Delta\le \frac12-\frac{K_X^2-2c_2(X)}{12H^2},$$

where $\Delta$ is a normalized Chern-character expression of the pushed-forward sheaf. Its exact value is unimportant for the contradiction: the right-hand side must be nonnegative. This is the claimed surface obstruction.

Choose a surface that violates the bound

Begin with four general points of the projective plane and the six lines joining their pairs, the complete quadrangle. Blow up the four points. The six strict transforms and four exceptional curves form a branch divisor with ten components and fifteen ordinary crossing points.

Adjoin $n$-th roots of five independent ratios of the six line equations and normalize. For prime $n\ge3$, the manuscript obtains a smooth surface covering the blown-up plane with degree $n^5$. Its local branch coordinates are ordinary power maps, making the ramification explicit.

Canonical-divisor and Euler-characteristic calculations give

$$K_X^2-2c_2(X)=n^5\left(1-\frac{10}{n^2}\right), \qquad H^2=5n^3.$$

The polarization is constructed as an integral ample globally generated divisor; having the desired numerical square alone would not suffice. Taking $n=7$ gives the incompatible numbers in section 4. The resulting completed cone is the asserted ring.

7. Scope, limitations, and open questions

The claim is an existence counterexample, not a classification of all three-dimensional domains without small Cohen–Macaulay modules. Many rings still possess them, including regular and Cohen–Macaulay rings themselves.

The numerical inequality is a necessary condition for the chosen cone construction. Satisfying it is not asserted to guarantee a small module. The constant six also should not be identified with the sharper thresholds in earlier graded-bundle obstructions: those statements exclude different classes of objects.

Characteristic zero and the specified complex surface are part of this construction. The theorem does not negate positive-characteristic graded existence results or big Cohen–Macaulay existence. It also does not prove the absence of every asymptotic substitute, such as the module sequences used in lim Cohen–Macaulay theory.

The prior completed-local proposal is explicitly acknowledged. The significance assigned here concerns the mathematical scope of the claimed obstruction and example, without independently adjudicating novelty or priority.

8. Sources and further reading

Hochster’s Cohen–Macaulay modules (1973), Topics in the Homological Theory of Modules over Commutative Rings (1975), and Homological conjectures and lim Cohen–Macaulay sequences (2017) trace the finite and infinite existence questions. Bhatt’s On the non-existence of small Cohen–Macaulay algebras (2014) concerns the distinct algebra problem.

The closely related sources are Anghel’s surface construction, his graded-module obstruction, and Liang Chen’s completed-local proposal. Hirzebruch’s Arrangements of lines and algebraic surfaces (1983) is the geometric source of the covering construction.

Bhatt, Hochster, and Ma’s lim Cohen–Macaulay work explains asymptotic alternatives. These references supply context for the arbitrary finite-module claim in the manuscript below.

Source manuscripts

• M1. OpenAI, A Complete Local Domain without a Small Cohen–Macaulay Module (September 23, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.