ON THE ISOMORPHISM BETWEEN THE DUALIZING SHEAF AND THE CANONICAL SHEAF

MATTHEW H. BAKER AND JÁNOS A. CSIRIK

Abstract. We give a new proof of the isomorphism between the dualizing sheaf and the canonical sheaf of a non-singular projective variety X over a perfect field k. Our proof uses concepts and results from algebraic number theory.

1. Introduction

Recall the definition of a dualizing sheaf from [3, III, 7]. If X is a proper scheme of dimension n over a field k, a dualizing sheaf for X is a coherent sheaf ω on X, together with a trace morphism t : H^{n}(X, ω) → k such that for all coherent sheaves F on X, the natural pairing

Hom(F, ω) × H^{n}(X, F) → H^{n}(X, ω)

followed by t gives an isomorphism

Hom(F, ω)→^{∼}H^{n}(X, F)′.

If a dualizing sheaf for X exists, then it is unique up to canonical isomporphism by [3, III, 7.2].

An explicit calculation shows that ℙ^{n} has a dualizing sheaf ω_{ℙ^{n}}. If X is any projective variety over a perfect field k, then a dualizing sheaf ω_{X} exists and can be constructed as follows. By Noether normalization, there is a finite separable morphism f : X → ℙ^{n}. Then ω_{X} = f^{!}ω_{ℙ^{n}} is a dualizing sheaf for X, where f^{!} is a functor whose definition is given in Section 2. This construction readily leads to a proof of the Serre duality theorem [3, III, 7.6], different from the one given in [3, III, §7].

In Section 3, which is the main part of this paper, we give a direct proof that the dualizing sheaf ω_{X} constructed above is isomorphic to the canonical sheaf if X is nonsingular. The proof uses a number of facts about Dedekind domains and discrete valuation rings. In particular, the theory of the different plays an important role. The reader familiar with algebraic number theory will find the present approach an intriguing alternative to the homological methods used in [3, III, §7]. Yet another proof of our main result, Corollary 9, can be found in [4, 5].

All references in round brackets will be to [3].

1991 Mathematics Subject Classification. 14F10.

Key words and phrases. Serre Duality, Dualizing Sheaf, Differentials.

The authors would like to thank H. W. Lenstra, Jr. for his valuable assistance. Special debt is owed to R. Hartshorne for suggesting this project to us.

The first author was supported by an NDSEG fellowship.

2. Serre duality

Let f : X → Y be a finite morphism of noetherian schemes. For any quasicoherent sheaf G on Y, the sheaf Hom_{Y} (f_{∗}O_{X}, G) is a quasi-coherent f_{∗}O_{X}-module, so it corresponds to a quasi-coherent O_{X}-module that we shall call f^{!}G (see (III, Ex. 6.10(a))). Before we go on, let us consider the case where f : X = Spec B → Y = Spec A corresponds to f^{#} : A → B. For some terminology, if A and B are any commutative rings, with N ∈ A-𝔐𝔬𝔡 and M ∈ (A, B)-𝔅𝔦𝔐𝔬𝔡, we define \underline{Hom}_{A}(M, N) to be the usual Hom but with the obvious B-module structure derived from M. Then the functor f^{!} corresponds to \underline{Hom}_{A}(B, −).

Returning to the general (not necessarily affine) case, the functor f^{!} behaves like a right adjoint to f_{∗} in the sense that for any F ∈ ℭ𝔬𝔥(X) and G ∈ 𝔔𝔠𝔬(Y), there is a natural isomorphism

f_{∗}Hom_{X}(F, f^{!}G) → Hom_{Y} (f_{∗}F, G) (1)

(see (III, Ex. 6.10(b))). Taking global sections in (1), we obtain the isomorphism

Hom_{X}(F, f^{!}G) ≅ Hom_{Y} (f_{∗}F, G). (2)

For future reference, note that a simple calculation using (II, Ex. 5.2) gives the following lemma:

Lemma 1. Let f : X → Y be a finite morphism of noetherian schemes, and M a locally free sheaf on Y. Then

f^{!}M ≅ f^{!}O_{Y} ⊗_{O_{X}} f^{∗}M. (3)

We now sketch how the Serre duality theorem (III, 7.6) can be proved for an integral projective scheme X of dimension n over a perfect field k using the functor f^{!}. Let us first recall the following lemma.

Lemma 2 (Noether Normalization). Let X be an integral projective scheme of dimension n over a perfect field k. Then there exists a finite separable morphism f : X → ℙ^{n}.

Proof. By (I, 4.8A), K(X)/k is separably generated, so by (I, 4.7A) it contains a separating transcendence base, yielding an injection K(ℙ^{n}) = k(Y_{1}, …, Y_{n}) ↪ K(X). By (I, 4.4), this gives a dominant morphism f : X → ℙ^{n}, which is finite and separable, since K(X)/K(ℙ^{n}) is. ■

Now let X be as above. By Lemma 2, there exists a finite separable morphism f : X → ℙ^{n}. We define ω_{X} := f^{!}ω_{ℙ^{n}}, where ω_{ℙ^{n}} is the dualizing sheaf for ℙ^{n} (see (III, 7.1)). By (III, Ex. 7.2), ω_{X} is a dualizing sheaf for X. (Note that by (III, 7.2), any two dualizing sheaves for X are canonically isomorphic, so ω_{X} does not depend on the choice of f.) Since f is flat if and only if X is Cohen–Macaulay by [2, Exer. 18.17 (or Cor. 18.17)], the Serre duality theorem (III, 7.6) now follows from (III, Ex. 6.10).

3. The dualizing sheaf

Now we will show that the dualizing sheaf ω_{X} := f^{!}ω_{ℙ^{n}} is in fact the canonical sheaf on a nonsingular (noetherian) projective scheme X. Recall from (III, 7.1) that for ℙ^{n}, the canonical sheaf ^{⋀n}Ω_{ℙ^{n}} is the dualizing sheaf. Before stating our next lemma, we need to recall the following definition.

ON THE ISOMORPHISM BETWEEN THE DUALIZING AND THE CANONICAL SHEAVES 3

Definition 3. If T is a torsion sheaf on a normal noetherian scheme X, we define the ramification divisor of T to be

R = ∑_{Z} length_{O_{ζ}}(T_{ζ}) · Z,

where the sum ranges over all the irreducible closed subschemes Z of codimension one in X, and ζ denotes the generic point of Z.

Lemma 4. Let X be a normal noetherian scheme. Suppose F and G are locally free sheaves of rank n on X and that T is a torsion sheaf on X with ramification divisor R, such that the sequence

0 ⟶ F ⟶ G ⟶ T ⟶ 0

is exact. Then ^{⋀n}G ≅ ^{⋀n}F ⊗ L(R).

Proof. We make the following definition: if M is a module of finite length over some commutative ring A, then we denote by χ_{A}(M) the product of the (not necessarily distinct) primes occuring in a Jordan-Hölder series for M. (For noetherian integrally closed domains, this χ_{A} is the canonical map from the category of finite length A-modules to its Grothendieck group, which is the group of ideals of A. See [7, I §5] for details.) Also note that since X is normal, it makes sense to talk about Weil divisors on X.

We now begin the proof of the lemma. Taking nth exterior powers, we get an exact sequence

0 ⟶ ^{⋀n}F ⟶^{det} ^{⋀n}G ⟶ C ⟶ 0.

Since^{⋀n}G is locally free of rank 1, we can tensor by its dual to get an exact sequence

0 ⟶ ^{⋀n}F ⊗ (^{⋀n}G)^{−1} ⟶ O_{X} ⟶ C ⊗ (^{⋀n}G)^{−1} ⟶ 0.

But then ^{⋀n}F ⊗(^{⋀n}G)^{−1} equals (as subsheaves of O_{X}) I_{D}, the ideal sheaf of some locally principal closed subscheme of X corresponding to an (effective) Cartier divisor D, and hence C ≅ (^{⋀n}G) ⊗ O_{D}.

If we let C(X) be the group of Cartier divisors on X, then there is an injection C(X) ↪ Div(X), where Div(X) is the group of Weil divisors. We claim that R = D as Weil divisors. It suffices to show that n_{Z} = n′_{Z} for each prime divisor Z of X, where n_{Z} denotes the coefficient of Z in R, and n′_{Z} denotes the coefficient of Z in D. Fix one such Z, and denote by ζ its generic point. Then the local ring O_{ζ} is a discrete valuation ring, and we have the following exact sequences of O_{ζ}-modules:

0 ⟶ F_{ζ} ⟶^{u} G_{ζ} ⟶ T_{ζ} ⟶ 0

and

0 ⟶ ^{⋀n}F_{ζ} ⟶^{det u} ^{⋀n}G_{ζ} ⟶ C_{ζ} ⟶ 0.

By the theory of modules over a principal ideal domain, we have

n_{Z} = length_{O_{ζ}} T_{ζ} = length_{O_{ζ}}(coker u) = v_{Z} (χ_{O_{ζ}}(coker u)) ,

where v_{Z} denotes the valuation on the local ring O_{ζ}. By [7, I, §5, Lemma 3], we have χ_{O_{ζ}}(coker u) = β · O_{ζ}, where β · O_{ζ} is the image of det u. So n_{Z} = v_{Z}(β) = length_{O_{ζ}}(O_{D})_{ζ} = n′_{Z} as claimed. This shows that D = R as Weil divisors, and hence as Cartier divisors; in particular, R is a well-defined Cartier divisor! We conclude that

L(−R) ≅ I_{D} ≅ ⋀^{n}F ⊗ (⋀^{n}G)^{−1} ,

so ^{⋀n}G ≅ ^{⋀n}F ⊗ L(R). ■

Remark: Using the terminology of (II, Ex. 6.11(b)), we have proved that the determinant det T of our sheaf T is isomorphic to L(R).

We now want to prove the following:

Theorem 5. Let f : X → Y be a finite separable morphism between integral nonsingular schemes of dimension n. Then

Ω_{X}^{n} ≅ f^{!}Ω_{Y}^{n} .

Proof. What we will end up doing is proving in a slightly roundabout way that

Ω_{X}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R) (4)

and

f^{!}Ω_{Y}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R) (5)

for the same “ramification divisor” R.

Let us first try to prove (4). By (II, 8.11), we have an exact sequence

f^{∗}Ω_{Y} → Ω_{X} → Ω_{X/Y} → 0.

Now Ω_{X} is locally free of rank n on X, and Ω_{Y} is locally free of rank n on Y, so f^{∗}Ω_{Y} is locally free of rank n on X (since f^{∗}O_{Y} = O_{X}). As f is a finite separable morphism, K(X)/K(Y) is a finite separable extension so by [6, Thm. 25.3], we get Ω_{K(X)/K(Y)} = 0. Hence Ω_{X/Y} is a torsion sheaf. Now letting K be the kernel of f^{∗}Ω_{Y} → Ω_{X}, we get an exact sequence at each point P of X

0 → K_{P} → O_{P}^{n} → O_{P}^{n} → (Ω_{X/Y})_{P} → 0.

Tensoring this sequence with K(X), which is flat over O_{P}, we get

dim_{K(X)} K_{P} ⊗ K(X) = −n + n + dim_{K(X)}(Ω_{X/Y})_{P} ⊗ K(X) = 0,

so K_{P} is torsion. But K_{P} ⊆ O_{P}^{n}, and O_{P} is an integral domain, so we have K = 0, i.e.

0 → f^{∗}Ω_{Y} → Ω_{X} → Ω_{X/Y} → 0.

Now taking the ramification divisor R associated to Ω_{X/Y}, we have

Ω_{X}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R)

by Lemma 4, proving (4). Assertion (5) will be proved as Lemma 8. ■

In order to prove Lemma 8, some further preparation is needed. Let f : X → Y again be a finite separable morphism between nonsingular irreducible schemes X and Y. Then X is in particular Cohen-Macaulay, so f_{∗}O_{X} is a locally free O_{Y} - module (and also is an O_{Y} -algebra via f^{#}).

We want to define an O_{Y} -linear “trace” map Tr : f_{∗}O_{X} → O_{Y} as follows. On any sufficiently small affine open U = Spec A ∈ Y, we have f^{−1}U = Spec B affine, and B is a free A-module of rank d := deg f. Let {e_{1}, …, e_{d}} be a basis for B over A, and for b ∈ B write be_{i} = ∑ a_{ij}e_{j}. Then we can set Tr(b) := ∑ a_{ii} on B, which is independent of the chosen basis, so the local maps defined in this way glue properly to give a global map of sheaves on X. (For further discussion see [1, Chapter VI, Remark 6.5].)

By (2), the map Tr ∈ Hom_{Y} (f_{∗}O_{X}, O_{Y}) gives rise to a map \overline{Tr} ∈ Hom_{X}(O_{X}, f^{!}O_{Y}). We will be interested in the cokernel F of the morphism \overline{Tr}. We first collect some facts about Dedekind rings and then prove a proposition characterizing the sheaf F.

Lemma 6. Let A and B be Dedekind domains with fraction fields K and L, respectively. Suppose that L is finite separable over K, with B the integral closure of A in L. Then

(a) B ⊗_{A} K = L

(b) \underline{Hom}_{A}(B, A) ↪ \underline{Hom}_{K}(L, K)

(c) If I, J are fractional ideals of B, I/IJ ≅ B/J.

(d) If M is a torsion B-module, then M may be written uniquely in the form M = B/I_{1} ⊕⋯⊕B/I_{m}, where I_{1} ⊆ ⋯ ⊆ I_{m} is an ascending sequence of nontrivial ideals of B.

Proof. (a) Both L and B ⊗_{A} K can be characterized as the integral closure of K in L; (b) follows from (a); (c) is an exercise for the reader; (d) see [2, Ex. 19.6(c)]. ■

Proposition 7. \overline{Tr} is injective, and its cokernel F is a torsion sheaf. Moreover, the ramification divisor of F is equal to the ramification divisor of Ω_{X/Y}.

Proof. We begin with a sequence of reductions. Since the statement of the lemma is of local nature and f is a finite morphism, we can assume that Y = Spec A and X = Spec B. In fact, it suffices to prove the proposition after localizing at a height 1 prime of A; we may thus assume that (A, 𝔪) is a discrete valuation ring and B is a semilocal ring with maximal ideals {𝔭_{1}, …, 𝔭_{r}} dominating 𝔪. As f is surjective (and hence dominant), f^{#} : A → B is injective, so we may assume without loss of generality that A ⊆ B.

By the equivalence of categories between B-modules and quasicoherent sheaves on Spec B, we find that the exact sequence

O_{X} ⟶^{\overline{Tr}} f^{!}O_{Y} ⟶ F ⟶ 0 (6)

corresponds to an exact sequence of B-modules

B ⟶^{\overline{Tr}} B^{∗} ⟶ M ⟶ 0, (7)

where B^{∗} = \underline{Hom}_{A}(B, A).

The image of 1 ∈ B under \overline{Tr} is easily seen to be Tr ∈ B^{∗}. If we let K and L denote the fields of fractions of A and B respectively, then L/K is separable (since f is), so Tr ≠ 0 ∈ B^{∗} by [7, III, §3]. This shows that \overline{Tr} is injective.

Let K and L be the fraction fields of A and B, respectively. Note that K = K(Y) and L = K(X). By assumption A is Dedekind, and since B is finite over A it is integral over A. But X is nonsingular, hence normal, so B is integrally closed and thus is the integral closure of A in L. By (I, 6.3A) B is also a Dedekind ring, so applying Lemma 6 it follows that B ⊗_{A} K = L, and that we have an injection B^{∗} = \underline{Hom}_{A}(B, A) ↪ \underline{Hom}_{K}(L, K). But \underline{Hom}_{K}(L, K) can be canonically identified with L via

L →^{∼} \underline{Hom}_{K}(L, K)
x ↦ (l ↦ Tr_{L/K}(xl)).

Under this identification, B^{∗} corresponds to

B^{†} = {x ∈ L : Tr_{L/K}(xB) ⊆ A} ⊆ L.

Here B^{†} is a fractional ideal of B containing B; its inverse, the different 𝔇_{B/A}, is an integral ideal of B [7, III, §3]. By Lemma 6(c), we have

B^{†}/B ≅ B/𝔇_{B/A}.

Note also that since B is a Dedekind ring, B^{∗} is locally free of rank 1 as a B-module. Hence f^{!}O_{Y} is a locally free sheaf of rank 1, and we in fact see that F is a torsion sheaf in (6):

0 ⟶ O_{X} ⟶ f^{!}O_{Y} ⟶ F ⟶ 0. (8)

We now claim that length_{O_{ζ}} (Ω_{X/Y})_{ζ} = length_{O_{ζ}} F_{ζ} for all ζ corresponding to generic points of prime divisors Z on X. So we have to show that

(9) length_{B_{𝔭}} M_{𝔭} = length_{B_{𝔭}} (Ω_{B/A})_{𝔭}

for height 1 primes 𝔭 ⊆ B, i.e. 𝔭_{1}, … 𝔭_{r}. But this follows from [8, Theorem 4.1(1)]. (Note that since L/K is separable, Ω_{B/A}⊗_{B} L = Ω_{L/K} = 0, hence Ω_{B/A} is a torsion B-module.) ■

Remark: By Lemma 6(d), we must have Ω_{B/A} = ⨁_{1}^{t} B/I_{i} for some ideals I_{i} in B. Then we can see from [8] that 𝔇_{B/A} can be expressed as 𝔇_{B/A} = ∏_{1}^{t} I_{i}.

Remark: Note that if we knew that all k(ζ)/k(η) were separable (where η ∈ Y corresponds to 𝔪 ∈ Spec A), then we could apply [7, III, §7 Prop. 14] to obtain this result.

We have all the components in place now for the proof of (5):

Lemma 8. Let R be the ramification divisor of F (which, by Proposition 7, is the same as the ramification divisor of Ω_{X/Y}). Then the following isomorphism holds.

f^{!}Ω_{Y}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R).

Proof. Using the short exact sequence (8), we obtain, by Lemma 4,

f^{!}O_{Y} ≅ L(R).

Tensoring this isomorphism on both sides by the sheaf f^{∗}Ω_{Y}^{n}, we see that

f^{!}O_{Y} ⊗_{O_{X}} f^{∗}Ω_{Y}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R).

But since Ω_{Y}^{n} is locally free, by Lemma 1 we get

f^{!}Ω_{Y}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R).
■

This establishes (5), so the proof of Theorem 5 is now complete.

Corollary 9. Let X be a nonsingular projective variety of dimension n over k. Then ω_{X} ≅ Ω_{X}^{n}.

Proof. Let f : X → ℙ^{n} be a finite separable morphism whose existence is guaranteed by Lemma 2. By definition, ω_{X} ≅ f^{!}ω_{ℙ^{n}}. But ω_{ℙ^{n}} ≅ Ω_{ℙ^{n}}^{n}. The result now follows from Theorem 5. ■

References

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[2] David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, 1995

[3] Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1977

[4] Ernst Kunz, Holomorphe Differentialformen auf algebraischen Varietäten mit Singularitäten I, Manuscripta math., 15 (1975), 91–108

[5] Ernst Kunz, Differentialformen auf algebraischen Varietäten mit Singularitäten II, Abh. Math. Sem. Univ. Hamburg, 47 (1978), 42–70

[6] Hideyuki Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge University Press, 1986

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Edinburgh Math. Soc. (2), 40 (1997), no. 2, 353–365

   E-mail address: baker@math.berkeley.edu
   E-mail address: janos@math.berkeley.edu

Department of Mathematics, University of California, Berkeley, CA 94720-3840