A ring is local when it has exactly one maximal ideal; write (R,m) and
k = R/m.  It is Noetherian when every ideal is finitely generated, and
complete when R maps isomorphically onto the inverse limit of the rings
R/m^j.

Take a nonzero finitely generated module M over a Noetherian local ring
(R,m).  An element x of m is M-regular when multiplication by x is
injective on M; a list x_1,...,x_t is M-regular when each x_i is regular
on M/(x_1,...,x_{i-1})M.  The depth of M is the largest length of an
M-regular sequence, and does not depend on choices.  The dimension of M
is dim(R/Ann M), the Krull dimension of its support.  Always
depth M <= dim M.

M is Cohen-Macaulay (CM) when depth M = dim M.  A small CM module over
R is a nonzero finitely generated M with depth M = dim R.  A regular
local ring is CM over itself, and if R is CM and x is a nonzerodivisor
then R/(x) is CM; so C[[x,y,z]]/(f), for any nonzero f, is a
two-dimensional CM ring.  Over a three-dimensional ring, a small CM
module is a nonzero finite module of depth three.

Hochster asked in the early 1970s whether every complete Noetherian
local domain has a small CM module.  In dimension at most two the
normalization of a complete local domain is a finite CM module.  In
three dimensions, finitely generated nonnegatively graded domains whose
degree-zero part is a perfect field of positive characteristic have
graded small CM modules, by Hartshorne, Peskine-Szpiro and Hochster.  In
general the answer is negative.

pdf: https://sites.lsa.umich.edu/hochster/wp-content/uploads/sites/1337/2024/08/DSlim2.pdf

Dropping finite generation, a big CM 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.  Hochster and Huneke proved
that the absolute integral closure of an excellent local domain of
positive characteristic is such an algebra.  Perfectoid methods gave big
CM algebras in mixed characteristic, completing existence for Noetherian
local rings.  None of these bounds finite generation, and none produces
an ordinary (finitely generated) witness.

pdf: https://www.jstor.org/stable/2946563
pdf: https://arxiv.org/pdf/1710.06160

For a normal local domain containing the rationals, trace makes the ring
a direct summand of every module-finite extension domain, so a CM finite
extension would force the ring itself to be CM.  Bhatt built
positive-characteristic complete normal local domains with no finite CM
extension, by a Witt-vector cohomological obstruction; this excludes
finite algebra extensions but not arbitrary finite modules.  Bhatt,
Hochster and Ma study asymptotic substitutes and record the existence
problem as open even for local rings at maximal ideals of affine
three-dimensional domains over algebraically closed fields.

pdf: https://arxiv.org/pdf/2410.18372

Fix a smooth integral projective surface X over C with an ample,
globally generated divisor H.  Let K_X be the canonical divisor and
c_2(X) the second Chern number.  Write T(X,H) for the graded ring, sum
over j >= 0 of H^0(X, O_X(jH)), and R(X,H) for its completion at the
irrelevant ideal T(X,H)_{>0}.  Two statements are proved.

Theorem A.  There is a three-dimensional complete Noetherian normal
local domain R over C, with residue field C, such that every nonzero
finitely generated R-module has depth at most two; in particular R has
no small CM module.

Theorem B.  If R(X,H) has a small CM module, then
K_X^2 - 2 c_2(X) <= 6 H^2.

Theorem A is Theorem B applied to one surface for which the inequality
fails.

The obstruction uses only that a finite module of depth three exists:
no grading, no algebra structure, no semistability, no Ulrich condition.
What replaces these is a comparison of coherent extensions across the
exceptional divisor of the completed cone.  Earlier obstructions cover
restricted witnesses: extension algebras (excluded by trace when the
ring is normal over the rationals), Ulrich sheaves, and graded modules.
Anghel's complete-quadrangle Hirzebruch-Kummer surfaces supply the
example; Anghel excludes arithmetically CM bundles when
K_X^2 - 2c_2(X) > 3H^2 and excludes graded maximal CM modules over these
section rings, while Chen proposes a local-cohomology bound when K_X is
proportional to H with ratio four.  The statement here needs no
proportionality, and the exponent-seven polarization has ratio five.

pdf: https://arxiv.org/pdf/2609.02718
pdf: https://arxiv.org/pdf/2609.15589
pdf: https://arxiv.org/pdf/2609.24142

A surface inequality K_X^2 - 2c_2(X) <= 6H^2 for Ulrich sheaves follows
from Bogomolov's inequality and the Hodge index theorem, and appears in
Beauville's lecture slides and in Anghel's first preprint.  On X an
Ulrich sheaf has trivial pushforward under a finite linear projection to
P^2, by Eisenbud-Schreyer.  Replacing that triviality by a comparison of
extensions across the exceptional divisor keeps the actual bundle on
the punctured cone instead of assuming a graded or Ulrich descent, and
that is what lets the argument reach arbitrary nongraded modules.

pdf: https://math.univ-cotedazur.fr/u/beauvill/conf/Ulrich.pdf
pdf: https://arxiv.org/pdf/math/0111040

Idea, in four sentences.  A small CM module over a completed cone is
free over the power series ring, hence a vector bundle on the punctured
cone; pushing it to the exceptional surface gives a torsion-free sheaf
P.  Two inequalities then fight over one number attached to P.
Triviality of the bundle at infinity forces that number nonnegative,
while narrow slope of the pieces of P forces it at most
1/2 - (K_X^2 - 2c_2(X))/(12H^2).  Comparing gives the bound.

The same object with numbers: for a coherent sheaf F of positive rank N
on the plane P^2 put v(F) = 2 ch_2(F)/N - (c_1(F)/N)^2, where c_1(F) is
the integer coefficient of the line class and ch_2(F) the degree of the
second Chern character.  On a direct sum of N line bundles
O(a_1),...,O(a_N), the number c_1(F)/N is the mean of the a_i and v(F)
is their variance.  Variance is nonnegative, and at most 1/4 when all
a_i lie in an interval of length one.  That is the whole mechanism: the
variance is nonnegative because it comes from a divisor, and it is small
because semistability bounds it.

Now turn any small CM module into a sheaf on the exceptional surface.
Let X, H be as in Theorem B and h = H^2.

Finite map.  Global generation gives sections s_0,s_1,s_2 of O_X(H) with
no common zero: take s_0 nonzero, then s_1 not vanishing on any component
of div(s_0), then s_2 nonzero on the finite set Z where s_0 and s_1
vanish together.  The ratio p |-> [s_0(p):s_1(p):s_2(p)] defines
f: X -> P^2 = D with f^*O_D(1) = O_X(H).  Since X is projective f is
proper; a positive-dimensional fiber would contain a curve C with
H.C = f^*O_D(1).C = 0, against ampleness, so f is finite.  For finite f
of degree e the pullback formula gives H^2 = (f^*O_D(1))^2 = e, so
deg f = h.

The ring.  Let S = C[x_0,x_1,x_2] with x_i acting as s_i, let
A = C[[x_0,x_1,x_2]], and let R = T(X,H) tensor over S with A.  Since f
is surjective, S -> T is injective.  The ring T is a domain: a rational
trivialization of H by a section tau sends a section of O_X(jH) to an
element of C(X) after division by tau^j, embedding T in C(X)[t,t^{-1}].
It is a finite graded S-module: T_j = H^0(D, F(j)) with F = f_*O_X
coherent, and a surjection O_D(-a)^p -> F together with vanishing of the
first cohomology of the kernel in large twists shows T_j lies in
S_{j-a} T_a, so the finitely many degrees up to a generate.  Then R is
the completion of T, a complete Noetherian ring, and (below) a normal
local domain with residue field C.

The model.  Let V_0 = Tot(O_D(-1)), the total space of O_D(-1), which is
the blowup of A^3 = Spec S at the origin, with exceptional divisor D.
Let W_0 = X x_D V_0 = Tot(O_X(-H)).  The map W_0 -> Spec T is an
isomorphism off the vertex: on X_i = {s_i != 0} clearing poles along
div(s_i) gives T[1/s_i] = O(X_i)[s_i, s_i^{-1}], the coordinate ring of
the punctured line bundle over X_i.  Base change along S -> A produces
V = V_0 x_S Spec A and W = W_0 x_S Spec A, with g: W -> V finite and
p: W -> Spec R proper, both isomorphisms off the closed point.  The zero
sections X in W and D in V are effective Cartier divisors with ideal
sheaves J = g^*I and I satisfying I|_D = O_D(1) and J|_X = O_X(H)
(the normal bundle of a blowup's exceptional divisor is O_D(-1); the
normal bundle of the zero section of Tot(O_X(-H)) is O_X(-H)).  Both V
and W are regular: at a closed point of W a local equation t of X is a
nonzerodivisor with regular quotient, so the local ring is regular, and
properness over the local base makes every component meet the integral
closed fiber, so both are integral.  Global functions: O(W) equals
O(W_0) tensor over S with A, which is R, by flat base change over a
finite affine cover.  A fraction of R integral over R is integral over
each local ring O_{W,w}, hence lies in R, so R is normal.  Therefore R
is a three-dimensional complete Noetherian normal local domain with
residue field C.

Module to sheaf.  Let M be a nonzero finitely generated R-module of
depth three.  The elements x_0,x_1,x_2 form an M-regular sequence, so M
has depth three over A, and by Auslander-Buchsbaum over the regular
local ring A, M is free over A, say A^N with N > 0.  For a prime q of R
other than m the localization inequality gives
depth M_q >= 3 - dim(R/q) >= dim R_q; by the model R_q is regular, so a
second Auslander-Buchsbaum makes M_q free, and M gives a vector bundle on
the punctured spectrum.  Put E = (p^* of the sheaf M)^{**}, coherent and
torsion-free on W, equal to M off X, and let P = E|_X.  A reflexive
sheaf on the regular three-dimensional W has depth at least two
everywhere, and the local equation of X is a nonzerodivisor on E, so P
is torsion-free of positive rank r (positive at the generic point of X,
a discrete valuation ring where E is free).  Finally g_*E is
torsion-free, a subsheaf of the function-field sheaf of V, and agrees
with O_V^N off D by M = A^N.  The result is a torsion-free sheaf E on W,
trivial of rank N away from D, whose restriction P = E|_X is
torsion-free of positive rank.