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A COMPLETE LOCAL DOMAIN WITH NO SMALL COHEN-MACAULAY MODULE
plain text edition, version 2
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Everything is over the complex numbers C. The background needed is:
rings, ideals, modules, localization, and the first definitions about
projective varieties. Everything else (depth, Cohen-Macaulay,
intersection numbers, Chern classes, Harder-Narasimhan filtrations) is
defined where it is first used, and the references at the end are
pointers, never a dependency of the proof.
CONTENTS
1. The question and the answer
2. The cone of a polarized surface
3. Balancing the slopes
4. The inequality forced by an ample divisor
5. The numerical obstruction
6. The surface
7. The counterexample
8. References
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1. THE QUESTION AND THE ANSWER
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1.1 Depth, dimension, Cohen-Macaulay
A ring is local if it has exactly one maximal ideal; write (R,m) and
k = R/m for the residue field. It is Noetherian if every ideal is
finitely generated, and complete if the natural map from R to the
inverse limit of R/m^n is an isomorphism.
Let M be a nonzero finitely generated module over a Noetherian local
ring (R,m). An element x in m is M-regular if "multiply by x" is an
injective map M -> M. A list x_1,...,x_t is an M-regular sequence if
each x_i is regular on M/(x_1,...,x_{i-1})M. Then
depth M := the maximal length of an M-regular sequence,
independent of all choices, and
dim M := dim(R/Ann(M)),
the Krull dimension of the support of M. Always depth M <= dim M.
M is Cohen-Macaulay (CM) if depth M = dim M. A small CM module over R
is a nonzero finitely generated module M with depth M = dim R. (The
word "small" contrasts with "big" CM modules, which may be infinitely
generated.)
Examples. A regular local ring is CM over itself; if R is CM and x in
m is a nonzerodivisor then R/(x) is CM. Hence C[[x,y,z]]/(f), for any
nonzero f, is a 2-dimensional CM ring. Over a 3-dimensional ring,
"small CM module" means "nonzero finite module of depth 3".
1.2 The question
Hochster asked in the early 1970s: does every complete Noetherian local
domain have a small CM module? Some answers are positive. In
dimension at most 2 the normalization of a complete local domain is a
finite CM module. For finitely generated graded domains of positive
characteristic there are graded small CM modules. In general the
answer is no.
Theorem A. There is a 3-dimensional complete Noetherian normal local
domain R containing C, with residue field C, such that every nonzero
finitely generated R-module M has depth M <= 2. In particular R has
no small CM module.
Theorem B. Let X be a smooth integral projective surface over C and
let H be an ample, globally generated divisor on X. Write K_X for the
canonical divisor of X, c_2(X) for the second Chern number of X, and
R(X,H) for the completed cone of the pair (X,H) (Section 2). If
R(X,H) has a small CM module then
K_X^2 - 2 c_2(X) <= 6 H^2. (1.1)
Theorem A is Theorem B applied to one surface for which (1.1) fails.
1.3 The idea in four sentences
A small CM module over a completed cone is a free module over the
power series ring, hence a vector bundle on the punctured cone. Push
that bundle onto the exceptional surface X: it becomes a torsion-free
sheaf P on X. Two inequalities then fight over one number attached
to P. The triviality of the bundle at infinity forces that number to
be >= 0 (Section 4), while semistability of the pieces of P forces it
to be <= 1/2 - (K_X^2 - 2c_2(X))/(12H^2) (Section 5); comparing the
two gives (1.1).
1.4 The same idea 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, (1.2)
where c_1(F) is the integer coefficient of the line class and ch_2(F)
is the degree of the second Chern character. If F is a direct sum of
N line bundles O(a_1),...,O(a_N), then c_1(F)/N is the average of the
numbers a_i and v(F) is exactly the variance of the numbers a_i.
Variance is never negative, and it is at most 1/4 when all the a_i lie
in an interval of length 1. That is the whole mechanism: the
variance is nonnegative because it comes from a divisor (Section 4),
and it is small because semistability bounds it (Section 5).
1.5 What is new here
Earlier obstructions concern restricted classes of witnesses. If R
contains Q and is normal, trace shows that a CM finite extension
algebra would force R itself to be CM, so an extension algebra is a
special kind of witness. Ulrich sheaves and graded modules are also
special. The argument below uses only that some finite module of
depth 3 exists: no grading, no algebra structure, no semistability and
no Ulrich condition on the module. What replaces those hypotheses is
a comparison of coherent extensions across the exceptional divisor of
the cone (Lemma 4.1).
1.6 Plan of the paper
Section 2 builds the completed cone R of (X,H), a regular model W of
Spec R, and turns a small CM module into a torsion-free sheaf E on W.
Section 3 modifies E along the exceptional surface so that the
Harder-Narasimhan slopes of its restriction P lie in an interval of
length at most 1. Section 4 proves that triviality of the pushforward
of E away from the exceptional divisor forces v(f_*P) >= 0. Section
5 combines that with Bogomolov's inequality for the semistable factors
and deduces (1.1). Section 6 constructs a surface with
K_X^2 - 2c_2(X) > 6H^2. Section 7 assembles Theorems A and B and the
uncompleted form.
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2. THE CONE OF A POLARIZED SURFACE
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Fix for the rest of the paper:
X : a smooth integral projective surface over C,
H : an ample divisor on X that is globally generated,
h : H^2, a positive integer,
D : the plane P^2.
2.1 A finite map to the plane
Lemma 2.1. There are sections s_0,s_1,s_2 of O_X(H) with no common
zero. They define a morphism f: X -> D with f^*O_D(1) = O_X(H); this
morphism is finite and surjective, and deg f = h.
Proof. Choose s_0 != 0. Its zero divisor has finitely many irreducible
components; choose s_1 not vanishing identically on any of them. Then
the zero divisors of s_0 and s_1 share no component, so their
intersection Z is finite. Choose s_2 not vanishing at any point of Z.
Now s_0,s_1,s_2 have no common zero, and the ratio
p |-> [s_0(p):s_1(p):s_2(p)] defines f.
X is projective, so f is proper. A fiber of positive dimension would
contain a curve contracted to a point, and then
H·C = (f^*O_D(1))·C = 0, contradicting ampleness of H. So f is finite.
For a finite map of degree e, the pullback formula for intersection
numbers gives
h = H^2 = (f^*O_D(1))^2 = e (O_D(1))^2 = e.
QED
2.2 The ring and the model
Let S = C[x_0,x_1,x_2], where x_i acts on sections as s_i, and let
T = T(X,H) = direct sum over j >= 0 of H^0(X, O_X(jH)),
a graded ring containing S. Let A = C[[x_0,x_1,x_2]] and let
n = T_{>0} be the irrelevant ideal. The completed cone of (X,H) is
R = R(X,H) = T (tensor over S) A,
which is also the completion of T at n; that identification is step (c)
of Proposition 2.3.
To resolve Spec R we use the following two schemes. Let V_0 be the
total space of O_D(-1), which is also the blowup of A^3 = Spec S at the
origin, with exceptional divisor D. Let W_0 = X x_D V_0, the total
space of O_X(-H). Base change along Spec S -> Spec A:
V = V_0 x_S Spec A, W = W_0 x_S Spec A.
Lemma 2.2. With this notation:
(i) W_0 -> Spec T is an isomorphism over the complement of the vertex,
and W_0 -> X is a line bundle;
(ii) g: W -> V is finite, p: W -> Spec R is proper, and p is an
isomorphism over the complement of the closed point of Spec R;
(iii) X in W and D in V are effective Cartier divisors with ideal
sheaves J and I satisfying
J = g^*I, I|_D = O_D(1), J|_X = O_X(H);
(iv) V and W are regular and integral;
(v) the ring of global functions on W is R.
Proof. (i) On X_i = {s_i != 0} the line bundle O_X(H) is trivialized by
s_i, and clearing poles along div(s_i) identifies
T[1/s_i] = O(X_i)[s_i, s_i^{-1}].
Both the punctured line bundle over X_i and the corresponding open part
of Spec T have this ring of functions, so W_0 -> Spec T is an
isomorphism over the complement of the vertex. Also the global
functions on the total space of a line bundle L are the sum of the
H^0 of all powers of L; for L = O_X(-H) this sum is T.
(ii) Spec T -> Spec S is finite, so W_0 -> V_0 is finite by base
change, and hence g: W -> V is finite. V -> Spec A is proper, being a
blowup, so W -> Spec A is proper; Spec R -> Spec A is separated, hence
p: W -> Spec R is proper. Over the punctured base, p and V -> Spec A
are the open immersions described in (i), by flat base change.
(iii) X in W_0 is the zero section of a line bundle, hence an effective
Cartier divisor, and base change preserves that. The ideal I of D in V
satisfies I|_D = O_D(1), because the normal bundle of the exceptional
divisor of a blowup is O_D(-1). Base change gives J = g^*I, and the
normal bundle of the zero section of the total space of O_X(-H) is
O_X(-H), so J|_X = O_X(H).
(iv) V is the blowup of A^3 at the origin, base changed to the
completion, hence regular. Let w be a closed point of W and let t be a
local equation of X at w. Then t is a nonzerodivisor on O_{W,w} and
O_{W,w}/(t) is the local ring of X at a closed point, which is regular.
Lifting generators of the quotient maximal ideal to O_{W,w} shows that
the embedding dimension of O_{W,w} is at most its dimension: there is
one more generator and, t being a nonzerodivisor, one more dimension.
Hence O_{W,w} is regular. Every point of W is a localization of such a
closed point: take a closed point in its closure, which exists because
closures meet the closed fiber. Localizations of regular local rings
are regular, so W is regular, and the same argument gives V regular.
A regular Noetherian scheme has disjoint irreducible components, and
every component of W meets the closed fiber of W -> Spec A, which is X
and is integral. So W is integral; the same holds for V with closed
fiber D.
(v) W_0 -> Spec S is separated and quasi-compact, so flat base change
applied to the Cech complex of a finite affine cover gives
O(W) = O(W_0) (tensor over S) A = T (tensor over S) A = R.
QED
2.3 The ring is a complete normal local domain
Proposition 2.3. R is a 3-dimensional complete Noetherian normal local
domain containing C, with residue field C, and R is a finite A-module.
Proof. (a) T is a domain. Trivialize O_X(H) rationally by a nonzero
rational section tau. Dividing a section of O_X(jH) by tau^j embeds
each graded piece in the field C(X) of rational functions, and the
multiplication is compatible, so T embeds in C(X)[t, t^{-1}], which is
a domain.
(b) T is a finite graded S-module. By the projection formula the
graded piece T_j is H^0(D, F(j)), where F = f_*O_X is a coherent sheaf
on D. Choose a surjection O_D(-a)^p -> F, with kernel K. For j >= a
the sequence
0 -> K(j) -> O_D(j-a)^p -> F(j) -> 0
has H^1(D, O_D(j-a)) = 0, so H^0(D, F(j)) is the image of H^0(D,
O_D(j-a))^p = (S_{j-a})^p. That is, T_j is contained in S_{j-a} T_a.
Hence T is generated as an S-module by the finite-dimensional space
T_0 + T_1 + ... + T_a.
(c) R is the completion of T at n. Let a = (x_0,x_1,x_2)T. If the
homogeneous S-module generators of T have degrees at most a', then
a^j is contained in T_{>= j}, T_{>= j+a'} is contained in a^j,
so the a-adic and n-adic filtrations are cofinal. Therefore
R = lim T/a^j = lim T/n^j = the product of the T_j for j >= 0,
with the Cauchy product as multiplication. In particular R is
complete. Moreover T_0 = H^0(X,O_X) = C and T_{>0} = n, so the
residue field of R is C.
(d) R is a domain, finite over A. Let u = sum u_j and v = sum v_j be
nonzero, with u_j and v_k the first nonzero homogeneous parts. The
first nonzero homogeneous part of uv is u_j v_k, which is nonzero
because T is a domain. So R is a domain. Since T is a finite
S-module, R = T (tensor over S) A is a finite A-module. The map
S -> T is injective because f is surjective, and A is flat over S, so
A -> R is injective. Hence dim R = dim A = 3, and R is Noetherian
because it is finite over the Noetherian ring A.
(e) R is normal. By Lemma 2.2, W is regular and integral, so W is
normal, and O(W) = R. If a fraction of R is integral over R then it is
integral over every local ring O_{W,w}, hence lies in every O_{W,w},
hence lies in R.
QED
2.4 From a module to a sheaf on the model
Proposition 2.4. Let M be a nonzero finitely generated R-module of
depth 3. Then:
(a) M is a free A-module of some rank N > 0;
(b) M is a free R_q-module for every prime q of R different from m;
(c) the sheaf E = (p^*(M-tilde))^{**} is coherent and torsion-free on
W, agrees with M away from X, and its restriction P = E|_X is
torsion-free of positive rank r;
(d) g_*E is torsion-free and agrees with O_V^N away from D.
Proof. (a) M has depth 3 over R, so it is Cohen-Macaulay over R of
dimension 3, hence over A. A system of parameters of A is therefore an
M-regular sequence, in particular x_0,x_1,x_2 is. By the
Auslander-Buchsbaum formula, projective dimension plus depth equals the
depth of the ring, so M has projective dimension 0 over the regular
ring A: M is free.
(b) Let q be a prime of R other than m. The localization inequality
gives depth_{R_q} M_q >= depth_R M - dim(R/q), and since depth_R M = 3
and dim R = 3 this is at least 3 - dim(R/q). Concatenating a chain of
primes below q with one above q gives dim R >= dim(R/q) + dim R_q, so
3 - dim(R/q) >= dim R_q. Thus depth_{R_q} M_q >= dim R_q. By Lemma
2.2 the local ring R_q is regular, so another application of
Auslander-Buchsbaum shows that M_q is free.
(c) By (b), M-tilde is a vector bundle on the punctured spectrum of
Spec R, which p identifies with W - X. So p^*(M-tilde) is a vector
bundle there and E is reflexive, agreeing with M away from X. On a
regular integral scheme of dimension 3 a reflexive sheaf has depth at
least 2 at every point: dualizing a finite presentation writes the
sheaf as the kernel of a map between locally free sheaves, and the
depth lemma applies.
Let x be a point of X with dim O_{W,x} >= 2. The local equation t of X
is a nonzerodivisor on E because E is torsion-free, so depth P_x >= 2-1
= 1; that is, P is torsion-free there. At the generic point of X the
ring O_{W,x} is a discrete valuation ring, E_x is free of some positive
rank r, and t is again a nonzerodivisor, so P_x is free of rank r.
Hence P is torsion-free of rank r > 0.
(d) g_*E is a subsheaf of the constant sheaf of the function field of
V, so it is torsion-free. Off D, g is an isomorphism onto V - D, and E
is trivial there of rank N by (a), so g_*E agrees with O_V^N away from
D.
QED
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3. BALANCING THE SLOPES
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3.1 Slopes, semistability, Harder-Narasimhan
Let B be a nonzero torsion-free coherent sheaf on X. Its normalized
slope is
mu(B) = (c_1(B) · H) / (rank(B) * h). (3.1)
Because h = H^2, twisting by O_X(H) raises every slope by exactly 1.
We say B is slope-semistable if no nonzero subsheaf B' of smaller rank
has mu(B') > mu(B).
Every nonzero torsion-free B has a Harder-Narasimhan filtration
0 = B_0 subset B_1 subset ... subset B_m = B
whose quotients G_i = B_i/B_{i-1} are torsion-free and semistable and
whose slopes strictly decrease. Here is the construction, since we use
it below. A torsion-free sheaf embeds in a sum of sufficiently positive
line bundles, so by taking determinants the slopes of all subsheaves are
bounded above. The denominators are bounded because the ranks are
bounded. Choose a subsheaf of maximal slope and, among those, one of
maximal rank, and saturate it. Saturating does not lower the degree,
because the quotient is torsion and a torsion sheaf on a surface has an
effective divisorial first Chern class. The chosen subsheaf is
semistable, and every subsheaf of its torsion-free quotient has strictly
smaller slope, so induction on the rank produces the filtration.
3.2 The modification lemma
Lemma 3.1. Let W be an integral Noetherian scheme and let X in W be an
effective Cartier divisor that is a smooth integral projective surface
over C, with ideal J satisfying J|_X = O_X(H) for an ample divisor H.
Let E be a coherent torsion-free sheaf on W whose restriction
P = E|_X is torsion-free of positive rank r. Then there is a coherent
torsion-free sheaf E' on W, agreeing with E away from X, such that
P' = E'|_X is torsion-free and the slopes of its Harder-Narasimhan
factors lie in an interval of length at most 1.
Proof. Suppose the largest and smallest slopes in the filtration of P
differ by more than 1. Let G be the last factor, so G is a semistable
quotient of P of minimal slope, and let B be the kernel of P -> G.
Set E' = ker(E -> i_*G), where i: X -> W is the inclusion. It is a
subsheaf of the torsion-free sheaf E, hence torsion-free, and it equals
E away from X because i_*G is supported on X. Restricting the exact
sequence
0 -> E' -> E -> i_*G -> 0
to the Cartier divisor X is right exact, and Tor_1^W(E, O_X) = 0 because
the local equation of X is a nonzerodivisor on E. So there is an exact
sequence
0 -> Tor_1^W(i_*G, O_X) -> E'|_X -> P -> G -> 0,
and Tor_1^W(i_*G, O_X) = G (tensor) O_X(J|_X) = G(H). Hence
0 -> G(H) -> P' -> B -> 0. (3.2)
So P' is an extension of torsion-free sheaves, hence torsion-free, and
rank P' = rank P = r.
Two monotonicity statements. First, the maximal slope of an extension
of torsion-free sheaves is at most the larger of the two maximal slopes:
if S is a subsheaf of the extension, intersect it with the kernel and
take its image in the quotient, and mu(S) is a weighted average of the
two resulting slopes. Now mu(G(H)) = mu(G) + 1 < mu(G_1), where G_1 is
the first factor, because the interval has length more than 1. Hence
mu_max(P') <= mu_max(P). (3.3)
Second, additivity of rank and first Chern class in (3.2) gives
mu(P') = mu(P) + rank(G)/r >= mu(P) + 1/r. (3.4)
The average slope never exceeds the maximal slope, and mu_max is
non-increasing by (3.3) while the average climbs by at least 1/r per
step by (3.4). So the procedure stops after finitely many steps. At
that point the interval of slopes has length at most 1. QED
Apply Lemma 3.1 to the sheaf E of Proposition 2.4, and keep the names E
and P for the results. The rank r and the bundle away from X are
unchanged, so g_*E is still trivial of rank N away from D.
The point of the lemma is not semistability of P but the width of its
slope interval. That width is what makes the weighted variance in
Section 5 at most 1/4, and it is available because modifying along X
costs nothing on the punctured cone.