PFAFFIAN SUBSCHEMES

CHARLES H. WALTER

Abstract. A subscheme X ⊂ ℙ^{n+3} of codimension 3 is Pfaffian if it is the degeneracy locus of a skew-symmetric map f : E^{∨}(−t) → E with E a locally free sheaf of odd rank on ℙ^{n+3}. It is shown that a codimension 3 subscheme X ⊂ ℙ^{n+3} is Pfaffian if and only if it is locally Gorenstein, subcanonical (i.e. ω_{X} ≅ O_{X}(ℓ) for some integer ℓ), and the following parity condition holds: if n ≡ 0 (mod 4) and ℓ is even, then χ(O_{X}(ℓ/2)) is also even. A local analogue is also proved.

Also included is a modern version of a 1964 theorem of Horrocks stating that the derived category intermediate cohomology functor provides an equivalence between the stable category of vector bundles on ℙ^{N} and a certain full subcategory FinL ⊂ D^{b}(S-Mod) of the derived category of graded modules over S := k[X_{0}, …, X_{N}] with inverse the syzygy bundle functor.

In commutative algebra there exist a number of structural theorems describing the form of minimal free resolutions of particular types of ideals or modules over (to keep things simple) regular local rings or polynomial rings. For instance the Hilbert-Burch theorem (when translated to sheaves on projective space) says that any arithmetically Cohen-Macaulay subscheme Y ⊂ ℙ^{N} of codimension 2 has a locally free resolution of the form

0 → F ⟶^{α} G ⟶^{Λ^{r}α^{∨}} O_{ℙ^{N}} → O_{Y} (1)

with F = ⨁_{j=1}^{r} O_{ℙ^{N}}(b_{j}) and G = ⨁_{i=1}^{r+1} O_{ℙ^{N}}(a_{i}) direct sums of line bundles, with α a matrix of homogeneous polynomials, and such that the entries of the row vector Λ^{r}α^{∨} are the maximal minors of α. In codimension 3 the best known structural theorem is due to Buchsbaum and Eisenbud ([BE2] Theorem 2.1) and says that any arithmetically Gorenstein subscheme X ⊂ ℙ^{N} of codimension 3 has a locally free resolution of the form

0 → O_{ℙ^{N}}(−t) ⟶^{h} E^{∨}(−t) ⟶^{f} E ⟶^{g} O_{ℙ^{N}} → O_{X} (2)

1991 Mathematics Subject Classification. Primary 14M05, 14F05, 13C05. Key words and phrases. Codimension 3, subcanonical, Pfaffian, stable category,

derived category.

The author was supported in part by NSA research grant MDA904-92-H-3009.

with E = ⨁_{i=1}^{2p+1} O_{ℙ^{N}}(a_{i}) a direct sum of an odd number of line bundles, t an integer, f a skew-symmetric matrix of homogeneous polynomials, and g (resp. h = g^{∨}) a row vector (resp. column vector) whose entries are the Pfaffians of order 2p of f.

Unfortunately the hypotheses of these and similar theorems are too restrictive for many applications. After all an n-dimensional subscheme Y ⊂ ℙ^{N} is arithmetically Cohen-Macaulay only if H^{1}(I_{Y} (r)) = 0 for all r, and H^{i}(O_{Y} (r)) = 0 for all r and 0 < i < n. The subscheme Y is arithmetically Gorenstein only if it is an arithmetically Cohen-Macaulay and subcanonical, i.e. if its canonical sheaf satisfies ω_{Y} ≅ O_{Y} (ℓ) for some integer ℓ. These are very restrictive conditions. For instance embedded abelian varieties of dimension at least 2 are subcanonical but not arithmetically Cohen-Macaulay. So these theorems from commutative algebra will never apply to them nor to many other interesting varieties.

As a result, one would like to have generalizations of these and similar theorems which apply to subschemes of ℙ^{N} which are locally Cohen-Macaulay (or locally Gorenstein and subcanonical). Such generalizations would say that such subschemes would have locally free resolutions of the form of (1) or (2) with E, F, or G vector bundles but not necessarily direct sums of line bundles.

In codimension 2 such an algebro-geometric generalization of the Hilbert-Burch theorem is already well known and actually not difficult. Indeed given a locally Cohen-Macaulay subscheme Y ⊂ ℙ^{N} of codimension 2, one may construct a locally free resolution as in (1) as follows. Define G → O_{ℙ^{N}} by composing the inclusion I_{Y} ⊂ O_{ℙ^{N}} with a surjection β : G = ⨁ O_{ℙ^{N}}(a_{i}) ↠ I_{Y} corresponding to a minimal set of generators of the homogeneous ideal I(Y). Then set F = ker β. The vector bundle F is not in general a direct sum of line bundles. Instead it is an important algebraic invariant of Y. Indeed the entire method of studying and trying to classify codimension 2 subvarieties of ℙ^{N} using liaison may be considered to be based on a theorem of Rao ([R] Theorem 1.3) which says that the vector bundle F, up to twist and stable equivalence (i.e. adding and removing line bundle direct summands, see §2), describes precisely the even liaison class of the codimension 2 subscheme Y ⊂ ℙ^{N}. Moreover, the internal structure of an individual even liaison class is strongly influenced by properties of the bundle F according to the theory developed by Martin-Deschamps and Perrin [MDP].

In codimension 3 a similar program has yet to be carried out. Such a program would begin with an algebro-geometric generalization of the theorem of Buchsbaum and Eisenbud mentioned above. The need for such a generalization was raised in a recent paper of Okonek [O]. He introduced the terminology Pfaffian subscheme for a codimension 3 subscheme of ℙ^{N} (or of another ambient variety) which is the degeneracy locus of a skew-symmetric map f : E^{∨}(−t) → E such that E is a locally free sheaf of odd rank 2p + 1 on ℙ^{N}, and f is generically of rank 2p and degenerates to rank 2p − 2 in the expected codimension 3. The scheme X then has a locally free resolution of the form

0 → O_{ℙ^{N}}(−t − 2s) ⟶^{h} E^{∨}(−t − s) ⟶^{f} E(−s) ⟶^{g} O_{ℙ^{N}} → O_{X} (3)

where s = c_{1}(E) + pt, and where g and h = g^{∨}(−t − 2s) are given locally by the Pfaffians of order 2p of f. This resolution is the same as (2) up to twist and is just a patching together of the local version studied in [BE2]. The self-duality of the resolution (3) implies that X is locally Gorenstein with canonical sheaf ω_{X} ≅ O_{X}(t + 2s − N − 1). Thus Pfaffian subschemes are always locally Gorenstein of codimension 3 and subcanonical.

Hence Okonek asked: Under what conditions is a locally Gorenstein subcanonical subscheme of codimension 3 in ℙ^{N} (or in another ambient variety) a Pfaffian subscheme? We will give the following answer for projective space.

Theorem 0.1. Let k be an algebraically closed field of characteristic different from 2. Suppose X ⊂ ℙ_{k}^{n+3} is a locally Gorenstein subscheme of equidimension n > 0 such that ω_{X} ≅ O_{X}(ℓ) for some integer ℓ. Then X is a Pfaffian subscheme if and only if the following parity condition holds: if n ≡ 0 (mod 4) and ℓ is even, then χ(O_{X}(ℓ/2)) is also even.

As Okonek pointed out, the Barth-Lefschetz theorems imply that every smooth subvariety of codimension 3 in ℙ^{8} and ℙ^{9} is subcanonical. Hence we have a corollary:

Corollary 0.2. Every smooth subvariety of codimension 3 in ℙ^{8} and ℙ^{9} is a Pfaffian subscheme.

Of course the corollary holds for ℙ^{N} with N ≥ 10 as well (with certain conditions if N ≡ 3 (mod 4)). But in this range all smooth subvarieties of codimension 3 in ℙ^{N} are supposed to be complete intersections according to Hartshorne’s Conjecture.

The parity condition in Theorem 0.1 may be deduced as follows. Suppose X is Pfaffian and n and ℓ are both even. We may twist the resolution (3) by ℓ/2 to get

0 → ω_{ℙ}(−ℓ/2) → E^{∨} ⊗ ω_{ℙ}(s − ℓ/2) → E(ℓ/2 − s) → O_{ℙ}(ℓ/2) → O_{X}(ℓ/2) → 0.

From this sequence and Serre duality on ℙ^{n+3}, it now follows that:

χ(O_{X}(ℓ/2)) = 2χ(O_{ℙ^{n+3}}(ℓ/2)) − 2χ(E(ℓ/2 − s)) ≡ 0 (mod 2).

The main theorem has the following local analogue. Call an unmixed ideal I of height 3 in a regular local ring (R, 𝔪, k) Pfaffian if R/I has a resolution of the form

0 → R ⟶^{g^{∨}} E^{∨} ⟶^{f} E ⟶^{g} R → R/I

with E is a reflexive R-module of odd rank such that E_{𝔭} is a free R_{𝔭}-module for all prime ideals 𝔭 ≠ 𝔪, and f skew-symmetric. The canonical module of any unmixed ideal I of height 3 is by definition ω_{R/I} = Ext_{R}^{3}(R/I, R). It is always saturated. We prove the following:

Theorem 0.3. Let (R, 𝔪, k) be a regular local ring of dimension n > 4 with algebraically closed residue field not of characteristic 2. Let I be an unmixed ideal of R of height 3. Then I is Pfaffian if and only if the following three conditions hold:

(a) (R/I)_{𝔭} is Gorenstein for all prime ideals 𝔭 ≠ 𝔪,

(b) ω_{R/I} ≅ (R/I)^{sat}, and

(c) if n ≡ 0 (mod 4), then H_{𝔪}^{n/2}(I) is of even length.

In both theorems we have stated the parity condition only for n ≡ 0 (mod 4) rather than for all even n. This is because the perfect pairing of Serre duality

H^{n/2}(O_{X}(ℓ/2)) × H^{n/2}(O_{X}(ℓ/2)) → H^{n}(O_{X}(ℓ)) ⟶^{tr} k (4)

(or its local analogue) is defined using the cup product in cohomology. Since the cup product is graded commutative, the pairing is (−1)^{n/2}- symmetric. Hence if n ≡ 2 (mod 4), then H^{n/2}(O_{X}(ℓ/2)) admits a non-degenerate skew-symmetric bilinear form and so is of even dimension. Thus subcanonical varieties with n ≡ 2 (mod 4) and ℓ even automatically have

χ(O_{X}(ℓ/2)) ≡ h^{n/2}(O_{X}(ℓ/2)) ≡ 0 (mod 2).

Both main theorems remain valid with an arbitrary ground field (or residue field) if one rephrases the parity conditions appropriately. Thus for Theorem 0.1 the parity condition should read in general: “if n and ℓ are even, then the vector space H^{n/2}(O_{X}(ℓ/2)) contains a subspace of exactly half its dimension which is totally isotropic for the pairing (4).” Similarly condition (c) in Theorem 0.3 should read in general: “if n is even, then the module H_{𝔪}^{n/2}(I) contains a submodule of half its length which is totally isotropic for the duality pairing.” Although we leave the cases of non-algebraically closed fields to the reader, we will prove both theorems, as modified, in the case where k is an algebraically closed field of characteristic 2.

Outline of the paper. Our proof of Theorem 0.1 will follow Buchsbaum and Eisenbud’s original proof of their structural theorem to a substantial extent. We therefore analyze their method in §1. We find that their method takes as “input” a locally free resolution

0 → O_{ℙ^{n+3}}(−ℓ − n − 4) ⟶^{d_{3}} F_{2} ⟶^{d_{2}} F_{1} ⟶^{d_{1}} O_{ℙ^{n+3}} → O_{X} (5)

satisfying certain technical conditions (cf. Proposition 1.2). It then returns as “output” an isomorphism F_{2} ≅ F_{1}^{∨}(−ℓ − n − 4) with respect to which the middle arrow d_{2} is skew-symmetric. Hence something extra is needed in order to find the “input” resolution.

In §2 we therefore update an old paper of Horrocks [Ho] where, loosely speaking, he showed that a vector bundle on ℙ^{N} can be recognized by its intermediate cohomology modules H_{∗}^{i}(E) := ⨁_{t∈ℤ} H^{i}(E(t)) for 0 < i < N together with some additional data. More precisely, one considers two categories. The first category, to be denoted \underline{VB}, is the stable category of vector bundles on ℙ^{N}. Its objects are the vector bundles on ℙ^{N}, but Hom\underline{VB}(E, F) is the space of morphisms (in the ordinary sense) from E to F modulo the subspace of morphisms which factor through a direct sum of line bundles. The second category, to be denoted FinL ⊂ D^{b}(S-Mod), is the full subcategory of the derived category of bounded complexes of graded modules over S := k[X_{0}, …, X_{N}] whose objects are those complexes C^{∗} such that H^{i}(C^{∗}) is a module of finite length for 0 < i < N, and H^{i}(C^{∗}) = 0 for all other i. Much of §2 is then devoted to giving the following modern statement and a modern proof (using derived categories) of the main result of Horrocks’ paper. (See §2 for the precise definitions of the functors.)

Theorem 0.4 (Horrocks). The functor τ_{>0}τ_{<N}RΓ_{∗} : \underline{VB} → FinL, which associates to any vector bundle E its intermediate cohomology in the derived category sense, is an equivalence of categories. Its inverse is the syzygy bundle functor Syz : FinL → \underline{VB}.

At the end of §2 we use the same techniques to prove several related but new lemmas which are necessary for the proof of Theorem 0.1. One of them (Proposition 2.10) allows the transfer of a chosen portion of the cohomology of O_{X} to prescribed parts of a locally free resolution of O_{X}. Two others (Lemmas 2.11 and 2.12) give cohomological criteria for a morphism between vector bundles on ℙ^{N} to be an isomorphism.

In §3 we use Horrocks’ theorem and the related lemmas to construct the resolution (5) by assigning the intermediate cohomology of F_{1} and F_{2} explicitly. In doing so, we are constrained by two facts: the intermediate cohomology modules of F_{1} and F_{2} must induce those of O_{X} because of (5), while the intermediate cohomology modules of F_{2} must be dual to those of F_{1} if one is to have F_{2} ≅ F_{1}^{∨}(−ℓ − n − 4). The strategy we choose is essentially to assign the lower half of the intermediate cohomology of O_{X} to F_{1} and the upper half to F_{2}. If n is even, this involves splitting the module H_{∗}^{n/2}(O_{X}) in half and assigning a maximal isotropic submodule to F_{1} and the quotient module to F_{2}. (It is here that one uses the parity condition.) Horrocks’ theorem and Proposition 2.10 permit us to construct such bundles F_{1} and F_{2} as well as the morphisms d_{1}, d_{2}, and d_{3} which together form the resolution (5). Lemma 2.12 then allows us to show that the resolution we have constructed is self-dual in a very strong way: i.e. any chain map from the resolution to its dual which extends the identity on O_{X} is necessarily an isomorphism of complexes. The locally free resolution is required to have this property for the Buchsbaum-Eisenbud proof to work.

In §4 we show that our locally free resolution of O_{X} can be endowed with a commutative differential graded algebra structure. This is a matter of calculating the obstruction to the lifting of a certain map. This property is also necessary in order for the Buchsbaum-Eisenbud proof to work. This will complete the proof of Theorem 0.1.

In §5 we consider Theorem 0.1 in characteristic 2. Essentially, certain lemmas in the fourth section fail in characteristic 2 and must be replaced by analogues which are slightly different.

Finally in §6 we consider the results for regular local rings. Theorem 0.1 concerning projective spaces has an obvious analogue (Theorem 6.1) for the punctured spectrum of a regular local ring. We show that this analogue is equivalent to Theorem 0.3.

Acknowledgment. The author would like to thank R. M. Mirò-Roig who brought the problem to his attention and with whom he had several discussions concerning it. The paper was written in the context of the Space Curves group of Europroj.

1. The Buchsbaum-Eisenbud Proof

In this section we review Buchsbaum and Eisenbud’s proof of the local version of Theorem 0.1. Transposing their proof to the context of sheaves on ℙ^{n+3}, we describe the two conditions that a locally free resolution of O_{X} must satisfy in order for their proof to show that a subcanonical subscheme X ⊂ ℙ^{n+3} is Pfaffian (Proposition 1.2). We begin by recalling the statement of Buchsbaum and Eisenbud’s theorem.

Theorem 1.1 ([BE2] Theorem 2.1). Let R be a regular local ring and I an ideal of R of height 3 such that R/I is a Gorenstein ring. Then I has a minimal projective resolution of the form

0 → R ⟶^{g^{∨}} F^{∨} ⟶^{f} F ⟶^{g} R → R/I

such that F of odd rank 2p + 1, the map f is skew-symmetric, and g is composed of the Pfaffians of order 2p of f.

Sketch of Buchsbaum and Eisenbud’s proof of Theorem 1.1. Since R/I is Gorenstein, the minimal projective resolution of R/I is of the form

P^{∗} : 0 → R ⟶^{d_{3}} F_{2} ⟶^{d_{2}} F_{1} ⟶^{d_{1}} R

We now seek to find a way of identifying F_{2} ≅ F_{1}^{∨} so that d_{2} becomes skew-symmetric.

The first step is to endow P^{∗} with the structure of a commutative associative differential graded algebra ([BE2] pp. 451–453). To define the multiplication, they define S_{2}(P^{∗}) = (P^{∗} ⊗ P^{∗})/M^{∗} where M^{∗} is the graded submodule of P^{∗} ⊗ P^{∗} generated by

{a ⊗ b − (−1)^{(deg a)(deg b)}b ⊗ a | a, b homogeneous elements of P^{∗}}.

Using universal properties of projective modules, they then construct a map of complexes Φ : S_{2}(P^{∗}) → P^{∗} which covers the multiplication R/I ⊗ R/I → R/I and which is the identity on the subcomplex R ⊗ P^{∗} ⊂ S_{2}(P^{∗}). This makes P^{∗} into a commutative differential graded algebra. The associativity of this algebra follows from the fact that it of length 3, i.e. P_{n} = 0 for n ≥ 4.

The next step (p. 455) is to note that the multiplication F_{i} ⊗ F_{3−i} → F_{3} = R induces maps s_{i} : F_{i} → F_{3−i}^{∨} and a commutative diagram:

\begin{aligned}P^{∗} : & 0 & ⟶ & R & ⟶^{d_{3}} & F_{2} & ⟶^{d_{2}} & F_{1} & ⟶^{d_{1}} & R \\ & & & ∥ & & ↓^{s_{2}} & & ↓^{s_{1}} & & ∥ \\ (P^{∗})^{∨} : & 0 & ⟶ & R & ⟶^{d_{1}^{∨}} & F_{1}^{∨} & ⟶^{−d_{2}^{∨}} & F_{2}^{∨} & ⟶^{d_{3}^{∨}} & R \end{aligned} (6)

This map of complexes is a covering of the Gorenstein duality isomorphism R/I ≅ ω_{R/I} = Ext_{R}^{3}(R/I, R) to the minimal projective resolutions of R/I and ω_{R/I}. Since any map between minimal projective resolutions which covers an isomorphism in degree 0 must be an isomorphism, it follows that the s_{i} are all isomorphisms.

We can therefore use the identification s_{2} : F_{2} ≅ F_{1}^{∨}. A very simple computation (p. 465) shows that with this identification, the commutativity and associativity of the differential graded algebra structure on P^{∗} imply the skew-symmetry of d_{2}. In particular d_{2} must have even rank (say 2p), and F_{2} must have odd rank 2p + 1. The identification of d_{1} and d_{3} with the vectors of Pfaffians of order 2p of d_{2} is a lengthy but unproblematic computation (pp. 458–464).

Now let X be a locally Gorenstein subcanonical subscheme of codimension 3 in ℙ^{n+3} with ω_{X} ≅ O_{X}(ℓ). We wish to repeat the proof we have just sketched only with P^{∗} replaced by a locally free resolution of O_{X}:

P^{∗} : 0 → L ⟶^{d_{3}} F_{2} ⟶^{d_{2}} F_{1} ⟶^{d_{1}} O_{ℙ^{n+3}} (7)

where we will write L in place of ω_{ℙ^{n+3}}(−ℓ) in order to simplify our diagrams.

A careful reading yields only two places where the fact that P^{∗} is a minimal projective resolution of R/I was used in a way that does not immediately carry over to the locally free resolution P^{∗}. The first place was in the definition of the map Φ : S_{2}(P^{∗}) → P^{∗} which made P^{∗} into a commutative differential graded algebra. Therefore we will need to show directly the existence of a map of complexes

S_{2}(P^{∗}) : ⋯ ⟶ L ⊕ [F_{2} ⊗ F_{1}] ⟶^{σ} F_{2} ⊕ Λ^{2}F_{1} ⟶ F_{1} ⟶ O_{ℙ^{n+3}}
↓^{ϕ_{3}}  ↓^{ϕ_{2}}  ∥  ∥
P^{∗} :  0 ⟶  L  ⟶^{d_{3}} F_{2}  ⟶^{d_{2}} F_{1} ⟶^{d_{1}} O_{ℙ^{n+3}}

The critical problem in defining the morphism of complexes is the following. Let ψ : Λ^{2}F_{1} → ker(d_{1}) be defined by ψ(a∧b) = d_{1}(a)b−d_{1}(b)a. We then must lift

\begin{aligned}& & & & & & Λ^{2}F_{1} \\ ↓ & ψ \\ 0 & ⟶ & L & ⟶ & F_{2} & ⟶ & ker(d_{1}) & ⟶ & 0 \end{aligned} (8)

to a ϕ ∈ Hom(Λ^{2}F_{1}, F_{2}). Once that is done, the rest of the chain map follows. For one may define ϕ_{2} = (1_{F_{2}}, ϕ). Then

ϕ_{2} ∘ σ(L ⊕ [F_{2} ⊗ F_{1}]) ⊂ ker(d_{2}) = L.

So ϕ_{2} ∘ σ factors through L, allowing one to define ϕ_{3}. Thus one can put a commutative associative differential graded algebra structure on P^{∗} provided ψ can be lifted. The obstruction to lifting ψ lies in Ext^{1}(Λ^{2}F_{1}, L) ≅ H^{n+2}(Λ^{2}F_{1}(ℓ))^{∗}.

Once we have the commutative differential graded algebra structure on P^{∗}, we may use it to define maps s_{i} : F_{i} → F_{3−i}^{∨} ⊗ L and a commutative diagram analogous to (6):

\begin{aligned}P^{∗} : & 0 & ⟶ & L & ⟶^{d_{3}} & F_{2} & ⟶^{d_{2}} & F_{1} & ⟶^{d_{1}} & O_{ℙ} \\ & & & ∥ & & ↓^{s_{2}} & & ↓^{s_{1}} & & ∥ \\ (P^{∗})^{∨} : & 0 & ⟶ & L & ⟶^{d_{1}^{∨}} & F_{1}^{∨} ⊗ L & ⟶^{−d_{2}^{∨}} & F_{2}^{∨} ⊗ L & ⟶^{d_{3}^{∨}} & O_{ℙ} \end{aligned} (9)

The vertical maps cover the isomorphism

O_{X} ≅ ω_{X}(−ℓ) = Ext^{3}(O_{X}, L).

We now run into the second problem with locally free resolutions. Namely, a morphism of locally free resolutions which covers an isomorphism in degree 0 is not automatically an isomorphism between the resolutions. But we reach the conclusion:

Proposition 1.2. Suppose X is a locally Gorenstein subcanonical subscheme of codimension 3 in ℙ^{n+3} with ω_{X} ≅ O_{X}(ℓ). Then X will be a Pfaffian scheme if O_{X} has a locally free resolution P^{∗} as in (7) satisfying the following two conditions:

(a) Any morphism of complexes P^{∗} → (P^{∗})^{∨} as in (9) which covers the identity of O_{X} is an isomorphism of complexes, and

(b) The morphism ψ of (8) lifts to a map ϕ ∈ Hom(Λ^{2}F_{1}, F_{2}).

We will now construct locally free resolutions P^{∗} satisfying the conditions of the proposition. Our method involves the Horrocks correspondence.

2. The Horrocks Correspondence

In this section we give a modern proof using derived categories of Horrocks’ theorem (Theorem 0.4) on the equivalence of the stable category \underline{VB} of vector bundles on ℙ^{N} with a category FinL which captures the intermediate cohomology of the vector bundles. We will also use the same techniques to prove several related lemmas which are needed for the proof of Theorem 0.1.

After the introduction of some notation and terminology and some preliminary lemma (Lemmas 2.1–2.3), the proof of Theorem 0.4 will divide into two parts. The first part consists of Lemmas 2.4–2.6, which identify the categories VB and FinL with equivalent categories of complexes of free modules VB^{𝔓} and FinL^{𝔓}. In the second part, Proposition 2.7 one shows the desired equivalence between \underline{VB}^{𝔓} and FinL^{𝔓}. The syzygy bundle functor, which is identified with the inverse equivalence, is defined after Proposition 2.7.

For Horrocks’ original proof of the theorem, see [Ho] §§6–7. (The equivalence of categories is asserted at the top of p. 703.) However, Horrocks’ definition of the category FinL and the functor τ_{>0}τ_{<N}RΓ_{∗} are different from ours, and it seemed better to redo the theorem in a modern way than to refer the reader to Horrocks’ paper.

Notation and Generalities. We first recall some generalities about complexes. If 𝔄 is an abelian category, let C(𝔄) (resp. K(𝔄), D(𝔄)) denote the category (resp. homotopy category, derived category) of complexes of objects of 𝔄, and let C^{b}(𝔄), C^{−}(𝔄), C^{+}(𝔄), etc., denote the corresponding complexes of bounded (resp. bounded above, bounded below) complexes of objects of 𝔄. When speaking of complexes, we will generally reserve the word “isomorphism” for isomorphisms in C(𝔄). Isomorphisms in K(𝔄) (resp. D(𝔄)) are referred to as homotopy equivalences (resp. quasi-isomorphisms).

If r is an integer, then any complex C^{∗} of objects of 𝔄 has two canonical truncations at r and a naive truncation:

\begin{aligned}τ_{≤r}(C^{∗}) : & ⋯ & → & C^{r−2} & → & C^{r−1} & → & ker(δ^{r}) & → & 0 & → & 0 & → & ⋯, \\ τ_{>r}(C^{∗}) : & ⋯ & → & 0 & → & 0 & → & C^{r}/ ker(δ^{r}) & → & C^{r+1} & → & C^{r+2} & → & · · · . \\ σ_{≥r}(C^{∗}) : & ⋯ & → & 0 & → & 0 & → & C^{r} & → & C^{r+1} & → & C^{r+2} & → & · · · . \end{aligned}

We will often find it more convenient to write τ_{<r+1} instead of τ_{≤r}. All the truncations are functorial in C(𝔄). The canonical truncations are functorial in K(𝔄) and D(𝔄) as well. The moral of Proposition 2.7 will be that naive truncations are also functorial in stable homotopy categories of projectives.

Suppose now that 𝔄 has enough projectives. Every bounded above complex C^{∗} of objects in 𝔄 admits a projective resolution, i.e. a quasiisomorphism P^{∗} → C^{∗} with P^{∗} a complex of projectives ([Ha1] Proposition I.4.6). The projective resolution of a complex is unique up to homotopy equivalence. If C^{∗} and E^{∗} are bounded above complexes of objects in 𝔄, and if P^{∗} → C^{∗} is a projective resolution of C^{∗}, then there is a natural isomorphism Hom_{D^{−}(𝔄)}(C^{∗}, E^{∗}) ≅ Hom_{K^{−}(𝔄)}(P^{∗}, E^{∗}). Thus if 𝔓 denotes the full subcategory of projective objects of 𝔄, then the inclusion K^{−}(𝔓) → D^{−}(𝔄) is an equivalence of triangulated categories ([Ha1] Proposition I.4.7).

Let S = k[X_{0}, …, X_{N}] be the homogeneous coordinate ring of ℙ^{N}, and let 𝔪 = (X_{0}, …, X_{N}) be its irrelevant ideal. Let S-Mod be the category of graded S-modules. Then S-Mod has enough projectives, namely the free modules. We will call a complex P^{∗} of projective Smodules minimal if all its objects P^{i} are free of finite rank and its differential δ^{∗} satisfies δ^{i}(P^{i}) ⊂ 𝔪P^{i+1} for all i. If C^{∗} is a bounded above complex of graded S-modules whose cohomology modules H^{i}(C^{∗}) are all finitely generated, then C^{∗} has a minimal projective resolution, i.e. a projective resolution by a minimal complex of projectives. (The fact that such a C^{∗} has a resolution by a bounded above complex Q^{∗} of projectives of finite rank may be found in [Ha2] Lemma III.12.3. Moreover, such a complex of projectives of finite rank can always be split into a direct sum Q^{∗} ≅ P^{∗} ⊕ R^{∗} with P^{∗} a minimal projective complex and R^{∗} an exact complex, which may be dropped.) The next lemma, which is a well known consequence of Nakayama’s lemma, says that minimal projective resolutions are unique up to isomorphism and not merely up to homotopy equivalence:

Lemma 2.1. Let ϕ : P^{∗} → Q^{∗} be a homotopy equivalence between minimal complexes of free graded S-modules of finite rank. Then ϕ is an isomorphism.

Let O-Mod be the category of sheaves of O_{ℙ^{N}}-modules. For E a sheaf of O_{ℙ^{N}}-modules, let Γ_{∗}(E) = ⨁_{t∈ℤ} Γ(E(t)). Then Γ_{∗} defines a left exact functor from O-Mod to S-Mod. It has a right derived functor RΓ_{∗} : D^{b}(O-Mod) → D^{b}(S-Mod) whose cohomology functors we denote H_{∗}^{i}(E) = ⨁_{t∈ℤ} H^{i}(E(t)). The functor Γ_{∗} has an exact left adjoint , the functor of associated sheaves.

Let Γ_{𝔪} : S-Mod → S-Mod be the functor associating to a graded S-module M the maximal submodule Γ_{𝔪}(M) ⊂ M supported at the origin 0 of 𝔸^{N+1}. This functor is also left exact and has a right derived functor RΓ_{𝔪} : D^{b}(S-Mod) → D^{b}(S-Mod). Its cohomology functors are denoted H_{𝔪}^{i}.

Some Equivalences of Categories. The first part of the proof of Theorem 0.4 consists of replacing the category VB of vector bundles and the category FinL by more concrete but equivalent categories VB^{𝔓} and FinL^{𝔓} of complexes of projectives. It will then be easier to compare the latter categories. We begin with a fairly general lemma, which is actually true for Cohen-Macaulay local rings.

Lemma 2.2. Let F^{∗} be a bounded complex of free graded S-modules of finite rank where S = k[X_{0}, …, X_{N}]. If F^{∗} is minimal, then

max{i | F^{i} ≠ 0} = max{i | H^{i}(F^{∗}) ≠ 0},
min{i | F^{i} ≠ 0} = min{i | H_{𝔪}^{i}(F^{∗}) ≠ 0} − N − 1.

Proof. The assertion about maxima is well known and reduces to saying that if m := max{i | F^{i} ≠ 0}, then H^{m}(F^{∗}) = F^{m}/δ^{m−1}(F^{m−1}) ≠ 0 since δ^{m−1}(F^{m−1}) ⊂ 𝔪F^{m} by minimality.

For the assertion about minima, write F^{i} = S^{⊕r_{i}} for each i. (For simplicity we suppress all notation related to the S-module grading.) Then H_{𝔪}^{q}(F^{i}) = 0 if q ≠ N + 1, while H_{𝔪}^{N+1}(F^{i}) ≅ E^{⊕r_{i}} where E is the injective envelope of the residue field. Therefore the spectral sequence of hypercohomology with supports with E_{1}^{pq} = H_{𝔪}^{q}(F^{p}) and abutment H_{𝔪}^{p+q}(F^{∗}) degenerates. So H_{𝔪}^{i}(F^{∗}) is the cohomology (in the middle) of the complex

⋯ → E^{⊕r_{i−N−2}} ⟶^{δ^{i−N−2}} E^{⊕r_{i−N−1}} ⟶^{δ^{i−N−1}} E^{⊕r_{i−N}} → ⋯

where each δ^{j} is given by a matrix of polynomials which is the same (up to sign) as the original δ^{j} : S^{⊕r_{j}} → S^{⊕r_{j+1}}.

If now μ := min{i | F^{i} ≠ 0}, then this complex vanishes in degrees less than μ + N + 1. So H_{𝔪}^{i}(F^{∗}) = 0 for i < μ + N + 1. Moreover, H_{𝔪}^{μ+N+1}(F^{∗}) is the kernel of δ^{μ} : E^{⊕r_{μ}} → E^{⊕r_{μ+1}}. Now by minimality all entries in the matrix δ^{μ} lie in 𝔪. Therefore δ^{μ} kills the socle of E^{⊕r_{μ}}. So H_{𝔪}^{μ+N+1}(F^{∗}) contains the socle of E^{⊕r_{μ}} and is therefore nonzero. This completes the proof of the lemma.

In the next lemma and in Lemma 2.5 it is essential that S have finite global dimension. It is these lemmas which restrict Horrocks’ equivalences (in the form given) to projective spaces and the punctured spectra of regular local rings.

Lemma 2.3. (a) Suppose

P^{∗} : ⋯ → 0 → P^{0} → P^{1} → ⋯ → P^{N−1} → 0 → ⋯

is a complex of free graded S-modules of finite rank such that H^{i}(P^{∗}) is a module of finite length for 0 < i < N. Let E = H^{0}(P^{∗})^{∼}. Then P^{∗} is quasi-isomorphic to τ_{<N}RΓ_{∗}(E).

(b) Conversely, if E is a vector bundle on ℙ^{N}, then the minimal projective resolution of τ_{<N}RΓ_{∗}(E) is of the above form.

Proof. (a) Note that the complex P̃^{∗} of coherent sheaves on ℙ^{N} has vanishing cohomology in degrees different from 0. So it is quasi-isomorphic to H^{0}(P̃^{∗}) = E. Hence the triangle of functors (cf. [W] Proposition 1.1)

RΓ_{𝔪} → Id → RΓ_{∗}∘^{∼} → RΓ_{𝔪}[1],

when applied to P^{∗}, yields a triangle

RΓ_{𝔪}(P^{∗}) → P^{∗} ⟶^{β} RΓ_{∗}(E) → RΓ_{𝔪}(P^{∗})[1]. (10)

By Lemma 2.2 we have H_{𝔪}^{i}(P^{∗}) = 0 for i ≤ N. So H^{i}(β) : H^{i}(P^{∗}) → H_{∗}^{i}(E) is an isomorphism for i < N. Therefore β induces a quasiisomorphism of P^{∗} onto τ_{<N}RΓ_{∗}(E).

(b) Conversely, if E is a vector bundle on ℙ^{N}, then H_{∗}^{i}(E) is finitely generated for i < N ([S] p. 269, Théorème 2). Hence τ_{<N}RΓ_{∗}(E) has a minimal projective resolution P^{∗}. The complex P^{∗} is bounded below (as well as above) because S has finite global dimension. For 0 < i < N the module H^{i}(P^{∗}) = H_{∗}^{i}(E) is of finite length because E is locally free. By construction H^{i}(P^{∗}) = H^{i}(τ_{<N}RΓ_{∗}(E)) = 0 for i ≥ N. So we have P^{i} = 0 for i ≥ N by Lemma 2.2. Looking again at the triangle (10), we see by the construction of P^{∗} that H^{i}(β) is an isomorphism for i < N and an injection for i = N. So H_{𝔪}^{i}(P^{∗}) = 0 for i ≤ N. So by Lemma 2.2 we see that P^{i} = 0 for i ≤ −1. Thus P^{∗} has the form asserted by the lemma.

We wish to interpret the last lemma categorically. Let 𝔓 ⊂ S-Mod be the full subcategory of projective S-modules, so that the natural morphism K^{−}(𝔓) → D^{−}(S-Mod) is an equivalence of triangulated categories. Let VB^{𝔓} ⊂ K^{b}(𝔓) be the full subcategory of complexes of free graded S-modules of finite rank the form

P^{∗} : ⋯ → 0 → P^{0} → P^{1} → ⋯ → P^{N−1} → 0 → ⋯

such that H^{i}(P^{∗}) is a module of finite length for 0 < i < N and the differentials satisfy δ^{i}(P^{i}) ⊂ 𝔪P^{i+1} for all i. The previous lemma may be interpreted as follows.

Lemma 2.4. The functor P^{∗} : VB → VB^{𝔓} which associates to an E ∈ ob(VB) the minimal projective resolution of τ_{<N}RΓ_{∗}(E) is an equivalence of categories with inverse given by C^{∗} ↦ H^{0}(C^{∗})^{∼}.

Proof. The functor P^{∗} : VB → K^{−}(𝔓) is the composition of τ_{<N}RΓ_{∗} : VB → D^{−}(S-Mod) with the functor D^{−}(S-Mod) → K^{−}(𝔓) of projective resolutions. Since the first functor τ_{<N}RΓ_{∗} has a left inverse H^{0}(−)^{∼}, it is full and faithful. And the projective resolution functor is an equivalence. Therefore P^{∗} : VB → K^{−}(S-Mod) is full and faithful. By Lemma 2.3 the image of P^{∗} is contained in VB^{𝔓}, and conversely any object in VB^{𝔓} is isomorphic to an object in the image of P^{∗}. Therefore P^{∗} induces an equivalence between VB and VB^{𝔓}. Its inverse is H^{0}(−)^{∼}.

The complex P^{∗}(E) may be computed effectively by taking a minimal projective resolution of the graded module E^{∨} = Γ_{∗}(E^{∨}) and dualizing it. This is indeed what Horrocks did.

The graded module associated to a vector bundle E on ℙ^{N} has a minimal projective resolution:

0 → Q^{−(N−1)} → ⋯ → Q^{−1} → Q^{0} → Γ_{∗}(E)

For any E we now define Q^{∗}(E) by

Q^{∗}(E) : ⋯ → 0 → Q^{−(N−1)} → ⋯ → Q^{−1} → Q^{0} → 0 → · · · . We then let R^{∗}(E) be the natural concatenation of Q^{∗}(E) with P^{∗}(E) induced by the composition Q^{0} ↠ Γ_{∗}(E) ↪ P^{0}:

R^{∗}(E) : ⋯ → 0 → Q^{−(N−1)} → ⋯ → Q^{0} → P^{0} → ⋯ → P^{N−1} → 0 → ⋯

Thus R^{i}(E) = P^{i}(E) for i ≥ 0, and R^{i}(E) = Q^{i+1}(E) for i < 0. Note that although the projective complexes P^{∗}(E) and Q^{∗}(E) are minimal, R^{∗}(E) need not be minimal, because there could be a direct summand of Q^{0}(E) which is mapped isomorphically onto a direct summand of P^{0}(E). However, one may write R^{∗}(E) as the direct sum of a minimal complex of projectives R_{min}^{∗}(E)

R_{min}^{∗}(E) :  ⋯ → Q^{−2} → Q^{−1} → Q_{min}^{0} → P_{min}^{0} → P^{1} → P^{2} → ⋯

and of an exact complex of projectives

(11)  ⋯ → 0 → L ⟶^{1} L → 0 → · · · .

These operations define functors Q^{∗}, R^{∗}, and R_{min}^{∗} on VB with values in K^{b}(𝔓). Moreover, we may use the identification between the categories VB and VB^{𝔓} to define complexes Q^{∗}(P^{∗}), R^{∗}(P^{∗}), and R_{min}^{∗}(P^{∗}) for P^{∗} in VB^{𝔓}. Namely, Q^{∗}(P^{∗}) is the minimal projective resolution of H^{0}(P^{∗}), R^{∗}(P^{∗}) is the concatenation of Q^{∗}(P^{∗}) with P^{∗}, etc.

We now define a category of complexes of type R_{min}^{∗}. More formally, let FinL^{𝔓} ⊂ K^{b}(𝔓) be the full subcategory of minimal complexes of projective modules of finite rank of the form

⋯ → 0 → R^{−N} → ⋯ → R^{−1} → R^{0} → ⋯ → R^{N−1} → 0 → ⋯ (12)

such that the cohomology modules H^{i}(R^{∗}) are of finite length for 0 < i < N and vanish for all other i.

Lemma 2.5. The natural functor FinL^{𝔓} → FinL is an equivalence of categories.

Proof. Clearly the natural functor is full and faithful, since FinL^{𝔓} is a full subcategory of K^{−}(𝔓) and we are considering its image under the natural equivalence K^{−}(𝔓) → D^{−}(S-Mod). Hence it remains only to show that the minimal projective resolution R^{∗} of any object C^{∗} in FinL lies in FinL^{𝔓}. Since S is of finite global dimension, R^{∗} is a bounded complex to which we may apply Lemma 2.2. In particular, since H^{i}(R^{∗}) = H^{i}(C^{∗}) = 0 for i ≥ N, we have R^{i} = 0 for i ≥ N. Moreover, all the H^{i}(C^{∗}) are of finite length, so H_{𝔪}^{i}(C^{∗}) = H^{i}(C^{∗}) for all i. In particular, H_{𝔪}^{i}(R^{∗}) = H_{𝔪}^{i}(C^{∗}) = 0 for i ≤ 0. So R^{i} = 0 for i ≤ −N − 1 by Lemma 2.2. Thus the minimal projective resolution of any object of FinL is in FinL^{𝔓}.

Lemma 2.6. The equivalences of Lemmas 2.4 and 2.5 identify the intermediate cohomology functor τ_{>0}τ_{<N}RΓ_{∗} : VB → FinL with the functor R_{min}^{∗} : VB^{𝔓} → FinL^{𝔓}.

Proof. Suppose P^{∗} is an object of VB^{𝔓} and that E = H^{0}(P^{∗})^{∼} is the corresponding vector bundle. By construction P^{∗} ≃ τ_{<N}RΓ_{∗}(E). So τ_{>0}(P^{∗}) ≃ τ_{>0}τ_{<N}RΓ_{∗}(E). We are supposed to end up with a minimal projective resolution of τ_{>0}τ_{<N}RΓ_{∗}(E). This is thus the same as the minimal projective resolution of

τ_{>0}(P^{∗}) : ⋯ → 0 → P^{0}/H^{0}(P^{∗}) → P^{1} → ⋯ → P^{N−1} → 0 → ⋯,

which is R_{min}(P^{∗}).

Stable categories. Let \underline{VB} be the stable category of vector bundles on ℙ^{N}. Its objects are the same as those of VB (i.e. all vector bundles on ℙ^{N}) but its morphisms are given by

Hom\underline{VB}(E, F) = Hom_{O_{ℙ^{N}}}(E, F)/ Hom_{𝔓}(E, F)

where Hom_{𝔓}(E, F) is the set of all morphisms which factor through a direct sum of line bundles. Two vector bundles are isomorphic in the stable category \underline{VB} if and only they are stably equivalent, i.e. if and only if there exist sets of integers {n_{i}} and {m_{j}} such that E ⊕⨁_{i} O_{ℙ^{N}}(n_{i}) ≅ F ⊕ ⨁_{j} O_{ℙ^{N}}(m_{j}). Any stable equivalence class has a minimal element E_{0}, unique up to isomorphism, such that any other member of the class is isomorphic to E_{0} ⊕⨁_{i} O_{ℙ^{N}}(a_{i}) for some integers a_{i} (cf. [Ho] Theorem 6.2 and Lemma 7.1).

Direct sums of line bundles correspond to objects of VB^{𝔓} of the form

⋯ → 0 → L → 0 → ⋯ (13)

with L a free graded S-module of finite rank appearing in degree 0. We may therefore define the stable category \underline{VB}^{𝔓} as the category with the same objects as VB^{𝔓} by with morphisms

Hom\underline{VB}_{𝔓}(E^{∗}, F^{∗}) = Hom_{K^{b}(𝔓)}(E^{∗}, F^{∗})/ Hom_{𝔓}(E^{∗}, F^{∗})

where Hom_{𝔓}(E^{∗}, F^{∗}) is the set of all morphisms which factor through a complex of the form (13). The equivalence between VB and VB^{𝔓} induces an equivalence between \underline{VB} and \underline{VB}^{𝔓}.

The next proposition is the heart of the proof of Theorem 0.4.

Proposition 2.7. The functor R_{min}^{∗} : VB^{𝔓} → FinL^{𝔓} factors as the quotient functor VB^{𝔓} → \underline{VB}^{𝔓} followed by an equivalence of categories \underline{R}_{min}^{∗} : \underline{VB}^{𝔓} → FinL^{𝔓}. The inverse equivalence is the naive truncation σ_{≥0} : FinL^{𝔓} → \underline{VB}^{𝔓}.

Proof. We first show that R_{min}^{∗} factors through \underline{VB}^{𝔓}. This amounts to saying that for any pair of objects E^{∗} and F^{∗} in VB^{𝔓}, the kernel of the natural map

(14) Hom_{VB}_{𝔓}(E^{∗}, F^{∗}) → Hom_{FinL}_{𝔓}(R_{min}^{∗}(E^{∗}), R_{min}^{∗}(F^{∗}))

contains the subspace Hom_{𝔓}(E^{∗}, F^{∗}) of maps factoring through complexes of the form (13). But if α does factor through such a complex L^{∗}, then R_{min}^{∗}(α) factors through R_{min}^{∗}(L^{∗}) = 0 and so vanishes. This proves the factorization.

Next we show that the induced functor \underline{R}_{min}^{∗} is faithful, i.e. that the kernel of (14) is exactly Hom_{𝔓}(E^{∗}, F^{∗}). So suppose α is in the kernel of (14). Since α is a morphism in the full subcategory VB^{𝔓} ⊂ K^{b}(𝔓), it is a homotopy class of chain maps. We may choose a representative β in this homotopy class. We may then complete β to a chain map ρ : R^{∗}(E^{∗}) → R^{∗}(F^{∗}).

R^{∗}(E^{∗})  ⋯ → 0 → \overline{E}^{−N} → ⋯ → \overline{E}^{−1} → E^{0} → ⋯ → E^{N−1} → 0 → ⋯
↓^{ρ}  ↓  ↓  ↓  ↓^{β}  ↓^{β}  ↓
R^{∗}(F^{∗})  ⋯ → 0 → \overline{F}^{−N} → ⋯ → \overline{F}^{−1} → F^{0} → ⋯ → F^{N−1} → 0 → ⋯

Since R_{min}^{∗}(α) = 0, we have R^{∗}(α) = 0 as well because R^{∗} and R_{min}^{∗} are homotopy equivalent. Therefore ρ, which is a representative of R^{∗}(α), is homotopic to 0. Concretely, if we now write δ^{i} for the differentials of R^{∗}(E^{∗}), and ϵ^{i} for the differentials of R^{∗}(F^{∗}), then there is a chain homotopy h = (h^{i}) such that ρ^{i} = h^{i+1}δ^{i} + ϵ^{i−1}h^{i} for all i. Now restrict h to a chain homotopy ĥ = (ĥ^{i}) with ĥ^{i} : E^{i} → F^{i−1} defined by defined by ĥ^{i} = h^{i} for all i ≥ 1, and ĥ^{i} = 0 for all i ≤ 0. Then β is homotopic to a morphism whose components are

β^{i} − (ĥ^{i+1}δ^{i} + ϵ^{i−1}ĥ^{i}) = \begin{cases}ρ^{i} − (h^{i+1}δ^{i} + ϵ^{i−1}h^{i}) = 0 & if i ≥ 1, \\ ρ^{0} − h^{1}δ^{0} = ϵ^{−1}h^{0} & if i = 0, \\ 0 & if i ≤ −1. \end{cases}

Hence the homotopy class α of β factors through the complex

⋯ → 0 → \overline{F}^{−1} → 0 → ⋯

of type (13). So the kernel of (14) is as asserted, and the functor \underline{R}_{min}^{∗} : \underline{VB}^{𝔓} → FinL^{𝔓} is faithful.

Next we show that the functor \underline{R}_{min}^{∗} is full, i.e. that (14) is always surjective. Suppose ϕ ∈ Hom_{FinL}_{𝔓}(R_{min}^{∗}(E^{∗}), R_{min}^{∗}(F^{∗})). Since FinL^{𝔓} is a full subcategory of K^{b}(𝔓), the morphism ϕ is a homotopy class of chain maps. Let f be a representative of the homotopy class. Then f may be extended to a chain map \overline{f} : R^{∗}(E^{∗}) → R^{∗}(F^{∗}) by defining it to be 0 on the exact factor of the type (11). Then σ_{≥0}\overline{f} maps E^{∗} to F^{∗}, and its homotopy class is a morphism in VB^{𝔓} with image ϕ in FinL^{𝔓}. Thus (14) is always surjective, and the functor is full.

Now if we apply the naive truncation σ_{≥0} to an object C^{∗} of FinL^{𝔓} and follow it with R_{min}^{∗}, we just end up just replacing the portion of C^{∗} we truncated off. Hence R_{min}^{∗}(σ_{≥0}C^{∗}) ≅ C^{∗}. Thus every object of FinL^{𝔓} is isomorphic to an object in the image of \underline{R}_{min}^{∗} : \underline{VB}^{𝔓} → FinL^{𝔓}. This functor is therefore an equivalence, all the necessary properties being verified.

Next we wish to show that σ_{≥0} : FinL^{𝔓} → \underline{VB}^{𝔓} is a well-defined functor. It is clearly well-defined on objects. To prove it is well-defined on morphisms, we need to show that if ρ : C^{∗} → D^{∗} is a null-homotopic chain map between objects of FinL^{𝔓}, then σ_{≥0}(ρ) is homotopic to a chain map which factors through a complex of the type (13). But this may be shown by repeating the calculation with chain homotopies given above.

The arguments given above to show the fullness and surjectivity of \underline{R}_{min}^{∗} actually show that \underline{R}_{min}^{∗}σ_{≥0} ≅ 1_{FinL}_{𝔓}. Therefore it remains only to consider σ_{≥0}\underline{R}_{min}^{∗}. If we start off with an object P^{∗} of \underline{VB}^{𝔓}, then applying first \underline{R}_{min}^{∗} and then σ_{≥0} has the effect of chopping off a direct summand L of P^{0} and replacing P^{∗} with

P_{min}^{∗} : ⋯ → 0 → P_{min}^{0} → P^{1} → P^{2} → ⋯

(cf. (11)). Since what we have removed is a direct summand of the complex P^{∗} of the form (13), the projection P^{∗} → P_{min}^{∗} is an isomorphism in the stable category \underline{VB}^{𝔓}. It is easy to verify that these projections actually defines an invertible natural transformation 1\underline{VB}_{𝔓} ⟶^{≅} σ_{≥0}\underline{R}_{min}^{∗}. Therefore σ_{≥0} is an equivalence inverse to \underline{R}_{min}^{∗}.

The end of the proof of the proposition also proves the following corollary.

Corollary 2.8. Any object P^{∗} of VB^{𝔓} is isomorphic in the category VB^{𝔓} to σ_{≥0}R_{min}^{∗}(P^{∗}) ⊕ L^{∗} where L^{∗} is a complex as in (13) which vanishes except in degree 0 where it is a free module of finite rank.

To complete the proof of Theorem 0.4 we define the syzygy bundle functor Syz : FinL → \underline{VB}. It is of course the functor which corresponds to the naive truncation σ_{≥0} : FinL^{𝔓} → \underline{VB}^{𝔓} under the equivalences of Lemmas 2.4 and 2.5 between FinL and FinL^{𝔓} and between \underline{VB} and \underline{VB}^{𝔓}. Concretely, given an “intermediate cohomology complex” C^{∗} in FinL, one takes its minimal projective resolution

⋯ → 0 → P^{−N} → ⋯ → P^{0} ⟶^{d^{0}} P^{1} → ⋯ → P^{N−1} → 0 → ⋯

and lets Syz(C^{∗}) be the vector bundle ker(d^{0})^{∼}. The operation of Syz on morphisms is defined similarly. Note that although Syz of an object of FinL is a vector bundle, Syz of a morphism is only a stable morphism of vector bundles.

As an example, suppose C^{∗} has cohomology only in a single degree i where H^{i}(C^{∗}) = M. Then P^{∗} is a translation of a minimal projective resolution of M, and Syz(C^{∗}) is the sheaf associated to the (i + 1)-th syzygy module of M.

Proof of Theorem 0.4. We may use the equivalences of Lemmas 2.4– 2.6 to identify the categories and functors appearing in Proposition 2.7 with those appearing in Theorem 0.4. Under this identification, Proposition 2.7 says that the intermediate cohomology functor τ_{>0}τ_{<N}RΓ_{∗} : VB → FinL factors as the quotient functor VB → \underline{VB} followed by an equivalence \underline{VB} → FinL with inverse equivalence Syz : FinL → \underline{VB}. This completes the proof of Theorem 0.4.

Corollary 2.8 also has a translation in terms of the categories and functors of Theorem 0.4.

Corollary 2.9. Any vector bundle E on ℙ^{N} is isomorphic to a bundle Syz(τ_{>0}τ_{<N}RΓ_{∗}(E)) ⊕ ⨁ O_{ℙ^{N}}(n_{i}) for some integers n_{i}.

An Insertion Lemma. We now prove several other lemmas on themes related to Theorem 0.4. The first will allow us to construct good locally free resolutions of coherent sheaves.

Proposition 2.10. Let Q be a quasi-coherent sheaf on ℙ^{N}, let C^{∗} be an object of FinL, and let β : C^{∗} → τ_{>0}τ_{<N}RΓ_{∗}(Q) be a morphism in D^{b}(S-Mod). Then there exists a morphism of quasi-coherent sheaves β̃ : Syz(C^{∗}) → Q such that β = τ_{>0}τ_{<N}RΓ_{∗}(β̃). In particular, the induced morphisms H_{∗}^{i}(Syz(C^{∗})) → H_{∗}^{i}(Q) are the same as H^{i}(β) for 1 ≤ i ≤ N − 1.

Proof. Let R^{∗} be a minimal projective resolution of C^{∗}, and let I^{∗} be an injective resolution of Q. Then β may be identified with an actual chain map

⋯ → R^{−2} → R^{−1} → R^{0} ⟶^{λ} R^{1} → ⋯ → R^{N−2} → R^{N−1} → 0
↓ ↓ ↓ ↓ ↓ ↓
⋯ → 0 → Γ_{∗}(Q) → Γ_{∗}(I^{0}) ⟶^{μ} Γ_{∗}(I^{1}) → ⋯ → Γ_{∗}(I^{N−2}) → ker(δ^{N−1}) → 0

Thus β induces a morphism β̃ from Syz(C^{∗}) = ker(λ)^{∼} to Q = ker(μ)^{∼}. We now need to calculate RΓ_{∗}(β̃). Consider the complex

P^{∗} : ⋯ → 0 → R^{0} → R^{1} → ⋯ → R^{N−1} → 0 → · · · .

The previous diagram induces a new commutative diagram

P̃^{∗}:  ⋯ → 0 → R̃^{0} → R̃^{1} → ⋯ → R̃^{N−2} → R̃^{N−1} → 0 → 0 → ⋯
\sqrt{β} ↓ ↓ ↓ ↓ ↓ ↓ ↓
I^{∗}:  ⋯ → 0 → I^{0} → I^{1} → ⋯ → I^{N−2} → I^{N−1} → I^{N} → 0 → ⋯

between resolutions of Syz(C^{∗}) and Q extending β̃. Let γ : P^{∗} → J^{∗} be an injective resolution of P^{∗}. Then \overline{β} factors through γ̃ as P̃^{∗} → J̃^{∗} → I^{∗}. Applying Γ_{∗} now gives a factorization

P^{∗} → Γ_{∗}(J̃^{∗}) → Γ_{∗}(I^{∗}). (15)

Now \overline{β} is a map between resolutions of Syz(C^{∗}) and Q, respectively, which extends β̃ : Syz(C^{∗}) → Q, while γ̃ is a quasi-isomorphism. So the map J̃^{∗} → I^{∗} is a map between injective resolutions of Syz(C^{∗}) and Q extending β̃. So by definition, the second arrow of (15) is RΓ_{∗}(β̃) : RΓ_{∗}(Syz(C^{∗})) → RΓ_{∗}(Q). On the other hand, the proof of Lemma 2.3(a) shows that the first arrow of (15) can be identified with the truncation τ_{<N}(RΓ_{∗}(Syz(C^{∗}))) → RΓ_{∗}(Syz(C^{∗})) because it induces isomorphisms H^{i}(P^{∗}) ≅ H^{i}(Γ_{∗}(J̃^{∗})) = H_{∗}^{i}(Syz(C^{∗})) for i < N. Hence Γ_{∗}(\overline{β}) : P^{∗} → Γ_{∗}(I^{∗}) can be identified with the composition of the truncation τ_{<N}(RΓ_{∗}(Syz(C^{∗}))) → RΓ_{∗}(Syz(C^{∗})) with RΓ_{∗}(β̃). Thus τ_{<N}RΓ_{∗}(β̃) may be identified with the diagram

⋯ → 0 → R^{0} → R^{1} → ⋯ → R^{N−2} → R^{N−1} → 0 → ⋯
↓ ↓ ↓ ↓ ↓ ↓
⋯ → 0 → Γ_{∗}(I^{0}) → Γ_{∗}(I^{1}) → ⋯ → Γ_{∗}(I^{N−2}) → ker(δ^{N−1}) → 0 → ⋯

induced by β. Truncating on the left, we reach a diagram equivalent to the first diagram of the proof of the proposition. So β = τ_{>0}τ_{<N}RΓ_{∗}(β̃).

Isomorphism Criteria. We now need two homological criteria for maps of vector bundles to be isomorphisms. The first is a generalization of [Ho] Theorem 7.5 and Proposition 9.5.

Lemma 2.11. Let E and F be vector bundles on ℙ^{N} with neither containing a line bundle as a direct summand. If α : E → F is a map such that H_{∗}^{i}(α) : H_{∗}^{i}(E) → H_{∗}^{i}(F) is an isomorphism for 0 < i < N, then α is an isomorphism.

Proof. We use the functors R^{∗} and R_{min}^{∗} of Lemmas 2.4–2.6. The hypothesis on α is that τ_{>0}τ_{<N}RΓ_{∗}(α) is a quasi-isomorphism, i.e. an isomorphism in FinL. Since R_{min}^{∗} is the composition of τ_{>0}τ_{<N}RΓ_{∗} with the minimal projective resolution functor, this means that R_{min}^{∗}(α) is an isomorphism in FinL^{𝔓} ⊂ K^{b}(𝔓), i.e. a homotopy equivalence. But since R_{min}^{∗}(α) is a homotopy equivalence between minimal projective resolutions, it must be an isomorphism of complexes by Lemma 2.1.

The hypotheses that E and F have no line bundle direct summands mean that R^{∗}(E) and R^{∗}(F) are minimal complexes and hence that R^{∗}(E) = R_{min}^{∗}(E) and R^{∗}(F) = R_{min}^{∗}(F). Thus R^{∗}(α) is an isomorphism of complexes. However, E, F, and α can be recovered from R^{∗}(E), R^{∗}(F), and R^{∗}(α) because E is isomorphic to the kernel of R^{0}(E)^{∼} → R^{1}(E)^{∼} and similarly for F. But since R^{∗}(α) was an isomorphism, the morphism α recovered from it is also an isomorphism.

We will also need a criterion which will work in some cases where the bundles E and F have line bundles as direct summands, but where the line bundles are chosen minimal so that Γ_{∗}(E) and Γ_{∗}(F) both cover the global sections of a common coherent sheaf Q.

Lemma 2.12. Let E = E_{0} ⊕ ⨁ O_{ℙ^{N}}(n_{i}) and F be vector bundles on ℙ^{N} with E_{0} not containing a line bundle as a direct summand. Let Q be a coherent sheaf on ℙ^{N}. Suppose that there exist morphisms α : E → F and β : F → Q such that

(i) H_{∗}^{i}(α) : H_{∗}^{i}(E) → H_{∗}^{i}(F) is an isomorphism for 0 < i < N, (ii) βα takes the generators of the summands S(n_{i}) of Γ_{∗}(E) onto a

minimal set of generators of the module \overline{Q} := Γ_{∗}(Q)/βα(Γ_{∗}(E_{0})), (iii) E and F have the same rank. Then α is an isomorphism.

Proof. Write F = F_{0} ⊕ ⨁ O_{ℙ^{N}}(m_{j}) with F_{0} not containing a line bundle as a direct summand. The splittings of E and of F into direct summands are not canonical. But choosing such splittings gives an injection E_{0} ↪ E and a projection F ↠ F_{0}. Then the composition

\overline{α} : E_{0} → E ⟶^{α} F → F_{0}

is, like α, an isomorphism on H_{∗}^{i} for 0 < i < N. So \overline{α} is an isomorphism by Lemma 2.11. Hence by identifying F_{0} with α(E_{0}) ⊂ F, we see that α induces a morphism of diagrams

\begin{aligned}0 & ⟶ & E_{0} & ⟶ & E & ⟶ & ⨁ O_{ℙ^{N}}(n_{i}) & ⟶ & 0 \\ & & ∥ & & ↓^{α} & & ↓^{α_{1}} \\ 0 & ⟶ & F_{0} & ⟶ & F & ⟶ & ⨁ O_{ℙ^{N}}(m_{j}) & ⟶ & 0 \end{aligned} (16)

The morphisms α and β therefore induce maps

⨁ S(n_{i}) ⟶^{Γ_{∗}(α_{1})} ⨁ S(m_{j}) ⟶^{\overline{β}} \overline{Q} = Γ_{∗}(Q)/βα(Γ_{∗}(E_{0})).

The composition is a surjection corresponding to a minimal set of generators of \overline{Q} by hypothesis (ii). Hence the righthand map \overline{β} must be a surjection corresponding to a set of generators of \overline{Q}. However, the two free modules have the same rank by hypothesis (iii). Hence \overline{β} also corresponds to a minimal set of generators, and Γ_{∗}(α_{1}) must be an isomorphism. So returning to diagram (16), α_{1} and hence α are isomorphisms.

3. The Self-Dual Resolution

Let X ⊂ ℙ^{n+3} be a locally Gorenstein subcanonical subscheme of equicodimension 3 satisfying the parity condition. In this section we use the Horrocks correspondence and especially Proposition 2.10 to construct a locally free resolution of O_{X}. We then use Lemma 2.12 to show that the resolution satisfies condition (a) of Proposition 1.2.

In the course of the construction we will need a more refined variant of the canonical truncation. Namely, suppose D^{∗} is a complex of objects in an abelian category with differentials δ^{i} : D^{i} → D^{i+1}. Suppose r is an integer, and W ⊂ H^{r}(D^{∗}) a subobject. Then W may be pulled back to a \overline{W} satisfying

im(δ^{r−1}) ⊂ \overline{W} ⊂ ker(δ^{r}) ⊂ D^{r}

We then define:

τ_{≤r,W} (D^{∗}) : ⋯ → D^{r−2} → D^{r−1} → \overline{W} → 0 → 0 → · · · .

The cohomology of this complex is given by

H^{i}(τ_{≤r,W} (D^{∗})) = \begin{cases}H^{i}(D^{∗}) & if i < r, \\ W & if i = r, \\ 0 & if i > r. \end{cases}

We will also use the following conventions. If E is a coherent sheaf on ℙ^{N} and α, β ∈ ℚ are not both integers, then we define H^{α}(E(β)) = 0. Also if D^{∗} is a complex and α ∈ ℚ, we define τ_{≤α}(D^{∗}) = τ_{≤[α]}(D^{∗}).

Definition of the Locally Free Resolution. Suppose X ⊂ ℙ^{n+3} is a locally Gorenstein subscheme of equidimension n > 0 such that ω_{X} ≅ O_{X}(ℓ) for some integer ℓ and such that h^{n/2}(O_{X}(ℓ/2)) is even.

Let ν = n/2 and ℓ′ = ℓ/2. By hypothesis H^{ν}(O_{X}(ℓ′)) is an evendimensional vector space (zero if n or ℓ is odd) equipped with a nondegenerate (−1)^{ν}-symmetric bilinear form

H^{ν}(O_{X}(ℓ′)) × H^{ν}(O_{X}(ℓ′)) → H^{n}(O_{X}(ℓ)) ⟶^{tr} k.

Let U ⊂ H^{ν}(O_{X}(ℓ′)) be an isotropic subspace of maximal dimension h^{ν}(O_{X}(ℓ′))/2. Let

W = U ⊕ ⨁_{t>ℓ′} H^{ν}(O_{X}(t)) ⊂ H_{∗}^{ν}(O_{X}) (17)

We begin the construction of the locally free resolution with the short exact sequence

0 → I_{X} → O_{ℙ^{n+3}} → O_{X} → 0. (18)

Since H_{∗}^{i}(O_{X}) ≅ H_{∗}^{i+1}(I_{X}) for 0 < i < n + 2, we have W ⊂ H_{∗}^{ν+1}(I_{X}).

Now since X is locally Cohen-Macaulay of equidimension n, the modules H_{∗}^{i}(I_{X}) are of finite length for 0 < i < n + 1. Hence the truncated complex C_{X}^{∗} = τ_{>0}τ_{≤ν+1,W} RΓ_{∗}(I_{X}) has cohomology modules H^{i}(C_{X}^{∗}) of finite length for 0 < i ≤ ν + 1, while H^{i}(C_{X}^{∗}) = 0 for all other i. Hence C_{X}^{∗} is in FinL.

The definition of C_{X}^{∗} as a truncation means that it is endowed with a natural map β : C_{X}^{∗} → τ_{>0}τ_{<n+3}RΓ_{∗}(I_{X}). By Proposition 2.10 this map induces a morphism β̃: Syz(C_{X}^{∗}) → I_{X}. Let Q be the cokernel

H_{∗}^{0}(Syz(C_{X}^{∗})) ⟶^{H_{∗}^{0}(β̃)} H_{∗}^{0}(I_{X}) → Q → 0.

Let d_{1}, …, d_{r} be the degrees of a minimal set of generators of Q. These generators lift to H_{∗}^{0}(I_{X}), allowing us to define a surjection

γ : F_{1} := Syz(C_{X}^{∗}) ⊕ ⨁ O_{ℙ^{n+3}}(−d_{i}) ↠ I_{X}. (19)

By construction, F_{1} is locally free.

Let K = ker(γ). We may then attach the short exact sequence 0 → K → F_{1} → I_{X} → 0 to the short exact sequence (18) to get an exact sequence

0 → K → F_{1} → O_{ℙ^{n+3}} → O_{X} → 0. (20)

The construction described above leads immediately to the following conclusions about the cohomology of F_{1} and about the induced morphisms H_{∗}^{i}(γ) : H_{∗}^{i}(F_{1}) → H_{∗}^{i}(I_{X}) (cf. Proposition 2.10).

• H_{∗}^{i}(γ) is surjective (resp. an isomorphism) for i = 0 (resp. 0 < i < ν + 1).

• H_{∗}^{ν+1}(γ) : H_{∗}^{ν+1}(F_{1}) ≅ W ↪ H_{∗}^{ν+1}(I_{X}) is injective.

• H_{∗}^{i}(F_{1}) = 0 for ν + 1 < i < n + 3.

One may now draw the following conclusions about the cohomology of K.

• H_{∗}^{i}(K) = 0 for 0 < i < ν + 2.

• H_{∗}^{ν+2}(K) ≅ H_{∗}^{ν}(O_{X})/W.

• H_{∗}^{i}(K) ≅ H_{∗}^{i−2}(O_{X}) for ν + 2 < i < n + 3.

To finish the definition of the locally free resolution, consider the isomorphisms

Ext^{1}(K, ω_{ℙ^{n+3}}(−ℓ)) ≅ H^{n+2}(K(ℓ))^{∗} ≅ H^{n}(O_{X}(ℓ))^{∗} ≅ H^{0}(O_{X}).

The extension class corresponding to 1 ∈ H^{0}(O_{X}) gives a short exact sequence

0 → ω_{ℙ^{n+3}}(−ℓ) → F_{2} → K → 0 (21)

which we may attach to (20) to get a complex of the type (7) resolving O_{X}

P^{∗} : 0 → ω_{ℙ^{n+3}}(−ℓ) → F_{2} → F_{1} → O_{ℙ^{n+3}}. (22)

Lemma 3.1. The sheaves F_{1} and F_{2} in the resolution (22) satisfy H_{∗}^{i}(F_{2}) ≅ (H_{∗}^{n+3−i}(F_{1}))^{∗} (ℓ) for 0 < i < n + 3.

Proof. If 0 < i < ν+2, then H_{∗}^{i}(F_{2}) ≅ H_{∗}^{i}(K) = 0 and H_{∗}^{n+3−i}(F_{1}) = 0. So the lemma holds for these values of i.

If i = ν + 2, then H_{∗}^{ν+2}(F_{2}) ≅ H_{∗}^{ν+2}(K) ≅ H_{∗}^{ν}(O_{X})/W, while H_{∗}^{ν+1}(F_{1}) ≅ W. However, the submodule W ⊂ H_{∗}^{ν}(O_{X}) has been constructed so that it is an isotropic submodule with respect to the perfect pairing of Serre duality

H_{∗}^{ν}(O_{X}) × H_{∗}^{ν}(O_{X}) → H_{∗}^{n}(O_{X}) ⟶^{tr} k(−ℓ).

Moreover the length of W is half the length of H_{∗}^{ν}(O_{X}). Hence W = W^{⊥}, and the duality isomorphism H_{∗}^{ν}(O_{X}) ≅ (H_{∗}^{ν}(O_{X}))^{∗} (ℓ) carries the submodule W onto (H_{∗}^{ν}(O_{X})/W)^{∗} (ℓ).

If ν + 2 < i < n + 2, then H_{∗}^{i}(F_{2}) ≅ H_{∗}^{i}(K) ≅ H_{∗}^{i−2}(O_{X}), while H_{∗}^{n+3−i}(F_{1}) ≅ H_{∗}^{n+3−i}(I_{X}) ≅ H_{∗}^{n+2−i}(O_{X}). The asserted duality is then simply the Serre duality pairing

H_{∗}^{i−2}(O_{X}) × H_{∗}^{n+2−i}(O_{X}) → H_{∗}^{n}(O_{X}) ⟶^{tr} k(−ℓ).

Finally if i = n + 2, we have an exact sequence

0 → H_{∗}^{n+2}(F_{2}) → H_{∗}^{n+2}(K) → H_{∗}^{n+3}(ω_{ℙ^{n+3}}(−ℓ)).

Now H_{∗}^{n+2}(K) ≅ H_{∗}^{n}(O_{X}). Moreover, the fact that the extension class defining F_{2} corresponded under the Serre duality identifications to 1 ∈ H_{∗}^{0}(O_{X}) ≅ (H_{∗}^{n+2}(K))^{∗} (ℓ) implies that the last exact sequence dualizes to

H_{∗}^{0}(O_{ℙ^{n+3}}) ⟶^{1} H_{∗}^{0}(O_{X}) → (H_{∗}^{n+2}(F_{2}))^{∗} (ℓ) → 0.

Hence (H_{∗}^{n+2}(F_{2}))^{∗} (ℓ) ≅ H_{∗}^{1}(I_{X}) ≅ H_{∗}^{1}(F_{1}). Dualizing now gives the last of the asserted isomorphisms.

Corollary 3.2. The coherent sheaf F_{2} in the resolution (22) is locally free.

Proof. Since F_{1} is locally free, H_{∗}^{i}(F_{1}) is of finite length for 0 < i < n+3. So by the lemma, H_{∗}^{i}(F_{2}) is also of finite length for 0 < i < n+3. In addition F_{2} is torsion-free, so H_{∗}^{0}(F_{2}) is also finitely generated. But this implies that F_{2} is locally free by [S], p. 269, Théorème 2.

Proposition 3.3. The locally free resolution (22) satisfies condition (a) of Proposition 1.2.

Proof. We write L = ω_{ℙ^{n+3}}(−ℓ). We have to show that if there is a commutative diagram

\begin{aligned}0 & ⟶ & L & ⟶^{d_{3}} & F_{2} & ⟶^{d_{2}} & F_{1} & ⟶^{d_{1}} & O_{ℙ^{n+3}} \\ & & ∥ & & ↓^{s_{2}} & & ↓^{s_{1}} & & ∥ \\ 0 & ⟶ & L & ⟶^{d_{1}^{∨}} & F_{1}^{∨} ⊗ L & ⟶^{−d_{2}^{∨}} & F_{2}^{∨} ⊗ L & ⟶^{d_{3}^{∨}} & O_{ℙ^{n+3}} \end{aligned} (23)

such that the vertical maps extend the identity on O_{X}, then s_{1} and s_{2} are isomorphisms.

By exactness, the image of d_{3}^{∨} is I_{X}. We will show that s_{1} is an isomorphism by applying Lemma 2.12 to the composition

F_{1} ⟶^{s_{1}} F_{2}^{∨} ⊗ L ↠ I_{X}.

Note that this composition is exactly the surjection γ : F_{1} ↠ I_{X} of (19). Hence the composition

H_{∗}^{i}(F_{1}) ⟶^{H_{∗}^{i}(s_{1})} H_{∗}^{i}(F_{2}^{∨} ⊗ L) → H_{∗}^{i}(I_{X})

is injective for 0 < i < n + 3. A fortiori, H_{∗}^{i}(s_{1}) is also injective for 0 < i < n + 3.

However, by Serre duality H_{∗}^{i}(F_{2}^{∨} ⊗ L) ≅ (H_{∗}^{n+3−i}(F_{2}))^{∗} (ℓ) for all i. So by Lemma 3.1, we have H_{∗}^{i}(F_{2}^{∨} ⊗ L) ≅ H_{∗}^{i}(F_{1}) for 0 < i < n + 3.

Hence for each 0 < i < n + 3, the morphism H_{∗}^{i}(s_{1}) is an injection of modules of the same finite length. Hence H_{∗}^{i}(s_{1}) is an isomorphism for 0 < i < n + 3. Thus condition (i) of Lemma 2.12 holds.

Condition (ii) of Lemma 2.12 holds because of the method of construction of F_{1} and of the surjection γ in (19). Finally exactness in the resolution implies that F_{1} and F_{2} have the same rank. Hence F_{1} and F_{2}^{∨}⊗L also have the same rank, which is condition (iii) of Lemma 2.12. Hence all three conditions of Lemma 2.12 hold, and we may conclude that s_{1} is an isomorphism.

The map s_{2} must now also be an isomorphism by the five-lemma. This completes the proof of the proposition.

4. The Differential Graded Algebra Structure

In this section we finish the proof of Theorem 0.1 by showing that the locally free resolution (22) defined in the previous section satisfies condition (b) of Proposition 1.2. That is to say, we show that the locally free resolution (22) admits a commutative, associative differential graded algebra structure.

Throughout this section we assume that the characteristic is not 2. We recall what needs to be proven. In the previous section we defined

a locally free resolution (22) of O_{X}

0 → L ⟶^{d_{3}} F_{2} ⟶^{d_{2}} F_{1} ⟶^{d_{1}} O_{ℙ^{n+3}} → O_{X}.

Let K = ker(d_{1}). There is then a morphism ψ : Λ^{2}F_{1} → K defined by ψ(a ∧ b) = d_{1}(a)b − d_{1}(b)a. We also have a long exact sequence

⋯ → Hom(Λ^{2}F_{1}, F_{2}) → Hom(Λ^{2}F_{1}, K) → Ext^{1}(Λ^{2}F_{1}, L) → · · · .

According to diagram (8), the problem is to lift ψ ∈ Hom(Λ^{2}F_{1}, K) to a ϕ ∈ Hom(Λ^{2}F_{1}, F_{2}). The obstruction to doing this is simply the image of ψ in

Ext^{1}(Λ^{2}F_{1}, L) = H^{n+2}(Λ^{2}F_{1}(ℓ))^{∗}.

Our first goal will therefore be to compute H^{n+2}(Λ^{2}F_{1}(ℓ)). We begin by considering a complex of locally free sheaves on ℙ^{N}.

G^{∗} : 0 → G^{0} → G^{1} → ⋯ → G^{r} → 0. (24)

There is an involution

T : G^{∗} ⊗ G^{∗} → G^{∗} ⊗ G^{∗}
a ⊗ b ↦ (−1)^{(deg a)(deg b)}b ⊗ a

interchanging the factors of G^{∗} ⊗ G^{∗}. Since the characteristic is not 2, the complex G^{∗} ⊗ G^{∗} splits into a direct sum of subcomplexes on which T acts as multiplication by ±1, viz. G^{∗} ⊗ G^{∗} = S_{2}(G^{∗}) ⊕ Λ^{2}(G^{∗}). The complex Λ^{2}(G^{∗}) is of the form

Λ^{2}(G^{∗}) : 0 → H^{0} → H^{1} → ⋯ → H^{2r} → 0 (25)

where (cf. [BE2] p. 452)

(26) H^{i} ≅ ⨁_{q<i/2} (G^{q} ⊗ G^{i−q}) ⊕ \begin{cases}0 & if i is odd, \\ Λ^{2}(G^{i/2}) & if i ≡ 0 (mod 4), \\ S_{2}(G^{i/2}) & if i ≡ 2 (mod 4). \end{cases}

Lemma 4.1. Suppose G^{∗} is a complex of locally free sheaves on ℙ^{N} as in (24) which is exact except in degree 0. Let E = H^{0}(G^{∗}). Then Λ^{2}(G^{∗}) is an exact sequence of locally free sheaves which is exact except in degree 0, and H^{0}(Λ^{2}(G^{∗})) = Λ^{2}E.

Proof. The standard spectral sequences of the double complex G^{∗} ⊗ G^{∗} degenerate to show that the simple complex G^{∗} ⊗ G^{∗} is exact except in degree 0, and H^{0}(G^{∗} ⊗ G^{∗}) = E ⊗ E. Thus the augmented complex 0 → E ⊗ E → G^{∗} ⊗ G^{∗} is exact, and consequently its direct summand 0 → Λ^{2}E → Λ^{2}(G^{∗}) is also exact.

Lemma 4.2. Suppose E is a locally free sheaf on ℙ^{N}. Let r < N/2 be an integer. Suppose that H_{∗}^{i}(E) = 0 for r < i < N. Then

(a) H_{∗}^{i}(Λ^{2}E) = 0 for 2r < i < N,

(b) H_{∗}^{2r}(Λ^{2}E) ≅ S_{2}(H_{∗}^{r}(E)) if r is odd, and H_{∗}^{2r}(Λ^{2}E) ≅ Λ^{2}(H_{∗}^{r}(E)) if r is even.

(c) If H^{r}(E(t)) = 0 for t < q for some integer q, then H^{2r}(Λ^{2}E(t)) = 0 for t < 2q, while H^{2r}(Λ^{2}E(2q)) ≅ S_{2}(H^{r}(E(q))) if r is odd, and H^{2r}(Λ^{2}E(2q)) ≅ Λ^{2}(H^{r}(E(q))) if r is even.

Proof. By Lemma 2.3(b), the minimal projective resolution P^{∗} of the truncation τ_{<N}RΓ_{∗}(E) is a complex of free graded S-modules such that P^{i} = 0 unless 0 ≤ i ≤ N − 1. Indeed, since H^{i}(τ_{<N}RΓ_{∗}(E)) = 0 for all i > r, Lemma 2.2 indicates that P^{i} = 0 unless 0 ≤ i ≤ r, i.e. P^{∗} is of the form

P^{∗} : 0 → P^{0} → ⋯ → P^{r−1} → P^{r} → 0.

We now consider the complex of free graded S-modules

Λ^{2}(P^{∗}) : 0 → Λ^{2}P^{0} → ⋯ → P^{r−1} ⊗ P^{r} → T_{2}(P^{r}) → 0.

where T_{2}(P^{r}) = Λ^{2}(P^{r}) if r is even, and T_{2}(P^{r}) = S_{2}(P^{r}) if r is odd (cf. (25) and (26)). According to Lemma 2.3, the complex of sheaves P̃^{∗} associated to P^{∗} is exact except in degree 0 where the homology is E. So Lemma 4.1 implies that the complex of sheaves Λ^{2}(P̃^{∗}) is also exact except in degree 0 where the homology is Λ^{2}E. The complex Λ^{2}(P^{∗}) of graded S-modules therefore has homology of finite length except in degree 0. Moreover, the complex Λ^{2}(P^{∗}) vanishes except in degrees between 0 and 2r < N, and the coefficients of its differentials lie in 𝔪 because it those of P^{∗} and therefore P^{∗} ⊗P^{∗} do. It now follows from Lemma 2.3(a) that Λ^{2}(P^{∗}) is the minimal projective resolution of τ_{<N}RΓ_{∗}(Λ^{2}E).

Therefore H_{∗}^{i}(Λ^{2}E) ≅ H^{i}(Λ^{2}(P^{∗})) for all i < N. In particular, since Λ^{2}(P^{∗}) is concentrated in degrees between 0 and 2r by (25), we see that H_{∗}^{i}(Λ^{2}E) = 0 for 2r < i < N. This is part (a) of the lemma.

For (b) note that H_{∗}^{r}(E) and H_{∗}^{2r}(Λ^{2}E) has respective presentations

\begin{aligned}P^{r−1} & ⟶^{δ} & P^{r} & → & H_{∗}^{r}(E) & → & 0, \\ P^{r−1} ⊗ P^{r} & ⟶^{δ_{1}} & T_{2}(P^{r}) & → & H_{∗}^{2r}(Λ^{2}E) & → & 0, \end{aligned}

where δ_{1}(e ⊗ f) = δ(e)f ∈ T_{2}(P^{r}). But since the presentation of T_{2}(H_{∗}^{r}(E)) is of exactly this form, we see that H_{∗}^{2r}(Λ^{2}E) ≅ T_{2}(H_{∗}^{r}(E)), as asserted by the lemma.

For (c) write H = H^{r}(E(q)). The hypothesis that H^{r}(E(t)) = 0 for t < q implies that P^{r} = (H ⊗_{k} S(−q)) ⊕ F with F = ⨁ S(−n_{i}) for some n_{i} > q. Then T_{2}(P^{r}) = (T_{2}H ⊗_{k} S(−2q)) ⊕ G with G = (H ⊗_{k} F (−q)) ⊕ T_{2}F = ⨁ S(−m_{j}) for some m_{j} > 2q. Thus T_{2}(P^{r}) vanishes in degrees less than 2q, as does its quotient H_{∗}^{2r}(Λ^{2}E). In other words H^{2r}(Λ^{2}E(t)) = 0 for t < 2q. Next note that since P^{∗} is a minimal projective resolution of τ_{<N}RΓ_{∗}(E), there are no nonzero constant entries in the matrix of δ and therefore none in the matrix of δ_{1} either. Hence no direct summand of P^{r−1}⊗P^{r} is mapped surjectively onto a summand of T_{2}(P^{r}). Thus the part of the image of P^{r−1} ⊗ P^{r} in degree 2q vanishes. Therefore the graded pieces of T_{2}(P^{r}) and of H_{∗}^{2r}(Λ^{2}E) in degree 2q are isomorphic, i.e. H^{2r}(Λ^{2}E(2q)) ≅ T_{2}H.

Corollary 4.3. Let n, ℓ, and X ⊂ ℙ^{n+3} be as in Theorem 0.1. Suppose that U ⊂ H^{n/2}(O_{X}(ℓ/2)) is the maximal isotropic subspace defined in (17), and that F_{1} is the locally free sheaf defined in (19). Then

H^{n+2}(Λ^{2}F_{1}(ℓ)) ≅ \begin{cases}0 & if n or ℓ is odd, \\ S_{2}U & if ℓ is even, and n ≡ 0 (mod 4), \\ Λ^{2}U & if ℓ is even, and n ≡ 2 (mod 4). \end{cases}

Proof. If n is odd, then H_{∗}^{i}(F_{1}) = 0 for (n + 1)/2 < i < n + 3. So Lemma 4.2(a) applies with r = (n + 1)/2. Therefore H_{∗}^{i}(Λ^{2}F_{1}) = 0 for n + 1 < i < n + 3, i.e. H^{n+2}(Λ^{2}F_{1}(t)) = 0 for all t.

If n is even but ℓ is odd, then Lemma 4.2(c) applies with r = (n+2)/2 and q = (ℓ + 1)/2. Then H^{n+2}(Λ^{2}F_{1}(t)) = 0 for all t < ℓ + 1.

If ℓ and n are even, then Lemma 4.2(c) applies with r = (n+2)/2 and q = ℓ/2. Since H^{(n+2)/2}(F_{1}(ℓ/2)) ≅ U, it follows that H^{n+2}(Λ^{2}F_{1}(ℓ)) ≅ Λ^{2}U if r is even, and H^{n+2}(Λ^{2}F_{1}(ℓ)) ≅ S_{2}U if r is odd. The corollary follows.

Lemma 4.4. If F_{1} is the locally free sheaf defined in (19), then the image of the map ψ of (8) in Ext^{1}(Λ^{2}F_{1}, L) ≅ H^{n+2}(Λ^{2}F_{1}(ℓ))^{∗} vanishes.

Proof. If n or ℓ is odd, then H^{n+2}(Λ^{2}F_{1}(ℓ)) = 0 according to Corollary 4.3, so the image of ψ is evidently zero.

If n and ℓ are even, then we claim that the image of ψ in the space H^{n+1}(Λ^{2}F_{1}(ℓ))^{∗} is the map

{S_{2}U or Λ^{2}U} → k

which is the restriction to U of the pairing

H^{n/2}(O_{X}(ℓ/2)) × H^{n/2}(O_{X}(ℓ/2)) → k

of (4). Since U was chosen isotropic, this map vanishes.

In order to prove the claim, we consider the diagonal i : ℙ^{n+3} = Δ ⊂ ℙ^{n+3} × ℙ^{n+3}. Then there is a natural inclusion i(X) ⊂ X × X which corresponds to a restriction map

O_{X×X} → i_{∗}O_{X}. (27)

This map is essentially the multiplication O_{X} ⊗ O_{X} → O_{X}. In any case applying RΓ_{∗} to (27) gives the cup product map

RΓ_{∗}(O_{X}) ⊗_{k} RΓ_{∗}(O_{X}) → RΓ_{∗}(O_{X}). (28)

Now consider the “resolution” of O_{X} given in (20)

K^{∗} : 0 → K → F_{1} ⟶^{d_{1}} O_{ℙ^{n+3}} → 0.

The complex K^{∗} is quasi-isomorphic to O_{X}. Hence the restriction to the diagonal map (27) corresponds to a morphism in the derived category

p_{1}^{∗}K^{∗} ⊗ p_{2}^{∗}K^{∗} → i_{∗}K^{∗}.

In fact this morphism in the derived category is represented by an actual map of complexes of sheaves

\begin{aligned}⋯ & → & p_{1}^{∗}K ⊕ (p_{1}^{∗}F_{1} ⊗ p_{2}^{∗}F_{1}) ⊕ p_{2}^{∗}K & → & p_{1}^{∗}F_{1} ⊕ p_{2}^{∗}F_{1} & → & O_{ℙ×ℙ} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & i_{∗}K & → & i_{∗}F_{1} & → & i_{∗}O_{ℙ} & → & 0 \end{aligned} (29)

All the vertical maps are straightforward restrictions to the diagonal except for the component p_{1}^{∗}F_{1} ⊗ p_{2}^{∗}F_{1} → i_{∗}(K) which is defined (like ψ of (8)) by noting that the composition

\begin{matrix}p_{1}^{∗}F_{1} ⊗ p_{2}^{∗}F_{1} & → p_{1}^{∗}F_{1} ⊕ p_{2}^{∗}F_{1} → & i_{∗}F_{1} \\ p_{1}^{∗}(a) ⊗ p_{2}^{∗}(b) & ↦ & i_{∗} (d_{1}(a)b − d_{1}(b)a) \end{matrix} (30)

is contained in the kernel of i_{∗}F_{1} → i_{∗}O_{ℙ}.

Now since K^{∗} is quasi-isomorphic to O_{X}, if we apply RΓ_{∗} to (29) we get a morphism RΓ_{∗}(p_{1}^{∗}K ⊗ p_{2}^{∗}K) → RΓ_{∗}(i_{∗}K^{∗}) in D^{b}(S ⊗ S-Mod) which is quasi-isomorphic to (28). In particular, the maps of hypercohomology are quasi-isomorphic to the cup product

H_{∗}^{n}(p_{1}^{∗}O_{X} ⊗ p_{2}^{∗}O_{X}) ≅ ⨁_{i} H_{∗}^{i}(O_{X}) ⊗_{k} H_{∗}^{n−i}(O_{X}) → H_{∗}^{n}(O_{X}).

The hypercohomology H_{∗}^{n}(K^{∗}) ≅ H_{∗}^{n}(O_{X}) is of course the same as the H^{n} of the total complex of the double complex

0 → RΓ_{∗}(K) → RΓ_{∗}(F_{1}) → RΓ_{∗}(O_{ℙ^{n+3}}) → 0.

According to the calculations at the beginning of the previous section, this H_{∗}^{n} is all attributable to K, i.e. the truncation K^{∗} → K[2] induces an isomorphism H_{∗}^{n}(O_{X}) ≅ H_{∗}^{n}(K^{∗}) ≅ H_{∗}^{n+2}(K).

Similarly the hypercohomology H_{∗}^{n}(p_{1}^{∗}K^{∗} ⊗ p_{2}^{∗}K^{∗}) is the same as the H^{n} of the total complex of RΓ_{∗} of the first row of (29). The submodule W ⊗_{k} W ⊂ H_{∗}^{n}(p_{1}^{∗}O_{X} ⊗ p_{2}^{∗}O_{X}) is attributable as the H_{∗}^{n+2} of the summand p_{1}^{∗}F_{1} ⊗ p_{2}^{∗}F_{1} in the first row of (29). Therefore H_{∗}^{n+2} of the vertical map p_{1}^{∗}F_{1} ⊗ p_{2}^{∗}F_{1} → K is simply the cup product map W ⊗_{k} W → H_{∗}^{n}(O_{X}).

Now the fact that i_{∗}(K) is supported on Δ, plus the symmetry of the product map imply that the vertical map of (29) factors as

\begin{matrix}p_{1}^{∗}F_{1} ⊗ p_{2}^{∗}F_{1} & → & i_{∗}(F_{1} ⊗ F_{1}) & → & i_{∗}(Λ^{2}F_{1}) & ⟶^{i_{∗}(ψ)} & i_{∗}(K). \\ p_{1}^{∗}(a) ⊗ p_{2}^{∗}(b) & ↦ & i_{∗}(a ⊗ b) & ↦ & i_{∗}(a ∧ b) & ↦ & d_{1}(a)b − d_{1}(b)a \end{matrix} (31)

We wish to calculate H_{∗}^{n+2} of the above morphisms. Let

P^{∗} : 0 → P^{0} → ⋯ → P^{(n+2)/2} → 0

be a minimal projective resolution of τ_{<n+3}RΓ_{∗}(F_{1}) (cf. Lemma 2.3). Then if one applies τ_{<n+3}RΓ_{∗} to the first two morphisms of (31), one gets the natural maps

P^{∗} ⊗_{k} P^{∗} → P^{∗} ⊗_{S} P^{∗} → Λ^{2}(P^{∗})

(cf. the proof of Lemma 4.2). All three complexes are supported in degrees between 0 and n + 2, and applying H^{n+2} gives surjections

W ⊗_{k} W ↠ W ⊗_{S} W ↠ {S_{2}W or Λ^{2}W }.

It therefore follows that H_{∗}^{n+2}(ψ) : H_{∗}^{n+2}(Λ^{2}F_{1}) → H_{∗}^{n+2}(K) is isomorphic to the cup product map {S_{2}W or Λ^{2}W } → H_{∗}^{n}(O_{X}). In particular, in degree ℓ the morphism H^{n+2}(Λ^{2}F_{1}(ℓ)) → H^{n+2}(K(ℓ)) is the same as {S_{2}U or Λ^{2}U} → H^{n}(O_{X}(ℓ)).

We now have to consider the extension of (21)

0 → ω_{ℙ^{n+3}}(−ℓ) → F_{2} → K → 0.

(Recall L = ω_{ℙ^{n+3}}(−ℓ).) In the associated long exact sequence of cohomology

⋯ → H^{n+2}(F_{2}(ℓ)) → H^{n+2}(K(ℓ)) ⟶^{tr} H^{n+3}(ω_{ℙ^{n+3}}) ≅ k,

the differential is the element of H^{n+2}(K(ℓ))^{∗} ≅ Ext^{1}(K(ℓ), ω_{ℙ^{n+3}}) corresponding to the extension class. So by construction the differential is the trace map tr ∈ H^{n+2}(K(ℓ))^{∗} ≅ H^{n}(O_{X}(ℓ))^{∗} which corresponds under Serre duality to 1 ∈ H^{0}(O_{X}).

Now the image of ψ in Ext^{1}(Λ^{2}F_{1}(ℓ), ω_{ℙ^{n+3}}) ≅ H^{n+2}(Λ^{2}F_{1}(ℓ))^{∗} is exactly the composition

H^{n+2}(Λ^{2}F_{1}(ℓ)) ⟶^{H^{n+2}(ψ)} H^{n+2}(K(ℓ)) ⟶^{tr} H^{n+3}(ω_{ℙ^{n+3}}) ≅ k.

By our previous calculations, this is the composition of the cup product map {S_{2}U or Λ^{2}U} → H^{n}(O_{X}(ℓ)) with the trace map H^{n}(O_{X}(ℓ)) → k. Therefore this composition is the restriction to S_{2}U or Λ^{2}U of the Serre duality pairing H^{n/2}(O_{X}(ℓ/2)) ⊗ H^{n/2}(O_{X}(ℓ/2)) → k. This is what was claimed at the beginning of the proof of the lemma. Since U was chosen isotropic, this composition vanishes, i.e. the image of ψ in Ext^{1}(K, L) vanishes.

Proof of Theorem 0.1. According to Proposition 1.2, in order to prove Theorem 0.1 it suffices to find a locally free resolution

0 → L → F_{2} → F_{1} → O_{ℙ^{n+3}} → O_{X}

which satisfies two conditions. But the locally free resolution defined in (22) was shown to satisfy the first of these conditions in Proposition 3.3. Moreover, this resolution was just shown to satisfy the second condition in Lemma 4.4. Hence Theorem 0.1 holds.

5. Characteristic 2 Computations

In the introduction, we asserted that Theorem 0.1 even if the base field is of characteristic 2 or is not algebraically closed provided that the parity condition is suitably modified. Over an algebraically closed field of characteristic 2, however, the only change in the statement is that all cases with n even must be considered in the parity condition. But the changes needed in the proof of the theorem are less trivial, since Lemmas 4.1 and 4.2 and Corollary 4.3 require assumptions that the characteristic is not 2. In this section we prove analogues of these lemmas and corollaries in characteristic 2, thereby extending Theorem 0.1 to characteristic 2.

Throughout this section we assume that the base field is of characteristic is 2. In Lemma 5.4 we will assume in addition that it is algebraically closed.

We recall certain simple facts from modular representation theory. Let R be a commutative algebra over a field of characteristic 2, and let V be a free R-module. Let t ∈ End(V ⊗ V) be the endomorphism t(a ⊗ b) = a ⊗ b − b ⊗ a. Set D_{2}V = ker(t), and Λ^{2}V = im(t), and S_{2}V = coker(t). Since t^{2} = 0 in characteristic 2, there are inclusions

0 ⊂ Λ^{2}V ⊂ D_{2}V ⊂ V ⊗ V

and corresponding surjection of quotients of V ⊗ V

V ⊗ V ↠ S_{2}V ↠ Λ^{2}V → 0.

The subquotient D_{2}V/Λ^{2}V is F (V), the Frobenius pullback of V. It is a free module of the same rank as V. This F (V) is also the kernel of the surjection S_{2}V ↠ Λ^{2}V.

Note that the natural map from S_{2}V to V ⊗ V given by xy ↦ x ⊗ y + y ⊗ x is not injective because any x^{2} ↦ 0. The map is the composition S_{2}V ↠ Λ^{2}V ↪ V ⊗ V with kernel F (V).

The operations D_{2}, Λ^{2}, S_{2}, and F are all functorial. Therefore we may define D_{2}E, Λ^{2}E, S_{2}E, and F (E) for any locally free sheaf E on any scheme X over a field of characteristic 2.

In some ways F has better properties than the others. If M = (m_{ij}) : R^{n} → R^{m} is a morphism of free R-modules, then F (M) = (m_{ij}^{2}) : R^{n} → R^{m}. One may use this formula together with the Buchsbaum-Eisenbud exactness criterion [BE1] to show that F of an exact sequence of locally free sheaves is exact. As a result of this we get the following lemma.

Lemma 5.1. Let E be a locally free sheaf on ℙ^{N}. If for some integer r one has H_{∗}^{i}(E) = 0 for r < i < N, then H_{∗}^{i}(F (E)) = 0 for r < i < N also.

Proof. Let P^{∗} be the minimal projective resolution of τ_{<N}RΓ_{∗}(E). By Lemmas 2.3 and 2.2, P^{∗} has the form

P^{∗} : 0 → P^{0} → P^{1} → ⋯ → P^{r} → 0.

Moreover, P^{∗} is exact except in degree 0 away from the irrelevant ideal 𝔪 ⊂ S, and H^{0}(P̃^{∗}) = E.

The functoriality and exactness of F now imply that

F (P^{∗}) : 0 → F (P^{0}) → F (P^{1}) → ⋯ → F (P^{r}) → 0

is exact except in degree in degree 0 away from the irrelevant ideal, and has H^{0}(F (P̃^{∗})) = F (E). Applying Lemma 2.3 again, we conclude that F (P^{∗}) is the minimal projective resolution of τ_{<N}RΓ_{∗}(F (E)).

So if r < i < N, then H_{∗}^{i}(F (E)) = H^{i}(F (P^{∗})) = 0 since F (P^{∗}) vanishes in degrees greater than r.

Corollary 5.2. Let E be a locally free sheaf on ℙ^{N} over a field of characteristic 2 such that H_{∗}^{N−1}(Λ^{2}E) = 0. Suppose r < N is an integer such that H_{∗}^{i}(E) = 0 for r < i < N. Then H_{∗}^{i}(S_{2}E) ≅ H_{∗}^{i}(Λ^{2}E) for r < i < N.

Proof. We consider the exact sequence 0 → F (E) → S_{2}E → Λ^{2}E → 0 and the associated long exact sequence

⋯ → H_{∗}^{i}(F (E)) → H_{∗}^{i}(S_{2}E) → H_{∗}^{i}(Λ^{2}E) → H_{∗}^{i+1}(F (E)) → ⋯

The hypothesis H_{∗}^{i}(E) = 0 for r < i < N implies also H_{∗}^{i}(F (E)) = 0 for r < i < N by Lemma 5.1. Hence the long exact sequence implies that H_{∗}^{i}(S_{2}E) ≅ H_{∗}^{i}(Λ^{2}E) for r < i < N − 1 and that H_{∗}^{N−1}(S_{2}E) ↪ H_{∗}^{N−1}(Λ^{2}E) is injective. But by hypothesis H_{∗}^{N−1}(Λ^{2}E) = 0. Therefore H_{∗}^{N−1}(S_{2}E) = 0 as well.

We now prove the analogue of Lemma 4.2.

Lemma 5.3. Suppose E is a locally free sheaf on ℙ^{N} over a field of characteristic 2. Let 0 < r < N/2 be an integer such that H_{∗}^{i}(E) = 0 for r < i < N. Then

(a) H_{∗}^{i}(Λ^{2}E) = 0 for 2r < i < N,

(b) H_{∗}^{2r}(Λ^{2}E) ≅ S_{2}(H_{∗}^{r}(E)).

(c) If H^{r}(E(t)) = 0 for t < q for some integer q, then H^{2r}(Λ^{2}E(t)) = 0 for t < 2q, while H^{2r}(Λ^{2}E(2q)) ≅ S_{2}(H^{r}(E(q))).

Proof. Let P^{∗} be the minimal projective resolution of τ_{<N}RΓ_{∗}(E)

P^{∗} : 0 → P^{0} ⟶^{δ^{0}} P^{1} ⟶^{δ^{1}} P^{2} → ⋯ → P^{r} → 0

Let P = P̃^{0}, and let F = ker(δ^{1})^{∼}. Then we have an exact sequence 0 → E → P → F → 0 such that P is a direct sum of line bundles, H_{∗}^{i}(F) = H_{∗}^{i+1}(E) for 0 < i < N − 1, and H_{∗}^{N−1}(F) = 0.

It is easy to see that there is a natural exact complex

0 → Λ^{2}E → Λ^{2}P → P ⊗ F → S_{2}F → 0. (32)

We now prove parts (a) and (b) of the lemma by induction on r. If r = 1, then F = P̃^{1} is a direct sum of line bundles, and the complex (32) is just the augmented complex

0 → Λ^{2}E → Λ^{2}(P̃^{∗})

which is still exact in this case. So we may conclude just as in Lemma 4.2 that H_{∗}^{i}(Λ^{2}E) = 0 for 2 < i < N, and that H_{∗}^{2}(Λ^{2}E) = S_{2}(H_{∗}^{1}(E)).

If r > 1, then H_{∗}^{i}(F) = 0 for r − 1 < i < N. So by induction H_{∗}^{i}(Λ^{2}F) = 0 for 2r − 2 < i < N, and also H_{∗}^{2r−2}(Λ^{2}F)) ≅ S_{2}(H_{∗}^{r−1}(F)) ≅ S_{2}(H_{∗}^{r}(E)). It now follows from Corollary 5.2 that H_{∗}^{i}(S_{2}F) ≅ H_{∗}^{i}(Λ^{2}F) = 0 for r − 1 < i < N. So in particular H_{∗}^{i}(S_{2}F) = 0 for 2r − 2 < i < N and that H_{∗}^{2r−2}(S_{2}F) ≅ S_{2}(H_{∗}^{r}(E)). Now since P is a direct sum of line bundles, we have H_{∗}^{i}(Λ^{2}P) = 0 for 0 < i < N, and H_{∗}^{i}(P ⊗ F) = 0 for r − 1 < i < N. So if we break up (32) into short exact sequences and take its graded cohomology, we can deduce that H_{∗}^{i}(Λ^{2}E)) ≅ H_{∗}^{i−2}(S_{2}F) for r + 1 < i < N. Since r + 1 < 2r, this gives parts (a) and (b) of the lemma.

Part (c) of the lemma follows from part (b) by the same argument as in Lemma 4.2.

We have the following corollary in analogy with Corollary 4.3.

Corollary 5.4. Let n, ℓ, and X ⊂ ℙ_{k}^{n+3} be as in Theorem 0.1 with k an algebraically closed field of characteristic 2. Suppose that U ⊂ H^{n/2}(O_{X}(ℓ/2)) is the maximal isotropic subspace defined in (17), and that F_{1} is the locally free sheaf defined in (19). Then

H^{n+2}(Λ^{2}F_{1}(ℓ)) ≅ \begin{cases}0 & if n or ℓ is odd, \\ S_{2}U & if n and ℓ are even. \end{cases}

The proof of Lemma 4.4 in characteristic 2 is essentially the same as in the previous section, only with Lemma 5.3 and Corollary 5.4 replacing their analogues, and with T_{2} = S_{2} always. Hence Theorem 0.1 also holds in characteristic 2 as long as one treats all even n the same.

6. The Local Version of the Main Theorem

In this section we consider Theorem 0.3, the local version of our main result. We state a variant version which is clearly a local analogue of Theorem 0.1 with an identical proof, and then show that this variant version is equivalent to Theorem 0.3.

Let (R, 𝔪, k) be a regular local ring, and let U = Spec(R) − {𝔪} be the punctured spectrum of R. We say that a closed subscheme Y ⊂ U of pure codimension 3 is Pfaffian if O_{X} has a locally free resolution on U

0 → O_{U} ⟶^{h} E^{∨} ⟶^{f} E ⟶^{g} O_{U} → O_{X}

where E is a locally free O_{U}-module of odd rank 2p + 1, f is skewsymmetric, and g and h = g^{∨} are given locally by the Pfaffians of order 2p of f. The following theorem is the obvious local analogue of Theorem 0.1.

Theorem 6.1. Let (R, 𝔪, k) be a regular local ring of dimension n + 4 > 4 with an algebraically closed residue field not of characteristic 2, and let U = Spec(R) − {𝔪}. Let X ⊂ U be a closed subscheme of pure codimension 3. Then X is Pfaffian if and only if the following three conditions hold:

(a) X is locally Gorenstein,

(b) ω_{X} ≅ O_{X}, and

(c) if n ≡ 0 (mod 4), then H^{n/2}(O_{X}) is of even length.

Theorem 6.1 may be proven in exactly the same manner as Theorem 0.1. All results concerning the Buchsbaum-Eisenbud proof, the Horrocks correspondence, duality, and the cohomology of H^{n+2}(Λ^{2}F_{1}) work identically for graded modules over polynomial rings over k and for modules over regular local k-algebras. There is only one point which is in any way more subtle in the local case. Namely, if n is even, then one has a duality pairing of R-modules of even finite length H^{n/2}(O_{X}) × H^{n/2}(O_{X}) → k. This pairing is perfect in the sense that for any submodule M ⊂ H^{n/2}(O_{X}) one has

length(M) + length(M^{⊥}) = length(H^{n/2}(O_{X})).

In order to be able to define C_{X}^{∗} and F_{1} as in (19) one must choose an isotropic submodule W of length equal to half that of H^{n/2}(O_{X}). But it is not difficult to show that this is possible when the residue field is algebraically closed.

We now compare Theorems 0.3 and 6.1. First of all, E = Γ(E) gives a bijective correspondence between locally free sheaves E on U and reflexive R-modules E such that E_{𝔭} is a free R_{𝔭}-module for all prime ideals 𝔭 ≠ 𝔪. There is also bijective correspondence between closed subschemes X ⊂ U of pure codimension 3 and unmixed ideals I ⊂ R of height 3 given by I = Γ(I_{X}). Hence an ideal I is Pfaffian in the sense of Theorem 0.3 if and only if the corresponding subscheme X ⊂ U is Pfaffian in the sense of Theorem 6.1.

The three conditions (a), (b), and (c) of the two theorems also correspond. In the case of (a) this is obvious. For (b) note that ω_{R/I} ≅ Γ(ω_{X}) since for all 𝔭 ∈ U one has ω_{R/I,𝔭} = Ext_{R_{𝔭}}^{3}((R/I)_{𝔭}, R_{𝔭}) = ω_{X,𝔭}, and ω_{R/I} is saturated. Similarly (R/I)^{sat} ≅ Γ(O_{X}). This gives the equivalence of the two conditions (b).

As for the conditions (c), first note that the dimension n in Theorem 0.3 corresponds to n + 4 in Theorem 6.1. But if one uses n as in the latter theorem, one has

H^{n/2}(U, O_{X}) ≅ H^{(n+2)/2}(U, I_{X}) ≅ H_{𝔪}^{(n+4)/2}(I).

Hence the two conditions (c) correspond.

Therefore the two theorems 0.3 and 6.1 are equivalent, as claimed. In equicharacteristic 2 the computations of Section 5 remain true in

the local case. So Theorems 0.3 and 6.1 are true in equicharacteristic 2 (with an algebraically closed residue field) provided one changes the phrase “n ≡ 0 (mod 4)” in the parity condition to “n is even.” If R is a regular local ring with residue field of characteristic 2 and quotient field of characteristic 0, a different set of calculations is needed. These are unfortunately somewhat involved, and we do not reproduce them here.

References

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[W] C. Walter, Algebraic Methods for the Cohomology of Normal Bundles of Algebraic Space Curves, preprint.

URA 168, Mathématiques, Université de Nice, Parc Valrose, F-06108 Nice Cedex 02, France

  E-mail address: walter@math.unice.fr