A REALIZATION THEOREM FOR MODULES OF CONSTANT JORDAN TYPE AND VECTOR BUNDLES

DAVE BENSON AND JULIA PEVTSOVA^{∗}

Abstract. Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors ℱ_{i} from finitely generated kE-modules of constant Jordan type to vector bundles over projective space ℙ^{r−1}. The fibers of the functors ℱ_{i} encode complete information about the Jordan type of the module.

We prove that given any vector bundle ℱ of rank s on ℙ^{r−1}, there is a kE-module M of stable constant Jordan type [1]^{s} such that ℱ_{1}(M) ≅ ℱ if p = 2, and such that ℱ_{1}(M) ≅ F^{∗}(ℱ) if p is odd. Here, F: ℙ^{r−1} → ℙ^{r−1} is the Frobenius map. We prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type [1]^{s} then the Chern numbers c_{1}, …, c_{p−2} of ℱ_{1}(M) are divisible by p.

1. Introduction

The class of modules of constant Jordan type was introduced by Carlson, Friedlander and the second author [5], and then consequently studied in [1, 2, 4, 6, 8, 7]. The connection between modules of constant Jordan type and algebraic vector bundles on projective varieties was first observed and developed by Friedlander and the second author in [8] in the general setting of an arbitrary infinitesimal group scheme. In the present paper, we study this connection for an elementary abelian p-group.

Let k be a field of characteristic p and let E be an elementary abelian p-group of rank r. We define functors ℱ_{i} (1 ≤ i ≤ p) from finitely generated kE-modules of constant Jordan type to vector bundles on projective space ℙ^{r−1}, capturing the sum of the socles of the length i Jordan blocks. The following is the main theorem of this paper.

Theorem 1.1. Given any vector bundle ℱ of rank s on ℙ^{r−1}, there exists a finitely generated kE-module M of stable constant Jordan type [1]^{s} such that

(i) if p = 2, then ℱ_{1}(M) ≅ ℱ.

(ii) if p is odd, then ℱ_{1}(M) ≅ F^{∗}(ℱ), the pullback of ℱ along the Frobenius morphism F: ℙ^{r−1} → ℙ^{r−1}.

The kE-modules produced this way are usually large. For example, in [1], the first author showed how to produce a finitely generated kE-module M of constant Jordan type such that ℱ_{2}(M) is isomorphic to the rank two Horrocks–Mumford bundle on ℙ^{4}. In this case, the construction used to prove our main theorem produces a module M of dimension many hundred times p^{5} plus two such that ℱ_{1}(M) ≅ F^{∗}(ℱ_{HM}), whereas the construction in [1] produces a module of dimension 30p^{5} of stable constant Jordan type [p − 1]^{30}[2]^{2}[1]^{26} such that applying ℱ_{2} gives ℱ_{HM}(−2).

The theorem for p = 2 may be thought of as a version of the Bernstein–Gelfand–Gelfand correspondence [3], since the group algebra of an elementary abelian 2-group in characteristic two is isomorphic to an exterior algebra. But for p odd it says something new and interesting. In particular, it is striking that the p odd case of Theorem 1.1 cannot be strengthened to say that ℱ_{1}(M) ≅ ℱ. The following theorem, which is proved in Section 6, gives limitations on the vector bundles appearing as ℱ_{1}(M) with M of stable constant Jordan type [1]^{s}.

∗ partially supported by the NSF.

Theorem 1.2. Suppose that M has stable constant Jordan type [1]^{s}. Then p divides the Chern numbers c_{m}(ℱ_{1}(M)) for 1 ≤ m ≤ p − 2.

The paper is organized as follows. In Section 2 we give basic definitions of the functors ℱ_{i} and show that applied to modules of constant Jordan type, they produce algebraic vector bundles. Section 3 together with Lemmas 5.1 and 5.2 analyze behavior of the functors ℱ_{i} with respect to Heller shifts and duals. This analysis plays a key role in the proof of our main theorem. In Section 4 we construct a functor from the bounded derived category of coherent 𝒪_{ℙ^{r−1}}–modules to the stable module category of kE. This functor is employed in the proof of Theorem 1.1 which occupies Section 5. Finally, in Section 6 we discuss Chern classes of the functors ℱ_{i} and prove Theorem 1.2.

2. Definition of the functors ℱ_{i}

Let k be a perfect field of characteristic p. Let E = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian p-group of rank r, and set X_{i} = g_{i} − 1 ∈ kE for 1 ≤ i ≤ r. Let J(kE) = ⟨X_{1}, …, X_{r}⟩ be the augmentation ideal of kE. The images of X_{1}, …, X_{r} form a basis for J(kE)/J^{2}(kE), which we think of as affine space 𝔸_{k}^{r} over k. Let K/k be a field extension. If 0 ≠ α = (λ_{1}, …, λ_{r}) ∈ 𝔸_{K}^{r}, we define

X_{α} = λ_{1}X_{1} + ⋯ + λ_{r}X_{r} ∈ KE.

This is an element of J(KE) satisfying X_{α}^{p} = 0. If M is a finitely generated kE-module then X_{α} acts nilpotently on M_{K} = M ⊗ K, and we can decompose M_{K} into Jordan blocks. They all have eigenvalue zero, and length between 1 and p. We say that M has constant Jordan type [p]^{a_{p}} … [1]^{a_{1}} if there are a_{p} Jordan blocks of length p, …, a_{1} blocks of length 1, independently of the choice of α. Since a_{p} is determined by a_{p−1}, …, a_{1} and the dimension of M, we also say that M has stable constant Jordan type [p − 1]^{a_{p−1}} … [1]^{a_{1}}. Note that the property of having constant Jordan type and the type itself do not depend on the choice of generators ⟨g_{1}, …, g_{r}⟩ (see [5]).

We write k[Y_{1}, …, Y_{r}] for the coordinate ring k[𝔸^{r}], where the Y_{i} are the linear functions defined by Y_{i}(X_{j}) = δ_{ij} (Kronecker delta). We write ℙ^{r−1} for the corresponding projective space. Let 𝒪 be the structure sheaf on ℙ^{r−1}. If ℱ is a sheaf of 𝒪-modules and j ∈ ℤ, we write ℱ(j) for the jth Serre twist ℱ ⊗_{𝒪} 𝒪(j). If M is a finitely generated kE-module, we write M̃ for the trivial vector bundle M ⊗_{k} 𝒪, so that M̃(j) = M ⊗_{k} 𝒪(j). Friedlander and the second author [8, §4] define a map of vector bundles θ_{M}: M̃ → M̃(1) by the formula

θ_{M}(m ⊗ f) = ∑_{i=1}^{r} X_{i}(m) ⊗ Y_{i}f.

By abuse of notation we also write θ_{M} for the twist θ_{M}(j): M̃(j) → M̃(j + 1). With this convention we have θ_{M}^{p} = 0.

We define functors ℱ_{i,j} for 0 ≤ j < i ≤ p from finitely generated kE-modules to coherent sheaves on ℙ^{r−1} by taking the following subquotients of M̃:

ℱ_{i,j}(M) = \frac{Ker θ_{M}^{j+1} ∩ Im θ_{M}^{i−j−1}}{(Ker θ_{M}^{j+1} ∩ Im θ_{M}^{i−j}) + (Ker θ_{M}^{j} ∩ Im θ_{M}^{i−j−1})}

We then define

ℱ_{i}(M) = ℱ_{i,0}(M) = \frac{Ker θ_{M} ∩ Im θ_{M}^{i−1}}{Ker θ_{M} ∩ Im θ_{M}^{i}}

For a point 0 ≠ α ∈ 𝔸^{r} and the corresponding operator X_{α}: M → M, we also define

ℱ_{i,α}(M) = \frac{Ker X_{α} ∩ Im X_{α}^{i−1}}{Ker X_{α} ∩ Im X_{α}^{i}}

Note that ℱ_{i,α}(M) is evidently well-defined for ᾱ ∈ ℙ^{r−1}.

In the next Proposition we show that functors ℱ_{i} take modules of constant Jordan type to algebraic vector bundles (equivalently, locally free sheaves), and that they commute with specialization.

Proposition 2.1.

(1) Let M be a kE-module of constant Jordan type [p]^{a_{p}} … [1]^{a_{1}}. Then the sheaf ℱ_{i}(M) is locally free of rank a_{i}.

(2) Let f: M → N be a map of modules of constant Jordan type. For any point ᾱ = [λ_{1} : ⋯: λ_{r}] ∈ ℙ^{r−1} with residue field k(ᾱ) we have a commutative diagram

\xymatrix{ℱ_{i}(M) ⊗_{𝒪} k(ᾱ) \ar[d]^{≃} \ar[r]^{ℱ_{i}(f)} & ℱ_{i}(N) ⊗_{𝒪} k(ᾱ) \ar[d]^{≃} \\ ℱ_{i,α}(M) \ar[r] & ℱ_{i,α}(N)}

Proof. (1). Since the module M is fixed throughout the proof, we shall use θ to denote θ_{M}. Note that Ker θ ∩ Im θ^{i} = Ker {θ: Im θ^{i} → Im θ^{i+1}}. Hence, we have a short exact sequence

(2.1.1)  \xymatrix{0 \ar[r] & Ker θ ∩ Im θ^{i} \ar[r] & Im θ^{i} \ar[r]^{θ} & Im θ^{i+1} \ar[r] & 0}

Since M has constant Jordan type, Im θ^{i} is locally free by [8, 4.13]. Therefore, specialization of the sequence (2.1.1) at any point ᾱ = [λ_{1} : ⋯: λ_{r}] of ℙ^{r−1} yields a short exact sequence of vector spaces

(2.1.2)  0 → (Ker θ ∩ Im θ^{i}) ⊗_{𝒪} k(ᾱ) → Im θ^{i} ⊗_{𝒪} k(ᾱ) → Im θ^{i+1} ⊗_{𝒪} k(ᾱ) → 0.

By [8, 4.13], Im θ^{i} ⊗_{𝒪} k(ᾱ) ≃ Im {X_{α}^{i}: M → M}. In particular, the dimension of fibers of

Im θ^{i} is constant and equals ∑_{j=i+1}^{p} a_{j}(j − i). We can rewrite the sequence (2.1.2) as

\xymatrix{0 \ar[r] & (Ker θ ∩ Im θ^{i}) ⊗_{𝒪} k(ᾱ) \ar[r] & Im X_{α}^{i} \ar[r]^{X_{α}} & Im X_{α}^{i+1} \ar[r] & 0}.

3

Hence, the fiber of Ker θ∩Im θ^{i} at a point ᾱ equals Ker X_{α} ∩Im X_{α}^{i}. In particular, Ker θ∩Im θ^{i} has fibers of constant dimension equal to

∑_{j=i+1}^{p} a_{j}(j − i) − ∑_{j=i+2}^{p} a_{j}(j − i − 1) = ∑_{j=i+1}^{p} a_{j}.

Applying [8, 4.11] (see also [10, V. ex. 5.8]), we conclude that Ker θ ∩ Im θ^{i} is locally free of

rank ∑_{j=i+1}^{p} a_{j}.

Consider the short exact sequence that defines ℱ_{i}(M):

\xymatrix{0 \ar[r] & Ker θ ∩ Im θ^{i} \ar[r] & Ker θ ∩ Im θ^{i−1} \ar[r] & ℱ_{i}(M) \ar[r] & 0.}

Specializing at ᾱ, we get

[figure: (Ker θ ∩ Im θ^{i}) ⊗_{𝒪} k(ᾱ); (Ker θ ∩ Im θ^{i−1}) ⊗_{𝒪} k(ᾱ); ℱ_{i}(M) ⊗_{𝒪} k(ᾱ); 0; Ker X_{α} ∩ Im X_{α}^{i}; Ker X_{α} ∩ Im X_{α}^{i−1}; ℱ_{i}(M) ⊗_{𝒪} k(ᾱ); 0]

The first arrow of the bottom row is clearly an injection. Hence,

dim(ℱ_{i}(M) ⊗_{𝒪} k(ᾱ)) = ∑_{j=i}^{p} a_{j} − ∑_{j=i+1}^{p} a_{j} = a_{i}

for any point α ∈ ℙ^{r−1}. Applying [8, 4.11] again, we conclude that ℱ_{i}(M) is locally free (of rank a_{i}).

Statement (2) follows immediately by applying the last diagram to both M and N. □

Lemma 2.2. M̃ has a filtration in which the filtered quotients are isomorphic to ℱ_{i,j}(M) for 0 ≤ j < i ≤ p.

Proof. We consider two filtrations on M̃, the “kernel filtration” and the “image filtration”:

0 ⊂ Ker θ_{M} ⊂ … ⊂ Ker θ_{M}^{p−1} ⊂ M̃
0 = Im θ_{M}^{p} ⊂ Im θ_{M}^{p−1} ⊂ … ⊂ Im θ_{M} ⊂ Im θ_{M}^{0} = M̃

To simplify notation, we set 𝒦_{j} = Ker θ_{M}^{i} and ℐ_{i} = Im θ_{M}^{p−i}. Using the standard refinement procedure, we refine the kernel filtration by the image filtration:

𝒦_{j} ⊂ (𝒦_{j+1} ∩ ℐ_{1}) + 𝒦_{j} ⊂ … ⊂ (𝒦_{j+1} ∩ ℐ_{ℓ}) + 𝒦_{j} ⊂ (𝒦_{j+1} ∩ ℐ_{ℓ+1}) + 𝒦_{j} ⊂ … ⊂ 𝒦_{j+1}

For any three sheaves A, B, C with B ⊂ A, the second isomorphism theorem and the modular law imply that

\frac{A + C}{B + C} ≃ \frac{A + (B + C)}{B + C} ≃ \frac{A}{A ∩ (B + C)} ≃ \frac{A}{B + (A ∩ C)}.

Hence, we can identify the subquotients of the refined kernel filtration above as

\frac{(𝒦_{j+1} ∩ ℐ_{ℓ+1}) + 𝒦_{j}}{(𝒦_{j+1} ∩ ℐ_{ℓ}) + 𝒦_{j}} ≃ \frac{𝒦_{j+1} ∩ ℐ_{ℓ+1}}{(𝒦_{j+1} ∩ ℐ_{ℓ}) + (𝒦_{j} ∩ ℐ_{ℓ+1})}

Setting i = p − ℓ + j, we get that the latter quotient is precisely ℱ_{i,j}(M) (note that when j > ℓ, the corresponding subquotient is trivial). □

Lemma 2.3. For 0 ≤ j < i, we have a natural isomorphism ℱ_{i,j}(M) ≅ ℱ_{i}(M)(j).

Proof. For 0 < j < i, the map θ_{M}: M̃ → M̃(1) induces a natural isomorphism ℱ_{i,j}(M) → ℱ_{i,j−1}(M)(1). Since ℱ_{i,0} = ℱ_{i}, the result follows by induction on j. □

Remark 2.4. It follows from the proof of Proposition 2.1 that the subquotient functors ℱ_{i,j} are linked as follows:

[figure: ℱ_{p,p−1}; ℱ_{p−1,p−2}; ℱ_{p,p−2}; ℱ_{p−2,p−3}; ℱ_{p−1,p−3}; …; …; …; …; …; ℱ_{1,0}; ℱ_{p−1,1}; …; ℱ_{p,1}; ℱ_{p−2,0}; ℱ_{p−1,0}; ℱ_{p,0}]

We finish this section with an example.

Example 2.5. Let M = kE/J^{2}(kE). Then M has constant Jordan type [2][1]^{r−1}. In the short exact sequence of vector bundles

0 → { \atop M/RadM} ⟶^{θ} { \atop RadM}(1) → ℱ_{1}(M)(1) → 0

the map θ (induced by θ_{M}) is equal to the map defining the tangent bundle (or sheaf of derivations) 𝒯 of ℙ^{r−1}:

0 → 𝒪 → 𝒪(1)^{r} → 𝒯 → 0.

It follows that ℱ_{1}(M) ≅ 𝒯(−1). On the other hand we have ℱ_{2,1}(M) ≅ 𝒪, and hence ℱ_{2}(M) ≅ 𝒪(−1).

3. Twists and syzygies

We need a general lemma whose proof we provide for completeness.

Lemma 3.1. Let X be a Noetherian scheme over k, and let M, N be locally free 𝒪_{X}-modules. Let f: M → N be a morphism of 𝒪_{X}-modules such that

f ⊗_{𝒪_{X}} k(x): M ⊗_{𝒪_{X}} k(x) → N ⊗_{𝒪_{X}} k(x)

is an isomorphism for any x ∈ X. Then f is an isomorphism.

Proof. It suffices to show that f induces an isomorphism on stalks. Hence, we may assume that X = Spec R, where R is a local ring with the maximal ideal 𝔪, and M, N are free modules. Since specialization is right exact, f is surjective by Nakayama’s lemma. Hence, we have an exact sequence of R-modules:

0 → ker f → M → N → 0.

Since N is free, Tor_{1}^{R}(N, R/𝔪) vanishes, and hence ker f ⊗_{R} R/𝔪 = 0. By Nakayama’s lemma, ker f = 0; therefore, f is injective. □

Theorem 3.2. Let M be a finite dimensional kE-module and let 1 ≤ i ≤ p − 1. Then there is a natural isomorphism

ℱ_{i}(M)(−p + i) ≅ ℱ_{p−i}(ΩM).

Proof. Consider the diagram

(3.2.1)  \xymatrix{0 \ar[r] & Ω̃M \ar[r] \ar[d]^{θ_{ΩM}} & P̃_{M} \ar[r] \ar[d]^{θ_{P_{M}}} & M̃ \ar[r] \ar[d]^{θ_{M}} & 0 \\ 0 \ar[r] & Ω̃M(1) \ar[r] & P̃_{M}(1) \ar[r] & M̃(1) \ar[r] & 0,}

where P_{M} is a projective cover of M. Let

δ: Ker θ_{M} → Coker θ_{ΩM}

be the switchback map. A simple diagram chase in conjunction with the fact that θ_{P_{M}}^{p} = 0 yields that the restriction of δ to Ker θ_{M} ∩ Im θ_{M}^{i−1} lands in

\frac{Ker θ_{ΩM}^{p−i}}{Ker θ_{ΩM}^{p−i} ∩ Im θ_{Ω(M)}} (1).

Projecting the latter onto

ℱ_{p−i,p−i−1}(ΩM)(1) = \frac{Ker θ_{ΩM}^{p−i}}{Ker θ_{ΩM}^{p−i−1} + Ker θ_{ΩM}^{p−i} ∩ Im θ_{Ω(M)}} (1),

we get a map of bundles:

δ′: Ker θ_{M} ∩ Im θ_{M}^{i−1} → ℱ_{p−i,p−i−1}(ΩM)(1).

Since δ′ evidently kills Ker θ_{M} ∩ Im θ_{M}^{i}, we conclude that δ′ factors through ℱ_{i}(M). Hence, we have an induced map

δ̄: ℱ_{i}(M) → ℱ_{p−i,p−i−1}(ΩM)(1).

A simple block count shows that this is an isomorphism at each fiber. Hence, by Lemma 3.1, this is an isomorphism of bundles. Thus using Lemma 2.3 (i.e., applying θ_{ΩM} a further p − i − 1 times), we have

ℱ_{i}(M) ≅ ℱ_{p−i,p−i−1}(ΩM)(1) ≅ ℱ_{p−i}(ΩM)(p − i).

Twisting by 𝒪(−p + i), we get the desired isomorphism.

Let f: M → N be a map of kE-modules. The naturality of the isomorphism

ℱ_{i}(M)(−p + i) ≅ ℱ_{p−i}(ΩM)

is equivalent to the commutativity of the diagram

(3.2.2)  \xymatrix{ℱ_{i}(M)(−p + i) \ar[d]^{≃} \ar[r]^{ℱ_{i}(f)(−p+i)} & ℱ_{i}(N)(−p + i) \ar[d]^{≃} \\ ℱ_{p−i}(ΩM) \ar[r]^{ℱ_{p−i}(Ωf)} & ℱ_{p−i}(ΩN).}

The commutativity follows from the construction of the map δ̄ and naturality of the “shifting isomorphism” of Lemma 2.3. □ Corollary 3.3. Let M be a finite dimensional kE-module and let 1 ≤ i ≤ p − 1. Then ℱ_{i}(Ω^{2}M) ≅ ℱ_{i}(M)(−p). Proof. Apply the theorem twice. □ Corollary 3.4. We have ℱ_{1}(Ω^{2n}k) ≅ 𝒪(−np), and ℱ_{p−1}(Ω^{2n−1}k) ≅ 𝒪(1 − np). Proof. This follows from the theorem and the corollary, using the isomorphism ℱ_{1}(k) ≅ 𝒪. □ Remark 3.5. If p = 2 then Theorem 3.2 and Corollary 3.4 reduce to the statements that ℱ_{1}(ΩM) ≅ ℱ_{1}(M)(−1) and ℱ_{1}(Ω^{n}k) ≅ 𝒪(−n).

For a coherent sheaf ℰ, we denote by ℰ^{∨} = ℋom_{𝒪} (ℰ, 𝒪) the dual sheaf. Theorem 3.6. Let M^{∗} be the k-linear dual of M, as a kE-module. Then

ℱ_{i}(M^{∗}) ≅ ℱ_{i}(M)^{∨}(−i + 1).

Proof. We shall first establish the following isomorphism:

(3.6.1)  ℱ_{i,i−1}(M^{∗}) ≅ ℱ_{i,0}(M)^{∨}

By definition, ℱ_{i,i−1}(M^{∗}) ≃ \frac{Ker Θ_{M^{∗}}^{i}}{Ker Θ_{M^{∗}}^{i}∩Im Θ_{M^{∗}}+Ker Θ_{M^{∗}}^{i−1}} . Let ϕ ∈ Ker Θ_{M^{∗}}^{i}. Hence ϕ : M̃ → 𝒪 is an 𝒪-linear map such that ϕ ∘ Θ_{M}^{i} = 0. Restricting ϕ to Ker Θ_{M} ∩ Im Θ_{M}^{i−1} we obtain a linear map

ϕ′ : Ker Θ_{M} ∩ Im Θ_{M}^{i−1} → 𝒪

The condition ϕ ∘ Θ_{M}^{i} = 0 immediately implies that ϕ′ vanishes on Ker Θ_{M} ∩ Im Θ_{M}^{i}. Hence, it induces a linear map on the quotient:

\xymatrix{\overline{ϕ} : ℱ_{i}(M) = \frac{Ker Θ_{M} ∩Im Θ_{M}^{i−1}}{Ker Θ_{M} ∩Im Θ_{M}^{i}} \ar[r] & 𝒪}

We define

\xymatrix{Φ : Ker Θ_{M}^{i} \ar[r] & Hom_{𝒪} (ℱ_{1}(M), 𝒪) = ℱ_{1}(M)^{∨}}

via

ϕ ↦ ϕ̄.

Suppose that ϕ ∈ Ker Θ_{M^{∗}}^{i−1}. Then ϕ ∘ Θ_{M}^{i−1} = 0 and, hence, ϕ′ = 0. Now suppose that ϕ ∈ Ker Θ_{M^{∗}}^{i} ∩ Im Θ_{M^{∗}}. Then ϕ = ψ ∘ Θ_{M} for some linear map ψ : M̃ → 𝒪. This implies that ϕ ↓_{Ker Θ_{M}} = 0. Therefore, ϕ′ = 0. We have thus estabslished that Φ vanishes on Ker Θ_{M^{∗}}^{i−1} + Ker Θ_{M^{∗}}^{i} ∩ Im Θ_{M^{∗}}. Hence, it induces a map

\xymatrix{\overline{Φ} : ℱ_{i,i−1}(M^{∗}) = \frac{Ker Θ_{M^{∗}}^{i}}{Ker Θ_{M^{∗}}^{i−1}+Ker Θ_{M^{∗}}^{i}∩Im Θ_{M^{∗}}} \ar[r] & ℱ_{1}(M)^{∨}}

It follows from construction that the map is 𝒪-linear. A simple block count shows that it induces an isomorphism on fibers. Hence, \overline{Φ} is an isomorphism.

The theorem now follows from 3.6.1 together with Lemma 2.3. □

We finish this section with exactness properties of the functors ℱ_{i} which will be essential in the proof of the main theorem.

Let 𝒞(kE) be the exact category of modules of constant Jordan type as introduced in [4]. This is an exact category in the sense of Quillen: the objects are finite dimensional kEmodules of constant Jordan type, and the admissible morphisms are morphisms which can be completed to a locally split short exact sequence. We call a sequence of kE-modules

\xymatrix{0 \ar[r] & M_{1} \ar[r] & M_{2} \ar[r] & M_{3} \ar[r] & 0}

locally split if it is split upon restriction to k[X_{α}]/X_{α}^{p} for any 0 ≠ α ∈ 𝔸^{r}.

Proposition 3.7. The functor ℱ_{i}: 𝒞(kE) → Coh(ℙ_{k}^{r−1}) is exact for 1 ≤ i ≤ p − 1.

Proof. Let

\xymatrix{0 \ar[r] & M_{1} \ar[r] & M_{2} \ar[r] & M_{3} \ar[r] & 0}

be a locally split short exact sequence of modules of constant Jordan type. Then the Jordan type of the middle term is the sum of Jordan types of the end terms. Hence, rk ℱ_{i}(M_{2}) = rk ℱ_{i}(M_{1})+rk ℱ_{i}(M_{3}) for any i. Consider the map ℱ_{i}(M_{2}) → ℱ_{i}(M_{3}). By Proposition 2.1, the specialization ℱ_{i}(M_{2})⊗_{𝒪} k(ᾱ) → ℱ_{i}(M_{3})⊗_{𝒪} k(ᾱ) is surjective at any point ᾱ ∈ ℙ^{r−1}. Arguing as in Lemma 3.1, we conclude that ℱ_{i}(M_{2}) → ℱ_{i}(M_{3}) is surjective. Similarly, we show that ℱ_{i}(M_{1}) → ℱ_{i}(M_{2}) is injective. Finally, the equality rk ℱ_{i}(M_{2}) = rk ℱ_{i}(M_{1}) + rk ℱ_{i}(M_{3}) implies exactness in the middle term. □

4. The construction

In this section we construct a functor

𝒢 : D^{b}(𝒪-mod) → kE-stmod

from the bounded derived category of coherent 𝒪-modules to the stable module category of kE.

Since H^{1}(E, k) is the vector space dual of J(kE)/J^{2}(kE), there are elements y_{1}, …, y_{r} forming a vector space basis for H^{1}(E, k) and corresponding to the linear functions Y_{1}, …, Y_{r} on J(kE)/J^{2}(kE) introduced in Section 2. For p > 2, we write β: H^{1}(E, k) → H^{2}(E, k) for the Bockstein map, and we set x_{i} = β(y_{i}). The cohomology ring H^{∗}(E, k) has the following structure.

H^{∗}(E, k) = \{ \begin{matrix}k[y_{1}, …, y_{r}] & p = 2 \\ Λ(y_{1}, …, y_{r}) ⊗ k[x_{1}, …, x_{r}] & p odd \end{matrix}

where deg(y_{i}) = 1 and deg(x_{i}) = 2.

We shall write \underline{Hom} for the Hom in the stable module category. The isomorphism H^{1}(E, k) ≃ \underline{Hom}(Ω^{n+1}k, Ω^{n}k) implies that for a generator y_{i} ∈ H^{1}(E, k) we have a corresponding map y_{i} : Ω^{n+1}k → Ω^{n}k in the stable module category, and similarly for x_{i} for p odd. For our construction, we need to chose canonical representatives of these maps so that they strictly commute, that is, commute as maps in kE-mod, not just in the stable module category. We proceed to describe such a choice.

Let C = ⟨g⟩ be a cyclic group of order p, and let C be the periodic projective resolution of the trivial kC–module k which is given by

⋯ kC ⟶^{1−g} kC ⟶^{1+g+···+g^{p−1}} kC ⟶^{1−g} kC → 0.

If p = 2 this is periodic of period one, and there is a degree one map ŷ: C → C representing y ∈ H^{1}(C, k). If p is odd it has period two, and there is a degree two map x̂: C → C representing x ∈ H^{2}(C, k).

Now if E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r}, we let

P = \underbrace{C ⊗ ⋯ ⊗ C}_{r} .

Then P is a minimal projective resolution of k as a kE-module. Let

ŷ_{i} = 1 ⊗ ⋯ ⊗ x̂ ⊗ ⋯ ⊗ 1: P → P (p = 2)
x̂_{i} = 1 ⊗ ⋯ ⊗ ŷ ⊗ ⋯ ⊗ 1: P → P (p odd)

Here, all terms in the tensor product are equal to one except the i^{th} term which is ŷ, respectively x̂. The elements ŷ_{i} and x̂_{i} represent the elements y_{i} ∈ H^{1}(E, k) (p = 2), respectively x_{i} ∈ H^{2}(E, k) (p odd). These maps commute by construction, and so we get a ring homomorphism

k[y_{1}, …, y_{r}] → Hom_{kE}^{∗}(P, P) (p = 2)
k[x_{1}, …, x_{r}] → Hom_{kE}^{∗}(P, P) (p odd).

Since Ω^{n}(k) is the n^{th} kernel in the complex P, we get induced maps which by abuse of notation we write as

(4.0.1)  \begin{matrix}y_{i}: Ω^{n+1}(k) → Ω^{n}(k) (p = 2) \\ x_{i}: Ω^{n+2}(k) → Ω^{n}(k) (p odd) \end{matrix}

for all n ≥ 0. Dualizing, we get these maps for all n ∈ ℤ. They strictly commute, in the sense that y_{i}y_{j} and y_{j}y_{i} give the same map Ω^{n+2}(k) → Ω^{n}(k) (p = 2); and x_{i}x_{j} and x_{j}x_{i} give the same map Ω^{n+4}(k) → Ω^{n}(k) (p odd).

Let C be a complex in D^{b}(𝒪-mod). By the Hilbert syzygy theorem, the Cartan-Eilenberg

resolution of C can be taken to have entries of the form ⨁_{j=1}^{ℓ} 𝒪(i_{j}). Hence, we can replace C

with a quasi-isomorphic complex which has the form

(4.0.2)  ⨁_{j=1}^{m_{n}} 𝒪(a_{n,j}) → ⋯ → ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) → ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j}).

Each of the maps in this complex is a matrix whose entries are homogeneous polynomials in Y_{1}, …, Y_{r}. Replacing each Y_{i} with y_{i} if p = 2 and with x_{i} if p is odd where y_{i} and x_{i} are the maps defined in (4.0.1) gives matrices of homogeneous polynomials on y_{i} for p = 2 (and x_{i} for p odd) which we may use to form the following sequence of objects and maps in kE-mod:

(4.0.3)  ⨁_{j=1}^{m_{n}} Ω^{−εa_{n,j}}(k) → ⋯ → ⨁_{j=1}^{m_{1}} Ω^{−εa_{1,j}}(k) → ⨁_{j=1}^{m_{0}} Ω^{−εa_{0,j}}(k)

where ε = 1 if p = 2 and ε = 2 if p is odd. Since the maps y_{i} (p = 2), respectively x_{i} (p odd) strictly commute, we get a complex in kE-mod and, hence, an object in D^{b}(kE-mod). We call the resulting complex 𝒢̃(C).

Note that chain maps between bounded complexes of coherent 𝒪–modules which are direct sums of Serre twists of the structure sheaf and a chain homotopy between such chain maps are all given by matrices of polynomials on Y_{i}, so that they are realized as chain maps and chain homotopies in D^{b}(kE-mod). This observation implies the following proposition a detailed proof of which we leave as an exercise for an interested reader.

Proposition 4.1. The association

𝒢̃ : D^{b}(𝒪-mod) → D^{b}(kE-mod)

which sends a bounded complex of coherent 𝒪–modules quasi-isomorphic to

⨁_{j=1}^{m_{n}} 𝒪(a_{n,j}) → ⋯ → ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) → ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j})

to the complex

⨁_{j=1}^{m_{n}} Ω^{−εa_{n,j}}(k) → ⋯ → ⨁_{j=1}^{m_{1}} Ω^{−εa_{1,j}}(k) → ⨁_{j=1}^{m_{0}} Ω^{−εa_{0,j}}(k)

in D^{b}(kE-mod) as defined in (4.0.3) is a triangulated functor.

We denote by 𝕂^{b}(proj) the category of perfect complexes over kE, that is, the thick subcategory of D^{b}(kE-mod) consisting of complexes quasi–isomorphic to bounded complexes of free kE-modules. There is a localization sequence

\xymatrix{𝕂^{b}(proj) \ar[r] & D^{b}(kE-mod) \ar[r] & kE-stmod}

(see [12]). Denote the corresponding localization functor by ℛ:

ℛ : D^{b}(kE-mod) → kE-stmod.

We define 𝒢 : D^{b}(𝒪-mod) → kE-stmod as a composition 𝒢 = ℛ ∘ 𝒢̃:

\xymatrix{𝒢 : D^{b}(𝒪-mod) \ar[r]^{𝒢̃} & D^{b}(kE-mod) \ar[r]^{ℛ} & kE-stmod}

Since ℛ is a triangulated functor, Proposition 4.1 implies that 𝒢 is a triangulated functor.

The proof of the main theorem (Theorem 1.1) will rely on the following two properties of the functor 𝒢 to be proved in Section 5:

(1) Let ℱ be an algebraic vector bundle of rank s on ℙ^{r−1}. Consider ℱ as a complex in D^{b}(𝒪-mod) concentrated in degree 0. Then 𝒢(ℱ) is a module of stable constant Jordan type [1]^{s}.

(2) Under the assumption of (1), we have

ℱ_{1}(𝒢(ℱ)) ≃ \{ \begin{matrix}ℱ & if p = 2 \\ F^{∗}(ℱ) & if p is odd \end{matrix}

where F : ℙ^{r−1} → ℙ^{r−1} is the Frobenius map.

5. Proof of the main theorem.

We start with several technical lemmas. Because of the difference in structure of the cohomology ring, we divide the initial discussion into two cases, according to p = 2 or p odd.

Case 1: p = 2. We define a k-algebra homomorphism

ρ: H^{∗}(E, k) = k[y_{1}, …, y_{r}] → k[Y_{1}, …, Y_{r}]

by ρ(y_{i}) = Y_{i}. Recall that we have an isomorphism 𝒪(−n) = ℱ_{1}(k)(−n) ≃ ℱ_{1}(Ω^{n}k) by Remark 3.5.

Lemma 5.1. If ζ ∈ H^{n}(E, k) is represented by a cocycle ζ̂: Ω^{n+j}k → Ω^{j}k (with j ∈ ℤ) then the diagram

\xymatrix{𝒪(−n − j) \ar[d]^{≅} \ar[r]^{ρ(ζ)} & 𝒪(−j) \ar[d]^{≅} \\ ℱ_{1}(Ω^{n+j}k) \ar[r]^{ℱ_{1}(ζ̂)} & ℱ_{1}(Ω^{j}k)}

commutes.

Proof. Consider ζ̂: Ω^{n}k → k. The commutative diagram 3.2.2 applied to ζ̂ and iterated j times becomes

\xymatrix{ℱ_{1}(Ω^{n}k)(−j) \ar[d]^{≃} \ar[r]^{ℱ_{1}(ζ̂)(−j)} & ℱ_{1}(k)(−j) \ar[d]^{≃} \\ ℱ_{1}(Ω^{n+j}k) \ar[r]^{ℱ_{1}(Ω^{j} ζ̂)} & ℱ_{1}(Ω^{j}k).}

Hence, it suffices to assume that j = 0. Additivity of the functor ℱ_{1} allows us to assume that ζ is a monomial on generators y_{1}, …, y_{r}. Finally, since multiplication in cohomology corresponds to composition of the corresponding maps on Heller shifts of k, it suffices to prove our statement for a degree one generator ζ = y_{i}.

In the case j = 0, ζ = y_{i}, we need to show that the following diagram commutes (this is the diagram above twisted by 𝒪(1))

(5.1.1)  [figure: ℱ_{1}(k); ≃; Y_{i}; ℱ_{1}(k)(1); ℱ_{1}(Ωk)(1); ℱ_{1}(ŷ_{i}); ℱ_{1}(k)(1).]

Let E_{i} be the subgroup of index two in E such that y_{i} is inflated from E/E_{i} to E, namely the subgroup generated by all of g_{1}, …, g_{r} except g_{i}. Then y_{i} represents the class of the extension

0 → k → M_{i} → k → 0

where M_{i} is the permutation module on the cosets of E_{i}. This is a length two module on which X_{1}, …, X_{r} act as zero except for X_{i}, which acts as a Jordan block of length two. We have a commutative diagram of kE-modules

(5.1.2)  [figure: 0; Ωk; y_{i}; P_{0}; k; 0; 0; k; M_{i}; k; 0]

The left vertical isomorphism \overline{δ}: ℱ_{1}(k) →^{∼} ℱ_{1}(Ωk)(1) of the diagram 5.1.1 is given by the switchback map for the short exact sequence

0 → Ωk → P_{0} → k → 0

as in diagram (3.2.1). Applying θ to the commutative diagram on free 𝒪-modules induced by the module diagram (5.1.2), we get a commutative diagram

[figure: 0; Ω̃k; ỹ_{i}; P̃_{0}; 𝒪; 0; 0; Ω̃k(1); ỹ_{i}(1); P̃_{0}(1); 𝒪(1); 0; 0; 𝒪; M̃_{i}; 𝒪; 0; 0; 𝒪(1); M̃_{i}(1); 𝒪(1); 0]

where all horizontal arrows going back to front are given by the operator θ on the corresponding module. The map ỹ_{i}: Ω̃k → 𝒪 is induced by y_{i}: Ωk → k. To compute the composite ℱ_{1}(ŷ_{i}) ∘ \overline{δ} we first do the switchback map of the top layer and then push the result down via ỹ_{i}(1). Since the diagram is commutative, we can first push down via the identity map of the right vertical back arrow and then do the switchback of the bottom layer. Hence, the composite ℱ_{1}(ŷ_{i}) ∘ \overline{δ} is given by the switchback map of the bottom layer; that is, of the diagram

\xymatrix{0 \ar[r] & 𝒪 \ar[r] \ar[d]^{θ_{k}} & M̃_{i} \ar[r] \ar[d]^{θ_{M_{i}}} & 𝒪 \ar[r] \ar[d]^{θ_{k}} & 0 \\ 0 \ar[r] & 𝒪(1) \ar[r] & M̃_{i}(1) \ar[r] & 𝒪(1) \ar[r] & 0}

The left and right hand vertical maps here are zero. Hence, the switchback map \overline{δ}: 𝒪 → 𝒪(1) is given by multiplication by θ_{M_{i}}, which is given by multiplication by Y_{i} in this situation. □

Case 2: p is odd. We define a k-algebra homomorphism

ρ: H^{∗}(E, k) ↠ k[x_{1}, …, x_{r}] → k[Y_{1}, …, Y_{r}]

by ρ(x_{i}) = Y_{i}^{p}.

Lemma 5.2. Let p be an odd prime. Let ζ be a degree n polynomial in k[x_{1}, …, x_{r}], regarded as an element of H^{2n}(E, k). If ζ is represented by a cocycle ζ̂: Ω^{2(n+j)}k → Ω^{2j}k then the diagram

\xymatrix{𝒪(−p(n + j)) \ar[r]^{ρ(ζ)} \ar[d]^{≅} & 𝒪(−pj) \ar[d]^{≅} \\ ℱ_{1}(Ω^{2(n+j)}k) \ar[r]^{ℱ_{1}(ζ̂)} & ℱ_{1}(Ω^{2j}k)}

commutes.

Proof. The proof is similar to the proof in the case p = 2, but more complicated. Again it suffices to treat the case where ζ = x_{i} and j = 0. In other words, we need to compute the composite ℱ_{1}(x̂_{i}) ∘ f in the diagram

[figure: ℱ_{1}(k); ≃; f; Y_{i}^{p}; ℱ_{1}(k)(p); ℱ_{1}(Ω^{2}k)(p); ℱ_{1}(x̂_{i}); ℱ_{1}(k)(p).]

where f: ℱ_{1}(k) → ℱ_{1}(Ω^{2}k)(p) is the isomorphism of Corollary 3.3. Let

\xymatrix{0 \ar[r] & Ω^{2}k \ar[r] & P_{1} \ar[r] & P_{0} \ar[r] & k \ar[r] & 0}

be a truncated projective resolution of k. Tracing through the proof of Theorem 3.2, we see that f: ℱ_{1}(k) → ℱ_{1}(Ω^{2}k)(p) is a composite of three maps:

(1) the switchback of the top two rows of the diagram 5.2.1 below which gives the isomorphism ℱ(k) → ℱ_{p−1,p−2}(Ωk)(1),

(2) followed by the isomorphism θ_{Ωk}^{p−2} : ℱ_{p−1,p−2}(Ωk)(1) →^{∼} ℱ_{p−1}(Ωk)(p−1) of Lemma 2.3;

(3) followed by another switchback map, now for the bottom two rows of diagram 5.2.1, which gives the isomorphism ℱ_{p−1}(Ωk)(p − 1) ≃ ℱ_{1}(Ω^{2}k)(p).

(5.2.1)  \xymatrix{ &  & 0 \ar[r] & Ω̃k \ar[r] \ar[d] & P̃_{0} \ar[d] \ar[r] & 𝒪 \ar[r] \ar[d] & 0 \\  &  & 0 \ar[r] & Ω̃k(1) \ar[d]^{θ_{Ωk}^{p−2}} \ar[r] & P̃_{0}(1) \ar[r] & 𝒪(1) \ar[r] & 0 \\ 0 \ar[r] & Ω̃^{2}k(p − 1) \ar[r] \ar[d] & P̃_{1}(p − 1) \ar[r] \ar[d] & Ω̃k(p − 1) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ω̃^{2}k(p) \ar[r] & P̃_{1}(p) \ar[r] & Ω̃k(p) \ar[r] & 0}

Let E_{i} be the subgroup of index p such that x_{i} is inflated from E/E_{i}, namely the subgroup generated by all of g_{1}, …, g_{r} except for g_{i}. We let M_{i} be the permutation module on the cosets of E_{i}. This is a length p module on which X_{1}, …, X_{r} act as zero except for X_{i}, which acts as a Jordan block of length p. Then x_{i} represents the class of the 2-fold extension

\xymatrix{0 \ar[r] & k \ar[r] & M_{i} \ar[r] & M_{i} \ar[r] & k \ar[r] & 0}

13

where the middle map is multiplication by X_{i}. We construct a diagram analogous to (5.2.1) for this extension:

(5.2.2) \xymatrix{ &  & 0 \ar[r] & Ñ_{i} \ar[r] \ar[d] & M̃_{i} \ar[d] \ar[r] & 𝒪 \ar[r] \ar[d] & 0 \\  &  & 0 \ar[r] & Ñ_{i}(1) \ar[d]^{θ_{N_{i}}^{p−2}} \ar[r] & M̃_{i} \ar[r] & 𝒪(1) \ar[r] & 0 \\ 0 \ar[r] & 𝒪(p − 1) \ar[r] \ar[d] & M̃_{i}(p − 1) \ar[r] \ar[d] & Ñ_{i}(p − 1) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 𝒪(p) \ar[r] & M̃_{i}(p) \ar[r] & 𝒪(p) \ar[r] & 0.}

Here, N_{i} = Im {X_{i}: M_{i} → M_{i}}. Just as in the proof of Lemma 5.2, the module diagram

[figure: 0; Ω^{2}k; x_{i}; P_{1}; P_{0}; k; 0; 0; k; M_{i}; M_{i}; k; 0]

induces a commutative diagram of vector bundles with (5.2.1) on top and (5.2.2) at the bottom. Arguing as in the proof of Lemma 5.1, we compute the composite ℱ_{1}(x̂_{i}) ∘ f by first mapping the rightmost 𝒪 of diagram 5.2.1 identically to the rightmost 𝒪 of diagram 5.2.2, and then applying our composite of a switchback, followed by θ_{N_{i}}^{p−2}, followed by another switchback in the diagram 5.2.2. The maps θ_{M_{i}}: M̃_{i} → M̃_{i}(1) and θ_{N_{i}}: Ñ_{i} → Ñ_{i}(1) are simply multiplication by Y_{i}. Since the leftmost and rightmost vertical arrows in (5.2.2) are zero, to compute the composite of the three maps involved in diagram 5.2.2, we have to multiply first by Y_{i}, then by Y_{i}^{p−2}, then by Y_{i} again. Hence, ℱ_{1}(x̂_{i}) ∘ f = Y_{i}^{p}. □

Let

F : ℙ^{r−1} → ℙ^{r−1}

be the Frobenius morphism, that is, the morphism given on the level of homogeneous coordinate algebras by

\xymatrix{k[Y_{1}, …, Y_{r}] \ar[r]^{Y_{i}↦Y_{i}^{p}} & k[Y_{1}, …, Y_{r}]}

Then the pull-back functor

F^{∗} : 𝒪-mod → 𝒪-mod

is exact and induces a functor

F^{∗} : D^{b}(𝒪-mod) → D^{b}(𝒪-mod).

In the next proposition we shall use the functor 𝒢̃ : D^{b}(𝒪-mod) → D^{b}(kE-mod) defined in Section 4.

Proposition 5.3. Let

C = ⨁_{j=1}^{m_{n}} 𝒪(a_{n,j}) → ⋯ → ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) → ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j})

be a complex of 𝒪–modules. Then

ℱ_{1}(𝒢̃(C)) = \{ \begin{matrix}C & p = 2 \\ F^{∗}(C) & p odd. \end{matrix}

Proof. By definition,

𝒢̃(C) = ⨁_{j=1}^{m_{n}} Ω^{−εa_{n,j}}(k) → ⋯ → ⨁_{j=1}^{m_{1}} Ω^{−εa_{1,j}}(k) → ⨁_{j=1}^{m_{0}} Ω^{−εa_{0,j}}(k).

Applying ℱ_{1} term by term, we get

⨁_{j=1}^{m_{n}} ℱ_{1}(Ω^{−εa_{n,j}}(k)) → ⋯ → ⨁_{j=1}^{m_{1}} ℱ_{1}(Ω^{−εa_{1,j}}(k)) → ⨁_{j=1}^{m_{0}} ℱ_{1}(Ω^{−εa_{0,j}}(k)).

The statement now follows from Lemma 5.1 for p = 2 and Lemma 5.2 for p odd. □

The following lemma will be applied to modules over K[X_{α}]/(X_{α}^{p}), where projective and injective modules coincide.

Lemma 5.4. Let ϕ = (\begin{matrix}a & b \\ c & d \end{matrix}) : A ⊕ P → B ⊕ Q be an injective map of modules, with

a: A → B a split injection, P and Q projective and injective. Then ϕ is a split injection.

Proof. Let B = B_{1} ⊕ B_{2}, where B_{1} = Im a, and let

a = (\begin{matrix}a_{1} \\ 0 \end{matrix}) : A → B_{1} ⊕ B_{2}, b = (\begin{matrix}b_{1} \\ b_{2} \end{matrix}) : P → B_{1} ⊕ B_{2}.

With this notation, we have

ϕ = (\begin{matrix}a_{1} & b_{1} \\ 0 & b_{2} \\ c & d \end{matrix}) : A ⊕ P → B_{1} ⊕ B_{2} ⊕ Q.

Since a_{1} is an isomorphism, there exist maps u : B_{1} → Q, v : P → A such that c = ua_{1} and b_{1} = a_{1}v. Hence,

(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ −u & 0 & 1 \end{matrix}) (\begin{matrix}a_{1} & b_{1} \\ 0 & b_{2} \\ c & d \end{matrix}) (\begin{matrix}1 & −v \\ 0 & 1 \end{matrix}) = (\begin{matrix}a_{1} & 0 \\ 0 & b_{2} \\ 0 & d − ub_{1} \end{matrix}) : A ⊕ P → B_{1} ⊕ B_{2} ⊕ Q.

By assumption, this map is injective. Hence, the map

(\begin{matrix}b_{2} \\ d − ub_{1} \end{matrix}) : P → B_{2} ⊕ Q

is injective and therefore split. □

Lemma 5.5. Let 0 → A → B → C → 0 be a short exact sequence of kE-modules, such that A and B have trivial stable constant Jordan type and ℱ_{1}(A) → ℱ_{1}(B) is injective. Then C is of trivial stable constant Jordan type and ℱ_{1}(C) is the cokernel of ℱ_{1}(A) → ℱ_{1}(B).

Proof. Applying Lemma 5.4 to the restriction of the sequence to K[X_{α}]/(X_{α}^{p}) for each 0 ≠ α ∈ 𝔸^{r}(k), we conclude that this sequence is locally split. Hence, C is a module of trivial constant Jordan type. By Proposition 3.7, applying ℱ_{1} gives an exact sequence

0 → ℱ_{1}(A) → ℱ_{1}(B) → ℱ_{1}(C) → 0.  □

We are now ready to prove the main theorem.

Theorem 5.6. Given any vector bundle ℱ of rank s on ℙ^{r−1}, there exists a finitely generated kE-module M of stable constant Jordan type [1]^{s} such that

(i) if p = 2, then ℱ_{1}(M) ≅ ℱ.

(ii) if p is odd, then ℱ_{1}(M) ≅ F^{∗}(ℱ), the pullback of ℱ along the Frobenius morphism F: ℙ^{r−1} → ℙ^{r−1}.

Proof. Given a vector bundle ℱ on ℙ^{r−1}, using the Hilbert syzygy theorem we can form a resolution by sums of twists of the structure sheaf:

0 → ⨁_{j=1}^{m_{r}} 𝒪(a_{r,j}) → ⋯ → ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) → ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j}) → ℱ → 0.

Equivalently, ℱ, considered as a complex concentrated in degree 0, is quasi-isomorphic to the complex

(5.6.1)  C = ⨁_{j=1}^{m_{r}} 𝒪(a_{r,j}) → ⋯ → ⨁_{j=1}^{m_{1}} 𝒪(a_{1,j}) → ⨁_{j=1}^{m_{0}} 𝒪(a_{0,j})

Let

M = 𝒢(ℱ),

where 𝒢 : D^{b}(𝒪-mod) → kE-stmod is the functor defined towards the end of Section 4. We shall prove that M satisfies the conclusions of the theorem.

Let

C_{≥i} = ⨁_{j=1}^{m_{r}} 𝒪(a_{r,j}) → ⋯ → ⨁_{j=1}^{m_{i}} 𝒪(a_{i,j})

be a truncation of C, 0 ≤ i ≤ r. We set M_{i} = 𝒢(C_{≥i}). In particular, M_{0} = M. We shall show by the downward induction on i that M_{i} has trivial stable constant Jordan type and that ℱ_{1}(M_{i}) is the cokernel of the rightmost map in the complex C_{≥i} for p = 2 or F^{∗}(C_{≥i}) for p odd. Applied to i = 0, this will give us the desired result about M_{0} = M.

By definition, 𝒢 = ℛ ∘ 𝒢̃. Furthermore, we have

(5.6.2)  𝒢̃(C) = ⨁_{j=1}^{m_{r}} Ω^{−εa_{r,j}}(k) → ⋯ → ⨁_{j=1}^{m_{1}} Ω^{−εa_{1,j}}(k) → ⨁_{j=1}^{m_{0}} Ω^{−εa_{0,j}}(k)

Since 𝒢 preserves triangles, the triangle

C_{≥i+1} → C_{i} → C_{≥i}

in D^{b}(𝒪-mod) gives rise to the triangle

M_{i+1} → ⨁_{j=1}^{m_{i}} Ω^{−εa_{i,j}}(k) → M_{i}

in stmod(kE) for any i, 0 ≤ i ≤ r − 1.

By Proposition 5.3, applying ℱ_{1} to the complex (5.6.2), we obtain either the original complex (5.6.1) if p = 2 or its Frobenius twist if p is odd.

Since M_{r} = 𝒢(C_{r}) = ⨁ Ω^{−εa_{r,j}}(k), we immediately get that M_{r} is a module of trivial stable constant Jordan type. Moreover, ℱ_{1}(M_{0}) → ℱ_{1}(⨁ Ω^{−εa_{r−1,j}}(k)) is an injection since it is either the original leftmost map of the complex C (p = 2) or of the complex F^{∗}(C) (p odd). Therefore, we may apply Lemma 5.5 to the triangle

M_{r} → ⨁_{j=1}^{m_{r−1}} Ω^{−εa_{r−1,j}}(k) → M_{r−1}.

It implies that M_{r−1} has trivial stable constant Jordan type and that ℱ_{1}(M_{r−1}) is the cokernel of the rightmost map of the complex C_{≥(r−1)} (p = 2) or the complex F^{∗}(C_{≥(r−1)}) (p odd). Proceeding in the same manner by downwards induction on i, we show that M_{i} has trivial stable constant Jordan type and that ℱ_{1}(M_{i}) is the cokernel of the rightmost map the complex C_{≥i} (p = 2) or the complex F^{∗}(C_{≥i}) (p odd) for any i, 0 ≤ i ≤ r − 1. Hence, for i = 0 we get that M = M_{0} has trivial stable constant Jordan type and that ℱ_{1}(M) is the cokernel of C, the resolution of ℱ (p = 2) or the cokernel of F^{∗}(C) (p odd) which is a resolution of F^{∗}(ℱ) since F^{∗} is exact. Therefore, ℱ_{1}(M) ≃ ℱ for p = 2 and ℱ_{1}(M) ≃ F^{∗}(ℱ) for p odd. The fact that the stable type of M is [1]^{s} now follows from Proposition 2.1. □

6. Chern numbers

Recall that the Chow ring of ℙ^{r−1} is

A^{∗}(ℙ^{r−1}) ≅ ℤ[h]/(h^{r}).

If ℱ is a vector bundle on ℙ^{r−1}, we write

c(ℱ, h) = ∑_{j≥0} c_{j}(ℱ)h^{j} ∈ A^{∗}(ℙ^{r−1})

for the Chern polynomial, where c_{0}(ℱ) = 1 and the c_{i}(ℱ) ∈ ℤ (1 ≤ i ≤ r − 1) are the Chern numbers of ℱ.

If 0 → ℱ → ℱ′ → ℱ′′ → 0 is a short exact sequence of vector bundles then we have the Whitney sum formula

c(ℱ′, h) = c(ℱ, h)c(ℱ′′, h).

Lemma 6.1. The formula for Chern numbers of twists of a rank s vector bundle is

c_{m}(ℱ(i)) = ∑_{j=0}^{m} i^{j}\binom{s − m + j}{j}c_{m−j}(ℱ).

Equivalently, the total Chern class of the twists is given by

(6.1.1)  c(ℱ(i), h) = ∑_{n=0}^{s} c_{n}(ℱ)h^{n}(1 + ih)^{s−n}

Proof. See Fulton [9, Example 3.2.2]. □

More explicitly,

c_{1}(ℱ(i)) = c_{1}(ℱ) + is
c_{2}(ℱ(i)) = c_{2}(ℱ) + i(s − 1)c_{1}(ℱ) + i^{2}\binom{s}{2}
c_{3}(ℱ(i)) = c_{3}(ℱ) + i(s − 2)c_{2}(ℱ) + i^{2}\binom{s − 1}{2}c_{1}(ℱ) + i^{3}\binom{s}{3}

and so on.

Lemma 6.2. For a vector bundle ℱ of rank s on ℙ^{r−1} we have

c(ℱ, h)c(ℱ(1), h) ⋯ c(ℱ(p − 1), h) ≡ 1 − sh^{p−1} (mod (p, h^{p})).

Proof. We write

c(ℱ) = ∏_{j=1}^{s}(1 + α_{j}h),

where the α_{j} are the Chern roots. Then the formula (6.1.1) is equivalent to

c(ℱ(i)) = ∏_{j=1}^{s}(1 + (α_{j} + i)h).

Thus we have

c(ℱ)c(ℱ(1)) ⋯ c(ℱ(p − 1)) = ∏_{j=1}^{s}(1 + α_{j}h)(1 + (α_{j} + 1)h) ⋯ (1 + (α_{j} + p − 1)h).

Now by Fermat’s little theorem, we have the identity

x(x + y) ⋯ (x + (p − 1)y) ≡ x^{p} − xy^{p−1} (mod p)

and so putting x = 1 + α_{j}h, y = h we obtain

c(ℱ)c(ℱ(1)) ⋯ c(ℱ(p − 1)) ≡ ∏_{j=1}^{s}((1 + α_{j}h)^{p} − (1 + α_{j}h))h^{p−1} (mod p)

≡ ∏_{j=1}^{s}(1 − h^{p−1} + (α_{j}^{p} − α_{j})h^{p}) (mod p)

≡ 1 − sh^{p−1} (mod (p, h^{p})).

A priori, this is a congruence between polynomials with algebraic integer coefficients. But if two rational integers are congruent mod p as algebraic integers then they are also congruent modulo p as rational integers. This is because their difference, divided by p, is both an algebraic integer and a rational number, therefore an integer. □

We now restate and prove Theorem 1.2.

Theorem 6.3. Suppose that M has stable constant Jordan type [1]^{s}. Then p divides the Chern numbers c_{m}(ℱ_{1}(M)) for 1 ≤ m ≤ p − 2.

Proof. Since M has stable Jordan type [1]^{s}, we have ℱ_{2}(M) = … = ℱ_{p−1}(M) = 0. Hence, the trivial vector bundle M̃ has a filtration with filtered quotients (not in order) ℱ_{1}(M), ℱ_{p}(M), ℱ_{p}(M)(1), …, ℱ_{p}(M)(p − 1). So we have

1 = c(M̃, h)
= c(ℱ_{1}(M), h)c(ℱ_{p}(M), h)c(ℱ_{p}(M)(1), h) ⋯ c(ℱ_{p}(M)(p − 1), h)
≡ c(ℱ_{1}(M), h) (mod (p, h^{p−1}))

by Lemma 6.2. It follows that the coefficients c_{m}(ℱ_{1}(M)) are divisible by p for 1 ≤ m ≤ p − 2. □

Remark 6.4. For p = 2 this theorem says nothing. But for p odd, it at least forces c_{1}(ℱ_{1}(M)) to be divisible by p. As an explicit example, the twists of the Horrocks–Mumford bundle ℱ_{HM}(i) have c_{1} = 2i + 5 and c_{2} = i^{2} + 5i + 10 ([11]). For p ≥ 7 these cannot both be divisible by p, and so there is no module M of stable constant Jordan type [1]^{2} and integer i such that ℱ_{1}(M) ≅ ℱ_{HM}(i).

Remark 6.5. The conclusion of the theorem is limited to the modules of stable constant Jordan type [1]^{s}. For example, if M_{n} is a “zig-zag” module of dimension 2n+1 for ℤ/p×ℤ/p, then ℱ_{1}(M_{n}) ≃ 𝒪(−n) and ℱ_{1}(M_{n}^{∗}) ≃ 𝒪(n) for any n ≥ 0 (see [8, §6]).

Acknowledgement. Both authors are grateful to MSRI for its hospitality during the 2008 program “Representation theory of finite groups and related topics” where much of this research was carried out. They also thank the referee for helpful suggestions.

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