GENERIC MODULE THEORY

Wayne W. Wheeler^{∗}

Abstract. The process of restricting modules to cyclic shifted subgroups is a fundamental technique in the modular representation theory of elementary abelian pgroups. If E is elementary abelian of p-rank r and k is an algebraically closed field of characteristic p, then each point in k^{r} − {0} determines a cyclic shifted subgroup. Because the restriction of a kE-module to this shifted subgroup depends only upon the corresponding point in projective space, it is often convenient to work with ℙ_{k}^{r−1} instead of k^{r} − {0}. Roughly speaking, this paper shows that if V is an irreducible subvariety of ℙ_{k}^{r−1} and M is a kE-module, then for almost all points in V the direct sum decomposition of M is the same upon restriction; moreover, this decomposition is completely determined by the behavior of M upon restriction to the cyclic shifted subgroup corresponding to the generic point of V. A similar idea provides a stratification of the rank variety of M into a disjoint union of locally closed subspaces. The closures of these subspaces are then described in terms of deformations of modules over a group of order p.

1. Introduction

If p is a prime, then one particularly useful way of studying modules over elementary abelian p-groups has been to look at restrictions to cyclic shifted subgroups. For example, this idea led to Carlson’s definition of the rank variety, which governs many of the homological properties of the module. Broadly, the purpose of this paper is to consider certain aspects of restrictions to cyclic shifted subgroups that do not seem to have been studied previously. In particular, we will show that there is a sense in which the direct sum decomposition of a finitely generated module is the same upon restriction to almost all cyclic shifted subgroups. Moreover, this decomposition is completely determined by the decomposition obtained by restricting to a certain cyclic shifted subgroup corresponding to a point defined over an appropriate field extension.

Begin by fixing an algebraically closed field k of characteristic p, and let E = ⟨g_{1}, …, g_{r}⟩ be an elementary abelian p-group of rank r. If α = (α_{1}, …, α_{r}) ∈ k^{r} − {0}, set

u_{α} = 1 + ∑_{i=1}^{r} α_{i}(g_{i} − 1) ∈ kE.

1991 Mathematics Subject Classification. 20J06. ∗Partially supported by a grant from the NSF.

Typeset by AMS-TEX

Then u_{α} is a unit that generates a cyclic group ⟨u_{α}⟩ of order p in kE, called a cyclic shifted subgroup, and the group algebra k⟨u_{α}⟩ is a subalgebra of kE. Thus for any finitely generated kE-module M there is a well-defined restriction M_{⟨u_{α}⟩} to the subalgebra k⟨u_{α}⟩. The rank variety V^{r}(M) of M is defined to be the subset of k^{r} given by

V^{r}(M) = {0} ∪ {α ∈ k^{r} − {0} | M_{⟨u_{α}⟩} is not projective}.

It is easy to check that if λ ∈ k, then k⟨u_{λα}⟩ = k⟨u_{α}⟩, and this fact implies that V^{r}(M) is a homogeneous subset of k^{r}. Carlson has shown that V^{r}(M) is actually a homogeneous affine variety [2].

For our purposes it will be more convenient to work with projective varieties, so we let V̄^{r}(M) denote the projective variety corresponding to V^{r}(M). If β ∈ ℙ_{k}^{r−1}, we write M_{⟨u_{β}⟩} for the restriction of M to the subalgebra k⟨u_{α}⟩ determined by choosing any point α = (α_{1}, …, α_{r}) ∈ k^{r} − {0} in the equivalence class of β. As we noted above, this subalgebra does not depend on the choice of α, even though u_{α} does. In fact, when no confusion is likely to arise, we will sometimes write u_{β} for the element obtained by choosing a particular representative (β_{1}, …, β_{r}) of β, even though u_{β} depends upon this choice; similarly, we may sometimes use the same symbol α to denote both an element (α_{1}, …, α_{r}) ∈ k^{r} − {0} and the corresponding element of ℙ_{k}^{r−1}.

As motivation for the ideas developed here, it will perhaps be useful to begin by considering a specific example. Let p = 3 and r = 2, and set M = kE/ Rad^{2} kE so that M is a 3-dimensional kE-module with basis m_{1}, m_{2}, m_{3} such that

(g_{1} − 1)m_{1} = m_{2}, (g_{2} − 1)m_{1} = m_{3},
(g_{1} − 1)m_{2} = (g_{2} − 1)m_{2} = (g_{1} − 1)m_{3} = (g_{2} − 1)m_{3} = 0.

If α = (α_{1}, α_{2}) ∈ k^{2} − {0}, then

M_{⟨u_{α}⟩} = km_{2} ⊕ span_{k}{m_{1}, α_{1}m_{2} + α_{2}m_{3}}

is a direct sum decomposition of M_{⟨u_{α}⟩} into indecomposable k⟨u_{α}⟩-modules whenever α_{2} ≠ 0; if α_{2} = 0, then we get such a decomposition by taking

M_{⟨u_{α}⟩} = km_{3} ⊕ span_{k}{m_{1}, m_{2}}.

Let us concentrate on the decomposition in the case when α_{2} ≠ 0. Because we are concerned with projective varieties, we rewrite the decomposition as

M_{⟨u_{α}⟩} = km_{2} ⊕ span_{k}\{m_{1}, \frac{α_{1}}{α_{2}} m_{2} + m_{3}\}

so that the bases of the summands do not depend upon the particular choice of α ∈ k^{2} − {0} but only on the corresponding element of ℙ_{k}^{1}. Then the summand km_{2} is a kE-module; the second summand, on the other hand, is certainly not a kE-module, although it is a k⟨u_{α}⟩-module for all α such that α_{2} ≠ 0. Our broad goal in this paper is, in effect, to study a category in which the objects have exactly this sort of structure: they are k⟨u_{α}⟩-submodules of some kE-module M for all α in some nonempty open subset of ℙ_{k}^{r−1}.

Sections 2 and 3 develop this idea more fully. We begin by fixing an irreducible variety V ⊆ ℙ_{k}^{r−1}, and the basic objective is to study restrictions of kE-modules to k⟨u_{α}⟩ for α ∈ V. In the approach presented here, the first step is to extend the field of coefficients from k to the homogeneous function field F of the variety V. Roughly speaking, an element c ∈ F can be considered as a function from V to k such that the value c(α) is not necessarily defined for all α ∈ V. Similarly, we will see in Section 2 that if M is any finite-dimensional vector space over k and X is an F -subspace of F ⊗_{k} M, then X determines a function from V to k-subspaces of M such that the subspace X(α) is not necessarily defined for all α ∈ V. If M is a finitely generated kE-module, then the particular subspaces of interest will be those with the property that X(α) is a k⟨u_{α}⟩-submodule of M_{⟨u_{α}⟩} whenever X(α) is defined.

One of the chief advantages of extending the coefficient field to F is that the projective space ℙ_{F}^{r−1} over F contains a point Γ that is generic for the variety V. The generic point Γ ∈ ℙ_{F}^{r−1} is uniquely determined by V, and one can consider the subalgebra F ⟨u_{Γ}⟩ of F E. In Section 3 we will see that for any finitely generated kE-module M there is a nonempty open set U ⊆ V such that if α ∈ U and 1 ≤ i ≤ p, then the multiplicity of the i-dimensional indecomposable k⟨u_{α}⟩-module as a summand of M_{⟨u_{α}⟩} is the same as the multiplicity of the i-dimensional indecomposable F ⟨u_{Γ}⟩-module as a summand of (F ⊗_{k} M)_{⟨u_{Γ}⟩}. In other words, the direct sum decomposition of (F ⊗_{k} M)_{⟨u_{Γ}⟩} completely determines that of M_{⟨u_{α}⟩} for almost all α ∈ V.

This result is closely related to certain stratification theorems for ℙ_{k}^{r−1} and for V̄^{r}(M) that are given in Section 4. In particular, we will show that any finitely generated kE-module M determines a finite decomposition of ℙ_{k}^{r−1} into disjoint subspaces that are locally closed in the Zariski topology. This decomposition in turn gives a stratification of V̄^{r}(M). Specifically, we will prove the following result.

Theorem 1.1. Suppose that (n_{1}, …, n_{p}) is a p-tuple of nonnegative integers and that M is a finitely generated kE-module. Let X(M; n_{1}, …, n_{p}) denote the set of all α ∈ ℙ_{k}^{r−1} such that n_{i} is the multiplicity of the indecomposable k⟨u_{α}⟩-module of dimension i as a summand of M_{⟨u_{α}⟩}. Then X(M; n_{1}, …, n_{p}) is locally closed in ℙ_{k}^{r−1}. Moreover, V̄^{r}(M) is the disjoint union of the sets X(M; n_{1}, …, n_{p}) over all p-tuples (n_{1}, …, n_{p}) such that at least one of n_{1}, …, n_{p−1} is nonzero.

In Section 4 we will also describe the closures of the sets X(M; n_{1}, …, n_{p}) in terms of deformations of modules over a cyclic group of order p.

Although the relationships between generic points and restrictions to cyclic shifted subgroups that are studied here seem to be interesting in their own right, they have been developed primarily with a view to possible applications. We have therefore endeavored to prove results in Sections 2 and 3 in the strongest form that might be useful, even when a somewhat weaker result is easier to prove. One possible application of this theory—and indeed the chief reason for its development—is to study quotient categories of the sort considered in [3]. We hope to return to this idea in the future.

2. Generic Vector Spaces

Throughout the remainder of this paper V will denote a fixed irreducible subvariety of ℙ_{k}^{r−1}, and the basic goal will be to study restrictions of kE-modules to cyclic shifted subgroups corresponding specifically to points in V. If M is a finitely generated kE-module, then we will typically be concerned with properties that are shared by the restrictions M_{⟨u_{α}⟩} for all α in some nonempty open subset of V.

Let F denote the homogeneous function field of V. There is more than one way to describe this field [6, 7], but it can be obtained by starting with the polynomial ring k[x_{1}, …, x_{r}] in r variables, where x_{i} is the linear functional on k^{r} given by x_{i}(t_{1}, …, t_{r}) = t_{i} for 1 ≤ i ≤ r. Let I(V) be the ideal generated by all homogeneous polynomials in k[x_{1}, …, x_{r}] that vanish at every point in V. Then the ring S(V) = k[x_{1}, …, x_{r}]/I(V) is an integral domain, and the field F consists of all elements c = P/Q of the field of fractions of S(V) such that P and Q can be represented by homogeneous elements of the same degree. Such an element determines a welldefined function at least on {α ∈ V | Q(α) ≠ 0}, and possibly on a larger open set; to put it less precisely, c determines a function on V such that c(α) is not necessarily defined for all α ∈ V.

For 1 ≤ i ≤ r let y_{i} denote the image of x_{i} in S(V); then y_{i} is the i^{th} coordinate function on the homogeneous affine variety corresponding to V. Now choose m such that 1 ≤ m ≤ r and y_{m} ≠ 0. Then the point

γ = (\frac{y_{1}}{y_{m}} , …, \frac{y_{r}}{y_{m}} ) ∈ F^{r} − \{0\}

determines a point Γ ∈ ℙ_{F}^{r−1}, called the generic point of the variety V. A good discussion of generic points can be found in [1].

If M is any finite-dimensional vector space over k, then for simplicity we will write F M for F ⊗_{k} M; similarly, if c ∈ F and m ∈ M, then we will write cm for c ⊗ m. If M is a kE-module, then we want to study the connections between the F ⟨u_{Γ}⟩-module (F M)_{⟨u_{Γ}⟩} and the k⟨u_{α}⟩-module M_{⟨u_{α}⟩} for all α ∈ V. We begin in this section by studying the relationships between the k-vector space M and the F -vector space F M, and in the next section we extend the theory to cover the situation in which M is a kE-module.

Lemma 2.1. Assume that M is a finite-dimensional vector space over k. Let x ∈ F M, and let α ∈ V. Suppose that m_{1}, …, m_{n} and m′_{1}, …, m′_{n} are k-bases of M, and let c_{1}, …, c_{n} and c′_{1}, …, c′_{n} be the elements of F such that

x = c_{1}m_{1} + ⋯ + c_{n}m_{n} = c′_{1}m′_{1} + ⋯ + c′_{n}m′_{n}.

Then c_{i}(α) is defined for 1 ≤ i ≤ n if and only if c′_{i}(α) is defined for 1 ≤ i ≤ n. Moreover, if this condition is satisfied, then

c_{1}(α)m_{1} + ⋯ + c_{n}(α)m_{n} = c′_{1}(α)m′_{1} + ⋯ + c′_{n}(α)m′_{n}.

Proof. Suppose that c_{j}(α) is defined for 1 ≤ j ≤ n. Write

m_{j} = ∑_{i=1}^{n} a_{ij}m′_{i}

for some a_{ij} ∈ k. Then

x = ∑_{j=1}^{n} c_{j}m_{j} = ∑_{i=1}^{n} ∑_{j=1}^{n} a_{ij}c_{j}m′_{i},

and it follows that

c′_{i} = ∑_{j=1}^{n} a_{ij}c_{j}.

Because c_{j}(α) is defined for all j, we see that c′_{i}(α) is defined. Moreover,

∑_{i=1}^{n} c′_{i}(α)m′_{i} = ∑_{j=1}^{n} ∑_{i=1}^{n} a_{ij}c_{j}(α)m′_{i} = ∑_{j=1}^{n} c_{j}(α)m_{j},

and this completes the proof. □

Let x ∈ F M, and let α ∈ V. If M has a k-basis m_{1}, …, m_{n} such that x = c_{1}m_{1} + ⋯ + c_{n}m_{n} and c_{i}(α) is defined for 1 ≤ i ≤ n, then we define x(α) ∈ M by

x(α) = c_{1}(α)m_{1} + ⋯ + c_{n}(α)m_{n}.

Lemma 2.1 shows that the vector x(α) does not depend upon the chosen basis m_{1}, …, m_{n}.

The proof of the next proposition is based in part upon the well-known fact that if V is an irreducible subvariety of ℙ_{r}^{k−1}, then every nonempty open subset of V is dense in V; equivalently, any two nonempty open subsets must intersect. This property of irreducible varieties will be used repeatedly throughout the remainder of the paper.

Proposition 2.2. Suppose that M is a finite-dimensional vector space over k. Let x ∈ F M, and choose a k-basis m_{1}, …, m_{n} of M. Let c_{1}, …, c_{n} ∈ F be the elements such that

x = c_{1}m_{1} + ⋯ + c_{n}m_{n},

and let C be the k-subspace of F spanned by c_{1}, …, c_{n}. For 1 ≤ i ≤ n set U_{i} = {α ∈ V | c_{i}(α) is defined}, and let U_{0} = U_{1} ∩ ⋯ ∩ U_{n}. Then

(1) U_{0} = {α ∈ V | c(α) is defined for all c ∈ C} = {α ∈ V | x(α) is defined}.

(2) U_{0} is a nonempty open set and is independent of the basis m_{1}, …, m_{n}.

Proof. (1) By definition U_{0} is the set of all points α ∈ V such that c_{1}(α), …, c_{n}(α) are all defined. Hence α ∈ U_{0} if and only if c(α) is defined for all c ∈ C. Moreover, if α ∈ U_{0}, then certainly x(α) is defined. On the other hand, if α ∈ V such that x(α) is defined, then there is some k-basis m′_{1}, …, m′_{n} of M such that

x = c′_{1}m′_{1} + ⋯ + c′_{n}m′_{n}

and c′_{i}(α) is defined for 1 ≤ i ≤ n. Then Lemma 2.1 shows that c_{i}(α) is defined for 1 ≤ i ≤ n, so α ∈ U_{0}.

(2) Because U_{i} is a nonempty open set for 1 ≤ i ≤ n, it follows that U_{0} is a nonempty open set. The fact that U_{0} is independent of the chosen basis is an easy consequence of (1) and Lemma 2.1. □

Suppose that M is a finite-dimensional vector space over k and x_{1}, …, x_{n} is an F -basis for F M. Then we will write U(x_{1}, …, x_{n}) for the set of points α ∈ V such that x_{1}(α), …, x_{n}(α) are all defined and span M.

Proposition 2.3. If M is a finite-dimensional vector space over k and x_{1}, …, x_{n} is an F -basis for F M, then U(x_{1}, …, x_{n}) is a nonempty open subset of V.

Proof. Choose a basis m_{1}, …, m_{n} of M, and let c_{ij} ∈ F be elements such that

x_{j} = ∑_{i=1}^{n} c_{ij}m_{i}

for 1 ≤ j ≤ n. It follows from Proposition 2.2 that there is a nonempty open set U such that the following conditions are equivalent:

(1) α ∈ U.

(2) x_{j}(α) is defined for all j.

(3) c_{ij}(α) is defined for all i and j. Let C = (c_{ij}), and let α ∈ U. Then x_{1}(α), …, x_{n}(α) span M if and only if the matrix C(α) = (c_{ij}(α)) is invertible. Because det C ∈ F, we see that

U(x_{1}, …, x_{n}) = U ∩ {α ∈ V | (det C)(α) ≠ 0}

is a nonempty open set, as desired. □

Proposition 2.4. Suppose that M is a finite-dimensional vector space over k and that x_{1}, …, x_{n} is an F -basis for F M. Let x ∈ F M, and let b_{i} ∈ F be the elements such that

x = b_{1}x_{1} + ⋯ + b_{n}x_{n}.

If α ∈ U(x_{1}, …, x_{n}) and x(α) is defined, then b_{i}(α) is defined for 1 ≤ i ≤ n, and

x(α) = b_{1}(α)x_{1}(α) + ⋯ + b_{n}(α)x_{n}(α).

Proof. For 1 ≤ i ≤ n set m_{i} = x_{i}(α). Because α ∈ U(x_{1}, …, x_{n}), we know that m_{1}, …, m_{n} is a basis for M. Let a_{ij} ∈ F be the elements such that

x_{j} = ∑_{i=1}^{n} a_{ij}m_{i}

for 1 ≤ j ≤ n, and set A = (a_{ij}). Then Proposition 2.2 implies that a_{ij}(α) is defined for all i and j. In fact, the matrix A(α) = (a_{ij}(α)) is the identity, so det A is defined at α and satisfies (det A)(α) = det A(α) = 1. If adj A denotes the adjoint of A, then

A^{−1} = \frac{1}{det A} adj A

and hence A^{−1}(α) is defined.

Now let B denote the column vector given by setting B = (b_{1}, …, b_{n})^{t}. Because

x = ∑_{j=1}^{n} b_{j}x_{j} = ∑_{i=1}^{n} ∑_{j=1}^{n} a_{ij}b_{j}m_{i}

and x(α) is defined, Proposition 2.2 implies that (AB)(α) is defined. Hence B(α) = A^{−1}(α)(AB)(α) is defined. Moreover,

x(α) = ∑_{i=1}^{n} ∑_{j=1}^{n} a_{ij}(α)b_{j}(α)m_{i} = ∑_{j=1}^{n} b_{j}(α)x_{j}(α),

as desired. □

Definition 2.5. Let M be a finite-dimensional vector space over k, and let X be an F -subspace of F M. Then X will be called a generic subspace of M. If we do not wish to specify the k-vector space M, then we will simply refer to X as a generic vector space.

Lemma 2.6. Suppose that M is a finite-dimensional vector space over k and that W and X are generic subspaces of M with W ⊆ X. Let α ∈ V, let w_{1}, …, w_{s} be an F -basis for W, and let x_{1}, …, x_{t} be an F -basis for X. Assume that these bases can be extended to F -bases w_{1}, …, w_{n} and x_{1}, …, x_{n} of F M such that α ∈ U(w_{1}, …, w_{n}) ∩ U(x_{1}, …, x_{n}). Then

span_{k}{w_{1}(α), …, w_{s}(α)} ⊆ span_{k}{x_{1}(α), …, x_{t}(α)}.

Proof. Let c_{ij} ∈ F be the elements such that

w_{j} = ∑_{i=1}^{n} c_{ij}x_{i}

for 1 ≤ j ≤ n. Because α ∈ U(w_{1}, …, w_{n}) ∩ U(x_{1}, …, x_{n}), Proposition 2.4 implies that c_{ij}(α) is defined for all i and j, and

w_{j}(α) = ∑_{i=1}^{n} c_{ij}(α)x_{i}(α).

But W ⊆ X, so c_{ij} = 0 whenever 1 ≤ j ≤ s and t < i ≤ n. Thus for 1 ≤ j ≤ s we have

w_{j}(α) = ∑_{i=1}^{t} c_{ij}(α)x_{i}(α).

Hence span_{k}{w_{1}(α), …, w_{s}(α)} ⊆ span_{k}{x_{1}(α), …, x_{t}(α)}, and this completes the proof. □

Let M be a finite-dimensional vector space over k, and let X be a generic subspace of M. Fix α ∈ V, and assume that X has an F -basis x_{1}, …, x_{t} that can be extended to an F -basis x_{1}, …, x_{n} of F M with α ∈ U(x_{1}, …, x_{n}). Set X(α) = span_{k}{x_{1}(α), …, x_{t}(α)}. Then Lemma 2.6 shows that X(α) depends only upon the point α and the generic vector space X, and not upon the chosen basis. Moreover, if U denotes the set of all points α ∈ V such that X(α) is defined, then U is the union of the sets U(x_{1}, …, x_{n}) over all F -bases x_{1}, …, x_{n} obtained by extending an F -basis x_{1}, …, x_{t} of X. Hence U is a nonempty open subset of V by Proposition 2.3.

Lemma 2.7. Let M and M′ be finite-dimensional vector spaces over k. Suppose that X is a generic subspace of M and X′ is a generic subspace of M′. Let f : X → X′ be an F -linear transformation. Choose F -bases x_{1}, …, x_{n} and x̃_{1}, …, x̃_{n} of F M such that x_{1}, …, x_{t} and x̃_{1}, …, x̃_{t} are F -bases of X. Let U be the set of all points α ∈ U(x_{1}, …, x_{n}) ∩ U(x̃_{1}, …, x̃_{n}) such that X′(α) is defined and f(x_{i})(α) and f(x̃_{i})(α) are defined for 1 ≤ i ≤ t. For any α ∈ U let f_{α}, f̃_{α} : X(α) → X′(α) be the unique linear transformations satisfying

f_{α}(x_{i}(α)) = f(x_{i})(α)

and

f̃_{α}(x̃_{i}(α)) = f(x̃_{i})(α)

for 1 ≤ i ≤ t. Then U is a nonempty open set, and f_{α} = f̃_{α} for all α ∈ U.

Proof. Let U_{0} be the set of all points α such that f(x_{i})(α) and f(x̃_{i})(α) are defined for 1 ≤ i ≤ t. Then U_{0} is nonempty and open by Proposition 2.2. If U_{1} is the set of all α ∈ V such that X′(α) is defined, then we know that U_{1} is also a nonempty open set. Moreover, U(x_{1}, …, x_{n}) and U(x̃_{1}, …, x̃_{n}) are nonempty and open by Proposition 2.3, so it follows that

U = U(x_{1}, …, x_{n}) ∩ U(x̃_{1}, …, x̃_{n}) ∩ U_{0} ∩ U_{1}

is a nonempty open set.

Now let α ∈ U, and let c_{ij} ∈ F be elements such that

x_{j} = ∑_{i=1}^{n} c_{ij}x̃_{i}

for 1 ≤ j ≤ n. Then c_{ij} = 0 if 1 ≤ j ≤ t < i ≤ n, and Proposition 2.4 shows that c_{ij}(α) is defined for all i and j. Thus for 1 ≤ j ≤ t we see that

f̃_{α}(x_{j}(α)) = f̃_{α} (∑_{i=1}^{t} c_{ij}(α)x̃_{i}(α))

= ∑_{i=1}^{t} c_{ij}(α)f(x̃_{i})(α)

= f (∑_{i=1}^{t} c_{ij}x̃_{i}) (α)

= f(x_{j})(α) = f_{α}(x_{j}(α)).

Hence f_{α} = f̃_{α}, and this completes the proof. □

Let M and M′ be finite-dimensional vector spaces over k. Suppose that X is a generic subspace of M and X′ is a generic subspace of M′, and let f : X → X′ be an F -linear transformation. Fix α ∈ V, and suppose that there is an F -basis x_{1}, …, x_{m} of F M such that x_{1}, …, x_{s} is an F -basis of X with α ∈ U(x_{1}, … x_{m}). Assume in addition that X′(α) is defined and that f(x_{i})(α) is defined for 1 ≤ i ≤ s. Then there is a unique k-linear transformation f_{α} : X(α) → X′(α) satisfying

f_{α}(x_{i}(α)) = f(x_{i})(α)

for 1 ≤ i ≤ s, and Lemma 2.7 shows that f_{α} is independent of the choice of basis x_{1}, …, x_{m}. It is easy to see that the points α ∈ V for which f_{α} is defined form a nonempty open set.

It will sometimes be useful for us to think of this definition in terms of matrices. Because X′(α) is defined, there is an F -basis x′_{1}, …, x′_{t} of X′ that extends to an F - basis x′_{1}, …, x′_{n} of F M′ such that α ∈ U(x′_{1}, …, x′_{n}). Let a_{ij} ∈ F be the elements such that

f(x_{j}) = ∑_{i=1}^{t} a_{ij}x′_{i}

for 1 ≤ j ≤ s. Then A = (a_{ij}) is the matrix representing f with respect to the bases x_{1}, …, x_{s} of X and x′_{1}, …, x′_{t} of X′. Because α ∈ U(x′_{1}, …, x′_{n}) and f(x_{j})(α) is defined for 1 ≤ j ≤ s by assumption, it follows from Proposition 2.4 that A(α) = (a_{ij}(α)) is defined, and

f_{α}(x_{j}(α)) = f(x_{j})(α) = ∑_{i=1}^{t} a_{ij}(α)x′_{i}(α).

Hence f_{α} : X(α) → X′(α) is just the linear transformation given by the matrix A(α) with respect to the bases x_{1}(α), …, x_{s}(α) of X(α) and x′_{1}(α), …, x′_{t}(α) of X′(α).

Now assume in addition that X′′ is a generic subspace of M′′ and that g : X′ → X′′ is an F -linear transformation such that g_{α} is defined. Then there is an F -basis x′′_{1}, …, x′′_{u} of X′′ that extends to an F -basis x′′_{1}, …, x′′_{l} of F M′′ with α ∈ U(x′′_{1}, …, x′′_{l}). Let B be the matrix of g with respect to the bases x′_{1}, …, x′_{t} and x′′_{1}, …, x′′_{u}. Because f_{α} and g_{α} are defined, it follows that A(α) and B(α) are defined. Hence (BA)(α) = B(α)A(α) is also defined and is the matrix of (gf)_{α} = g_{α}f_{α} with respect to the bases x_{1}(α), …, x_{s}(α) and x′′_{1}(α), …, x′′_{u}(α). Thus we get the following result.

Proposition 2.8. Let f : X → X′ and g : X′ → X′′ be F -linear transformations of generic vector spaces. If α ∈ V such that f_{α} and g_{α} are defined, then (gf)_{α} : X(α) → X′′(α) is defined, and (gf)_{α} = g_{α}f_{α}.

Proposition 2.9. Let f, g : X → X′ be F -linear transformations of generic vector spaces. Then the following conditions are equivalent:

(1) f = g.

(2) {α ∈ V | f_{α} = g_{α}} is a nonempty open set in V.

(3) There is a nonempty open set U ⊆ V such that f_{α} = g_{α} for all α ∈ U.

Proof. Suppose that f = g. Then {α ∈ V | f_{α} = g_{α}} = {α ∈ V | f_{α} is defined} is a nonempty open set, so (1) implies (2).

It is trivial that (2) implies (3), so assume that (3) holds. Let M and M′ be finite-dimensional vector spaces over k such that X is a generic subspace of M and X′ is a generic subspace of M′. Choose F -bases x_{1}, …, x_{s} of X and x′_{1}, …, x′_{t} of X′, and extend them to F -bases x_{1}, …, x_{m} of F M and x′_{1}, …, x′_{n} of F M′. Let b_{ij}, c_{ij} ∈ F be the elements such that

f(x_{j}) = ∑_{i=1}^{t} b_{ij}x′_{i}

and

g(x_{j}) = ∑_{i=1}^{t} c_{ij}x′_{i}

for 1 ≤ j ≤ s. Let U_{0} be the set of all points α ∈ U ∩ U(x_{1}, …, x_{m}) ∩ U(x′_{1}, …, x′_{n}) such that f(x_{j})(α) and g(x_{j})(α) are defined for 1 ≤ j ≤ s. Then U_{0} is a nonempty open set, and Proposition 2.4 implies that if α ∈ U_{0}, then b_{ij}(α) and c_{ij}(α) are defined for all i and j. But

f(x_{j})(α) = f_{α}(x_{j}(α)) = g_{α}(x_{j}(α)) = g(x_{j})(α)

by assumption, so it follows that b_{ij}(α) = c_{ij}(α) for all i and j. Because this equation holds for all α ∈ U_{0}, we conclude that b_{ij} = c_{ij} for all i and j. Hence f = g, and this completes the proof. □

Proposition 2.10. Let f : X → X′ be an F -linear transformation of generic vector spaces. Then

(1) (Ker f)(α) ⊆ Ker f_{α} and Im f_{α} ⊆ (Im f)(α).

(2) {α ∈ V | (Ker f)(α) = Ker f_{α}} and {α ∈ V | (Im f)(α) = Im f_{α}} are nonempty open sets.

Proof. We will prove the two statements involving kernels; the proofs of the corresponding statements for the images are similar.

Let M and M′ be finite-dimensional vector spaces over k such that X is a generic subspace of M and X′ is a generic subspace of M′. Let w_{1}, …, w_{s} be an F -basis for Ker f, and extend it to an F -basis w_{1}, …, w_{m} of F M; similarly, let x_{1}, …, x_{t} be an F -basis for X, and extend it to an F -basis x_{1}, …, x_{m} of F M. Finally, let x′_{1}, …, x′_{u} be an F -basis for X′, and extend it to an F -basis x′_{1}, …, x′_{n} of F M′. Set U = {α ∈ V | (Ker f)(α) = Ker f_{α}}, and let

U_{1} = U(w_{1}, …, w_{m}) ∩ U(x_{1}, …, x_{m}) ∩ U(x′_{1}, …, x′_{n}).

Let U_{0} = {α ∈ U_{1} | f(x_{i})(α) is defined for 1 ≤ i ≤ t}, and note that U is covered by open sets of this form for various choices of bases {w_{1}, …, w_{m}}, {x_{1}, …, x_{m}}, and {x′_{1}, …, x′_{n}}. Thus it suffices to show that U ∩ U_{0} is nonempty and open and that (Ker f)(α) ⊆ Ker f_{α} for all α ∈ U_{0}. We begin by showing that (Ker f)(α) ⊆ Ker f_{α}.

For 1 ≤ j ≤ m let c_{ij} ∈ F be the elements such that

w_{j} = ∑_{i=1}^{m} c_{ij}x_{i}.

Because Ker f ⊆ X, it follows that c_{ij} = 0 whenever 1 ≤ j ≤ s and t + 1 ≤ i ≤ m. If α ∈ U_{0}, then Proposition 2.4 implies that c_{ij}(α) is defined for all i and j. Thus if 1 ≤ j ≤ s, we have

f_{α}(w_{j}(α)) = ∑_{i=1}^{t} c_{ij}(α)f_{α}(x_{i}(α))

= ∑_{i=1}^{t} c_{ij}(α)f(x_{i})(α)

= f (∑_{i=1}^{t} c_{ij}x_{i}) (α)

= f(w_{j})(α)

= 0.

Because w_{1}(α), …, w_{s}(α) span (Ker f)(α), we conclude that (Ker f)(α) ⊆ Ker f_{α} for all α ∈ U_{0}.

Thus it remains to show that U ∩ U_{0} is a nonempty open set. Let b_{ij} ∈ F be the elements such that

f(x_{j}) = ∑_{i=1}^{u} b_{ij}x′_{i}

for 1 ≤ j ≤ t, and let B = (b_{ij}). If α ∈ U_{0}, then Proposition 2.4 implies that b_{ij}(α) is defined for all i and j. Moreover,

rank_{F} B = dim_{F} Im f = dim_{F} X − dim_{F} Ker f = dim_{k} X(α) − dim_{k}(Ker f)(α)

and

rank_{k} B(α) = dim_{k} Im f_{α} = dim_{k} X(α) − dim_{k} Ker f_{α}.

Hence (Ker f)(α) = Ker f_{α} if and only if rank_{F} B = rank_{k} B(α). Moreover, we know that rank_{F} B ≥ rank_{k} B(α) because (Ker f)(α) ⊆ Ker f_{α}. Let b = rank_{F} B. Then b is the largest integer such that the determinant of some b × b minor of B is nonzero in F. If a_{1}, …, a_{v} ∈ F are all of the determinants of b × b minors of B, then it follows that

U ∩ U_{0} = ⋃_{i=1}^{v}{α′ ∈ U_{0} | a_{i}(α′) ≠ 0}.

Hence U ∩ U_{0} is a nonempty open set, and this completes the proof. □

Corollary 2.11. Let f : X → X′ be an F -linear transformation of generic vector spaces. Then the following statements are equivalent.

(1) f is an isomorphism.

(2) {α ∈ V | f_{α} is an isomorphism} is a nonempty open set.

(3) There is a nonempty open set U such that f_{α} is an isomorphism for all α ∈ U.

Proof. Let M and M′ be finite-dimensional vector spaces over k such that X is a generic subspace of M and X′ is a generic subspace of M′. Let x_{1}, …, x_{m} ∈ F M and x′_{1}, …, x′_{n} ∈ F M′ be F -bases such that x_{1}, …, x_{s} is an F -basis for X and x′_{1}, …, x′_{t} is an F -basis for X′. Let U_{0} be the set of all points α ∈ U(x_{1}, …, x_{m}) ∩ U(x′_{1}, …, x′_{n}) such that f(x_{i})(α) is defined for 1 ≤ i ≤ s, and let U_{1} = {α ∈ V | f_{α} is an isomorphism}. To show that (1) implies (2), it suffices to prove that U_{0} ∩ U_{1} is a nonempty open set.

Suppose, then, that f is an isomorphism. If α ∈ U_{0}, then there are elements c_{ij} ∈ F such that

f(x_{j}) = ∑_{i=1}^{t} c_{ij}x′_{i}

for 1 ≤ j ≤ s = t, and Proposition 2.4 implies that c_{ij}(α) is defined for all i and j. Let C = (c_{ij}). Because f is an isomorphism, we know that det C ≠ 0. For any α ∈ U_{0} the map f_{α} is an isomorphism if and only if C(α) is nonsingular, so U_{0} ∩U_{1} = {α ∈ U_{0} | (det C)(α) ≠ 0} is a nonempty open set. Hence (1) implies (2).

It is trivial that (2) implies (3), so now assume that there is a nonempty open set U such that f_{α} is an isomorphism for all α ∈ U. By Proposition 2.10 there is a nonempty open set U′ ⊆ U such that if α ∈ U′, then (Ker f)(α) = Ker f_{α} = 0 and (Im f)(α) = Im f_{α} = X′(α). Thus Ker f = 0 and Im f = X′, so f is an isomorphism. This completes the proof. □

3. Modules and Generic Points

In this section we develop the ideas of the previous section further by introducing an action of the group E. If X is a generic vector space, then this action allows us to require that X(α) be a k⟨u_{α}⟩-module. This idea then provides a connection between the generic point of the variety V and the restrictions M_{⟨u_{α}⟩} of a finitely generated kE-module M to cyclic shifted subgroups corresponding to points α ∈ V.

As in Section 2, let y_{i} denote the i^{th} coordinate function on the homogeneous affine variety corresponding to V. Fix m such that 1 ≤ m ≤ r and U_{m} = {α ∈ V | α_{m} ≠ 0} is nonempty. Then y_{m} ≠ 0, and the point Γ ∈ ℙ_{F}^{r−1} determined by

γ = (\frac{y_{1}}{y_{m}} , …, \frac{y_{r}}{y_{m}} ) ∈ F^{r} − \{0\}

is the generic point of the variety V. Throughout this section we will always use the point γ ∈ F^{r} − {0} as a specific representative of the generic point Γ ∈ ℙ_{F}^{r−1}.

Proposition 3.1. Let M be a finitely generated kE-module, and let X be a generic subspace of M. Then the following statements are equivalent.

(1) X(α) is a k⟨u_{α}⟩-submodule of M_{⟨u_{α}⟩} whenever α ∈ V and X(α) is defined.

(2) X(α) is a k⟨u_{α}⟩-submodule of M_{⟨u_{α}⟩} for all α in some nonempty open set U ⊆ V.

(3) X is an F ⟨u_{Γ}⟩-submodule of (F M)_{⟨u_{Γ}⟩}.

Moreover, if these conditions are satisfied and h : X → X denotes the endomorphism given by the action of u_{γ} − 1, then h_{α} : X(α) → X(α) is given by the action of (u_{α} − 1)/α_{m} for all α ∈ U_{m} such that X(α) is defined.

Proof. Choose an F -basis x_{1}, …, x_{n} of F M such that x_{1}, …, x_{t} is an F -basis for X, and let U_{0} = U(x_{1}, …, x_{n}) ∩ U_{m}. Let m_{1}, …, m_{n} be a k-basis for M. Then there are unique constants a_{ijl} ∈ k such that

(g_{i} − 1)m_{j} = ∑_{l=1}^{n} a_{ijl}m_{l}

for 1 ≤ i ≤ r and 1 ≤ j ≤ n. In addition, there are unique elements c_{ij} ∈ F such that

x_{j} = ∑_{i=1}^{n} c_{ij}m_{i}

for 1 ≤ j ≤ n. Let C = (c_{ij}). If α ∈ U_{0}, then x_{1}(α), …, x_{n}(α) is a basis for M, and C(α) = (c_{ij}(α)) is defined by Proposition 2.2. Let B = C^{−1}, and set B = (b_{ij}). Then for any α ∈ U_{0} Proposition 2.4 implies that B(α) = (b_{ij}(α)) is also defined.

Let h : X → F M denote the F -linear transformation given by the action of u_{γ} − 1. Then for 1 ≤ i ≤ t we have

h(x_{i}) = (u_{γ} − 1)x_{i} = ∑_{j=1}^{r} \frac{y_{j}}{y_{m}} (g_{j} − 1) ∑_{l=1}^{n} c_{li}m_{l}

= ∑_{j,l} \frac{y_{j}}{y_{m}} c_{li} ∑_{u=1}^{n} a_{jlu}m_{u}

= ∑_{j,l,u} \frac{y_{j}}{y_{m}} c_{li}a_{jlu} ∑_{v=1}^{n} b_{vu}x_{v}.

If α ∈ U_{0}, set t_{α} = (u_{α} −1)/α_{m}. Then we see that h(x_{i})(α) is defined for 1 ≤ i ≤ t, and

h_{α}(x_{i}(α)) = h(x_{i})(α) = ∑_{j=1}^{r} \frac{α_{j}}{α_{m}} (g_{j} − 1) ∑_{l=1}^{n} c_{li}(α)m_{l} = t_{α} · x_{i}(α).

Thus h_{α} : X(α) → M is given by the action of t_{α} for all α ∈ U_{0}, and it easily follows that h_{α} : X(α) → M is given by the action of t_{α} for all α ∈ U_{m} such that X(α) is defined.

It is clear that (1) implies (2), so assume that (2) holds. If α ∈ U ∩ U_{0}, then X(α) is a k⟨u_{α}⟩-submodule of M_{⟨u_{α}⟩}. Thus for 1 ≤ i ≤ t we have t_{α} ·x_{i}(α) ∈ X(α), and it follows that

∑_{j,l,u} \frac{y_{j}}{y_{m}} (α)c_{li}(α)a_{jlu}b_{vu}(α) = 0

for t + 1 ≤ v ≤ n. But U ∩ U_{0} is a nonempty open subset of V, so we see that

∑_{j,l,u} \frac{y_{j}}{y_{m}} c_{li}a_{jlu}b_{vu} = 0

whenever 1 ≤ i ≤ t < v ≤ n. Hence h determines an endomorphism of X, and (3) holds.

Finally, suppose that X is an F ⟨u_{Γ}⟩-submodule of (F M)_{⟨u_{Γ}⟩}. Then h is an endomorphism of X, and h_{α} maps X(α) into X(α) for all α ∈ U_{m} such that X(α) is defined. Because h_{α} : X(α) → X(α) is given by the action of t_{α}, it is easy to see that X(α) must be a k⟨u_{α}⟩-submodule of M_{⟨u_{α}⟩} for all α ∈ U_{m} such that X(α) is defined. But V is covered by the open sets U_{i} = {α ∈ V | α_{i} ≠ 0} for 1 ≤ i ≤ r, so it follows that X(α) is a k⟨u_{α}⟩-submodule of M_{⟨u_{α}⟩} whenever X(α) is defined. This completes the proof. □

Proposition 3.2. Let M and M′ be finitely generated kE-modules. Suppose that X is an F ⟨u_{Γ}⟩-submodule of (F M)_{⟨u_{Γ}⟩} and X′ is an F ⟨u_{Γ}⟩-submodule of (F M′)_{⟨u_{Γ}⟩}, and let f : X → X′ be an F -linear transformation. Then the following statements are equivalent.

(1) f_{α} : X(α) → X′(α) is a k⟨u_{α}⟩-homomorphism whenever α ∈ V and f_{α} is defined.

(2) There is a nonempty open set U such that f_{α} : X(α) → X′(α) is a k⟨u_{α}⟩- homomorphism for all α ∈ U.

(3) f is an F ⟨u_{Γ}⟩-homomorphism.

Proof. It is easy to see that (1) implies (2), so assume that (2) holds. Let h : X → X and h′ : X′ → X′ be the F -endomorphisms given by the action of u_{γ} −1. If α ∈ U_{m}, set t_{α} = (u_{α} − 1)/α_{m}. Then Proposition 3.1 shows that h_{α} : X(α) → X(α) and h′_{α} : X′(α) → X′(α) are given by the action of t_{α} for all α ∈ U_{m} such that X(α) and X′(α) are defined. In particular, if α ∈ U_{m} ∩ U, then f_{α} is a k⟨u_{α}⟩-homomorphism so that f_{α}h_{α} = h′_{α}f_{α}. Because U_{m} ∩ U is a nonempty open set, Propositions 2.8 and 2.9 imply that fh = h′f. Hence f is an F ⟨u_{Γ}⟩-homomorphism, and (3) holds.

Finally, suppose that f is an F ⟨u_{Γ}⟩-homomorphism, and suppose that α is any element of V such that f_{α} is defined. Because V is covered by the open sets U_{i} = {α ∈ V | α_{i} ≠ 0} for 1 ≤ i ≤ r, we may assume without loss of generality that α ∈ U_{m}. If h : X → X and h′ : X′ → X′ are the F -endomorphisms given by the action of u_{γ} − 1, then fh = h′f. Moreover, h_{α} : X(α) → X(α) and h′_{α} : X′(α) → X′(α) are given by the action of t_{α} = (u_{α} − 1)/α_{m}. Then Proposition 2.8 implies that f_{α}h_{α} = h′_{α}f_{α}, so f_{α} is a k⟨u_{α}⟩-homomorphism. Thus (3) implies (1), and this completes the proof. □

For the remainder of this section we let E_{m} be the subgroup of E defined by E_{m} = ⟨g_{i} | i ≠ m⟩.

Lemma 3.3. Let X be any finitely generated F ⟨u_{Γ}⟩-module. Then there is a finitely generated k[E/E_{m}]-module M such that (F M)_{⟨u_{Γ}⟩} ≅ X.

Proof. Because ⟨u_{γ}⟩ is a cyclic p-group, Lemma 64.2 of [4] implies that there is a finitely generated k⟨u_{γ}⟩-module X′ such that F X′ ≅ X. Let ϕ : E → ⟨u_{γ}⟩ be the group homomorphism with ϕ(g_{m}) = u_{γ} and ϕ(g_{i}) = 1 for i ≠ m. Then X′ defines a kE-module M such that M = X′ as a vector space, and the action is given by g · x = ϕ(g)x for all g ∈ E and x ∈ M. In particular, M is a k[E/E_{m}]-module, and it is easy to check that (F M)_{⟨u_{γ}⟩} = F X′ as F ⟨u_{γ}⟩-modules. Thus (F M)_{⟨u_{γ}⟩} ≅ X, as desired. □

Lemma 3.4. Suppose that M and M′ are finitely generated kE-modules. Let X be an F ⟨u_{Γ}⟩-submodule of (F M)_{⟨u_{Γ}⟩}, and let X′ be an F ⟨u_{Γ}⟩-submodule of (F M′)_{⟨u_{Γ}⟩} such that X and X′ are isomorphic as F ⟨u_{Γ}⟩-modules. If there is a nonempty open set U ⊆ V such that X(α) is an indecomposable k⟨u_{α}⟩-module for all α ∈ U, then there is a nonempty open set U′ ⊆ V such that X′(α) is an indecomposable k⟨u_{α}⟩-module for all α ∈ U′.

Proof. Let f : X → X′ be an F ⟨u_{Γ}⟩-isomorphism, and suppose that X(α) is an indecomposable k⟨u_{α}⟩-module for all α ∈ U. By Corollary 2.11 and Proposition 3.2 there is a nonempty open set U′ ⊆ U such that f_{α} is a k⟨u_{α}⟩-isomorphism for all α ∈ U′. Hence X′(α) is indecomposable, as desired. □

Proposition 3.5. Let M be a finitely generated kE-module, and let X be an F ⟨u_{Γ}⟩-submodule of (F M)_{⟨u_{Γ}⟩}. Then X is indecomposable if and only if there is a nonempty open set U ⊆ V such that X(α) is an indecomposable k⟨u_{α}⟩-module for all α ∈ U.

Proof. Suppose that X is indecomposable. By Lemmas 3.3 and 3.4 we may assume that X = (F M′)_{⟨u_{Γ}⟩} for some indecomposable k[E/E_{m}]-module M′. Then U_{m} is a nonempty open set such that X(α) = M′_{⟨u_{α}⟩} is an indecomposable k⟨u_{α}⟩-module for all α ∈ U_{m}.

Conversely, suppose that X = X_{1} ⊕ X_{2} for some nonzero F ⟨u_{Γ}⟩-submodules X_{1} and X_{2} of X. Then there is a nonempty open set U_{0} ⊆ V such that X_{1}(α) and X_{2}(α) are nonzero k⟨u_{α}⟩-modules for all α ∈ U_{0}, and X(α) = X_{1}(α) ⊕ X_{2}(α). Because U_{0} is dense in V, it follows that every nonempty open set U ⊆ V contains some point α such that X(α) = X_{1}(α) ⊕ X_{2}(α) is a decomposable k⟨u_{α}⟩-module. □

We are now ready to prove the main result of this section, which shows that the direct sum decomposition of (F M)_{⟨u_{Γ}⟩} determines that of M_{⟨u_{α}⟩} for almost all α ∈ V.

Theorem 3.6. Let M be a finitely generated kE-module. Then there is a nonempty open subset U ⊆ V such that the multiplicity of the i-dimensional indecomposable F ⟨u_{Γ}⟩-module as a summand of (F M)_{⟨u_{Γ}⟩} is the same as the multiplicity of the i-dimensional indecomposable k⟨u_{α}⟩-module as a summand of M_{⟨u_{α}⟩} whenever 1 ≤ i ≤ p and α ∈ U.

Proof. For 1 ≤ i ≤ p let X_{i} be an indecomposable F ⟨u_{Γ}⟩-module of dimension i over F. By Lemma 3.3 we may assume that X_{i} = (F M_{i})_{⟨u_{Γ}⟩} for some kE-module M_{i} of dimension i over k. Then (F M)_{⟨u_{Γ}⟩} has a direct sum decomposition of the form

(F M)_{⟨u_{Γ}⟩} ≅ X_{1}^{n_{1}} ⊕ ⋯ ⊕ X_{p}^{n_{p}}

for some nonnegative integers n_{1}, …, n_{p}. Corollary 2.11 and Propositions 3.2 and 3.5 now imply that there is a nonempty open subset U ⊆ V such that if α ∈ U, then X_{i}(α) = (M_{i})_{⟨u_{α}⟩} is an indecomposable k⟨u_{α}⟩-module of dimension i, and

M_{⟨u_{α}⟩} ≅ X_{1}(α)^{n_{1}} ⊕ ⋯ ⊕ X_{p}(α)^{n_{p}}.

This completes the proof. □

Corollary 3.7. Let M be a finitely generated kE-module. Then V ⊆ V̄^{r}(M) if and only if Γ ∈ V̄^{r}(F M).

Proof. By Theorem 3.6 we know that Γ ∉ V̄^{r}(F M) if and only if there is a nonempty open subset U ⊆ V such that M_{⟨u_{α}⟩} is projective for all α ∈ U. But this condition is satisfied precisely when U ⊆ V − V̄^{r}(M). Because V − V̄^{r}(M) is itself an open subset of V, it follows that Γ ∉ V̄^{r}(M) if and only if V − V̄^{r}(M) ≠ ∅, as desired. □

4. Stratifying the Rank Variety of a Module

In this section we will obtain a number of results that are closely related to Theorem 3.6. In particular, we prove the theorem stated in the introduction, which provides a stratification of the rank variety of a finitely generated kE-module into disjoint locally closed subspaces. The final portion of the section is devoted to describing the closures of these subspaces in terms of deformations of modules for the cyclic group of order p.

For each p-tuple of nonnegative integers (n_{1}, …, n_{p}) let X(M; n_{1}, …, n_{p}) denote the set of all α ∈ ℙ_{k}^{r−1} such that the indecomposable k⟨u_{α}⟩-module of dimension i occurs with multiplicity n_{i} as a summand of M_{⟨u_{α}⟩} for 1 ≤ i ≤ p. Then it follows that ℙ_{k}^{r−1} is the disjoint union of the sets X(M; n_{1}, …, n_{p}) over all ptuples (n_{1}, …, n_{p}). Moreover, there can be only finitely many p-tuples such that X(M; n_{1}, …, n_{p}) is nonempty because such a p-tuple must satisfy

1 · n_{1} + 2n_{2} + ⋯ + pn_{p} = dim_{k} M.

The basic content of the following result is that the sets X(M; n_{1}, …, n_{p}) give a stratification of ℙ_{k}^{r−1} into disjoint locally closed subspaces. The proof is essentially a modification of Carlson’s proof that V^{r}(M) is a variety (see Theorem 4.3 of [2]).

Theorem 4.1. Let M be a finitely generated kE-module, and let (n_{1}, …, n_{p}) be a p-tuple of nonnegative integers. Then X(M; n_{1}, …, n_{p}) is a locally closed subset of ℙ_{k}^{r−1}. Moreover, if (m_{1}, …, m_{p}) is a p-tuple of nonnegative integers such that X(M; m_{1}, …, m_{p}) intersects the closure of X(M; n_{1}, …, n_{p}), then

∑_{j=i+1}^{p}(j − i)(n_{j} − m_{j}) ≥ 0

for 1 ≤ i ≤ p − 1.

Proof. We may assume that M ≠ 0. For any α ∈ k^{r} − {0} let A_{α} be the matrix representing the action of

u_{α} − 1 = ∑_{i=1}^{r} α_{i}(g_{i} − 1)

on M with respect to some fixed basis. By considering the Jordan canonical form of A_{α} + I, we see that the number of Jordan blocks of size i is the number of indecomposable summands of M_{⟨u_{α}⟩} of dimension i; moreover, one can easily check that for 1 ≤ i ≤ p the total number of indecomposable summands of M_{⟨u_{α}⟩} of dimension at least i is given by rank A_{α}^{i−1} − rank A_{α}^{i}.

Now assume that the indecomposable k⟨u_{α}⟩-module of dimension i has multiplicity n_{i} as a summand of M_{⟨u_{α}⟩} for 1 ≤ i ≤ p, and set

r_{i} = ∑_{j=i+1}^{p}(j − i)n_{j}.

It is easy to check that r_{i} = rank A_{α}^{i} for 0 ≤ i ≤ p. Thus r_{i} is the largest integer such that the determinant of some r_{i} × r_{i} minor of A_{α}^{i} is nonzero, and each of these determinants is a homogeneous polynomial in α_{1}, …, α_{r}. For 1 ≤ i ≤ p − 1 let P_{i} be the set of all polynomials given by determinants of (r_{i} + 1) × (r_{i} + 1) minors of A_{α}^{i}; let Q_{i} denote the set of all polynomials given by determinants of r_{i} × r_{i} minors of A_{α}^{i} if r_{i} > 0, and let Q_{i} = {1} if r_{i} = 0. Now let Q = {Q_{1} ⋯ Q_{p−1} | Q_{i} ∈ Q_{i} for 1 ≤ i ≤ p − 1}. Then α satisfies all the polynomial equations in P_{1}, …, P_{p−1}, but Q(α) ≠ 0 for some Q ∈ Q. Thus it follows that X(M; n_{1}, …, n_{p}) ⊆ V (P_{1}, …, P_{p−1}) − V (Q).

Conversely, suppose that β ∈ V (P_{1}, …, P_{p−1}) − V (Q). Then for 1 ≤ i ≤ p − 1 all of the (r_{i} + 1) × (r_{i} + 1) minors of A_{β}^{i} are singular, so rank A_{β}^{i} ≤ r_{i}. Moreover, there is a homogeneous polynomial Q ∈ Q such that Q(β) ≠ 0, and we can write Q = Q_{1} ⋯ Q_{p−1} with Q_{i} ∈ Q_{i} for 1 ≤ i ≤ p − 1. Then Q_{i}(β) ≠ 0, and it follows that if r_{i} > 0, then there is an r_{i} × r_{i} minor of A_{β}^{i} that is nonsingular. Hence rank A_{β}^{i} = r_{i} for 1 ≤ i ≤ p, and the total number of indecomposable summands of M_{⟨u_{β}⟩} of dimension at least i is given by

rank A_{β}^{i−1} − rank A_{β}^{i} = r_{i−1} − r_{i} = n_{i} + n_{i+1} + ⋯ + n_{p}.

Thus β ∈ X(M; n_{1}, …, n_{p}), and hence X(M; n_{1}, …, n_{p}) = V (P_{1}, …, P_{p−1}) − V (Q) is locally closed.

Now suppose that α′ ∈ X(M; m_{1}, …, m_{p}) lies in the closure of X(M; n_{1}, …, n_{p}). Then α′ ∈ V (P_{1}, …, P_{p−1}), and it follows that rank A_{α′}^{i} ≤ rank A_{α}^{i} for 1 ≤ i ≤ p−1. Hence

∑_{j=i+1}^{p}(j − i)(n_{j} − m_{j}) ≥ 0

for 1 ≤ i ≤ p − 1, as desired. □

The following result, which completes the proof of the theorem stated in the introduction, is an immediate consequence of Theorem 4.1 and the definition of the rank variety.

Corollary 4.2. Let M be a finitely generated kE-module, and let S be the set of all p-tuples of nonnegative integers (n_{1}, …, n_{p}) such that 1 · n_{1} + 2n_{2} + ⋯ + pn_{p} = dim_{k} M and at least one of n_{1}, …, n_{p−1} is nonzero. Then V̄^{r}(M) is the disjoint union of the locally closed subsets X(M; n_{1}, …, n_{p}) over all p-tuples (n_{1}, …, n_{p}) ∈ S.

The next result strengthens the conclusion of Theorem 3.6.

Corollary 4.3. Let M be a finitely generated kE-module, and let V be an irreducible subvariety of ℙ_{k}^{r−1} with homogeneous function field F and generic point Γ ∈ ℙ_{F}^{r−1}. For 1 ≤ i ≤ p let n_{i} be the multiplicity of the i-dimensional indecomposable F ⟨u_{Γ}⟩-module as a summand of (F M)_{⟨u_{Γ}⟩}. Then V ∩ X(M; n_{1}, …, n_{p}) is a nonempty open subset of V.

Proof. By Theorem 3.6 there is a nonempty open set U in V with U ⊆ V ∩ X(M; n_{1}, …, n_{p}). Because V is irreducible, it follows that the closure of V ∩ X(M; n_{1}, …, n_{p}) is all of V. Then by Theorem 4.1 the set V ∩ X(M; n_{1}, …, n_{p}) is locally closed with closure V, so it is open in V, as desired. □

If M is a finitely generated kE-module, then Theorem 4.1 gives some information about the closure of the set X(M; n_{1}, …, n_{p}). Our objective for the remainder of the section is to reinterpret this information in terms of the deformation theory studied by Donald and Flanigan [5]. We begin by recalling the relevant definitions in the special case of a module for a cyclic group C_{p} of order p.

Let ϕ : kC_{p} → End_{k}(M) be a homomorphism of k-algebras defining the structure of a finitely generated kC_{p}-module on M, and let K = k(x) be the field of rational functions over k. A generic deformation of ϕ is a homomorphism of K-algebras ϕ_{x} : KC_{p} → End_{K}(KM) of the form

ϕ_{x}(a) = ϕ(a) + xΦ_{1}(a) + x^{2}Φ_{2}(a) + ⋯,

where the maps Φ_{i} : KC_{p} → End_{K}(KM) are K-linear functions defined over k. Such a deformation ϕ_{x} defines a KC_{p}-module structure on the K-vector space KM. The resulting module is denoted by M_{x} and is said to be a generic deformation of M.

To state the next result, we note that the isomorphism type of any finitely generated kC_{p}-module M can be specified by a sequence of positive integers b_{1} ≥ b_{2} ≥ ⋯ ≥ b_{s} giving the dimensions of the indecomposable summands of M.

Lemma 4.4. Let M be a kC_{p}-module specified by b_{1} ≥ ⋯ ≥ b_{s}, and let N be a KC_{p}-module specified by c_{1} ≥ ⋯ ≥ c_{t}. Assume that dim_{k} M = dim_{K} N. Then the following statements are equivalent:

(1) M has a generic deformation isomorphic to N.

(2) t ≤ s and ∑_{i=1}^{j} c_{i} ≥ ∑_{i=1}^{j} b_{i} for 1 ≤ j ≤ t.

(3) dim_{K}(N/ soc^{i} N) ≥ dim_{k}(M/ soc^{i} M) for 1 ≤ i ≤ p − 1.

Proof. The equivalence of (1) and (2) is a special case of Theorem 19 of [5], so it suffices to show that (2) and (3) are equivalent. The proof of this result proceeds by induction on the length of the socle series of N. First suppose that soc N = N so that N is a trivial KC_{p}-module. Then (2) implies that N ≅ KM so that dim_{K}(N/ soc^{i} N) = dim_{k}(M/ soc^{i} M) for 1 ≤ i ≤ p − 1. Conversely, if dim_{K}(N/ soc^{i} N) ≥ dim_{k}(M/ soc^{i} M) for 1 ≤ i ≤ p − 1, then 0 = dim_{K}(N/ soc N) ≥ dim_{k}(M/ soc M). Hence soc M = M, and M is a trivial kC_{p}-module. It is now easy to see that (2) holds, as desired.

Now assume that soc N ≠ N. Set b_{i} = 0 for i > s and c_{i} = 0 for i > t. Let d = s − t so that

d = dim_{k}(soc M) − dim_{K}(soc N) = dim_{K}(N/ soc N) − dim_{k}(M/ soc M).

Thus d ≥ 0 if either (2) or (3) holds. Set N′ = N/ soc N and M′ = (M/ soc M)⊕k^{d} so that dim_{K} N′ = dim_{k} M′. Let b′_{1} ≥ b′_{2} ≥ ⋯ be the sequence of nonnegative integers giving the dimensions of the indecomposable summands of M′, and let c′_{1} ≥ c′_{2} ≥ ⋯ be the analogous sequence for N′.

If (3) holds, then dim_{K}(N′/ soc^{i} N′) ≥ dim_{k}(M′/ soc^{i} M′) for 1 ≤ i ≤ p − 1. Because the length of the socle series of N′ is less than that of N, it follows by induction that

∑_{i=1}^{j} c′_{i} ≥ ∑_{i=1}^{j} b′_{i}

for all j ≥ 1. If 1 ≤ j ≤ t, then c_{j} = c′_{j} + 1 and b_{j} ≤ b′_{j} + 1 so that

∑_{i=1}^{j} b_{i} ≤ ∑_{i=1}^{j} b′_{i} + j ≤ ∑_{i=1}^{j} c′_{i} + j = ∑_{i=1}^{j} c_{i},

and (2) follows.

Conversely, suppose that (2) holds. Let m = dim_{k}(soc(M/ soc M)) so that m is the number of indecomposable summands of M having dimension at least two. If 1 ≤ j ≤ m, then b_{j} = b′_{j} + 1 so that

∑_{i=1}^{j} b′_{i} = ∑_{i=1}^{j} b_{i} − j ≤ ∑_{i=1}^{j} c_{i} − j ≤ ∑_{i=1}^{j} c′_{i}.

If j > m, then 0 ≤ b′_{j} ≤ 1 and hence either c′_{j} = 0 or c′_{j} ≥ b′_{j}. But both of these possibilities imply that

∑_{i=1}^{j} b′_{i} ≤ ∑_{i=1}^{j} c′_{i}

for j > m, and it follows by induction that dim_{K}(N′/ soc^{i} N′) ≥ dim_{k}(M′/ soc^{i} M′) for 1 ≤ i ≤ p − 1. Hence dim_{K}(N/ soc^{i} N) ≥ dim_{k}(M/ soc^{i} M) for 2 ≤ i ≤ p − 1. Because d ≥ 0, this inequality also holds for i = 1. Thus (3) follows, and this completes the proof. □

Now let C_{p} be a cyclic group of order p generated by an element a, and let M be a finitely generated kE-module. If α ∈ k^{r} −{0}, then M_{⟨u_{α}⟩} becomes a kC_{p}-module with a · m = u_{α}m for all m ∈ M. Moreover, the isomorphism class of this kC_{p}- module depends only upon the equivalence class of α in ℙ_{k}^{r−1}. Thus if α ∈ ℙ_{k}^{r−1}, then M_{⟨u_{α}⟩} can be regarded as a kC_{p}-module, and we obtain the following result.

Theorem 4.5. Let M be a finitely generated kE-module, and let (m_{1}, …, m_{p}) and (n_{1}, …, n_{p}) be p-tuples of nonnegative integers such that X(M; m_{1}, …, m_{p}) intersects the closure of X(M; n_{1}, …, n_{p}). If α ∈ X(M; m_{1}, …, m_{p}) and β ∈ X(M; n_{1}, …, n_{p}), then the KC_{p}-module KM_{⟨u_{β}⟩} is a generic deformation of the kC_{p}-module M_{⟨u_{α}⟩}.

Proof. Because X(M; m_{1}, …, m_{p}) intersects the closure of X(M; n_{1}, …, n_{p}), Theorem 4.1 shows that

dim_{K}(KM_{⟨u_{β}⟩}/ soc^{i} KM_{⟨u_{β}⟩}) = ∑_{j=i+1}^{p}(j − i)n_{j}

≥ ∑_{j=i+1}^{p}(j − i)m_{j}

= dim_{k}(M_{⟨u_{α}⟩}/ soc^{i} M_{⟨u_{α}⟩})

for 1 ≤ i ≤ p−1. Thus KM_{⟨u_{β}⟩} is a generic deformation of M_{⟨u_{α}⟩} by Lemma 4.4. □

Acknowledgements

The author is grateful to D. J. Benson and J. F. Carlson for a number of helpful suggestions.

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Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA E-mail address: www@sloth.math.uga.edu