MODULES OF CONSTANT JORDAN TYPE AND A CONJECTURE OF RICKARD

DAVID J. BENSON

Abstract. We prove a special case of a conjecture of Rickard on modules of constant Jordan type over an elementary abelian p-group of rank at least 2. Namely, we show that if there are no Jordan blocks of length one, then the total number of Jordan blocks is divisible by p. We combine this with other techniques to rule out a large number of Jordan types.

1. Introduction

Let E = ⟨g_{1}, …, g_{r}⟩ ≅ (ℤ/p)^{r} (r ≥ 2) be an elementary abelian p-group and let k be an algebraically closed field of characteristic p. Set X_{i} = g_{i} − 1 as an element of the Jacobson radical J = J(kE). If α = (λ_{1}, …, λ_{r}) ∈ 𝔸^{r}(k), set X_{α} = ∑_{i=1}^{r} λ_{i}X_{i} ∈ J(kE). We say that a finite dimensional kE-module M has constant Jordan type [a_{1}] … [a_{t}] (with 0 < a_{i} ≤ p) if for every 0 ≠ α ∈ 𝔸^{r}(k) the Jordan canonical form of X_{α} on M has Jordan blocks of lengths a_{1}, …, a_{t}. The stable Jordan type is the same list but with the Jordan blocks of length p omitted. These concepts were introduced by Carlson, Friedlander and Pevtsova [6] and investigated further in [2, 3, 4, 5, 7, 8].

The following unpublished conjecture was formulated by Jeremy Rickard at MSRI in 2008.

Conjecture 1.1. Let M be a kE-module of constant Jordan type [a_{1}] … [a_{t}], and suppose that none of the a_{i} is equal to j. Then the total number of a_{i} that are greater than j is divisible by p.

This is one of a number of conjectures and questions about modules of constant Jordan type. Some others can be found in Section 9 of [6]. In particular, in Question 9.6 of that paper, Suslin asks whether the existence of a Jordan block of length j (2 ≤ j ≤ p − 1) implies the existence of a Jordan block of length either j − 1 or j + 1.

In this paper we prove a special case of Rickard’s Conjecture 1.1, namely the case j = 1.

Theorem 1.2. Let M be a kE-module of constant Jordan type [a_{1}] … [a_{t}], and suppose that none of the a_{i} is equal to 1. Then t is divisible by p.

This theorem has a number of consequences, starting with the case j = p − 1 of Rickard’s conjecture.

Corollary 1.3. Let M be a kE-module of constant Jordan type [a_{1}] … [a_{t}], and suppose that none of the a_{i} is equal to p − 1. Then the number of a_{i} that are equal to p is divisible by p.

Corollary 1.4. Let M be a kE-module of stable constant Jordan type [a_{1}] … [a_{t}], and suppose that none of the a_{i} is equal to 1 or p − 1. Then t is divisible by p.

Corollary 1.4 gives a new proof of the main theorem of [3]. Moreover, combining our main theorem with the methods of [3] gives new information:

Theorem 1.5. Suppose that M has constant Jordan type with all stable Jordan blocks of distinct odd length or all of distinct even length. Then the stable Jordan type is either [1] or [p − 1] or has at least four Jordan blocks.

Theorem 1.6. If p/2 < n < p − 1 there is no module of stable constant Jordan type [n] [1].

Of course, these theorems are probably not the best possible, and the gap between the theorems and the examples remains large. The smallest example where it is not known whether a Jordan type can occur is [4] [1] for p = 11 and r = 2 or 3. Such a module would, of course, give a negative answer to Suslin’s question [6, Question 9.6]. If r ≥ 4 then this type cannot occur, by a theorem of Baland [2].

The proof of Theorem 1.2 depend on the concept of generic kernel 𝔎M of a module M in the rank two case, introduced by Carlson, Friedlander and Suslin [7]. We examine in detail the subquotient J^{−1}𝔎M/J^{2}𝔎M, and show that if M has no blocks of length one then neither does J^{−1}𝔎M/J^{2}𝔎M, and that this subquotient has constant Jordan type provided M does, with the same number of Jordan blocks.

2. Generic kernels

It suffices to prove Theorem 1.2 in the case where E has rank two, so from now on, we make this assumption. In other words, we have E ≅ ℤ/p × ℤ/p. The proof of the theorem depends on the concept of generic kernel investigated in Section 7 of [7]. We write 𝔸^{2}(k) for affine 2-space, and ℙ^{1}(k) for the projective line. A point in ℙ^{1}(k) corresponds to a line through the origin in 𝔸^{2}(k). We write J for J(kE), namely the ideal generated by X_{1} and X_{2}.

Definition 2.1. Let M be a finite dimensional kE-module. If

0 ≠ α = (λ_{1}, λ_{2}) ∈ 𝔸^{2}(k),

then the kernel of X_{α} on M only depends on the image ᾱ of α in ℙ^{1}(k). We define the generic kernel 𝔎(M) to be the intersection over all cofinite subsets S of ℙ^{1}(k) of the sum of the kernels of X_{α} on M for ᾱ ∈ S.

Definition 2.2. A kE-module M has the constant image property if the image of X_{α} on M is independent of α for 0 ≠ α ∈ 𝔸^{2}(k).

Theorem 2.3. Let M be a kE-module.

(1) If M has the constant image property then for 0 ≠ α ∈ 𝔸^{2}(k) the image of X_{α} on M is equal to JM.

(2) The submodule 𝔎(M) has the constant image property.

(3) Every module with the constant image property has constant Jordan type.

(4) If M has constant Jordan type then for 0 ≠ α ∈ 𝔸^{2}(k) the kernel of X_{α} on M is contained in 𝔎(M).

Proof. See Propositions 2.5, 7.3, 2.8 and Corollary 7.7 of [7]. □

Lemma 2.4. Let M be a kE-module of constant Jordan type. If 0 ≠ α ∈ 𝔸^{2}(k) and m ∈ M such that X_{α}m ∈ 𝔎(M) then for all 0 ≠ β ∈ 𝔸^{2}(k) we have X_{β}m ∈ 𝔎(M). Thus we have

J.m ⊆ 𝔎(M).

Proof. If X_{α}m ∈ 𝔎(M) then X_{α}X_{β}m = X_{β}X_{α}m ∈ J𝔎(M). So by Theorem 2.3 (1) and (2) there exists m′ ∈ 𝔎(M) such that X_{α}m′ = X_{α}X_{β}m. Thus X_{α}(X_{β}m − m′) = 0. By Theorem 2.3 (4), this implies that X_{β}m − m′ ∈ 𝔎(M). Since also m′ ∈ 𝔎(M), this implies that X_{β}m ∈ 𝔎(M). □

3. Proof of Theorem 1.2

Throughout this section, we suppose that k is an algebraically closed field of characteristic p, E is an elementary abelian p-group of rank two, and M is a kEmodule of constant Jordan type with no Jordan blocks of length one.

Lemma 3.1. The number of Jordan blocks of X_{α} on M is the same as the number of Jordan blocks on 𝔎(M). This in turn is the same as the number of Jordan blocks on 𝔎(M)/J^{2}𝔎(M), which is the same as the dimension of 𝔎(M)/J𝔎(M).

Proof. This follows from Theorem 2.3 (1) and (4). □

Our goal is to show that for each summand N of 𝔎(M)/J^{2}𝔎(M), N/JN has dimension divisible by p, thereby proving Theorem 1.2.

Look at the maps of vector spaces induced by X_{α}

J^{−1}𝔎(M)/𝔎(M) ⟶^{X_{α}} 𝔎(M)/J𝔎(M) ⟶^{X_{α}} J𝔎(M)/J^{2}𝔎(M),

where J^{−1}𝔎(M) denotes {m ∈ M | J.m ⊆ 𝔎(M)}.

Lemma 3.2. For every 0 ≠ α ∈ 𝔸^{2}(k), the first of these maps is injective and the second is surjective.

Proof. This follows from Theorem 2.3 (1), (2) and (4). □

Lemma 3.3. J^{−1}𝔎(M)/J^{2}𝔎(M) has constant Jordan type with all Jordan blocks of length two or three, and the same number of Jordan blocks as M.

Proof. By Theorem 2.3 (4), the kernel of every X_{α} (α ≠ 0) is contained in 𝔎(M). So the number of Jordan blocks of X_{α} on any submodule of M containing 𝔎(M) is the same as the number of Jordan blocks on M.

If X_{α} has a Jordan block of length one on J^{−1}𝔎(M)/J^{2}𝔎(M) then there exists m ∈ J^{−1}𝔎(M) such that X_{α}m ∈ J^{2}𝔎(M) but m ∉ X_{α}J^{−1}𝔎(M). By Theorem 2.3 (4), we have m ∈ 𝔎(M). By Theorem 2.3 (1) there exists m′ ∈ J𝔎(M) such that X_{α}m = X_{α}m′, so X_{α}(m − m′) = 0. Since M has no Jordan blocks of length one, it follows that there exists m′′ ∈ M with X_{α}m′′ = m − m′. By Lemma 2.4 we have m′′ ∈ J^{−1}𝔎(M) and hence m ∈ X_{α}J^{−1}𝔎(M), giving a contradiction.

It follows that every Jordan block of X_{α} on J^{−1}𝔎(M)/J^{2}𝔎(M) has length two or three, and the total number is independent of α. If there are a Jordan blocks of length two and b of length three then a + b and 2a + 3b are independent of α, and hence a and b are independent of α. Thus J^{−1}𝔎(M)/J^{2}𝔎(M) has constant Jordan type. □

In order to prove the theorem, we may therefore replace M by J^{−1}𝔎(M)/J^{2}𝔎(M). Thus we may assume that J^{2}𝔎(M) = 0 and JM ⊆ 𝔎(M), which implies that J^{3}M = 0.

The indecomposable modules for kE/J^{2} are well understood in terms of the classification of modules for the Kronecker quiver. Since every X_{α} induces a surjective map from 𝔎(M)/J𝔎(M) to J𝔎(M), it follows that 𝔎(M) is a direct sum of “W - modules” for kE/J^{2}. Such a module has a basis v_{0}, …, v_{n}, w_{1}, …, w_{n} with

X_{1}v_{0} = 0, X_{1}v_{i} = w_{i}, X_{1}w_{i} = 0,
X_{2}v_{i−1} = w_{i}, X_{2}v_{n} = 0, X_{2}w_{i} = 0
(1 ≤ i ≤ n).

A diagram of such a module is as follows.

(3.4)  [figure: •^{v_{0}}; •^{v_{1}}; …; •^{v_{n}}; •_{w_{1}}; •_{w_{2}}; •_{w_{n}}]

Here, the action of X_{1} is downwards to the left, and the action of X_{2} is downwards to the right.

Let N be an indecomposable summand of 𝔎(M). Thus N is a module of the form (3.4) with n ≥ 0. Let 0 ≠ α ∈ 𝔸^{2}(k). Since X_{α} has no Jordan blocks of length one on M, the kernel of X_{α} on N is in the image of X_{α} on M. Indeed, this remains true after field extension, see [9]. Apply this to X_{α} = X_{1} + tX_{2} after extending the coefficients to the field of Laurent power series k[[t]][t^{−1}]. The kernel of this on N is one dimensional, spanned by

v_{0} − tv_{1} + t^{2}v_{2} − ⋯ ± t^{n}v_{n}.

It follows that there is a vector ∑_{i}(−1)^{i}t^{i}u_{i} whose image under X_{1} + tX_{2} is this vector. Since X_{1} and X_{2} are injective, comparing coefficients of powers of t shows that u_{i} is zero unless i ≥ 0. For i ≥ n we have X_{2}u_{i} = X_{1}u_{i+1}.

If u_{n} ≠ 0 then let m > n be as small as possible so that there is a linear relation of the form ∑_{i=n}^{m} λ_{i}u_{i} = 0. Such a linear relation exists because M is finite dimensional, and minimality of m implies that u_{n}, …, u_{m−1} are linearly independent and u_{m} is a linear combination of them. So

X_{2}u_{m} ∈ ⟨X_{2}u_{n}, …, X_{2}u_{m−1}⟩ = ⟨X_{1}u_{n+1}, …, X_{1}u_{m}⟩ ⊆ ⟨X_{1}u_{n}, …, X_{1}u_{m−1}⟩.

It follows that for all λ, μ not both zero,

λX_{1} + μX_{2} : ⟨u_{n}, …, u_{m−1}⟩ → ⟨X_{1}u_{n}, …, X_{1}u_{m−1}⟩

is injective, and hence an isomorphism. This is impossible because using X_{1} to identify these two spaces, X_{2} has to have a non-zero eigenvector for some eigenvalue. This contradiction shows that u_{n} = 0.

Again using the fact that X_{1} is injective and X_{2}u_{i} = X_{1}u_{i+1} for i ≥ n, it follows that u_{i} = 0 for i ≥ n. Thus we have

(X_{1} + tX_{2})(u_{0} − tu_{1} + t^{2}u_{2} − ⋯ ∓ t^{n−1}u_{n−1}) = v_{0} − tv_{1} + t^{2}v_{2} − ⋯ ± t^{n}v_{n}.

In particular it follows that n > 0. Comparing coefficients we get

X_{1}u_{0} = v_{0}, X_{1}u_{1} − X_{2}u_{0} = v_{1}, X_{1}u_{2} − X_{2}u_{1} = v_{2}, ⋯, −X_{2}u_{n−1} = v_{n}.

This shows that X_{1}u_{1} − 2v_{1} = X_{2}u_{0} − v_{1} is annihilated by X_{1}; for 2 ≤ i ≤ n − 1 we have

X_{2}(X_{1}u_{i−1} − iv_{i−1}) = X_{1}^{2}u_{i} − (i + 1)w_{i} = X_{1}(X_{1}u_{i} − (i + 1)v_{i}),

and

X_{2}(X_{1}u_{n−1} − nv_{n−1}) = −(n + 1)w_{n}.

We therefore have the following relationships between elements of 𝔎(M):

[figure: X_{1}u_{1} − 2v_{1}; X_{1}u_{2} − 3v_{2}; …; X_{1}u_{n−1} − nv_{n−1}; X_{1}^{2}u_{2} − 3w_{2}; X_{1}^{2}u_{3} − 4w_{3}; −(n + 1)w_{n}]

Here, we do not claim that the depicted elements are linearly independent, or even non-zero. However, it is easily checked that there can not be such a chain of elements in 𝔎M unless the coefficient of w_{n} on the lower rightmost vector is zero. This is because

w_{n} ∉ (X_{2}X_{1}^{−1})^{n−1}{0}_{𝔎M},

as can be seen by projecting onto the summand N of 𝔎M (the notation here is that if U is a subspace of 𝔎M we denote by X_{1}^{−1}U its inverse image under X_{1} and by X_{2}U its image under X_{2}).

It follows that −(n + 1) is zero in k, which implies that n ≡ −1 (mod p). It follows that the dimension of N/JN is divisible by p. Since this is true for every indecomposable summand N of 𝔎M, it follows that the dimension of 𝔎M/J𝔎M is divisible by p. But by Lemma 3.1 this dimension is the number of Jordan blocks in the Jordan type of M, which completes the proof of Theorem 1.2.

4. Consequences

Proof of Corollary 1.3. Let M be a module of constant Jordan type [a_{1}] … [a_{t}] with none of the a_{i} equal to p−1. Then Ω(M) is a module of constant Jordan type where the number of blocks of length p − j (j ≥ 1) is equal to the number of blocks of M of length j. The number of Jordan blocks of length p in Ω(M) is congruent mod p to minus the total number of Jordan blocks of M. Therefore, Corollary 1.3 for M follows by applying Theorem 1.2 to Ω(M). □

Proof of Corollary 1.4. If M has constant Jordan type with no blocks of length 1 or p − 1 then we may apply Theorem 1.2 to deduce that the total number of Jordan blocks is divisible by p, and we may also apply Corollary 1.3 to deduce that the number of Jordan blocks of length p is also divisible by p. Subtracting, we see that the number of Jordan blocks of length between 2 and p − 2 is divisible by p. □

For the proofs of Theorems 1.5 and 1.6 we need the following proposition.

Proposition 4.1. Let J_{a}, J_{b} be the indecomposable representations of dimensions a ≤ b ≤ p for a cyclic group of order p over k. Then

(1) J_{a} ⊗ J_{b} ≅ \begin{cases}J_{a+b−1} ⊕ J_{a+b−3} ⊕ J_{a+b−5} ⊕ ⋯ ⊕ J_{b−a+1} & p ≥ a + b − 1 \\ (a + b − p)J_{p} ⊕ J_{2p−a−b−1} \\ ⊕ J_{2p−a−b−3} ⊕ ⋯ ⊕ J_{b−a+1} & p ≤ a + b − 2 \end{cases}

(2) Λ^{2}(J_{a}) ≅ \begin{cases}J_{2a−3} ⊕ J_{2a−7} ⊕ J_{2a−11} ⊕ … & p ≥ 2a − 3 \\ (a − \frac{p+1}{2})J_{p} ⊕ J_{2p−2a−1} ⊕ J_{2p−2a−5} ⊕ … & p ≤ 2a − 5. \end{cases}

(3) S^{2}(J_{a}) ≅ Λ^{2}(J_{a+1}).

Proof. This follows from the computations given in Section III of Almkvist and Fossum [1]. □

Corollary 4.2. If a and b are distinct and odd then Λ^{2}(J_{a} ⊕ J_{b} ⊕ J_{1}) has at most p−1 non-projective indecomposable summands, and the dimension of each summand is odd and greater than one.

Proof. We have

Λ^{2}(J_{a} ⊕ J_{b} ⊕ J_{1}) ≅ Λ^{2}(J_{a}) ⊕ Λ^{2}(J_{b}) ⊕ J_{a} ⊗ J_{b} ⊕ J_{a} ⊕ J_{b}.

It follows from the Proposition that Λ^{2}(J_{a}) and Λ^{2}(J_{b}) each has at most (p − 3)/4 non-projective indecomposable summands, and J_{a} ⊗ J_{b} has at most (p − 3)/2. □

Proof of Theorem 1.5. Suppose that all the stable Jordan blocks of M are distinct and odd. Since Λ^{2}(M) is a direct summand of M ⊗ M, it follows from Theorem 3.7 and Corollary 4.3 of [6] that Λ^{2}(M) has constant Jordan type.

If M has one stable Jordan block then by Theorem 1.2 the length of the Jordan block is equal to one.

If M has two stable Jordan blocks then again applying Theorem 1.2, the restriction to a cyclic group is J_{a}⊕J_{1} plus projectives, with a > 1. So Λ^{2}(M) has stable constant Jordan type Λ^{2}(J_{a}⊕J_{1}) = Λ^{2}(J_{a})⊕J_{a}. By Proposition 4.1, this has at most (p−1)/4 non-projective summands, none of which is isomorphic to J_{1}, contradicting Theorem 1.2.

If M has three stable Jordan blocks then the restriction to a cyclic group is J_{a} ⊕ J_{b} ⊕ J_{1} plus projectives, with a < b. It follows from Corollary 4.2 that Λ^{2}(M) has at most p − 1 non-projective summands, and that none of them is isomorphic to J_{1}. Again this contradicts Theorem 1.2.

It follows that if M has more than one stable Jordan block then it has at least four.

Finally, if all the stable Jordan blocks are distinct and even, then Ω(M) has constant Jordan type with all stable Jordan blocks distinct and odd. So we apply the previous arguments to Ω(M). □

Proof of Theorem 1.6. If M has stable constant Jordan type [n] [1] then because of the restriction on the value of n, there are no Jordan blocks of length p − 1. So by Corollary 1.3, the number of Jordan blocks of length p is divisible by p. By Proposition 4.1, S^{2}(J_{n}) has n − (p − 1)/2 copies of J_{p}. So S^{2}(M) has no Jordan blocks of length p − 1 and the number of Jordan blocks of length p is congruent to n − (p − 1)/2 modulo p. This contradicts Corollary 1.3 for S^{2}(M). □

Remark 4.3. The same technique does not seem to dispose of stable constant Jordan type [n] [1] for 4 ≤ n < p/2 even, so for example [4] [1] for p = 11 is the smallest case that remains undecided.

Some further cases can be ruled out by the use of other Schur functors. For example, by applying the Schur functor S^{[2] [1]}, we can rule out the case of stable constant Jordan type [6] [1] for p = 13. More generally the Schur functor S^{[2] [1]} rules out modules of stable constant Jordan type [2n] [1] when p = 6n − 5 or p = 6n − 7. This is because for p = 6n − 5 we have

S^{[2] [1]}(J_{2n}) = 2J_{p} ⊕ J_{p−3} ⊕ ⋯

while for p = 6n − 7 we have

S^{[2] [1]}(J_{2n}) = 4J_{p} ⊕ J_{p−3} ⊕ ⋯

where the dots indicate smaller Jordan blocks.

References

1. G. Almkvist and R. Fossum, Decomposition of exterior and symmetric powers of indecomposable ℤ/pℤ-modules in characteristic p and relations to invariants, Séminaire d’Algèbre Paul Dubreil, 30ème année (Paris, 1976–1977), Lecture Notes in Mathematics, vol. 641, Springer-Verlag, Berlin/New York, 1978, pp. 1–111.

2. S. Baland, Modules of constant Jordan type with two non-projective blocks, J. Algebra 346 (2011), 343–350.

3. D. J. Benson, Modules of constant Jordan type with one non-projective block, Algebras and Representation Theory 13 (2010), 315–318.

4. D. J. Benson and J. Pevtsova, A realization theorem for modules of constant Jordan type and vector bundles, To appear in Trans. AMS.

5. J. F. Carlson and E. M. Friedlander, Exact category of modules of constant Jordan type, Algebra, arithmetic and geometry: Manin Festschrift, Progr. in Math., vol. 269, Birkhäuser Verlag, Basel, 2009, pp. 259–281.

6. J. F. Carlson, E. M. Friedlander, and J. Pevtsova, Modules of constant Jordan type, J. Reine & Angew. Math. 614 (2008), 191–234.

7. J. F. Carlson, E. M. Friedlander, and A. A. Suslin, Modules for ℤ/p × ℤ/p, Comment. Math. Helvetici 86 (2011), 609–657.

8. E. M. Friedlander and J. Pevtsova, Constructions for infinitesimal group schemes, Trans. Amer. Math. Soc. 363 (2011), 6007–6061.

9. E. M. Friedlander, J. Pevtsova, and A. Suslin, Generic and maximal Jordan types, Invent. Math. 168 (2007), 485–522.

Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE