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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.
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4. THE INEQUALITY FORCED BY AN AMPLE DIVISOR
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4.1 What the lemma says and why
Let V be an integral Noetherian scheme with an effective Cartier
divisor D isomorphic to the plane P^2, whose ideal I satisfies
I|_D = O_D(1). Lemma 4.1 says that a torsion-free sheaf on V that is a
trivial bundle of rank N away from D cannot behave arbitrarily along
D: the number v of (1.2) is nonnegative.
The mechanism is finite generation. Sections of a trivial bundle,
cleared by powers of the equation of D, sit between I^b L and I^{-b} L
for L = O_V^N. The quotient is then filtered by sheaves supported on
D whose successive layers are twisted by O_D(j), and those twists can
only push the variance up.
4.2 The lemma
Lemma 4.1. Let V be an integral Noetherian scheme with an effective
Cartier divisor D isomorphic to P^2, whose ideal I satisfies
I|_D = O_D(1). Let L' be a coherent torsion-free sheaf on V with
L' restricted to V - D trivial of rank N > 0. Then F = L'|_D has rank
N and v(F) >= 0.
Proof. Step 1: a lattice. Since D is Cartier, the local ring of V at
the generic point of D is a discrete valuation ring, and the
torsion-free sheaf L' is free of rank N there; hence F has rank N and
v(F) is defined. Let L = O_V^N and identify L with L' after inverting
the equation t of D. On a finite affine cover, clearing denominators
gives
I^b L subset L' subset I^{-b} L
for some b >= 0. Replacing L' by L' (tensor) I^b replaces F by F(b)
and does not change v, because twisting by O_D(b) sends c_1 to
c_1 + Nb and ch_2 to ch_2 + b c_1 + N b^2/2. So we may assume
I^{2b} L subset L' subset L. (4.1)
Step 2: the quotient. Let Q = L/L'. By (4.1) it is killed by I^{2b}.
Filter Q by Q_j = I^j Q / I^{j+1} Q for j >= 0. Each Q_j is a coherent
sheaf on D and, since I^j/I^{j+1} = O_D(j), it is a quotient of
O_D(j)^N. Let n_j = rank Q_j. Then
0 <= n_j <= N and c_1(Q_j) >= j n_j. (4.2)
For the second inequality: Q_j(-j) is a quotient of O_D^N, hence
globally generated, so the determinant of its torsion-free quotient is
a globally generated line bundle, of degree at least 0, while its
torsion part contributes an effective divisorial class.
Step 3: K-theory. For a coherent sheaf G on V let
r(G) = [G|_D] - [Tor_1^V(G, O_D)]
in the Grothendieck group of coherent sheaves on D. Derived
restriction shows that r is additive on short exact sequences, and
there is no higher Tor against a Cartier divisor. For G = L we get
r(L) = [O_D^N], since Tor_1 of a locally free sheaf vanishes. For
G = L' we get r(L') = [F], since L' is torsion-free, so t is a
nonzerodivisor and Tor_1 vanishes. For G = Q, which is supported on D,
we get
r(Q) = [Q] - [Q (tensor) I] = - sum over j of ([Q_j(1)] - [Q_j]).
Additivity applied to 0 -> L' -> L -> Q -> 0 gives
[F] = [O_D^N] + sum over j of ([Q_j(1)] - [Q_j]). (4.3)
Taking Chern characters in (4.3), and using that
ch(Q_j(1)) - ch(Q_j) has degree-1 part n_j and degree-2 part
c_1(Q_j) + n_j/2, we get
rank F = N, c_1(F) = sum_j n_j,
2 ch_2(F) = sum_j (2 c_1(Q_j) + n_j) >= sum_j (2j+1) n_j, (4.4)
the last step by (4.2).
Step 4: an elementary inequality. With 0 <= n_j <= N,
N sum_j (2j+1) n_j - (sum_j n_j)^2
= sum_j n_j (N - n_j) + 2 sum_{j > l} n_j (N - n_l) >= 0. (4.5)
The identity follows from sum_j (2j+1) n_j = sum_j n_j + 2 sum_j j n_j
together with sum_{j > l} n_j = sum_j j n_j, and the right hand side is
nonnegative because 0 <= n_j <= N. Combining (4.4) and (4.5),
N * 2 ch_2(F) - c_1(F)^2 >= 0,
that is, v(F) >= 0. QED
4.3 The consequence for the cone
Corollary 4.2. In the situation of Sections 2 and 3,
v(f_*P) >= 0.
Proof. Take V and D as in Lemma 2.2, and L' = g_*E. This L' is
torsion-free, and by Proposition 2.4(d) it is trivial of rank N away
from D. By Lemma 2.2(iii) and the exactness of finite pushforward,
L'|_D = f_*(E|_X) = f_*P.
So Lemma 4.1 applies and gives v(f_*P) >= 0. The integer N of the
lemma is the rank of f_*P, which is (rank P)(deg f) = r h. QED
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5. THE NUMERICAL OBSTRUCTION
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5.1 One semistable factor
Lemma 5.1. Let f: X -> D = P^2 be finite, with X a smooth integral
projective complex surface and H = f^*O_D(1) ample, h = H^2. Let G be
a nonzero torsion-free slope-semistable sheaf on X of rank u, and put
k = K_X * H. Then
v(f_*G) <= 1/4 - (K_X^2 - 2 c_2(X))/(12 h) (5.1)
and
c_1(f_*G)/rank(f_*G) = mu(G) - k/(2h) - 3/2. (5.2)
Proof. Normalize
m = c_1(f_*G)/(u h), q = ch_2(f_*G)/(u h),
so that rank f_*G = u h and v(f_*G) = 2q - m^2.
Riemann-Roch. For a coherent sheaf F on P^2 of rank N, first Chern
number c and second Chern character ch_2, the Euler characteristic of a
twist is
chi(F(t)) = ch_2 + t c + N t^2/2 + (3/2)(c + N t) + N,
because the Todd class of P^2 is 1 + (3/2)H + H^2. For a sheaf G on X,
chi(G(tH)) = ch_2(G) + t (c_1(G)*H) + u t^2 h/2
- (1/2)(c_1(G) + u t H)*K_X
+ u (K_X^2 + c_2(X))/12.
Finite pushforward and the projection formula identify chi(F(t)) with
chi(G(tH)) for F = f_*G. Comparing the coefficients of t gives (5.2).
Comparing the constant terms gives
1 + (3/2) m + q = chi(O_X)/h + ch_2(G)/(u h) - c_1(G)*K_X/(2 u h).
Put s = mu(G) - k/(2h), so that m = s - 3/2 and m^2 + 3m = s^2 - 9/4.
Then
2q - m^2 = 1/4 + 2 chi(O_X)/h
+ (2 ch_2(G)/u - c_1(G)*K_X/u)/h
- s^2. (5.3)
Bogomolov. For a torsion-free slope-semistable sheaf on a smooth
projective surface, with respect to any ample polarization,
2 (rank G) ch_2(G) <= c_1(G)^2. (5.4)
In the traditional form this is the inequality
2u c_2(G) >= (u - 1) c_1(G)^2; divide by u^2 to get (5.4). Set
C = c_1(G)/u - K_X/2,
a rational divisor class. By (5.4),
2 ch_2(G)/u - c_1(G)*K_X/u <= c_1(G)^2/u^2 - c_1(G)*K_X/u = C^2 - K_X^2/4,
and meanwhile s = (C * H)/h. So the last two terms of (5.3) are at most
(C^2 - K_X^2/4)/h - (C*H)^2/h^2 <= -K_X^2/(4h), (5.5)
by the Hodge index theorem, which says (C*H)^2 >= C^2 H^2 for every
rational divisor class C and every ample H. Therefore
2q - m^2 <= 1/4 + (2 chi(O_X) - K_X^2/4)/h.
Finally Noether's formula 12 chi(O_X) = K_X^2 + c_2(X) gives
2 chi(O_X) - K_X^2/4 = (K_X^2 + c_2(X))/6 - K_X^2/4
= -(K_X^2 - 2 c_2(X))/12,
which is (5.1). QED
5.2 Proof of Theorem B
Assume that R = R(X,H) has a small CM module. By Proposition 2.4 and
Lemma 3.1 there is a torsion-free sheaf P on X of positive rank r such
that
- P = E|_X, where E is torsion-free on W and g_*E is trivial of rank
N off D, so v(f_*P) >= 0 by Corollary 4.2;
- the Harder-Narasimhan factors G_1,...,G_m of P have slopes in an
interval of length at most 1.
Write u_i = rank G_i and w_i = u_i/r, so w_i > 0 and the w_i sum to 1.
Let F = f_*P. Finite pushforward is exact, so ch(F) is the sum of the
ch(f_*G_i); hence rank F = r h = N and
c_1(F)/N = sum_i w_i m_i, 2 ch_2(F)/N = sum_i w_i (2 q_i),
where m_i = c_1(f_*G_i)/(u_i h) and q_i = ch_2(f_*G_i)/(u_i h).
Therefore
v(F) = sum_i w_i v(f_*G_i) + sum_i w_i (m_i - mbar)^2, (5.6)
mbar = sum_i w_i m_i,
which is the identity "variance equals the mean of the variances plus
the variance of the means".
By (5.2) the numbers m_i differ from the mu(G_i) by one common
constant, so they too lie in an interval of length at most 1. A
distribution supported in an interval of length 1 has variance at most
1/4: the mean minimizes the mean square distance, and every point of the
interval is within 1/2 of its midpoint. Lemma 5.1 bounds each
v(f_*G_i) by 1/4 - (K_X^2 - 2c_2(X))/(12h). Putting these bounds into
(5.6),
v(F) <= 1/2 - (K_X^2 - 2 c_2(X))/(12h). (5.7)
Corollary 4.2 says v(F) >= 0. Comparing with (5.7) gives
K_X^2 - 2 c_2(X) <= 6h = 6 H^2.
QED
================================================================================
6. THE SURFACE
================================================================================
6.1 The branch divisor
Fix a prime n >= 3 and four points p_1,...,p_4 of the plane with no
three collinear. Let B be the blowup of P^2 at these four points, with
exceptional curves e_1,...,e_4, and let l be the class of a line. Then
l^2 = 1, l*e_i = 0, e_i*e_j = -delta_ij,
K_B = -3l + e_1 + e_2 + e_3 + e_4.
Let L_1,...,L_6 be the strict transforms of the six lines p_i p_j, and
let l_1,...,l_6 be linear forms on the plane defining them. Write T_a
for the total transform of the a-th line, so T_a = L_a + e_i + e_j when
the a-th line joins p_i and p_j, and T_a ~ l. Two of the six lines are
opposite if their pairs of endpoints are disjoint; there are three
opposite pairs. Put
Delta = L_1 + L_2 + L_3 + L_4 + L_5 + L_6 + e_1 + e_2 + e_3 + e_4.
Lemma 6.1. Delta is a reduced divisor with simple normal crossings on
B. It has 10 components, each meeting the others in 3 points, and 15
nodes in all. In the divisor class group, Delta = -2 K_B. Also
K_B^2 = 5.
Proof. Every component is a smooth rational curve. Two strict
transforms L_a and L_b meet exactly when the two lines share an
endpoint, in which case they meet on the corresponding exceptional
curve and nowhere else, or when the lines are opposite, in which case
they meet transversely at one point lying on no e_i. Each e_i meets
exactly the three lines through p_i, transversely, in three distinct
points. So every component meets exactly three others, all crossings
are transversal, and there are 4*3 + 3 = 15 of them. Also each point
p_i lies on exactly 3 of the six lines, so
sum_a L_a = 6l - 3(e_1 + e_2 + e_3 + e_4),
hence Delta = 6l - 2(e_1 + e_2 + e_3 + e_4). But -2K_B equals
6l - 2(e_1 + e_2 + e_3 + e_4). Finally
K_B^2 = (-3l)^2 + e_1^2 + e_2^2 + e_3^2 + e_4^2 = 9 - 4 = 5.
QED
6.2 The cover and its local structure
Work over K = C(B). Choose y_1,...,y_5 in an algebraic closure of K
with
y_a^n = l_a / l_6 (1 <= a <= 5),
and let K' = K(y_1,...,y_5). Let pi: X -> B be the normalization of B
in K'.
Lemma 6.2. X is a smooth integral projective surface and deg pi = n^5.
The map pi is unramified outside Delta and has ramification index n
along every component of Delta. At a point where exactly s components
of Delta meet, s = 0,1,2, the map pi is, after an etale base change and
up to disjoint copies, the standard map
(t_1,t_2) |-> (u_1,u_2) = (t_1^n, t_2) (s = 1),
(t_1,t_2) |-> (u_1,u_2) = (t_1^n, t_2^n) (s = 2),
and the identity for s = 0; the number of copies is n^{5-s}.
Proof. Degree. A ratio l_a/l_6 is a rational function on B whose
divisor is T_a - T_6. Taking valuations along L_1,...,L_5 shows that if
a product of powers of the five ratios is an n-th power in K then all
exponents are divisible by n, and then the valuation along L_6 gives the
same for the last exponent. So the five classes of the ratios in
K^*/K^{*n} generate a subgroup of order n^5. Since C contains the n-th
roots of unity, Kummer theory applies and shows that K'/K is Galois
with group of order n^5, that is, [K':K] = n^5 =: d.
X is integral and projective: it is the normalization of an integral
projective surface in a finite separable extension of function fields.
Valuations. For a curve C contained in Delta record the vector
v(C) = (v_C(y_1), ..., v_C(y_5)) in F_n^5.
Set e_6 = -(e_1 + e_2 + e_3 + e_4 + e_5). Because
div(y_a) = (1/n) pi^*(T_a - T_6), the coefficient of L_b in div(y_a) is
delta_ab, so the six lines correspond to the six vectors e_1,...,e_6.
For the exceptional curve e_i put
w_i = sum of e_a over the three lines a through p_i.
Then v(e_i) = w_i: the coefficient of e_i in div(y_a) is 1 when the line
a passes through p_i and 0 otherwise, except that the term T_6 subtracts
the same constant from all five coordinates, that is, adds a multiple of
e_1 + e_2 + e_3 + e_4 + e_5 = -e_6.
Independence at the nodes. At a node of two strict transforms the two
vectors are e_a and e_b with a != b, and these are independent.
At a node L_a and e_i, with a one of the three lines through p_i, we
must check that e_a and w_i are independent. Write w_i in coordinates.
If line 6 does not pass through p_i, then w_i has coordinate 1 exactly
at the three lines through p_i, so two of those coordinates are not the
coordinate a, and lambda e_a + mu w_i = 0 forces mu = 0 there, hence
lambda = 0. If line 6 does pass through p_i, then w_i equals minus the
sum of the three lines not through p_i, so its coordinates are 0 on the
three lines through p_i and -1 elsewhere; there are two lines b <= 5 not
through p_i, and at such a coordinate lambda e_a + mu w_i = 0 reads
-mu = 0, so mu = 0 and then lambda = 0. So the vectors at every node
are independent.
Local normal form. Let x be a closed point of B lying on s components
of Delta, and choose regular parameters u_1,u_2 with those components
given by u_1 = 0, ..., u_s = 0. Each ratio l_a/l_6 is, up to a unit, a
monomial in u_1,...,u_s. Adjoining n-th roots of the units is an etale
base change, and after dividing the y_a by the resulting roots we may
assume all radicands are monomials. Their exponent vectors modulo n are
exactly the vectors of the s branches through x, which are independent
by the previous paragraph. An invertible linear change of the five root
generators, with entries in Z/n, is legitimate because it is invertible
modulo n; lifting the matrix to Z realises it by multiplying the y_a by
monomials in the u_j. So the five equations can be brought to
Y_j^n = u_j (1 <= j <= s), Y_j^n = 1 (s < j <= 5).
The first s equations generate a field of degree n^s over the residue
field of the local ring, by the valuations along u_j = 0, and the
remaining equations split completely. Hence the generic algebra is a
product of n^{5-s} copies of that field. Over the completed local ring,
the normalization in one factor is
C[[t_1,...,t_s,v]] with u_j = t_j^n,
a power series ring, hence regular and normal, containing the completed
local ring, with the right fraction field: it is the integral closure.
The full normalization is the product of n^{5-s} copies. Every point of
B lies on at most two components of Delta, so this covers all cases. It
shows that X is smooth and that the stated local models hold. QED
6.3 The Chern numbers
Lemma 6.3. With d = n^5,
K_X = (1 - 2/n) pi^*(-K_B), K_X^2 = 5d (1 - 2/n)^2,
c_2(X) = d (2 - 10/n + 15/n^2),
K_X^2 - 2 c_2(X) = d (1 - 10/n^2) = n^3 (n^2 - 10).
Proof. Canonical divisor. At a smooth point of Delta the map pi is
(t,v) |-> (t^n, v) in suitable coordinates, so the pullback of a local
holomorphic 2-form is t^{n-1} times a unit times the form downstairs.
The zero divisor of that pullback is (n-1) times the reduced inverse
image of Delta. Writing D' for that reduced inverse image, so that
pi^*Delta = n D', we get
K_X = pi^*(K_B) + (n-1) D' = pi^*(K_B + (1 - 1/n) Delta).
Since Delta = -2K_B, this becomes K_X = pi^*((1 - 2/n)(-K_B)). Taking
squares and using pi^*(-K_B)^2 = d (-K_B)^2 = 5d,
K_X^2 = 5d (1 - 2/n)^2.
Euler number. For a smooth complex surface c_2 equals the topological
Euler characteristic. The surface B has Euler characteristic 3 + 4 = 7.
The smooth part of Delta is 10 copies of P^1 minus 3 points, of Euler
characteristic 10(2-3) = -10, and the 15 nodes contribute 15. By
additivity,
e(B - Delta) = 7 - (-10) - 15 = 2.
Over the stratum where exactly s branches meet, the reduced inverse
image is an etale cover of degree d/n^s. Additivity and
multiplicativity of the Euler characteristic give
e(X) = 2d + (-10)(d/n) + 15(d/n^2) = d (2 - 10/n + 15/n^2).
Finally, substituting d = n^5,
5n^5 (1 - 2/n)^2 - 2n^5 (2 - 10/n + 15/n^2)
= (5n^5 - 20n^4 + 20n^3) - (4n^5 - 20n^4 + 30n^3)
= n^5 - 10n^3 = n^3 (n^2 - 10).
QED
6.4 An ample, globally generated divisor
We need an integral divisor H with nH ~ pi^*(-K_B). Since pi is
ramified with index n along every component of Delta, the pullback of
each such component is divisible by n as an integral divisor. Put
R_a = (1/n) pi^*L_a, F_i = (1/n) pi^*e_i,
E = F_1 + F_2 + F_3 + F_4, J = (1/n) pi^*T_6,
and, for the a-th line joining p_i to p_j, J_a = R_a + F_i + F_j.
These are integral divisors because X is smooth. By construction
div(y_a) = J_a - J, so all six divisors J_a are linearly equivalent;
let z_a be the corresponding section of O_X(J), with div(z_a) = J_a.
(a) J is globally generated. The six total transforms T_a have empty
common intersection on B: away from the exceptional curves no point of
the plane lies on all six joining lines, and on e_i the three lines not
passing through p_i miss e_i entirely. Hence z_1,...,z_6 have no
common zero.
(b) 2J - E is globally generated. For an opposite pair a,b the four
endpoints are the four points, so J_a + J_b = R_a + R_b + E. Dividing
z_a z_b by the canonical section of O_X(E) gives a section of
O_X(2J - E) whose zero divisor is R_a + R_b. The three unions
R_a + R_b obtained from the three opposite pairs are pairwise disjoint,
so these three sections have no common zero.
(c) H = 3J - E is globally generated, being the sum of the globally
generated divisors J and 2J - E.
(d) H is ample. On B the divisor 2l - sum_i e_i is globally generated:
its sections are the conics through the four points, whose degenerate
members are the three unions of opposite lines. So 2l - sum_i e_i and l
are nef. Their sum -K_B = l + (2l - sum_i e_i) meets every integral
curve positively: on e_j we have l*e_j = 0 and
(2l - sum_i e_i)*e_j = 1, and on a nonexceptional curve C we have
l*C > 0 and (2l - sum_i e_i)*C >= 0. Also (-K_B)^2 = 5 > 0. So the
Nakai-Moishezon criterion makes -K_B ample.
Finally nH ~ pi^*(3l - sum_i e_i) = pi^*(-K_B) is the pullback of an
ample class, so nH is ample. Amplitudeness is numerical, so H itself is
ample as well. And
H^2 = (5d)/n^2 = 5n^3.
6.5 The invariants
Proposition 6.4. For every prime n >= 3 there is a smooth integral
projective surface X over C with an ample, globally generated divisor H
such that
H^2 = 5 n^3, K_X^2 - 2 c_2(X) = n^3 (n^2 - 10).
Proof. Take X and H from 6.1 to 6.4 and d = n^5. Lemma 6.3 gives the
Chern numbers, and 6.4(d) gives H^2 = 5d/n^2 = 5n^3. QED
================================================================================
7. THE COUNTEREXAMPLE
================================================================================
7.1 Theorem B gives Theorem A
Take n = 7 in Proposition 6.4. Then
H^2 = 5 * 7^3 = 1715,
6 H^2 = 10290,
K_X^2 - 2 c_2(X) = 7^3 (49 - 10) = 343 * 39 = 13377.
Since 13377 > 10290, inequality (1.1) fails for R = R(X,H). By Theorem
B this ring has no small CM module. By Proposition 2.3 it is a
3-dimensional complete Noetherian normal local domain containing C with
residue field C. A small CM module over a 3-dimensional ring is exactly
a nonzero finite module of depth 3, so R has no nonzero finite module of
depth 3. This is Theorem A.
The arithmetic is worth checking by hand once: 7^3 = 343, five times it
is 1715, twice 1715 is 3430 and three times 3430 is 10290, and 343 times
39 is 343*40 - 343 = 13720 - 343 = 13377.
7.2 The uncompleted ring
The same surface also gives a counterexample among localizations of
finitely generated C-algebras, without completing.
Corollary 7.1. Let X and H be as in Proposition 6.4 with n = 7, put
T = T(X,H) and n = T_{>0}. Then T is a finitely generated graded
3-dimensional C-domain whose vertex ideal n is maximal with T/n = C,
and the local domain T_n has no small CM module.
Proof. By Proposition 2.3, T is a domain finite over S, hence a
finitely generated 3-dimensional C-algebra, and T_0 = C, so n is maximal
with residue field C. Put R_0 = T_n. By construction the completion of
R_0 at its maximal ideal is the ring R = R(X,H) of Theorem A. Completion
preserves the dimension, so dim R_0 = 3.
Suppose that M is a nonzero finite R_0-module of depth 3, and let
z_1,z_2,z_3 be an M-regular sequence in the maximal ideal of R_0. Put
M_0 = M and M_i = M/(z_1,...,z_i)M for i = 1,2,3. Each z_i lies in the
maximal ideal, so by Nakayama M_i is nonzero: M_{i+1} is M_i/z_iM_i, and
this vanishes only if M_i vanishes, since M_i is finite over the local
ring R_0.
The completion map R_0 -> R is faithfully flat, and the completion of
the finite module M is M-hat = M (tensor over R_0) R, which is a nonzero
finite R-module. For i = 0,1,2 the sequence
0 -> M_i -> M_i -> M_{i+1} -> 0,
where the first map is multiplication by z_{i+1}, remains exact after
tensoring with R, since R is flat. Thus the images of z_1,z_2,z_3 lie in
the maximal ideal of R and form an M-hat-regular sequence of length 3.
So depth M-hat >= 3.
For a nonzero finite module over a Noetherian local ring, the depth is at
most the dimension of the support, which here is at most dim R = 3. So
depth M-hat = 3, contradicting Theorem A for R. QED
7.3 What the argument does and does not use
Nothing in Sections 2 to 5 uses that the surface comes from line
arrangements; any smooth integral projective surface with an ample,
globally generated H is allowed. Nothing uses a grading on the module,
an algebra structure, or any semistability of the module itself; only
the factor slopes of its restriction are used, and Lemma 3.1 arranges
those. What the proof does use, essentially, is that a finite module of
depth 3 over the cone is free over the power series ring, which lets a
comparison with a trivial bundle across the exceptional divisor produce
the nonnegativity in Section 4.
================================================================================
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