AN ALTERNATIVE APPROACH TO SERRE DUALITY FOR PROJECTIVE VARIETIES

MATTHEW H. BAKER AND JÁNOS A. CSIRIK

This paper was written in conjunction with R. Hartshorne’s Spring 1996 Algebraic Geometry course at UC Berkeley.

1. Introduction

The purpose of this paper is to develop some of the theory from R. Hartshorne’s Algebraic Geometry [5, III, §7] using different techniques. A sketch of our method, which is due to Professor Hartshorne, is as follows. By Noether normalization, an integral projective scheme X of dimension n over a perfect field k admits a finite separable morphism to ℙ_{k}^{n}. This morphism is flat if and only if X is Cohen-Macaulay. If f : X → Y is any finite morphism of noetherian schemes, we define a functor f^{!} : 𝔔𝔠𝔬(Y) → 𝔔𝔠𝔬(X) and show that Hom_{X}(F, f^{!}G) ≅ Hom_{Y} (f_{∗}F, G) (for F a coherent sheaf and G a quasi-coherent sheaf). We then prove that if f : X → ℙ^{n} is a finite morphism, then ω_{X} := f^{!}ω_{ℙ^{n}} is a dualizing sheaf for X, where ω_{ℙ^{n}} = O_{ℙ^{n}}(−n − 1). From this the Serre duality theorem follows.

Finally, we prove that the dualizing sheaf ω_{X} is isomorphic to the the canonical sheaf if X is nonsingular. This part of the paper utilizes a number of facts about Dedekind domains and discrete valuation rings. In particular, the theory of the different comes in at a crucial moment.

The present approach has several features which make it attractive. For one thing, our approach is very “geometric” in the sense that the dualizing sheaf of X is defined directly from a finite morphism f : X → ℙ^{n}. Also, we use a geometric characterization of the Cohen-Macaulay property as equivalent to the flatness of f. Finally, our approach to differentials provides an interesting link to some of the important concepts of algebraic number theory; the reader familiar with this theory (as found, for example, in [7]), may find the present method an intriguing alternative to the homological proof found in [5, III, Thm. 7.11].

All references in round brackets will be to [5].

2. Serre duality

Let f : X → Y be an affine morphism of schemes. Then f is quasicompact and separated; therefore, by (II, 5.8), f gives rise to a functor f_{∗} : 𝔔𝔠𝔬(X) → 𝔔𝔠𝔬(Y). In fact, we have that for any f ∈ 𝔔𝔠𝔬(X), f_{∗}F is an f_{∗}O_{X}-module.

We want to cook up a functor˜ : 𝔔𝔠𝔬(Y)∩f_{∗}O_{X}-𝔐𝔬𝔡 → 𝔔𝔠𝔬(X) going the other way. Given any G ∈ 𝔔𝔠𝔬(Y) ∩ f_{∗}O_{X}-𝔐𝔬𝔡, define G̃ ∈ 𝔔𝔠𝔬(X) in the following way. Pick an open affine U = Spec A ⊆ Y. Then since f is affine, f^{−1}U is affine, say Spec B. Since G is an f_{∗}O_{X}-module, G = Γ(U, G) is not only an A-module, but also a B-module via f^{#}. Now on Spec B we can define G̃|_{Spec B} to be G̃. It is easily seen that the various G̃’s defined on open affines in X patch to give a sheaf G̃ ∈ 𝔔𝔠𝔬(X). [For details, see [4, §1.4].]

Date: 8 May 1996 [Revised December 1997].

The authors would like to thank H. W. Lenstra, Jr. and A. Ogus for their valuable assistance.

Special debt is owed to R. Hartshorne for suggesting this project to us and outlining most of the results.

Observe that f_{∗} and ˜ define an equivalence of categories. For F ∈ 𝔔𝔠𝔬(X), if F = F̃ on Spec B then f_{∗}F is again F̃ on Spec A (with F viewed as an A-module), and so f̃_{∗}F ≅ F on Spec B, hence on all of X. Likewise, if G ∈ 𝔔𝔠𝔬(Y)∩f_{∗}O_{X}-𝔐𝔬𝔡, then G = G̃ on Spec A, and G also has a B-module structure making G̃ = G̃ on Spec B, and so f_{∗}(G̃) ≅ G.

It will be convenient to give a quick proof here of the following fact:

Lemma 1. Let Y be a noetherian scheme. If G = M̃ ∈ 𝔔𝔠𝔬(Y) and F = Ñ ∈ ℭ𝔬𝔥(Y) then we have Hom_{X}(F, G) ∈ 𝔔𝔠𝔬(Y). Furthermore, Hom_{X}(F, G) = Hom(N, M)˜ .

Proof. By (II, Ex. 5.4) we have locally an exact sequence

O_{Y} ^{m} ⟶ O_{Y} ^{n} ⟶ F ⟶ 0

to which we can apply the left exact functor Hom_{Y} (−, G) to obtain Hom_{X}(F, G) as the kernel of a map G^{n} → G^{m} of quasi-coherent sheaves. But Hom_{X}(F, G) = Hom_{X}(Ñ, M̃) ∈ 𝔔𝔠𝔬(Y) implies, by (II, 5.5) that this sheaf is just the twiddle of its global sections Hom(N, M), i.e., Hom_{X}(Ñ, M̃) ≅ Hom(N, M)˜ . ■

Construction of f^{!} Now let f : X → Y be a finite morphism of noetherian schemes. By (II, Ex. 5.5(c)), there is a corresponding functor f_{∗} : ℭ𝔬𝔥(X) → ℭ𝔬𝔥(Y). It will be important for what follows for us to define a sort of adjoint to f_{∗} which we will call f^{!} : 𝔔𝔠𝔬(Y) → 𝔔𝔠𝔬(X). First observe that any finite morphism is affine, so we can employ the operation ˜ defined above. Since f is finite, f_{∗}O_{X} is coherent, so for any G ∈ 𝔔𝔠𝔬(Y), the sheaf Hom_{Y} (f_{∗}O_{X}, G) is quasi-coherent by Lemma 1 and is also clearly an f_{∗}O_{X}-module. Thus it makes sense to define f^{!}G = Hom_{Y} (f_{∗}O_{X}, G)^{}˜ .

The Affine Case Before we go on, let us consider what our functors f_{∗},˜ , and f^{!} do in the case where f : X = Spec B → Y = Spec A corresponding 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) and N ⊗_{A} M to be the usual Hom and tensor product but with the obvious B-module structure derived from M. Observe then that

− ⊗_{B} M ⊣ \underline{Hom}_{A}(M, −) (1)

by [1, Thm. 7.2.20]. (The notation F ⊣ G denotes an adjunction between the functors F and G with F as the left adjoint and G as the right.)

The functor˜ is “for any A-module which is also a B-module in a manner compatible with f^{!}, consider it as a B-module.” f^{∗}, on the other hand, is the functor “consider B-modules as A-modules” . It can be described alternatively as − ⊗_{B} B (with the A-module structure derived from B) or \underline{Hom}_{B}(B, −). Thinking of f_{∗} in the latter interpretation, it has by (1) a left adjoint − ⊗_{A} B which is usually called f^{∗}.

Considering f_{∗} in its tensor product avatar gives it a right adjoint \underline{Hom}_{A}(B, −), which we recognize as corresponding exactly to our functor f^{!}. Since f is finite, B is finitely generated over A, so B̃ is a coherent sheaf on Y = Spec A. Again using Lemma 1 we get that indeed f^{!} corresponds to \underline{Hom}_{A}(B, −).

Note also that for locally free sheaves M on Y, f^{!} is particularly easy to compute:

Lemma 2. 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. (2)

Proof. Since f_{∗} and˜ define an equivalence of categories, we have

f^{!}O_{Y} ⊗_{O_{X}} f^{∗}M ≅ (f_{∗}(f^{!}O_{Y} ⊗_{O_{X}} f^{∗}M))˜ ,

but by the projection formula (II, Ex. 5.2(d))

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

Now by the definition of f^{!} and the definition of the dual of an invertible sheaf,

(f_{∗}(f^{!}O_{Y}) ⊗_{O_{Y}} M)˜ ≅ ((f_{∗}O_{X})^{∨} ⊗_{O_{Y}} M)˜ ,

and finally by (II, Ex. 5.2(b)),

((f_{∗}O_{X})^{∨} ⊗_{O_{Y}} M)˜ ≅ (Hom_{Y} (f_{∗}O_{X}, M))˜ ≅ f^{!}M,

so the formula (2) is valid. ■ We will now prove:

Lemma 3. For any F ∈ ℭ𝔬𝔥(X) and G ∈ 𝔔𝔠𝔬(Y), we have

f_{∗}Hom_{X}(F, f^{!}G) ≅ Hom_{Y} (f_{∗}F, G). (3)

Observe that by taking global sections of both sides in (3), we obtain the correspondence

Hom_{X}(F, f^{!}G) ≅ Hom_{Y} (f_{∗}F, G) (4)

mentioned in the introduction.

Proof. To prove (3), we will construct a map going from the left hand side to the right, and check locally that it is an isomorphism. We will build our map up in several steps. First of all, setting H = f^{!}G, we have the usual map

f_{∗}Hom_{X}(F, H) → Hom_{Y} (f_{∗}F, f_{∗}H), (5)

which on an open set U ⊆ Y is the map sending ϕ ∈ Hom_{O_{X}|_{f^{−1}U}} (F|_{f^{−1}U}, H|_{f^{−1}U}) to ψ ∈ Hom_{O_{Y} |_{U}} (f_{∗}F, f_{∗}H), where ψ_{W} : F(f^{−1}W) → H(f^{−1}W) is defined as ϕ_{f^{−1}W} for all open W ⊆ U.

We now see that

f_{∗}f^{!}G = f_{∗}((Hom_{Y} (f_{∗}O_{X}, G))^{}˜ ) ≅ Hom_{Y} (f_{∗}O_{X}, G)

by the equivalence of categories given by˜ and f_{∗}. This gives us

Hom_{Y} (f_{∗}F, f_{∗}f^{!}G)→^{∼}Hom_{Y} (f_{∗}F, Hom_{Y} (f_{∗}O_{X}, G)). (6)

Now f_{∗}O_{X} is an O_{Y} -algebra through the map f^{#} : O_{Y} → f_{∗}O_{X}. By the contravariant functoriality of Hom in the first variable, this gives us an evaluation map

Hom_{Y} (f_{∗}O_{X}, G) → Hom_{Y} (O_{Y} , G) = G.

By the covariant functoriality of Hom_{Y} in the second variable, we get a map

Hom_{Y} (f_{∗}F, Hom_{Y} (f_{∗}O_{X}, G)) → Hom_{Y} (f_{∗}F, G). (7)

Composing (5), (6) and (7) (with H = f^{!}G as before), we obtained the desired natural morphism

f_{∗}Hom_{X}(F, f^{!}G) → Hom_{Y} (f_{∗}F, G). (8)

To check that this is an isomorphism locally, we can restrict our attention to the case Y = Spec A, X = Spec B (with B an A-algebra via f^{!}), F = M̃, with M a finitely generated B-module, and G = Ñ, with N an A-module. Then the morphism (8) corresponds to the natural map

ψ : Hom_{B}(M, \underline{Hom}_{A}(B, N)) → Hom_{A}(M ⊗_{B} B, N)
ϕ ↦ (m ⊗ 1 ↦ ϕ(m)(1)).

(Recall that M ⊗_{B} B is just M considered as an A-module via the given map A → B.) But we know this map to be an isomorphism by (1). ■

Proposition 4. Let f : X → Y be a finite morphism of noetherian schemes. Then for all F ∈ ℭ𝔬𝔥(X), G ∈ 𝔔𝔠𝔬(Y), and all i ≥ 0, there is a natural map

Ext_{X}^{i}(F, f^{!}G) → Ext_{Y}^{i} (f_{∗}F, G).

Remark: The isomorphism of Lemma 3 is the case i = 0 here.

Proof. The map (5) is natural in the second variable H, so it can be considered a natural transformation of functors. Composing with the global sections functor Γ, we get the natural transformation of functors

Hom_{X}(F, −) → Hom_{Y} (f_{∗}F, f_{∗}−).

The second functor here is just the composition of the exact functor f_{∗} and the zeroeth component of the universal δ-functor Ext_{Y}^{i} (f_{∗}F, −). Hence Hom_{Y} (f_{∗}F, f_{∗}−) is the zeroeth component of the δ-functor Ext_{Y}^{i} (f_{∗}F, f_{∗}−). But since Hom_{X}(F, −) is the zeroeth component of its derived functor, the universal δ-functor Ext_{X}^{i}(F, −), the natural transformation above extends to a natural map of δ-functors

Ext_{X}^{i}(F, −) → Ext_{Y}^{i} (f_{∗}F, f_{∗}−).

We can now substitute f^{!}G to get a natural map

Ext_{X}^{i}(F, f^{!}G) → Ext_{Y}^{i} (f_{∗}F, f_{∗}f^{!}G).

In the proof of Lemma 3 we described the natural map f_{∗}f^{!}G ≅ Hom_{Y} (f_{∗}O_{X}, G) → G. Using the covariant functoriality of Ext_{Y}^{i} in the second variable, we get the natural map

Ext_{Y}^{i} (f_{∗}F, f_{∗}f^{!}G) → Ext_{Y}^{i} (f_{∗}F, G).

The composition of the last two displayed maps gives our desired map. ■

Proposition 5. With the conditions of Proposition 4, if in addition ℭ𝔬𝔥(X) has enough locally frees and f is flat, then the natural map

Ext_{X}^{i}(F, f^{!}G) → Ext_{Y}^{i} (f_{∗}F, G)

is an isomorphism for all F ∈ ℭ𝔬𝔥(X), G ∈ 𝔔𝔠𝔬(Y), i ≥ 0.

Remark: If X is a quasi-projective scheme, or a nonsingular scheme, then ℭ𝔬𝔥(X) has enough locally frees. Also, by (III, Prop. 9.2(e)), f is flat if and only if f satisfies the condition

(*) f_{∗}O_{X} is a locally free O_{Y} -module.

Proof. For F = O_{X} we have (III, 6.3(c))

Ext_{X}^{i}(O_{X}, f^{!}G) ≅ H^{i}(X, f^{!}G),

but since f is an affine map, we see using (III, Ex. 4.1) that

H^{i}(X, f^{!}G) ≅ H^{i}(Y, f_{∗}f^{!}G) = H^{i}(Y, Hom_{Y} (f_{∗}O_{X}, G)).

Now, using (II, Ex. 5.1(b)) and the fact that f_{∗}O_{X} is locally free, we see that

H^{i}(Y, Hom_{Y} (f_{∗}O_{X}, G)) ≅ H^{i}(Y, (f_{∗}O_{X})^{∨} ⊗ G),

and again by (III, 6.3(c)) we have

H^{i}(Y, (f_{∗}O_{X})^{∨} ⊗ G) ≅ Ext_{Y}^{i} (O_{Y} , (f_{∗}O_{X})^{∨} ⊗ G).

By (III, 6.7), again using condition (*), we have

Ext_{Y}^{i} (O_{Y} , (f_{∗}O_{X})^{∨} ⊗ G) ≅ Ext_{Y}^{i} (f_{∗}O_{X}, G).

It is an exercise for the reader to verify that the composite of these isomorphisms is our natural map! This shows that our claim is true for F = O_{X}.

It is clear now that the claim is also true if F is a free O_{X}-module of finite rank. In fact it is also true for F locally free of finite rank, since we are given a global map and verifying that it is an isomorphism may be done locally, which comes down to the calculation we have just done.

Since ℭ𝔬𝔥(X) has enough locally frees and F is coherent, we can find a locally free sheaf E of finite rank that surjects onto F. Calling the kernel R, we have a short exact sequence

0 → R → E → F → 0

of coherent sheaves on X. Taking f_{∗} of this sequence, we get a short exact sequence of coherent sheaves on Y:

0 → f_{∗}R → f_{∗}E → f_{∗}F → 0.

(Note that f_{∗} : ℭ𝔬𝔥(X) → ℭ𝔬𝔥(Y) is exact since it has a right adjoint f^{!}). Taking Hom_{X}(−, f^{!}G) of the first sequence and Hom_{Y} (−, G) of the second, and developing them into a long exact sequence of Ext’s using (III, 6.4), we can use our natural maps to get a morphism of long exact sequences:

0 ⟶ Hom_{X}(F, f^{!}G) ⟶ Hom_{X}(E, f^{!}G) ⟶ Hom_{X}(R, f^{!}G) ⟶ Ext_{X}^{1}(F, f^{!}G) ⟶ Ext_{X}^{1}(E, f^{!}G) ⟶ ⋯
↓^{ϕ_{1}}  ↓^{ϕ_{2}}  ↓^{ϕ_{3}}  ↓^{ϕ_{4}}  ↓^{ϕ_{5}}  ↓
0 ⟶ Hom_{Y} (f_{∗}F, G) ⟶ Hom_{Y} (f_{∗}E, G) ⟶ Hom_{Y} (f_{∗}R, G) ⟶ Ext_{Y}^{1} (f_{∗}F, G) ⟶ Ext_{Y}^{1} (f_{∗}E, G) ⟶ ⋯

(Note that this long exact sequence is in fact known to be infinitely long and therefore it is impossible to fit on a page of finite extent such as this one.) Now by (4), ϕ_{1}, ϕ_{2}, and ϕ_{3} are all isomorphisms. Since E is locally free of finite rank, ϕ_{5} is also an isomorphism by what we have just done. Applying the “subtle 5lemma” to the part of the diagram between ϕ_{2} and ϕ_{6}, we get that ϕ_{4} is injective. Since F was an arbitrary coherent sheaf, an analogous argument shows that ϕ_{6} is injective. Once again applying the “subtle 5-lemma”, we deduce that ϕ_{4} is in fact an isomorphism (as is ϕ_{6} by analogous reasoning).

We now have that ϕ_{1} to ϕ_{6} are all isomorphisms, and we can repeat the argument just given to show that ϕ_{7} to ϕ_{9} are also isomorphisms, etc. ■

Now that we have a duality theory for finite flat morphisms, we can prove the Serre Duality Theorem for projective schemes of dimension n. In doing so, we characterize those schemes which admit a finite flat morphism to ℙ^{n}.

The following lemma will allow us to apply the theory developed thus far to projective schemes.

Lemma 6 (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 recall the definition of a dualizing sheaf from (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)′.

By (III, 7.1), X = ℙ_{k}^{n} has a dualizing sheaf ω_{X} = O_{X}(−n − 1). (This is easily established using explicit calculations of the cohomology of projective space, see (III, 5).)

Theorem 7. Let X be a projective scheme of dimension n over a perfect field k. Then X has a dualizing sheaf.

Remark: We know from (III, 7.2) that any two dualizing sheaves are canonically isomorphic so we will refer to “the” dualizing sheaf.

Proof. Let f : X → ℙ^{n} be a finite morphism, and define

ω_{X} := f^{!}ω_{ℙ^{n}} = Hom_{ℙ^{n}}(f_{∗}O_{X}, ω_{ℙ^{n}})˜ .

Note that since ω_{ℙ^{n}} is locally free, Lemma 2 applies and ω_{X} = f^{!}O_{Y} ⊗_{O_{X}} f^{∗}ω_{ℙ^{n}}. Now ω_{X}, being the tensor product of two coherent sheaves, is coherent. Then we have by (4) that

Hom_{X}(F, ω_{X}) ≅ Hom_{ℙ^{n}}(f_{∗}F, ω_{ℙ^{n}})→^{∼}H^{n}(ℙ^{n}, f_{∗}F)′ ≅ H^{n}(X, F)′

where the last isomorphism comes from (III, Ex. 4.1). So Hom_{X}(F, ω_{X})→^{∼}H^{n}(X, F)′, and if we take F = ω_{X}, then the element 1 ∈ Hom(ω_{X}, ω_{X}) gives a homomorphism t : H^{n}(X, ω_{X}) → k which serves as a trace map. By functoriality, (ω_{X}, t) is a dualizing sheaf for X. ■

We’ll need the following lemma.

Lemma 8. Let f : X → Y be a finite morphism of noetherian projective schemes with X irreducible and Y nonsingular and irreducible. Then f is flat iff X is Cohen-Macaulay.

Proof. [3, Exer. 18.17 (or Cor. 18.17)]. ■

We can now state and prove the following theorem.

Theorem 9 (Serre duality). Let X be an irreducible projective scheme of dimension n over a perfect field k. Let ω_{X} be the dualizing sheaf on X.

(a) For all i ≥ 0 and all F ∈ ℭ𝔬𝔥(X), there are natural functorial maps

θ^{i} : Ext^{i}(F, ω_{X}) → H^{n−i}(X, F)′

such that θ^{0} is the isomorphism given in the definition of a dualizing sheaf.

(b) Moreover, the following conditions are equivalent: (i) there exists a finite flat morphism f : X → ℙ^{n}; (ii) every finite morphism f : X → ℙ^{n} is flat; (iii) X is Cohen-Macaulay.

(c) Finally, the above three equivalent conditions all imply: (iv) The maps θ^{i} of (a) are isomorphisms for all i ≥ 0 and all F ∈ ℭ𝔬𝔥(X).

Proof. The following commutative diagram defines θ^{i}:

Ext_{X}^{i}(F, ω_{X}) ⟶^{θ^{i}} H^{n−i}(X, F)′
ϕ^{i}↓  ↑≅
Ext_{ℙ^{n}}^{i}(f_{∗}F, ω_{ℙ^{n}}) ⟶^{≅} H^{n−i}(ℙ^{n}, f_{∗}F)′

where the ϕ^{i} are the maps coming from Proposition 4, and the bottom isomorphism comes from Serre duality for ℙ^{n} (III, 7.1). By Proposition 5, we know that the maps ϕ^{i} being isomorphisms for all i ≥ 0 and all F ∈ ℭ𝔬𝔥(X) is implied by the existence of a flat morphism f : X → ℙ^{n}. This proves (a) and (c).

(ii)⇒(i) is trivial, since Lemma 6 guarantees the existence of a finite morphism from X to ℙ^{n}. Then Lemma 8 tells us that (i)⇒(iii) and that (iii)⇒(ii). ■

3. The dualizing sheaf

Now that we have established Serre duality, 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 (II, Example 8.20.1) that for ℙ^{n}, ^{⋀n}Ω_{ℙ^{n}} is isomorphic to O_{ℙ^{n}}(−n−1) which we have already seen is the dualizing sheaf. We need a definition and a lemma first.

Definition 10. 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 1 in X, and ζ denotes the generic point of Z.

Lemma 11. 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 D = R 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 12. Let f : X → Y be a finite separable morphism of degree d 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) (9)

and

f^{!}Ω_{Y}^{n} ≅ f^{∗}Ω_{Y}^{n} ⊗ L(R) (10)

for the same “ramification divisor” R.

Let us first try to prove (9). 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 11, proving (9). (10) will be proved as Lemma 15. ■

In order to prove Lemma 15, 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 [2, Chapter VI, Remark 6.5].)

By (4), 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 13. 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 [3, Ex. 19.6(c)]. ■

Proposition 14. \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.

We now have f^{!}O_{Y} = \underline{Hom}_{A}(B, A)˜ , and 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 (11)

corresponds to an exact sequence of B-modules

B ⟶^{\overline{Tr}} B^{∗} ⟶ M ⟶ 0, (12)

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 13 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 13(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 (11):

0 ⟶ O_{X} ⟶ f^{!}O_{Y} ⟶ F ⟶ 0. (13)

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

(14) 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 13(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. The reader may find it amusing to try to come up with an example where the results of [7] are not sufficient.

We have all the components in place now for the proof of (10):

Lemma 15. Letting R be the ramification divisor of F (or Ω_{X/Y}),

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

Proof. Using the short exact sequence (13) we obtain, by Lemma 11,

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 2 we get

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

This establishes (10), so the proof of Theorem 12 is now complete.

Corollary 16. 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 6. Then ω_{X} ≅ f^{!}ω_{ℙ^{n}} by the proof of Theorem 7. But ω_{ℙ^{n}} ≅ Ω_{ℙ^{n}}^{n}. The result now follows from Theorem 12. ■

References

[1] William A. Adkins and Steven H. Weintraub: Algebra, An Approach via Module Theory, Graduate Texts in Mathematics 136, Springer-Verlag, 1992

[2] Allen Altman and Steven Kleiman, Introduction to Grothendieck Duality Theory, Lecture Notes in Mathematics 146, Springer-Verlag, 1970

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

[4] Alexander Grothendieck and Jean Dieudonné: Eléments de Géométrie Algébrique II, Publications Mathématiques 8, I.H.E.S., 1961

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

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

[7] Jean-Pierre Serre: Local Fields, Graduate Texts in Mathematics 67, Springer-Verlag, 1979 [8] Bart de Smit: The different and differentials of local fields with imperfect residue fields, to

appear in Edinburgh Math. Soc.

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

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