A NOTE ON THE NON-EXISTENCE OF SMALL COHEN-MACAULAY ALGEBRAS

arXiv:1207.5413v1 [] 23 Jul 2012

BHARGAV BHATT
A BSTRACT. By finding a p-adic obstruction, we construct many examples of complete noetherian local normal Fp -algebras
R such that no module-finite extension R ֒→ S is Cohen-Macaulay. These examples should be contrasted with a result of
Hochster-Huneke: the directed union of all such extensions is always Cohen-Macaulay.

1. I NTRODUCTION
A noetherian ring is called Cohen-Macaulay (CM) if there are no non-trivial relations between elements of a system
of parameters. These rings form an exceptionally well behaved class: local cohomology vanishes when it can, and
duality works out beautifully. This note studies rings realisable as subrings of CM rings:
Definition 1.1. A noetherian ring R admits a big CM algebra if there is an injective map R ֒→ S of rings with S CM.
If the ring S can be chosen to be finitely generated as an R-module, then we say that R admits a small CM algebra.
Under mild hypotheses, a fundamental result of Hochster and Huneke [HH92] shows that any ring R that contains
a field admits a big CM algebra; in fact, if R is an Fp -algebra, the absolute integral closure of R does the job. Thus,
one asks: does a ring R always admit a small CM algebra? The answer is “no” in characteristic 0 based on a local
cohomological obstruction1. In characteristic p, however, the results of loc. cit. immediately nullify any coherent
cohomological obstructions: any non-trivial relation can be trivialised after a finite extension. In fact, by [HL07],
there is a single finite extension R ֒→ S trivialising all unwanted relations. Nevertheless, our goal in this note is to
discuss a negative answer to the above question in characteristic p using a p-adic (rigid) cohomological obstruction.
b to be completion at the origin of
Theorem 1.2. Let (A, L) be polarised projective variety over a perfect field k; set R
0
n
i
b
⊕n≥0 H (A, L ). Assume H (A)<1 6= 0 for some 0 < i < dim(A). Then R does not admit a small CM algebra.
rig

The hypothesis on A is satisfied, for example, if A is an abelian surface. This theorem generalises a calculation of
Sannai-Singh [SS12, Example 5.3] (who showed the non-existence of a graded small CM algebra for a specific A).
This theorem should also be contrasted with a result of Hartshorne: if A is CM, then R admits a small CM module,
i.e., a module whose depth is dim(A) (see [Hoc75]). We end by noting that the proof of Theorem 1.2 shows a stronger
statement: any local Fp -algebra that admits a small CM algebra is “Witt Cohen-Macaulay,” see Remark 2.12.

A summary of the proof. Let us informally explain the proof in an example: A is an ordinary abelian surface. The
key idea is to work p-adically instead of modulo p until the end to track the divisibility properties of local cohomol2
b ét , Zp ).
ogy under finite extensions. More precisely, the p-rank of A contributes free Zp -summands to Hm
(Spec(R)
2
b
b
b
Trace formalism then shows: for any finite extension R → S, the group Hm (Spec(S)ét , Zp ) also contains free Zp 2 b
summands. Reducing these modulo p and using the Artin-Schreier sequence shows Hm
(S) 6= 0, so Sb is not CM. In
the body of the note, we work with Witt-vector cohomology instead of p-adic étale cohomology to apply the preceding
argument with fewer ordinarity constaints, see Remark 2.2.
Notation. For any Fp -scheme X, we write {Wn OX } of the projective system of (Zariski) sheaves defined by the
truncated Witt vector functors (see [Ill79, Ser58]), and W OX := limn Wn OX for its inverse limit (which is also
the derived limit: the transition maps Wn OX → Wn−1 OX are surjective for all n, and Wn OX has no higher cohomology on affines). For a closed subset Z ⊂ X and any abelian sheaf A, we write RΓZ (X, A) for the homotopykernel of RΓ(X, A) → RΓ(X − Z, A). In particular, since cohomology commutes with derived limits, we have
RΓZ (X, W OX ) ≃ R limn RΓZ (X, Wn OX ). Recall that RΓ(X, −) commutes with filtered colimits of sheaves if X
is quasi-compact and quasi-separated. In particular,
 if both X and X − Z are quasi-compact and quasi-separated, then
RΓZ (X, W OX,Q ) ≃ R limn RΓZ (X, Wn OX ) ⊗Z Q.
1A normal local noetherian Q-algebra R that admits a small CM algebra S is itself CM: H i (R) is a summand of H i (S) via the trace splitting.
m
m
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Acknowledgements. I am very grateful Manuel Blickle and Kevin Tucker for making me aware of the question
studied in this note, and several related discussions. Thanks are due to Johan de Jong, Mel Hochster and Anurag Singh
for their extremely helpful comments, questions, and encouragement.
2. P ROOF
2.1. Remarks on Witt vectors. We use k to denote a fixed perfect base field of characteristic p. We start by recalling
a fundamental result identifying the slope < 1 part of rigid cohomology with Witt vector cohomology; the smooth
case is due to Bloch-Deligne-Illusie (see [Blo77, Theorem 0.2] and [Ill79, §II.3]), while the singular case is newer:
i
Theorem 2.1 ([BBE07, Theorem 1.1]). Let A be a proper k-variety. Then H i (A, W OA )Q ≃ Hrig
(A)<1 .

Remark 2.2. Theorem 2.1 is the main reason we work with Witt-vector cohomology instead of p-adic étale cohomology in this note: the latter only describes the slope 0 part of rigid cohomology, while the former describes the
(potentially much larger) slope < 1 part. In particular, if A is an abelian variety, then H 1 (A, W OA,Q ) is always
non-zero for weight reasons, while H 1 (A, Zp ) vanishes if A is supersingular and k = k.
The next lemma allows deduction of modulo p consequences from rational assumptions.
Lemma 2.3. Let Y be a k-scheme with a closed subscheme Z. If HZi (Y, W OY )Q 6= 0 for some i, then HZj (Y, OY ) 6=
0 for j = i or j = i − 1.
Proof. The assumption on HZi (Y, W OY )Q and the formula


RΓZ (Y, W OY )Q ≃ R lim RΓZ (Y, Wn OY ) ⊗Z Q
n

HZj (Y, Wn OY

show that
) 6= 0 for some n > 0 and some j ∈ {i − 1, i} (as Ri lim = 0 for i > 1). The rest follows by
standard exact sequences expressing Wn OY as an iterated extension of copies of OY .

We need trace maps in Witt vector cohomology, so we recall a direct construction (essentially due to [SV96]).
Lemma 2.4. Let f : Y → X be a finite surjective morphism of noetherian normal schemes. For any abelian sheaf
A on Xét that is representable by an algebraic space, there is a functorial trace map Tr : f∗ f ∗ A → A such that the
f∗

Tr

composite A → f∗ f ∗ A → A is multiplication by the generic degree of f .
Proof. We refer the reader to [Bha, Proposition 6.2] for a proof.



Remark 2.5. The trace constructed in Lemma 2.4 is non-standard and slightly ad hoc. For example, if f : X → X
is the Frobenius map, then f ∗ : Shv(Xét ) → Shv(Xét ) is an equivalence, so the “correct” trace map should be an
≃
equivalence, while the one from Lemma 2.4 is multiplication by deg(f ) (composed with the inverse of id → f∗ f ∗ ).
In particular, the trace map constructed in Lemma 2.4 does not furnish a right adjoint to f! ≃ f∗ .
Using trace maps, we show that (rational) Witt vector cohomology cannot be killed by finite covers.
Corollary 2.6. Let f : Y → X be a finite surjective morphism of noetherian normal k-schemes, and let Z ⊂ X be a
closed subset. Then HZi (X, W OX )Q → Hfi −1 Z (Y, W OY )Q is a direct summand.
Proof. Let d be the generic degree of f . As Wn (−) is representable for each n > 0, Lemma 2.4 gives maps
f∗ Wn OY → Wn OX whose composition with the pullbacks Wn OX → f∗ Wn OY is multiplication by d. These maps
are compatible as n varies, so taking limits (and commuting them with f∗ ) gives a map f∗ W OY → W OX whose
composition with W OX → f∗ W OY is multiplication by d. The claim follows by applying HZi (X, − ⊗Z Q).

2.2. The main theorem. To prove Theorem 1.2, we first establish some notation.
Notation 2.7. Fix a polarised projective variety (A, L) of a perfect characteristic p field k; set R = ⊕n≥0 H 0 (A, Ln )
to be the section ring with m ⊂ R the homogeneous maximal ideal. Set X = Spec(R), Z = {m} ⊂ X and
b = Spec(R),
b U
b =X
b ×X U , and
U = X − Z with π : U → A realising U as the total space of L−1 over A. Set X
b
abusively let Z ⊂ X denote the closed point.
Next, we record the expected relation between the cohomology of A and U :
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Lemma 2.8. The natural map Wn OA → π∗ Wn OU is a direct summand for all n. In particular, H i (A, Wn OA ) →
H i (U, Wn OU ) is a direct summand for all i and n.
Proof. As π is a Gm -torsor, the natural map OA → π∗ OU realises the source as the weight 0 eigenspace of the target.
The rest follows by taking products and observing that Wn (R) ≃ Rn as sets functorially in R.

We can now prove the main theorem:
Proof of Theorem 1.2. As Wn OX is an extension of OX by Wn−1 OX , induction and the affineness of X show
H i (X, Wn OX ) = 0 for i > 0. Standard sequences then identify H i (U, Wn OU ) ≃ HZi+1 (X, Wn OX ) for all i, n > 0.
Since H 0 (U, Wn OU ) = Wn (H 0 (U, OU )), the system {H 0 (U, Wn OU )} has no lim1 (as Wn (R) → Wn−1 (R) is
surjective for any ring R), so
b W O b )Q
H i (U, W OU )Q ≃ H i+1 (X, W OX )Q ≃ H i+1 (X,
Z

Z

X

b Wn O b )}
for i > 0, where the last isomorphism comes from the excision identification {RΓZ (X, Wn OX )} ≃ {RΓZ (X,
X
i
i
of projective systems. Theorem 2.1 and Lemma 2.8 then show Hrig (A)<1 ≃ H (A, W OA )Q is a direct summand
i
b
b
b
of HZi+1
b )Q for i > 0. Choose 0 < i < dim(A) such that Hrig (A)<1 6= 0. Let f : Y → X be a finite
b (X, W OX
−1
b
surjective morphism of noetherian normal schemes; we will show that Y is not CM along f Z. Corollary 2.6 shows
j
b
b
H i+1
−1 (Y , W O b )Q 6= 0. Lemma 2.3 then gives H −1 (Y , O b ) 6= 0 for some j ∈ {i, i + 1}, which proves the claim
f

Z

Y

f

Y

Z

as i + 1 < dim(A) + 1 = dim(Yb ).



Remark 2.9. As rigid cohomology is a Weil cohomology theory, the hypothesis on A in Theorem 1.2 may be reformulated topologically, at least when A is smooth and k is finite, to say: for some 0 < i < dim(A), there is at least one
Frobenius eigenvalue on H i (Ak , Qℓ ) which is not divisible by p (for some auxilliary prime ℓ invertible on k). Indeed,
i
the eigenvalues occurring in Hrig
(A) coincide with those on H i (Ak , Qℓ ) and are algebraic integers.
Example 2.10. Some elementary examples of projective varieties A to which Theorem 1.2 applies include: any
projective variety of dimension ≥ 2 dominating a positive dimensional abelian variety, any projective variety of
dimension ≥ 3 dominating a K3 surface of finite height, etc.
We give an example showing that the presence of non-trivial middle cohomology of the structure sheaf does not
force the non-existence of a small CM algebra for the section ring, even for smooth projective varieties; this example
also shows the necessity of making a p-adic (rather than modulo p) assumption on A in Theorem 1.2.
Example 2.11. Assume k is algebraically closed, and let X be a smooth complete intersection in some Pnk with
dim(X) ≥ 2 with the property that a suitable subgroup G ≃ Fp ⊂ GLn+1 (k) preserves X and acts fixed point freely
on X; such examples were constructed by Serre, see [Ill05, §6]. Let Y = X/G denote the quotient. Then X → Y is a
finite étale G-torsor, and Y is a smooth projective k-variety; explicitly, the line bundle O(1) ∈ Pic(X) is equivariant
for the G-action, and hence descends to an ample line bundle L on Y . Let R and S denote the section rings of (Y, L)
and (X, O(1)) respectively. Then R → S is a finite extension. We will show that S is CM (even lci), but R is not. The
former follows immediately as (X, O(1)) is a complete intersection. On the other hand, the Lefschetz formalism (see
[Gro68, Corollary XII.3.5]) shows that X is simply connected. The Leray spectral sequence for X → Y then yields
Fp ≃ G ≃ π1ét (Y ) ≃ H 1 (Yét , Fp )∨ , and hence H 1 (Y, OY ) 6= 0 by the Artin-Schreier sequence. The latter group is a
2
direct summand of Hm
(R), so R is not CM (as dim(R) = dim(X) + 1 ≥ 3).
Remark 2.12. Let (R, m) be a noetherian local k-algebra. The proof of Theorem 1.2 shows that being “Witt Coheni
Macaulay” (i.e., having Hm
(Spec(R), W OR,Q ) = 0 for i < dim(R)) is necessary for the existence of small CM
algebras. We do not know if it is a sufficient condition. A (weakened) graded analogue asks: any projective variety
X with H i (X, W OX,Q ) = 0 for 0 < i < dim(X) admits an alteration π : Y → X with H i (Y, OY ) = 0 for
0 < i < dim(Y ). The simplest non-trivial instance of this question is when X = S × P1 with S a supersingular K3
surface, where a positive answer is implied by Artin’s conjecture on unirationality of supersingular K3 surfaces.
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D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF M ICHIGAN , A NN A RBOR 48109, USA
E-mail address: bhattb@umich.edu

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