J. ALGEBRAIC GEOMETRY 00 (XXXX) 000–000 S 1056-3911(XX)0000-0

REFLEXIVE PULL-BACKS AND BASE EXTENSION

BRENDAN HASSETT AND SÁNDOR J. KOVÁCS

Abstract We prove that Viehweg’s moduli functor of stable surfaces is locally closed.

1. Introduction

The moduli theory of curves has been studied extensively in the past few decades. A very important and useful feature of the theory is that the moduli space of smooth projective curves of genus g admits a geometrically meaningful compactification as the moduli space of stable curves of genus g. The success of moduli theory of curves leads naturally to a desire for a similar theory for higher dimensional varieties.

In recent years there has been great progress in the moduli theory of surfaces and higher dimensional varieties by Alexeev, Kollár, Shepherd-Barron, and Viehweg [1]; [11]; [13]; [14]. According to their work, moduli spaces exist for many moduli problems, in particular for smooth canonically polarized varieties. More generally it is established that if a moduli problem satisfies certain properties, then a corresponding (coarse) moduli space exists. The most important of these properties are separatedness, boundedness and local closedness. According to the above authors’ work the former two of these hold for the moduli problem of canonically polarized stable surfaces – the candidate for a geometrically meaningful compactification of the moduli space of smooth canonically polarized surfaces. Local closedness, however, has presented a very stubborn problem.

In fact, one of the main problems is that it is not entirely clear what the “right” definition of the moduli functor should be. This is a very delicate problem as one would like to make the functor large enough to obtain a compact moduli space, but enlargening the class too much could lead to a loss of separatedness and/or boundedness.

Received March 30, 2001. The first author was supported in part by an NSF Postdoctoral Fellowship, NSF Grant DMS-0070537, and the Institute of Mathematical Sciences of the Chinese University of Hong Kong. The second author was supported in part by NSF Grants DMS-019607 and DMS-0092165.

In addition, not only the admissible models have to be decided, but also the admissible families of those models. Experts generally agree on what models should be allowed (the semi log canonical models). However, the right notion of admissible families is still to be decided.

Both Kollár and Viehweg suggest reasonable definitions, but local closedness has yet to be established for either of their moduli functors. At this time it is not even clear whether their definitions differ. However, we should point out that Kollár’s moduli functor is known to satisfy a weak form of local closedness. Precisely, after passage to a formal or étale local ring, local closedness holds provided we restrict to base change morphisms arising from local ring homomorphisms [12], §14.

The goal of this paper is to prove that Viehweg’s moduli functor of canonically polarized varieties is locally closed.

Definitions and notation. Every scheme is considered to be of finite type over an algebraically closed field k unless specifically noted otherwise.

Let f : X → S be a morphism. Then X_{s} denotes the fibre of f over the point s ∈ S and f_{s} denotes the restriction of f to X_{s}. More generally, for a morphism α : T → S, let f_{T} : X_{T} = X ×_{S} T → T. In particular one has the following commutative diagram:

X_{T} = X ×_{S} T ⟶^{α_{X}} X
f_{T} ↓  ↓^{f}
T  ⟶^{α} S

For a coherent O_{X}-module F, F_{T} will denote α_{X}^{∗}F on X_{T}. Tensor products of O_{X_{T}} -modules are over O_{X_{T}}. These conventions will be used through the entire article.

We will write F^{∗} for the dual O_{X}-module Hom_{X}(F, O_{X}) when there is no risk of confusion. The double dual F^{∗∗} is called the reflexive hull of F and there is a natural O_{X}-module homomorphism

F → F^{∗∗};

F is said to be reflexive if this is an isomorphism. We shall also consider reflexive powers

F^{[m]} := (F^{⊗m})^{∗∗} F^{[−m]} := (F^{⊗m})^{∗}

for m > 0. In general, there exist natural maps

(F^{∗∗})_{T} → (F_{T})^{∗∗} and (F^{[m]})_{T} → (F_{T})^{[m]}

which need not be isomorphisms, even when F is reflexive. Of course, these maps are isomorphisms when F is locally free. Acknowledgements: We both owe a great debt to János Kollár for patiently answering our questions about the moduli problems and technical issues addressed in this paper. We would also like to thank Eckart Viehweg for many useful discussions and for inviting the second author to visit Universität Essen. The first author benefitted from conversations with Dan Abramovich, David Hyeon, and Rahul Pandharipande.

2. Moduli functors

Definition 2.1. Fix a base scheme B. The moduli functor of polarized proper schemes is the contravariant functor

𝔐𝔓 : B-schemes → Sets

given by

𝔐𝔓(S/B) := \begin{cases}pairs (f : X → S, L), where \\ f is a flat and proper morphism, \\ L is an f-ample line bundle on X \end{cases}\} / ∼

where two families (f_{1} : X_{1} → S, L_{1}) and (f_{2} : X_{2} → S, L_{2}) are equivalent [(f_{1} : X_{1} → S, L_{1}) ∼ (f_{2} : X_{2} → S, L_{2})] iff there is an isomorphism h : X_{1} → X_{2} such that f_{1} = f_{2} ∘ h and there is a line bundle M on S such that L_{1} ≅ h^{∗}L_{2} ⊗ f_{1}^{∗}M. For any morphism of B-schemes, α : T → S, we have

𝔐𝔓(α)(X → S, L) = (X_{T} → T, α_{X}^{∗}L).

In this article we will restrict to the case of B = k an algebraically closed field. Sch_{k} will denote the category of k-schemes.

Any subfunctor of this moduli functor will be called a moduli functor of polarized proper schemes.

Definition 2.2. A subfunctor 𝔉 ⊂ 𝔐𝔓 is called locally closed iff the following condition is satisfied:

For every (f : X → S, L) ∈ 𝔐𝔓(S) there is a locally closed subscheme j : S^{u} ↪ S such that if α : T → S is any morphism then

(f_{T} : X_{T} → T, α_{X}^{∗}L) ∈ 𝔉(T) iff there is a factorization T →^{α} S^{u} →^{j} S.

We say that 𝔉 ⊂ 𝔐𝔓 is open iff S^{u} ⊂ S is open for every S.

For the definition of bounded, separated, and complete moduli functors the reader is referred to [14], 1.15.

Definition 2.3. Fix a polynomial h ∈ ℚ[t] be such that h(ℤ) ⊆ ℤ. The moduli functor of polarized schemes with Hilbert polynomial h is the subfunctor 𝔐𝔓_{h} of 𝔐𝔓 given by:

𝔐𝔓_{h}(S) = {(f : X → S, L) ∈ 𝔐𝔓(S) | χ(L_{X_{s}}^{ν}) = h(ν) for all ν ∈ ℤ and s ∈ S}.

This is an open and closed subfunctor.

Definition 2.4. A subfunctor 𝔐^{[N]} ⊂ 𝔐𝔓 is called a moduli functor of canonically polarized ℚ-Gorenstein schemes of index N if, for each (f : X → S, L) ∈ 𝔐^{[N]}(S),

(2.4.1) X_{s} is connected, Cohen-Macaulay, and Gorenstein outside a closed subscheme of codimension at least two for each s ∈ S;

(2.4.2) f is equivalent to one of the form (f : X → S, ω_{X/S}^{[N]}).

Remark 2.5. Assumption 2.4.1 implies the fibers are equidimensional projective schemes. One can show that the relative Cohen-Macaulay condition is open (see [5] EGAIV_{3}, 12.2.1), as is the locus where the relative dualizing sheaf is locally free (the relative Gorenstein locus). Since the complement to the relative Gorenstein locus intersects the fibers in codimension ≤ 2 over an open subset of the base, it follows that Assumption 2.4.1 is open. Note that the singularity assumptions may also be expressed as a condition on the morphism f: its relative dualizing complex is supported in degree −dim(X/S) and the resulting dualizing sheaf is locally free over an open subset whose complement meets each fiber in codimension two. We refer the reader to [3] for a recent account of relative duality.

We emphasize that for families of canonically polarized ℚ-Gorenstein schemes of index N, ω_{X/S}^{[N]} is invertible by definition. This is a condition on the morphism, not just a condition on the fibers. Indeed, ω_{X/S}^{[N]} may fail to be invertible even when ω_{X_{s}}^{[N]} is invertible for each s ∈ S (see [13]). Also, it is not entirely obvious that Assumption 2.4.2 actually yields a subfunctor, i.e., that families of canonically polarized schemes pull back to families of canonically polarized schemes. This is proved in the following lemma:

Lemma 2.6. Given a family of canonically polarized ℚ-Gorenstein schemes of index N, f : X → S, and a morphism α : T → S, we have

α_{X}^{∗}ω_{X/S}^{[N]} ≃ ω_{X_{T} /T}^{[N]} .

Proof. Let U ⊂ X be the relative Gorenstein locus of f, i.e., the largest open subset U of X such that U_{s} is Gorenstein for all s ∈ S or equivalently the largest open subset U of X such that ω_{X/S}|_{U} is a line bundle. Then ω_{X/S}^{[N]}|_{U} ≃ ω_{U/S}^{N} and hence

α_{X}^{∗}ω_{X/S}^{[N]}|_{α_{X}^{−1}U} ≃ α_{X}^{∗}ω_{U/S}^{N} ≃ ω_{α_{X}^{−1}U/T}^{N} ≃ ω_{X_{T} /T}^{[N]} |_{α_{X}^{−1}U}.

Now codim(U_{s}, X_{s}) ≥ 2 for all s ∈ S, so codim((α_{X}^{−1}U)_{t}, (X_{T})_{t}) ≥ 2 for all t ∈ T. Finally α_{X}^{∗}ω_{X/S}^{[N]} and ω_{X_{T} /T}^{[N]} are reflexive, so since they are isomorphic on α_{X}^{−1}U, they are isomorphic on X_{T} (cf. 3.6.2). □

If 𝔐^{[N]} is a functor of canonically polarized ℚ-Gorenstein schemes of index N then we can consider 𝔐_{h}^{[N]} as well. An argument using 3.6 (and very similar to Lemma 2.6) implies

𝔐_{h}^{[N]}(S) = {(f : X → S) ∈ 𝔐^{[N]}(S) | χ(ω_{X_{s}}^{[ν·N]}) = h(ν)
for all ν ∈ ℤ and s ∈ S.}.

Remark 2.7. Note that we speak of “a” moduli functor and not “the” moduli functor. The reason is that in order to obtain a relatively nice moduli space one has to restrict to a smaller class than all the canonically polarized ℚ-Gorenstein schemes of index N. On the other hand one could consider “the” moduli space of smooth varieties, but in that case one would not obtain a compact moduli space. The “right” class of schemes will be somewhere between these two and part of the difficulty is to identify that class.

Assumptions 2.8. Assume the following:

(2.8.1) 𝔐_{h}^{[N]} is locally closed; (2.8.2) 𝔐_{h}^{[N]} is bounded; (2.8.3) 𝔐_{h}^{[N]} is separated; (2.8.4) 𝔐_{h}^{[N]} is complete; (2.8.5) for all smooth projective curves S, and for all (f : X → S) ∈

𝔐_{h}^{[N]}(S), the sheaf f_{∗}ω_{X/S}^{[ν·N]} is semi-positive for all ν sufficiently large and divisible.

Theorem 2.9. [10], 4.2.1; [13], 5.7; [11], 5.6; [14], 9.23, 9.30 Assume that k has characteristic zero. We retain the notation introduced above and assume that 𝔐_{h}^{[N]} satifies the conditions of 2.8. Let ν > 0 be a fixed integer such that ω_{X}^{[ν·N]} is very ample and without higher cohomology for all X ∈ 𝔐_{h}^{[N]}(k).

Then there exists a coarse algebraic moduli space M_{h}^{[N]} for 𝔐_{h}^{[N]} which is a projective scheme and for all μ ≫ 0 there exist a p > 0 and an ample invertible sheaf λ_{μ·ν·N}^{(p)} on M_{h}^{[N]}, such that for all (f : X → S) ∈ 𝔐_{h}^{[N]}(S) and for the

induced morphism ϕ : S → M_{h}^{[N]} one has ϕ^{∗}λ_{μ·ν·N}^{(p)} = (det f_{∗}ω_{X/S}^{[μ·ν·N]})^{p}.

Viehweg’s Functor. Property V^{[N]} Consider a family of polarized varieties, f : X → S, satisfying Assumption 2.4.1. It will be said that f satisfies property V^{[N]} if ω_{X/S}^{[N]} is invertible.

Note that families of canonically polarized ℚ-Gorenstein schemes of index N automatically satisfy property V^{[N]} (by Assumption 2.4.2).

Definition 2.10. Let 𝔙_{h}^{[N],d} be the moduli functor of canonically polarized ℚ-Gorenstein schemes of index N and Hilbert polynomial h satisfying the following:

(2.10.1) for each s ∈ S, X_{s} is a reduced scheme of dimension d and has

semi log canonical singularities. We emphasize that we are retaining Assumptions 2.4.1 and 2.4.2. Note

that each fiber X_{s} automatically has index N.

Let N′ = mN be a positive multiple of N and h′(t) = h(mt). There is a natural transformation,

𝔙_{h}^{[N],d} → 𝔙_{h′}^{[N′],d},

induced by taking the mth power of the canonical polarization.

Kollár’s Functor. Property K Consider a family of polarized varieties, f : X → S, satisfying Assumption 2.4.1. It will be said that f satisfies property K if

α_{X}^{∗}ω_{X/S}^{[j]} ≃ ω_{X_{T} /T}^{[j]}

for any morphism, α : T → S, and each j ∈ ℤ.

For canonically polarized ℚ-Gorenstein schemes of index N, it suffices to verify this for j = 1, …, N − 1. Indeed, 3.6 yields

ω_{X/S}^{[j+νN]} = ω_{X/S}^{[j]} ⊗ (ω_{X/S}^{[N]})^{ν}.

Definition 2.11. Let 𝔎_{h}^{[N],d} be the moduli functor of canonically polarized ℚ- Gorenstein schemes of index N and Hilbert polynomial h satisfying the following:

(2.11.1) for each s ∈ S, X_{s} is a reduced scheme of dimension d and has

semi log canonical singularities; (2.11.2) each family (f : X → S, ω_{X/S}^{[N]}) ∈ 𝔎_{h}^{[N],d}(S) satisfies property K.

If a family satisfies property K, then the family is in 𝔎_{h}^{[N],d} if the indices of the fibers all divide N. Let N′ = mN be a positive multiple of N and h′(t) = h(mt). Then the natural transformation,

𝔎_{h}^{[N],d} → 𝔎_{h′}^{[N′],d},

induced by taking the mth power of the canonical polarization, is an open immersion.

These conditions are stronger than those of Viehweg’s functor, so there is a natural transformation of moduli functors,

𝔎_{h}^{[N],d} → 𝔙_{h}^{[N],d},

inducing a bijection between 𝔎_{h}^{[N],d}(k) and 𝔙_{h}^{[N],d}(k).

Moduli of Surfaces: Smoothability and Boundedness.

Definition 2.12. Let 𝔙_{h,sm}^{[N],2}(k) denote the following subset of 𝔙_{h}^{[N],2}(k):

𝔙_{h,sm}^{[N],2}(k) = {X | X ∈ 𝔙_{h}^{[N],2}(k), and ∃(g : Y → C) ∈ 𝔙_{h}^{[N],2}(C), such that
C is an irreducible curve, X ≃ X_{c} for some c ∈ C, and

X_{gen} is a normal surface with at most rational double points.}

We define 𝔎_{h,sm}^{[N],2}(k) analogously.

Once we construct the moduli schemes V_{h}^{[N],2} and K_{h}^{[N],2}, we may realize 𝔙_{h,sm}^{[N],2}(k) and 𝔎_{h,sm}^{[N],2}(k) as the closed points of certain subvarieties. The points satisfying the smoothability condition form a union of irreducible components, and this union forms a closed subvariety. However, this subvariety does not necessarily admit a natural scheme structure.

Remark 2.13. Assume that k has characteristic zero for the remainder of this subsection. [1], 5.11 implies that there exists an N ∈ ℕ such that

𝔙_{h}(k) := ⋃_{m∈ℕ} 𝔙_{h(mt)}^{[m],2}(k) = 𝔙_{h(Nt)}^{[N],2}(k)

and

𝔙_{h,sm}(k) := ⋃_{m∈ℕ} 𝔙_{h(mt),sm}^{[m],2}(k) = 𝔙_{h(Nt),sm}^{[N],2}(k).

In order to construct moduli spaces for 𝔙_{h}^{[N],d} and 𝔎_{h}^{[N],d}, one has to verify the assumptions of 2.8. All the properties listed in 2.8, except 2.8.1, are the same for both 𝔙_{h}^{[N],d} and 𝔎_{h}^{[N],d}.

• [10], 2.1.2 implies 2.8.2.

• [13], 5.1 implies 2.8.3 and 2.8.4, at least for the irreducible components satisying the smoothability condition. For the general case, one has to construct a unique stable limit for a one-parameter family of nonnormal stable surfaces. Consider its normalization as a family of stable log surfaces with boundary equal to the conductor. Apply the log minimal model program and the results of [7] to obtain a unique limiting stable log surface. We glue back together along the conductor to recover the stable limit of our original family.

• [11], 4.12 implies 2.8.5.

That leaves us to verify 2.8.1, and in the rest of the article we will concentrate on this property.

Proof of Local Closedness. To prove that 𝔙_{h}^{[N],d} is locally closed, one would naturally list the properties of the functor and prove one by one that all of them are locally closed. However, this requires special attention. A potentially tricky part is that the order of this procedure matters. For instance, the requirement that ω_{X/S}^{[N]} be invertible should not be considered until only open properties are left, because it may very well happen that ω_{X/S}^{[N]} is not invertible along an admissible fiber X_{s}, but ω_{X_{T} /T}^{[N]} becomes invertible after restricting to some locally closed T ⊂ S containing s. In particular the locus where ω_{X/S}^{[N]} is invertible does not coincide with the locus where ω_{X_{s}}^{[N]} is invertible. The key problem is: taking reflexive powers does not generally commute with base extension.

The next theorem is the main result of this article. Here we reduce local closedness to a rather technical statement which will be proved in the next section.

Theorem 2.14. The moduli functor of canonically polarized ℚ-Gorenstein schemes of index N is locally closed.

Proof. In proving local closedness, we address the conditions imposed on the fibers of f : X → S separately from the conditions imposed on the morphism f itself. We have already observed in 2.5 that condition 2.4.1 is open. Now we turn to condition 2.4.2, i.e., ω_{X/S}^{[N]} is locally free. Suppose we are given an arbitrary family of polarized varieties (f : X → S, L) with fibers satisfying 2.4.1. We apply 3.11 with F = ω_{X/S}^{⊗N}. This sheaf may be terribly singular, perhaps even with torsion along certain fibers. However, ω_{X/S}^{⊗N} does have one salient property: it commutes with arbitrary base extensions α : T → S, i.e.,

α_{X}^{∗}ω_{X/S}^{⊗N} = ω_{X_{T} /T}^{⊗N} .

By 3.11 there exists a locally closed subscheme S^{u} ⊂ S with the following universal property. Given a morphism α : T → S, there exists an invertible sheaf N on T and an isomorphism

(ω_{X_{T} /T}^{⊗N})^{∗∗} ⟶^{≃} L_{T} ⊗ f_{T}^{∗}N

if and only if α factors through S^{u}. By definition we have

ω_{X_{T} /T}^{[N]} = (ω_{X_{T} /T}^{⊗N})^{∗∗},

so the proof that condition 2.4.2 is closed is complete. □

Theorem 2.15. If k is a field of characteristic zero then 𝔙_{h}^{[N],2} is a locally closed moduli functor. In particular, 𝔙_{h,sm}^{[N],2}(k) is locally closed.

Proof. It remains to verify that condition 2.10.1 is locally closed once conditions 2.4.1 and 2.4.2 are imposed. In particular, we may assume that we have families of canonically polarized ℚ-Gorenstein varieties of index N.

The condition that the geometric fibers X_{s} are reduced is open by [5] EGAIV_{3}, 12.2.1. The locus where the fibers have semi log canonical singularities is open by [8], 2.6 (see also [13],§5). □

Remark 2.16. (characteristic zero) If one assumes the existence of minimal models in dimension d + 1, the results of [8] imply that having semi log canonical singularities is an open condition for families of canonically polarized ℚ-Gorenstein varieties of index N. It follows that 𝔙_{h}^{[N],d} is locally closed.

3. Local closedness of reflexive pull-backs

We first recall the following result from [5] EGAIV, §6.3:

Proposition 3.1. Let A and B be noetherian local rings, k the residue field of A, ϕ : A → B a local homomorphism, M an A-module of finite type, and N a B-module of finite type. If N is a flat and nonzero as an A-module then

depth_{B}(M ⊗_{A} N) = depth_{A}(M) + depth_{B⊗_{A}k}(N ⊗_{A} k).

We shall assume that all schemes are noetherian.

Let f : X → S be a flat morphism of schemes and E a coherent O_{X}-module flat over S. We shall say E is S_{r} relative to f if the following holds: for each x ∈ X, s = f(x), F = X_{s} we have

depth_{O_{F,x}}(E|_{F}) ≥ min(r, dim O_{X,x} − dim O_{S,s}).

In other words, the restriction of E to each fiber is S_{r}. By 3.1, this is equivalent to

depth_{O_{X,x}}(E) ≥ depth_{O_{S,s}}(O_{S,s}) + min(r, dim O_{X,x} − dim O_{S,s}).

Two special cases deserve further attention. If E = O_{X} then our definition coincides with the ordinary definition of an S_{r} morphism (see [5] EGAIV, §6). If S = Spec(K), where K is a field, then we recover a notion of an S_{r} sheaf. For example, a sheaf is S_{1} provided it has no imbedded points. We can translate the definition of S_{r}-sheaves using the cohomological interpretation of depth (see [6], §III.3 Ex. 4).

Proposition 3.2. Let f : X → S be a flat morphism of schemes and E a coherent O_{X}-module flat over S. Then E is S_{r} relative to f if and only if, for each x ∈ X, s = f(x), we have

min{i : H_{x}^{i}(E) ≠ 0} ≥ depth_{O_{S,s}}(O_{S,s}) + min(r, dim O_{X,x} − dim O_{S,s}).

Here H_{x}^{i}(E) denotes cohomology on Spec(O_{X,x}) with support at the closed point and coefficients in E_{x}. If Z ⊂ X is closed, we use H_{Z}^{i} to denote the local cohomology sheaf associated to cohomology on X with support along Z.

Proposition 3.3. Let f : X → S be a flat morphism of schemes and E a coherent O_{X}-module flat over S and S_{r} relative to f. Let Z ⊂ X be a closed subscheme with ideal sheaf I_{Z}. Assume that codim(Z_{s}, X_{s}) ≥ r for each s ∈ S. Then we have depth_{O_{X}} (I_{Z}, E) ≥ r, or equivalently,

H_{Z}^{k}(E) = 0 for k = 0, …, r − 1

Proof. The cohomological interpretation of depth gives the equivalence of the two conclusions.

For each point x ∈ Z we have

H_{x}^{0} = H_{x}^{0} ∘ H_{Z}^{0}

which induces a spectral sequence

E_{2}^{p,q} := H_{x}^{p} ∘ H_{Z}^{q} ⟹ H_{x}^{p+q}.

The proof is by induction on k, starting with k = 0. Assume that H_{Z}^{0}(E) ≠ 0 and its support contains a point x ∈ Z. We have H_{x}^{0}(H_{Z}^{0}(E)) ≠ 0 and thus H_{x}^{0}(E) ≠ 0. Writing s = f(x), we obtain a contradiction of 3.2. Now assume that H_{Z}^{i}(E) = 0 for i = 0, …, k − 1 where k < r, but that H_{Z}^{k}(E) ≠ 0 and its support contains x ∈ Z. It follows that H_{x}^{k}(E) = H_{x}^{0}(H_{Z}^{k}(E)) ≠ 0, which again contradicts 3.2. □

Corollary 3.4. Let f : X → S be a flat S_{r} morphism, and Z ⊂ X a subscheme such that codim(Z_{s}, X_{s}) ≥ r for each s ∈ S. Then grade(I_{Z}) ≥ r.

Proposition 3.5. Let f : X → S be a flat morphism of schemes, E a coherent sheaf flat over S and S_{2} relative to f, and j : U ↪ X an open subscheme with complement Z. Assume that codim(Z_{s}, X_{s}) ≥ 2 for each s ∈ S. Then the natural map

E → j_{∗}(E|_{U})

is an isomorphism.

Proof. The long exact sequence

(3.5.1)  0 → H_{Z}^{0}(E) → E → j_{∗}(E|_{U}) → H_{Z}^{1}(E) → 0

yields the isomorphism. □

Proposition 3.6. Let f : X → S be a flat S_{2} morphism, F a reflexive coherent O_{X}-module. Let Z ⊂ X be a closed subscheme so that codim(Z_{s}, X_{s}) ≥ 2 for each s ∈ S, and let U be the complement of Z.

(3.6.1) Then H_{Z}^{k}(F) = 0 for k = 0, 1 and the natural map F → j_{∗}(F|_{U})

is an isomorphism. (3.6.2) Let F′ be another coherent O_{X}-module which is either S_{2} relative

to f or reflexive. If F|_{U}≃F′|_{U} then F ≃ F′. Proof. Consider a presentation of F^{∗} by locally free sheaves

E_{2} → E_{1} → F^{∗} → 0.

On dualizing we obtain

0 → F → E_{1}^{∗} → E_{2}^{∗}.

Since the E_{i}^{∗} are locally free and f is S_{2}, 3.3 yields

H_{Z}^{k}(E_{i}) = 0 for k = 0, 1.

Taking the associated long exact sequences, we obtain the desired vanishing for F. The long exact sequence in local cohomology (cf.3.5.1) yields the first isomorphism. The isomorphism between F and F′ is obtained by pushing forward. □

We obtain a criterion for when push-forwards of reflexive sheaves are reflexive:

Corollary 3.7. Let f : X → S be a flat S_{2} morphism and j : U ↪ X an open subscheme with complement Z. Assume that codim(Z_{s}, X_{s}) ≥ 2 for each s ∈ S. If G is a reflexive coherent sheaf on U then j_{∗}G is also reflexive and coherent.

Proof. Choose a coherent subsheaf E ⊂ j_{∗}G so the induced map E|_{U} → G is an isomorphism [6], §II.5 Ex.15. The reflexive hull E^{∗∗} is also coherent and we have E^{∗∗}|_{U} ≃ G. An application of 3.6.2 implies

E^{∗∗} → j_{∗}(E^{∗∗}|_{U}) ≃ j_{∗}G

is an isomorphism. □ We also obtain the following (cf. [2], 1.4.1): Corollary 3.8. Let f : X → S be a flat S_{2} morphism, E a coherent sheaf

flat over S and S_{2} relative to f, and j : U ↪ X an open subscheme with complement Z. Assume that codim(Z_{s}, X_{s}) ≥ 2 for each s ∈ S and E|_{U} is reflexive. Then E is reflexive, and furthermore, for each α : T → S the pull-back E_{T} is reflexive.

Proof. We apply 3.6.2 to show that the natural map E → E^{∗∗} is an isomorphism. Our hypotheses are preserved under base extension, so E_{T} is reflexive for each α : T → S. □

Suppose that f : X → S is a flat projective Cohen-Macaulay morphism of relative dimension d. Then the relative dualizing sheaf ω_{X/S} exists, commutes with base extension, and is S_{d} relative to f [9], §9,21; [3], 3.6.1; [4], 21.8. In light of our previous results, it is natural to compare the relative dualizing sheaf with the reflexive hull of a coherent sheaf.

Theorem 3.9. Let f : X → S be a flat projective Cohen-Macaulay morphism of relative dimension d with geometrically connected fibers. Let G be a coherent sheaf on X, and U ↪ X an open subset with complement Z so that codim(Z_{s}, X_{s}) ≥ 2 for each s ∈ S. Assume that ω_{X/S} and G are locally free on U. Then there exists a locally closed subset S^{u} ⊂ S with the following property. Given a morphism α : T → S, there exists an invertible sheaf N on T and an isomorphism

(3.9.1)  (G_{T})^{∗∗} ⟶^{≃} ω_{X_{T} /T} ⊗ f_{T}^{∗}N

if and only if α factors through S^{u}.

Proof. We produce a subset S^{u} ⊂ S over which the isomorphism 3.9.1 exists. Let W′ ⊂ S denote the subscheme supporting ℝ^{d}f_{∗}G. Note that W′ has a naturally defined scheme structure. Indeed, let P^{•} → G be a resolution of G by locally free O_{X}-modules. The proof of the cohomology and base change theorem [5] EGAIII, §7.7 produces a complex of locally free O_{S}-modules,

0 → E_{0} → E_{1} → … → E_{d−1} →^{ψ} E_{d} → 0

computing the direct image sheaves ℝ^{i}(f_{T})_{∗}P_{T}^{•} for any base extension T → S. Since the maximal fibre dimension of f is d, R^{j}(f_{T})_{∗} = 0 for all j > d. Furthermore, G is the highest non-zero cohomology sheaf of P^{•}, so

ℝ^{d}f_{∗}P^{•} ≃ ℝ^{d}f_{∗}G.

In particular, ℝ^{d}f_{∗}G is the cokernel of ψ and we define W′ using the rank(E_{d})- minors of ψ. It also follows that the formation of ℝ^{d}f_{∗}G commutes with base extension, i.e.,

(3.9.2)  (ℝ^{d}f_{∗}G)_{s} → H^{d}(X_{s}, G_{s})

is an isomorphism for each s ∈ S.

Let W ⊂ W′ be the locally closed subset obtained by removing points where rank(ψ) < rank(E_{d}) − 1 and f_{W} : X_{W} → W the corresponding morphism. Hence M := ℝ^{d}(f_{W})_{∗}G_{W} = (ℝ^{d}f_{∗}G)_{W} is locally free of rank one. The relative duality theorem of Kleiman [9], §10,21 gives an isomorphism of O_{W} -modules

(3.9.3)  Hom_{f_{W}} (G_{W} , ω_{X_{W} /W} ⊗ f_{W}^{∗} M) ⟶^{≃} Hom_{W} (M, M) = O_{W} .

The identity 1 ∈ O_{W} gives a natural homomorphism ϕ : G_{W} → ω_{X_{W} /W} ⊗ f_{W}^{∗} M which factors

G_{W}  ⟶^{ϕ} ω_{X_{W} /W} ⊗ f_{W}^{∗} M
↓  ∥
(G_{W})^{∗∗} ⟶^{ϕ^{∗∗}} ω_{X_{W} /W} ⊗ f_{W}^{∗} M

because ω_{X_{W} /W} is reflexive.

Let S^{u} ⊂ W be the open subset over which ϕ|_{U} is an isomorphism. The map ϕ^{∗∗} is an isomorphism over S^{u} by 3.6. For any S^{u}-scheme T, the map

(ϕ^{∗∗})_{T} : ((G_{W})^{∗∗})_{T} ⟶ (ω_{X_{W} /W})_{T} ⊗ f_{T}^{∗}M

induced by base extension is also an isomorphism. Since (ω_{X_{W} /W})_{T} = ω_{X_{T} /T} is flat and reflexive, the same holds for ((G_{W})^{∗∗})_{T}. Hence 3.6 guarantees that the natural map ((G_{W})^{∗∗})_{T} → (G_{T})^{∗∗} is an isomorphism.

It remains to show that S^{u} satisfies the universal property. Let T be an S-scheme, N an invertible sheaf on T, and

(3.9.4)  ρ^{∗∗} : (G_{T})^{∗∗} ⟶^{≃} ω_{X_{T} /T} ⊗ f_{T}^{∗}N

an isomorphism. For each t ∈ T, the natural map G_{t} → ((G_{T})^{∗∗})_{t} is an isomorphism over U_{t}, a subset with codimension ≥ 2 complement. We therefore obtain an isomorphism of cohomology groups

H^{d}(X_{t}, G_{t}) ⟶^{≃} H^{d}(X_{t}, ((G_{T})^{∗∗})_{t})

and the base-change isomorphism 3.9.2 yields

ℝ^{d}(f_{T})_{∗}G_{T} ⟶^{≃} ℝ^{d}(f_{T})_{∗}[(G_{T})^{∗∗}].

The composed morphism

ρ : G_{T} → (G_{T})^{∗∗} → ω_{X_{T} /T} ⊗ f_{T}^{∗}N

induces

ℝ^{d}(f_{T})_{∗}G_{T} ⟶^{≃} ℝ^{d}(f_{T})_{∗}[ω_{X_{T} /T} ⊗ f_{T}^{∗}N].

Since f_{T} is Cohen-Macaulay, relative duality yields an isomorphism

ℝ^{d}(f_{T})_{∗}ω_{X_{T} /T} ≃ Ext_{f_{T}}^{d} (O_{X_{T}} , ω_{X_{T} /T}) ⟶^{≃} Hom_{T} ((f_{T})_{∗}O_{X_{T}} , O_{T}) ≃ O_{T} ,

where the last isomorphism follows from the fact that f_{T} has geometrically connected fibers. It follows that

ℝ^{d}(f_{T})_{∗}G_{T} ≃ ℝ^{d}(f_{T})_{∗}[ω_{X_{T} /T} ⊗ f_{T}^{∗}N] ≃ ℝ^{d}(f_{T})_{∗}ω_{X_{T} /T} ⊗ N ≃ N ,

hence T → S factors as T → W ↪ S and N ≃ M_{T}. Applying duality again, we may regard ρ as an element of Hom_{T} (N , N) ≃ (Hom_{W} (M, M))_{T} and compare ρ and ϕ_{T} over T. The identification in the isomorphism 3.9.3 is functorial, hence ϕ_{T} corresponds to 1 ∈ Hom_{T} (N , N), ρ corresponds to some r ∈ O_{T}, and ρ = rϕ_{T}.

To complete the proof it suffices to check that r ∈ O_{T}^{∗}. Consider the restriction of the isomorphism 3.9.4 to the open subset U_{T} where the sheaves are all locally free

ρ|_{U_{T}} = ρ^{∗∗}|_{U_{T}} : (G_{T})|_{U_{T}} ⟶^{≃} (ω_{X_{T} /T} ⊗ f_{T}^{∗}N)|_{U_{T}} .

If r were not invertible at some t ∈ T then ρ|_{U_{T}} would have nontrivial cokernel over t, a contradiction. □

Corollary 3.10. Retain all the notation and hypotheses of 3.9. Assume in addition that G is S_{2}-relative to f. Then there exists a locally closed subset S^{u} ⊂ S with the following property. Given a morphism α : T → S, there exists an invertible sheaf N on T and an isomorphism

G_{T} ⟶^{≃} ω_{X_{T} /T} ⊗ f_{T}^{∗}N

if and only if α factors through S^{u}.

Proof. This follows from 3.9 and 3.8. □ Theorem 3.11. Let f : X → S be a flat projective Cohen-Macaulay mor-

phism of relative dimension d with geometrically connected fibers. Let L be an invertible sheaf on X, F a coherent sheaf on X, and U ↪ X an open subset with complement Z so that codim(Z_{s}, X_{s}) ≥ 2 for each s ∈ S. Assume that ω_{X/S} and F are locally free on U. Then there exists a locally closed subset S^{u} ⊂ S with the following property. Given a morphism α : T → S, there exists an invertible sheaf N on T and an isomorphism

(F_{T})^{∗∗} ⟶^{≃} L_{T} ⊗ f_{T}^{∗}N

if and only if α factors through S^{u}.

Proof. Without loss of generality we may assume that L is trivial (replace F by F ⊗ L^{−1}). Apply 3.9 with G = F ⊗ ω_{X/S}, so that (G_{T})^{∗∗} ≃ ω_{X_{T} /T} ⊗ f_{T}^{∗}N iff T factors through S^{u}. On the other hand, (G_{T})^{∗∗} ≃ ω_{X_{T} /T} ⊗ f_{T}^{∗}N if and only if G_{T} |_{U_{T}} ≃ (ω_{X_{T} /T} ⊗ f_{T}^{∗}N)|_{U_{T}}, which is the case exactly when F_{T} |_{U_{T}} ≃ f_{T}^{∗}N |_{U_{T}}, or equivalently, when (F_{T})^{∗∗} ≃ f_{T}^{∗}N (by 3.6). □

Remark 3.12. It is natural to try to generalize this result for more general sheaves L. The above argument is still valid provided L satisfies the following:

(3.12.1) L is S_{2} relative to f;

(3.12.2) L|_{U} is invertible.

Proof. We apply 3.9, with G = F ⊗ L^{∗} ⊗ ω_{X/S}. We obtain a locally closed subset S^{u} ⊂ S such that (G_{T})^{∗∗} ≃ ω_{X_{T} /T} ⊗ f_{T}^{∗}N iff T factors through S^{u}. Again, this is the case if and only if

(F ⊗ L^{∗} ⊗ ω_{X_{T} /T})_{T} |_{U_{T}} ≃ f_{T}^{∗}N |_{U_{T}} ,

which by Assumption 3.12.2 is equivalent to

F|_{U_{T}} ≃ L ⊗ f_{T}^{∗}N |_{U_{T}} ,

which in turn is equivalent to (F_{T})^{∗∗} ≃ (L_{T})^{∗∗} ⊗ f_{T}^{∗}N. Applying 3.8 along with Assumptions 3.12.1 and 3.12.2 yields that L_{T} is reflexive for each T, so the last isomorphism exists iff (F_{T})^{∗∗} ≃ L_{T} ⊗ f_{T}^{∗}N. □

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Math Department–MS 136, Rice University, 6100 S. Main St., Houston TX 77005-1892

E-mail address: hassett@math.rice.edu

University of Washington, Department of Mathematics, Box 354350, Seattle, WA 98103

E-mail address: kovacs@math.washington.edu