A Complete Local Domain without a Small Cohen–Macaulay Module

1 Introduction

A small Cohen–Macaulay module over a Noetherian local ring (R,\mathfrak m) is a nonzero finitely generated R-module M such that \depth_R M=\dim R. Such a module is also called a maximal Cohen–Macaulay module. The domain form of the small Cohen–Macaulay module conjecture asks whether every complete Noetherian local domain has one. We prove the following negative answer.

The distinction between modules and algebras matters here. A module-finite extension domain R\subseteq B that is Cohen–Macaulay supplies such a module, but the converse need not hold. Our obstruction applies to every finite module; it requires neither a grading nor an algebra structure. Hochster formulated the finite-module existence problem in his early work on Cohen–Macaulay modules and homological conjectures [13, 14]. In dimensions at most two, a complete local domain has a small Cohen–Macaulay module: its finite normalization is Cohen–Macaulay [15, Section 2]. There are also important positive results in the graded setting. The theorem of Hartshorne, Peskine–Szpiro, and Hochster gives graded small Cohen–Macaulay modules for three-dimensional finitely generated nonnegatively graded domains whose degree-zero part is a perfect field of positive characteristic; see [17, Theorem 5.2] for the statement and historical attribution. These results do not cover the characteristic-zero cones considered here. Hochster’s later account explicitly records both his early existence conjecture and his subsequent expectation of counterexamples [15, Section 2, Conjectures 2.1 and 2.2].

Dropping finite generation changes the problem. In the balanced convention, a big Cohen–Macaulay module is a possibly infinitely generated module on which every system of parameters is a regular sequence and whose quotient by the maximal ideal is nonzero. Hochster proved existence in equal characteristic [14]. Hochster–Huneke showed that the absolute integral closure of an excellent local domain of positive characteristic is such an algebra [16]. Perfectoid methods later yielded big Cohen–Macaulay algebras in mixed characteristic, completing the existence theorem for Noetherian local rings [1, Theorem 0.7.1]. These results impose no finite-generation conclusion.

For a normal local domain containing the rational numbers, trace makes the ring a direct summand of every module-finite extension domain. A Cohen–Macaulay extension would therefore force the original ring to be Cohen–Macaulay. Bhatt constructed positive-characteristic complete normal local domains with no finite Cohen–Macaulay extension, using a Witt-vector cohomological obstruction [6]. Neither obstruction excludes arbitrary finite modules. Bhatt, Hochster, and Ma instead study asymptotic substitutes for small Cohen–Macaulay modules. Their August 25, 2026 version describes the existence problem as open even for local rings at maximal ideals of affine three-dimensional domains over algebraically closed fields [7, Sections 1.1 and 1.3].

The geometric input is a smooth integral projective surface X over \C with an ample globally generated line bundle \cO_X(H). Write

Here the hat denotes completion at the maximal ideal. We write K_X for the canonical divisor and c_2(X) for the second Chern number. For the specific pair (X,H) used for Theorem 1.1, the uncompleted vertex localization T(X,H)_{T(X,H)_{>0}} also has no small Cohen–Macaulay module (Corollary 7.1). It is a three-dimensional local domain essentially of finite type over \C, so this supplies a counterexample in the affine-local class just described. The deduction uses faithful flatness of completion. Our main intermediate result is the following obstruction.

A numerical obstruction for Ulrich sheaves on surfaces follows from Bogomolov’s inequality and the Hodge index theorem. Beauville presented this route for surfaces of Néron–Severi rank one [4, pp. 18–20]; Anghel gives the surface inequality without that Néron–Severi rank restriction [3, Proposition 2.1 and Remark 2.3]. For a projective surface, an Ulrich sheaf has trivial vector-bundle pushforward under a finite linear projection to \PP^2 [10, Proposition 2.1]. The argument here replaces that triviality condition by a comparison of coherent extensions across the exceptional divisor of a completed cone. A module over the completed local ring need not carry a grading. We therefore retain its actual vector bundle on the punctured spectrum, rather than assume it descends to an Ulrich sheaf on X. It is this comparison that permits arbitrary nongraded modules. The explicit surface used below is the complete-quadrangle Hirzebruch–Kummer construction appearing in [3, Section 7.2]. We give its construction and all the required numerical and positivity checks directly.

There are two further recent comparisons. Anghel’s September 14, 2026 preprint excludes arithmetically Cohen–Macaulay bundles under the sharper inequality K_X^2-2c_2(X)>3H^2, and excludes graded maximal Cohen–Macaulay modules over the same complete-quadrangle section rings used here [2, Theorems 1.1 and 1.3, Corollary 4.3]. That statement does not assert nonexistence of arbitrary modules over the vertex completion. Chen’s September 21, 2026 preprint proposes a quantitative local-cohomology bound for every nonzero finite reflexive module over a completed cone when K_X\equiv4H and 15H^2/8-\chi(\cO_X)>0, and obtains a proposed completed-local counterexample from the exponent-six cover [8, Main claim 1.1 and Consequence 1.2]. Thus the existential completed-module claim overlaps with that work. Our obstruction applies without the proportionality hypothesis on K_X and H; the exponent-seven polarization used below has K_X\equiv5H.¹

Proof overview

Choose a finite morphism f:X\to\PP^2 with f^*\cO_{\PP^2}(1)=\cO_X(H), and put h=H^2=\deg f. The completed cone is finite over the regular local ring A=\C[[x_0,x_1,x_2]]. A small Cohen–Macaulay module is free over A. Its restriction to the punctured cone is a vector bundle, which extends to a coherent sheaf \cE on a regular model W\to\Spec R(X,H) that is an isomorphism away from the exceptional surface X.

For a nonzero torsion-free sheaf B on X, its normalized slope is

It is slope-semistable if no nonzero subsheaf of smaller rank has larger slope. Its Harder–Narasimhan filtration has torsion-free semistable factors with decreasing slopes. We choose \cE with torsion-free restriction P=\cE|_X. First, elementary modifications of \cE along X arrange that the slopes of the Harder–Narasimhan factors of P lie in an interval of length at most one. These modifications preserve the bundle away from X. Second, that bundle comes from an A-free module. Comparing the pushforward of \cE with a trivial sheaf on the blowup of \Spec A gives, for F=f_*P of rank N, the inequality 2N\ch_2(F)\geq c_1(F)^2, with Chern characters evaluated on \PP^2. Bogomolov’s inequality for the semistable factors bounds the same normalized Chern expression above. More precisely, if r=\rk P, then N=rh and the two comparisons give

This yields (1). Neither P nor f_*P is assumed semistable or Ulrich.

Section 2 constructs the resolution and extends an arbitrary module. Section 3 proves the slope modification lemma. Section 4 establishes the Chern-character inequality, and Section 5 combines it with surface theory. Finally, Section 6 constructs a surface with K_X^2-2c_2(X)>6H^2.

2 From a completed cone to its exceptional surface

Throughout this section, X and H satisfy the hypotheses of Theorem 1.2. We construct a regular model of the completed cone and explain how a small Cohen–Macaulay module produces a sheaf on it. We also prove that the completed ring is normal.

2.1 The finite map and the completed ring

Global generation supplies sections s_0,s_1,s_2 of \cO_X(H) with no common zero. Indeed, choose s_0\ne0, then s_1 avoiding the components of its zero divisor, and then s_2 nonvanishing at their finitely many common zeros. The resulting morphism

is finite and surjective. A positive-dimensional fiber would contain a curve on which H is both ample and trivial, which is impossible. Properness then gives finiteness, and the intersection formula gives \deg f=h:=H^2.

Put

where x_i acts as s_i. Surjectivity of f makes S\to T injective. The ring T is a domain, as seen by trivializing H rationally and embedding T in \C(X)[t]. It is a finite graded S-module: by the projection formula, its graded pieces are the nonnegative twisted sections of the coherent sheaf f_*\cO_X on D, and Serre vanishing implies finite generation. Explicitly, a surjection \cO_D(-a)^p\to f_*\cO_X and the vanishing of the first cohomology of its kernel in large twists generate all sufficiently large degrees; the remaining finitely many degrees complete a generating set.

Let \mathfrak n=T_{>0} and \mathfrak a=(x_0,x_1,x_2)T. The \mathfrak a-adic filtration is cofinal with the degree filtration: if homogeneous S-module generators have degrees at most a', then

Finiteness over S and completion of finite modules [19, Tags 00MA and 00MB] therefore identify

with multiplication by the graded Cauchy product. The product of the first nonzero homogeneous terms proves that R is a domain. An element with nonzero constant term is a unit. Moreover, T/\mathfrak a is a finite-dimensional positively graded algebra with degree-zero part \C, so \sqrt{\mathfrak a}=\mathfrak n. This justifies the last equality in (2) and shows that R is a complete Noetherian local ring with residue field \C. Flatness of S\to A gives an injection A\to R; this finite integral extension has dimension three.

2.2 The regular model and normality

Let V_0=\Tot(\cO_D(-1)), the blowup of \Spec S at the origin, and let

The map W_0\to V_0 is finite. Evaluation of homogeneous sections gives W_0\to\Spec T. It is an isomorphism off the zero section. To see this without any assumption that T is generated in degree one, set X_i=\{s_i\ne0\}. The set X_i is affine, and clearing poles along the zero divisor of s_i gives

Both the punctured line bundle over X_i and the corresponding open subset of \Spec T have this coordinate ring.

Base change along S\to A to obtain g:W\to V and p:W\to\Spec R. The zero sections remain X\subset W and D\subset V. Their ideal sheaves \cJ and \cI satisfy

The zero sections are Cartier divisors by flatness. The relevant maps fit into the diagram

The maps g and \Spec R\to\Spec A are finite, and V and W are proper over \Spec A. Since \Spec R is separated over \Spec A, the map p is proper as well. The maps p and V\to\Spec A identify W\setminus X and V\setminus D with the punctured spectra of R and A, respectively.

Both V and W are regular. At a point of the closed fiber, a local Cartier equation is a nonzerodivisor and its quotient local ring is regular, since D and X are smooth. Lifting generators of the quotient maximal ideal shows that the embedding dimension is at most one more than the quotient dimension; the nonzerodivisor raises dimension by one. The local ring is therefore regular. Properness over the local base ensures that the closure of every point meets the closed fiber. Regularity at all other points follows by localization. A regular Noetherian scheme has disjoint irreducible components; here every component meets the integral closed fiber, so each of V and W is integral.

This model also proves normality of R. The line bundle projection W_0\to X identifies its global functions with T. Flat base change [19, Tag 02KH] then gives

For this equality, take a finite affine cover of the separated scheme W_0 and tensor the kernel computing global sections with the flat S-algebra A. The global functions on an integral normal scheme form an integrally closed domain: a fraction integral over the global ring is integral over every local ring, hence regular everywhere. Since W is regular and integral, (4) proves that R is normal.

2.3 Extending an arbitrary module

Suppose that M is a small Cohen–Macaulay R-module. The parameters x_0,x_1,x_2 form an M-regular sequence, so M has depth three over A. The Auslander–Buchsbaum formula over the regular local ring A [19, Tags 00N6, 00O7 and 090V] shows that

The localization R\otimes_A\Frac(A) is a finite-dimensional domain over a field, hence a field. Thus M has positive generic rank over R and full support.

The restriction of M to the punctured spectrum is locally free. Indeed, Cohen–Macaulay modules of full support localize to Cohen–Macaulay modules. More explicitly, for a prime \mathfrak q the depth localization inequality [19, Tag 0FCC] gives

The second inequality follows by concatenating chains of primes below and above \mathfrak q. On the puncture R_{\mathfrak q} is regular by the model W, and Auslander–Buchsbaum gives freeness.

On W, take

where duals are over \cO_W. Then \cE is coherent and torsion-free, and agrees with M away from X. The restriction P is torsion-free of positive rank on X. Here is the local justification. On a regular integral scheme a coherent dual has depth at least \min(2,\dim\cO_{W,w}): dualizing a finite presentation expresses it as a kernel of a map between finite free modules, whose image is torsion-free, and the depth lemma applies. At any nongeneric point of X, quotienting \cE by the Cartier equation therefore leaves depth at least one. Thus P has no associated points except the generic point of X. At that generic point \cE is free over a discrete valuation ring, so P has the same positive rank as \cE.

Finally, g_*\cE is torsion-free and, by (5), agrees with \cO_V^N off D. We have obtained the precise input for the next two sections: a sheaf that can be modified along X, whose pushforward remains trivial away from D.

3 Balancing slopes by modifications

We now modify the extension along X so that its restriction has Harder–Narasimhan slopes in an interval of length at most one. Recall the normalization \mu(B)=c_1(B)\cdot H/(\rk(B)h), where h=H^2. Tensoring with \cO_X(H) increases slope by one.

Elementary modification to constrain the slopes of exceptional-divisor restrictions has an antecedent in Chen–Sun’s optimal extensions of reflexive analytic sheaves [9, Theorem 1.4(I), Proposition 2.6 and Remark 1.5]. Their analytic setting and reflexivity hypotheses differ from the Noetherian Cartier-divisor setting below; we give the needed algebraic modification and termination argument.

We recall briefly why the filtration exists in this setting. A torsion-free sheaf embeds in a sum of sufficiently positive line bundles, so its subsheaf slopes are bounded above by taking determinants. Their denominators are bounded because their ranks are bounded. Choose a subsheaf of maximum slope and, among these, of maximum rank; saturate it. Saturation does not decrease degree, since a torsion sheaf has effective divisorial first Chern class. The chosen subsheaf is semistable, and every subsheaf of its torsion-free quotient has strictly smaller slope. Induction on rank gives the filtration. This is the usual construction of the Harder–Narasimhan filtration [18, Section 1.6, Theorem 1.6.7 and Definition/Corollary 1.6.9].

Apply Lemma 3.1 to the sheaf constructed in Section 2, and again call the resulting sheaves \cE and P. The rank and the bundle on the puncture are unchanged. In particular, g_*\cE still agrees with \cO_V^N off D. The restriction P need not be semistable: it is the width of its factor slopes, rather than semistability, that we have arranged. This agreement on the puncture imposes the lower bound proved next.

4 A Chern-character inequality from a divisor

The pushforward of our extension agrees with a trivial bundle away from D. We show that this agreement forces a nonnegative expression in the Chern characters of its restriction to D. For a coherent sheaf F of positive rank N on D=\PP^2, identify c_1(F) with its integer coefficient in the line class, and let \ch_2(F) mean the degree of the second Chern character. Define

For F=\bigoplus_{i=1}^N\cO_D(a_i), this is the ordinary variance of the integers a_i. It need not be nonnegative for arbitrary sheaves. The expression is unchanged by twisting F by \cO_D(b).

For L'=g_*\cE, the equality \cJ=g^*\cI in (3) and finite pushforward give

Writing r=\rk P, this sheaf has rank rh by finite pushforward along f, and rank N by generic restriction along D in Lemma 4.1. Thus the two ranks agree: N=rh. The hypotheses of Lemma 4.1 hold by (3) and the construction of \cE. Consequently

It remains to bound this expression above using the narrow slope interval of P.

5 The numerical obstruction

We first compute the contribution of one semistable factor. We use surface Riemann–Roch and the Hodge index theorem in their standard forms; see [11, Appendix A, Theorem 4.1 and Example 4.1.2; Chapter V, Theorem 1.9]. On each smooth projective surface, finite locally free resolutions extend Riemann–Roch additively to coherent sheaves [11, Appendix A, Section 5]. This covers the possibly non-locally-free sheaves G and f_*G below; we compare their Euler characteristics, without requiring the finite morphism f to be smooth. All intersections in this section are on X, except for Chern characters explicitly taken on D=\PP^2.

6 An explicit surface and its polarization

We construct a family of smooth projective surfaces from which we will choose one violating the inequality established above. The construction uses the six lines joining four points of the plane, followed by a cover obtained by taking roots of their ratios. The same line arrangement provides the sections needed for an ample globally generated divisor on the cover. This choice of arrangement and polarization is the complete-quadrangle construction in [3, Section 7.2]; all properties needed here are verified below.

We prove the proposition by constructing the cover, checking its local coordinates, and then finding sections of the required polarization. The covering construction goes back to Hirzebruch’s work on line arrangements [12].

6.1 The branch divisor

Choose four points p_1,\ldots,p_4\in\mathbb P^2_{\mathbb C}, no three collinear, and let B be their blowup. Write \ell for the pullback of the line class and e_1,\ldots,e_4 for the exceptional curves. Thus

Number the six joining lines from 1 to 6, and choose defining linear forms l_1,\ldots,l_6. Denote the strict transform of line a by L_a and its total transform by T_a. If its endpoints are p_i,p_j, then

Two joining lines are called opposite if their endpoint pairs are disjoint. There are three opposite pairs.

Let

This reduced divisor has simple normal crossings. Each exceptional curve meets the three incident strict transforms in three distinct points. Strict transforms of lines with a common endpoint are disjoint, whereas each opposite pair meets transversely in one point away from the exceptional curves. The three latter points are distinct: an equality between two would force two lines with a common endpoint to meet away from that endpoint. Consequently \Delta has ten rational components, each meeting the others in three points, and fifteen nodes in total. Figure 1 records these incidences; in the figure L_{ij} denotes the strict transform of the line joining p_i and p_j. Since each point belongs to three joining lines,

6.2 The cover and its local coordinates

Fix a prime n\geq3, to be chosen at the end. In an algebraic closure of K=\mathbb C(B) choose elements y_a satisfying

and let \pi:X\to B be the normalization of B in K'=K(y_1,\ldots,y_5). The base B is normal and Noetherian, and this finite characteristic-zero field extension is separable. Its normalization is therefore finite [19, Tag 032L], so X is an integral projective surface.

6.3 The Chern numbers

The local coordinates in Lemma 6.2 give both the canonical divisor formula and the degrees of the coverings of the branch strata. Thus no further description of X is needed to compute its Chern numbers. These calculations recover Hirzebruch’s formulas [12, Section 2.2, Equations (5) and (7)] for this six-line arrangement.

The Jacobian of the coordinate power map gives

Using (12) and the degree of \pi, we obtain

For the second Chern number, let e_c denote compactly supported topological Euler characteristic. The surface B has Euler characteristic 3+4=7. Removing the three nodes from each rational component of \Delta gives the one-branch stratum \Delta^\circ, whose Euler characteristic is 10(2-3)=-10. The node set has fifteen points. Additivity therefore gives

Over the stratum with s branches, the reduced inverse image is a finite étale cover of degree d/n^s. By additivity and multiplicativity of e_c, followed by Chern–Gauss–Bonnet [5, Section 1.3 and Proposition 1.6],

Subtracting twice (14) from (13) yields

6.4 An ample globally generated divisor

To compare (15) with the obstruction, we seek an integral divisor H satisfying nH\sim\pi^*(-K_B). Such a divisor has square 5d/n^2, as required by Proposition 6.1. We first construct H and sections that generate \mathcal O_X(H), and then prove ampleness separately. The ramification makes the pullbacks of all branch components divisible by n as integral divisors. Put

These are effective Cartier divisors because X is smooth. If line a joins p_i,p_j, write

For 1\leq a\leq5, the root ratios give

Since J_6=J, all six J_a are linearly equivalent to J. Let z_a be the resulting sections of \mathcal O_X(J) with zero divisors J_a.

These six sections have no common zero. Away from the exceptional curves, no point of B lies on all six joining lines. On an exceptional curve e_i, the total transform of any line not incident with p_i is disjoint from e_i. Thus the six total transforms have empty common intersection on B, and the same holds for their reduced pullbacks on X.

For an opposite pair a,b, its endpoints are the four distinct points, and hence

Dividing z_az_b by the canonical section of \mathcal O_X(E) gives a section of \mathcal O_X(2J-E) with zero divisor R_a+R_b. These three sections have no common zero. In fact the unions L_a\cup L_b belonging to different opposite pairs are disjoint on B: a line in one pair shares an endpoint with every line in either other pair, so their strict transforms are disjoint. We have therefore proved that both J and 2J-E are globally generated. Their sum

is globally generated as well.

To check ampleness, first note that 2\ell-\sum_i e_i is globally generated on B by the three opposite-pair conics: after removing their exceptional components, their zero divisors are precisely the three disjoint unions just considered. Thus this divisor is nef, as is \ell. The class \ell has positive intersection with every nonexceptional curve, and (2\ell-\sum_i e_i)\cdot e_j=1 for every exceptional curve. Consequently

has positive intersection with every integral curve. Its square is 5, so the surface Nakai–Moishezon criterion [11, Chapter V, Theorem 1.10] proves that -K_B is ample. Since

finite pullback and passage to a positive tensor root [19, Tags 0892 and 01PT] show that H is ample. The intersection formula now gives

Equations (15) and (17), with d=n^5, prove Proposition 6.1.

7 The counterexample