TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 5, May 1996

ON HOMOMORPHISMS FROM A FIXED REPRESENTATION TO A GENERAL REPRESENTATION OF A QUIVER

WILLIAM CRAWLEY-BOEVEY

Abstract. We study the dimension of the space of homomorphisms from a given representation X of a quiver to a general representation of dimension vector β. We prove a theorem about this number, and derive two corollaries concerning its asymptotic behaviour as β increases. These results are related to work of A. Schofield on homological epimorphisms from the path algebra to a simple artinian ring.

In this paper we prove a number of results about hom(X, β), the dimension of the space of homomorphisms from a fixed representation X of a quiver to a general representation of dimension vector β. Our basic result relates this number to the dimension of a subset of a Grassmannian of submodules of X. This result is in the spirit of A. Schofield’s paper on general representations of quivers [S2], and generalizes one of his results. We derive two corollaries concerning the asymptotic behaviour of hom(X, rβ), of interest in themselves, and also because of their connection with another theorem of Schofield, that the homological epimorphisms from a path algebra to a simple artinian ring are in 1–1 correspondence with indivisible Schur roots. (A homological epimorphism is a ring homomorphism R → S with S ⊗_{R} S ≅ S and Tor_{i}^{R}(S, S) = 0 for i > 0. This correspondence was announced by Schofield in a lecture in March 1995 in Krippen, Germany, but actually dates from 1991.) In fact our second corollary is already known, proved by Schofield and used in the proof of the correspondence, but his proof of the corollary involves difficult results about semi-invariants of quivers, whereas the proof here is quite elementary.

Let K be an algebraically closed field, Q a finite quiver with vertex set I, and let ⟨−, −⟩ be the Ringel form on ℤ^{I}. If β ∈ ℕ^{I} we write

Rep_{KQ}(β) = ∏_{a:i→j} Hom(K^{β(i)}, K^{β(j)})

for the configuration space of representations of Q of dimension vector β, and if y ∈ Rep_{KQ}(β) we write K_{y} for the corresponding left KQ-module.

If X is a finitely generated left KQ-module and y ∈ Rep_{KQ}(β) then Hom(X, K_{y}) is a finite dimensional vector space. Moreover the function y ↦ dim Hom(X, K_{y}) is an upper semicontinuous function on Rep_{KQ}(β), and it follows that the minimum value of this function, denoted hom(X, β), is also its general value. The rank of a homomorphism X → K_{y} is the dimension vector of its image. For any α ∈ ℕ^{I} the set of homomorphisms of rank at most α is a closed subset of Hom(X, K_{y}). It follows that there is a unique maximal rank γ_{X,y} of homomorphisms from X to K_{y},

\begin{matrix}c \\ ◯ \end{matrix}1996 American Mathematical Society

Received by the editors July 21, 1995.

1991 Mathematics Subject Classification. Primary 16G20; Secondary 14M15.

and that the set of homomorphisms of rank γ_{X,y} is an open subset of Hom(X, K_{y}). The function y ↦ γ_{X,y} is constant on a non-empty open subset of Rep_{KQ}(β), and its general value is denoted γ_{X,β}, the general rank of a homomorphism from X to a representation of dimension β. (The general rank was introduced by Schofield [S2, Lemma 5.1] in case X is a general representation of some dimension vector. His argument adapts to prove that the function y ↦ γ_{X,y} is constant on a non-empty open subset, as claimed above, using the setup of our §4 in case X is an infinite dimensional finitely generated module.)

Our basic result is as follows. We hasten to say that its interest lies in its corollaries.

Theorem. If X is a finite dimensional KQ-module and β ∈ ℕ^{I} then

hom(X, β) = ⟨γ_{X,β}, β⟩ + dim U

for some non-empty open subset U of Gr_{KQ}(X, γ_{X,β}).

Here, if α is a dimension vector, then Gr_{KQ}(X, α) is the variety of submodules of X of codimension α (i.e. submodules Y ⊆ X with X/Y of dimension vector α). In fact the theorem holds more generally for X a finitely generated module. The main problem is to make sense of Gr_{KQ}(X, α). We discuss this in §4.

The theorem generalizes Theorem 5.2 of Schofield [S2] since for a sufficiently general representation X of dimension α the variety Gr_{KQ}(X, γ_{X,β}) has all irreducible components of dimension ⟨α − γ_{X,β}, γ_{X,β}⟩ (see the last line of [S2, p52]), so any non-empty open subset of this variety also has this dimension.

Let K_{0}(KQ) be the Grothendieck group of finitely presented left KQ-modules modulo short exact sequences (or equivalently of finitely generated projective left KQ-modules modulo direct sums). If β ∈ ℤ^{I} there is a homomorphism K_{0}(KQ) → ℤ, also denoted β, defined by β(KQe_{i}) = β(i) where e_{i} is the idempotent in the path algebra KQ corresponding to the vertex i. If X is finite dimensional then in fact β(X) = ⟨\underline{dim} X, β⟩. A finitely presented KQ-module X is said to be βsemistable if β(X) = 0, and β(Y) ≥ 0 for all finitely presented submodules Y of X. It is β-stable if in addition β(Y) > 0 whenever Y ≠ 0, X.

In §2 we prove the following corollaries in the special case when X is finite dimensional. We deduce the general case in §3 using a universal localization argument suggested by A. Schofield.

Corollary 1. If X is a finitely presented KQ-module and β ∈ ℕ^{I} then

lim_{r→∞} \frac{1}{r} hom(X, rβ) = max\{β(X/Y) | Y ⊆ X a finitely presented submodule\}.

If β is an indivisible Schur root then there is a corresponding homological epimorphism from KQ to a simple artinian ring given by Schofield’s theorem, and this has an associated Sylvester module rank function [S1, Theorem 7.12]. It can be constructed as follows: if χ_{β} is the function which sends a finitely presented module X to the number given by either side of the equation in Corollary 1, then the Sylvester module rank function is obtained by normalising χ_{β} so as to have value 1 at KQ.

Corollary 2. If X is a finitely presented KQ-module which is β-semistable with β ∈ ℕ^{I} then hom(X, rβ) = 0 for some integer r > 0.

As an example, taking KQ to be a free algebra, Corollary 2 states that if A is an n × n matrix with entries in K⟨x_{1}, …, x_{m}⟩ and which is full, meaning that it cannot be written as a product of an n × (n − 1) and an (n − 1) × n matrix, then the x_{i} can be specialized to r × r matrices over K in such a way that A becomes an invertible (rn) × (rn) matrix.

Some of these results were presented in lectures I gave at the Krippen meeting mentioned above, and I would like to thank the organizers, D. Happel, C. M. Ringel, and L. Unger. I would also particularly like to thank A. Schofield.

1. Proof of the theorem

Let X be a finite dimensional left KQ-module and let α, β ∈ ℕ^{I}. If e_{i} ∈ KQ is the idempotent corresponding to a vertex i ∈ I, then X has a decomposition X = ⨁_{i} X_{i} where X_{i} = e_{i}X. If V is a finite dimensional vector space we write Gr(V, d) for the Grassmannian of subspaces of codimension d in V. The variety Gr_{KQ}(X, α) of submodules of X of codimension α is the closed subset of

∏_{i} Gr(X_{i}, α(i))

consisting of the tuples (Y_{i}) which are subrepresentations of X. We define a vector space Hom(X, β) via

Hom(X, β) = ⨁_{i} Hom_{K}(X_{i}, K^{β(i)}).

We write Hom(X, β)_{α} for the locally closed subset consisting of those θ ∈ Hom(X, β) with each θ_{i} of rank α(i). It is standard that the map

ρ : Hom(X, β)_{α} ⟶ ∏_{i} Gr(X_{i}, α(i))

sending θ to (Ker θ_{i}) is a locally trivial bundle with fibre

∏_{i} Inj(K^{α(i)}, K^{β(i)}),

where Inj(U, V) is the set of injective linear maps U → V. Let Hom_{KQ}(X, β)_{α} be the subset of Hom(X, β)_{α} consisting of those θ with (Ker θ_{i}) a subrepresentation of X. It is a closed subset of Hom(X, β)_{α} since

Hom_{KQ}(X, β)_{α} = ρ^{−1}(Gr_{KQ}(X, β)).

The following lemma is immediate.

Lemma 1.1. The map κ : Hom_{KQ}(X, β)_{α} → Gr_{KQ}(X, α) sending θ to (Ker θ_{i}) is a locally trivial bundle with fibre ∏_{i} Inj(K^{α(i)}, K^{β(i)}).

We write RepHom_{KQ}(X, β) for the closed subset of Rep_{KQ}(β) × Hom(X, β) consisting of those pairs (y, θ) with θ a KQ-homomorphism from X to K_{y}. Similarly we define RepHom_{KQ}(X, β)_{α}.

Lemma 1.2. If α ≤ β then the map ϕ : RepHom_{KQ}(X, β)_{α} → Hom_{KQ}(X, β)_{α} sending (y, θ) to θ is a locally trivial bundle with fibre

∏_{a:i→j} Hom(K^{β(i)−α(i)}, K^{β(j)}).

Proof. Hom_{KQ}(X, β)_{α} has a covering by open subsets of the form

U_{ϵπ} = {θ ∈ Hom_{KQ}(X, β)_{α} | ∏_{i} det(π_{i}θ_{i}ϵ_{i}) ≠ 0}

where ϵ_{i} : K^{α(i)} → X are injective linear maps and π_{i} : K^{β(i)} → K^{α(i)} are surjective linear maps. We show that

ϕ^{−1}(U_{ϵπ}) ⟶ U_{ϵπ}

is a trivial bundle. Observe that the map sending θ to (π_{i}θ_{i}ϵ_{i})^{−1} is a regular map of varieties from U_{ϵπ} to the set of β(i) × β(i) matrices. One can choose a complement to Ker(π_{i}) in K^{β(i)}, and using this one can find maps

μ_{i} : K^{β(i)−α(i)} → K^{β(i)} and λ_{i} : K^{β(i)} → K^{β(i)−α(i)}

with λ_{i}μ_{i} the identity on K^{β(i)−α(i)} and Im(μ_{i}) = Ker(π_{i}). If a : i → j is an arrow, write x_{a} for the linear map X_{i} → X_{j} induced by multiplication by a. We show that the constructions

(1) z_{a} = y_{a}μ_{i}, and

(2) y_{a} = z_{a}λ_{i} + (θ_{j}x_{a} − z_{a}λ_{i}θ_{i})ϵ_{i}(π_{i}θ_{i}ϵ_{i})^{−1}π_{i} define a 1–1 correspondence between

(y, θ) ∈ ϕ^{−1}(U_{ϵπ}) and (θ, z) ∈ U_{ϵπ} × ∏_{a:i→j} Hom(K^{β(i)−α(i)}, K^{β(j)}).

This gives an isomorphism of varieties over U_{ϵπ}, so proves that ϕ^{−1}(U_{ϵπ}) → U_{ϵπ} is a trivial bundle, and the lemma follows.

First, given (θ, z), the element (y, θ) constructed according to (2) belongs to RepHom_{KQ}(X, β), and hence to ϕ^{−1}(U_{ϵπ}). To see this, we need to prove that y_{a}θ_{i} and θ_{j}x_{a} are equal. We have a decomposition X_{i} = Im ϵ_{i} ⊕ Ker θ_{i} since θ_{i} has rank α(i). Now y_{a}θ_{i} and θ_{j}x_{a} both vanish on Ker θ_{i}, for the fact that θ ∈ Hom_{KQ}(X, β) means that the spaces Ker θ_{i} form a subrepresentation of X, so x_{a}(Ker θ_{i}) ⊆ Ker θ_{j}. Also y_{a}θ_{i} and θ_{j}x_{a} agree on Im ϵ_{i} since they have the same composition with ϵ_{i}.

Next if (θ, z) is given, one constructs (y, θ) using (2) and (θ, z′) using (1), then

z′_{a} = z_{a}λ_{i}μ_{i} + (θ_{j}x_{a} − z_{a}λ_{i}θ_{i})ϵ_{i}(π_{i}θ_{i}ϵ_{i})^{−1}π_{i}μ_{i} = z_{a}.

Finally if (y, θ) is given, one constructs (θ, z) using (1) and (y′, θ) using (2), then the fact that θ_{j}x_{a} = y_{a}θ_{i} implies that

y′_{a} = (y_{a}μ_{i})λ_{i} + (y_{a}θ_{i} − (y_{a}μ_{i})λ_{i}θ_{i})ϵ_{i}(π_{i}θ_{i}ϵ_{i})^{−1}π_{i}
= y_{a}(μ_{i}λ_{i} + ξ_{i} − μ_{i}λ_{i}ξ_{i})

where ξ_{i} is the idempotent endomorphism θ_{i}ϵ_{i}(π_{i}θ_{i}ϵ_{i})^{−1}π_{i} of K^{β(i)}. Now ξ_{i}μ_{i} = 0 so K^{β(i)} = Im ξ_{i} ⊕ Im μ_{i} by dimensions, and by checking compositions with ξ_{i} and μ_{i} we obtain

μ_{i}λ_{i} + ξ_{i} − μ_{i}λ_{i}ξ_{i} = 1_{K^{β(i)}}.

Thus y′_{a} = y_{a}.

Lemma 1.3. Suppose that θ : Y → Z is a locally trivial map of varieties with fibre F, an irreducible variety. If U is an open subset of Y, then θ(U) is open in Z and dim θ(U) = dim U − dim F.

Proof. We may assume that θ is trivial, so Y = Z × F and θ is the projection. Then

θ(U) = ⋃_{f∈F} i_{f}^{−1}(U)

where i_{f} is the map Z → Z × F, z ↦ (z, f), so θ(U) is open in Z. Now every fibre of the map U → θ(U) is a non-empty open subset of F, so has dimension dim F. The claim follows.

Proof of the theorem. Let α = γ_{X,β}. There is a non-empty open subset R of Rep_{KQ}(β) with dim Hom(X, K_{y}) = hom(X, β) and γ_{X,y} = α for all y ∈ R. Let

V = {(y, θ) ∈ RepHom_{KQ}(X, β)_{α} | y ∈ R},

an open subset of RepHom_{KQ}(X, β)_{α}. Now all fibres of the projection V → R have dimension hom(X, β), so

dim V = dim R + hom(X, β) = dim Rep_{KQ}(β) + hom(X, β).

Applying Lemma 1.3 to the locally trivial map of Lemma 1.2 and the open subset V we have

dim ϕ(V) = dim V − dim ∏_{a:i→j} Hom(K^{β(i)−α(i)}, K^{β(j)}).

Now applying Lemma 1.3 to the locally trivial map of Lemma 1.1 and the open subset ϕ(V) we have

dim κ(ϕ(V)) = dim ϕ(V) − dim ∏_{i} Inj(K^{α(i)}, K^{β(i)}).

Combining these formulae we see that κ(ϕ(V)) is an open subset of Gr_{KQ}(X, α) of dimension

hom(X, β) + ∑_{a:i→j} β(i)β(j) − ∑_{a:i→j}(β(i) − α(i))β(j) − ∑_{i} α(i)β(i)

This simplifies to hom(X, β) − ⟨α, β⟩, which proves the theorem.

2. Proofs of the corollaries in the finite dimensional case

Proof of Corollary 1 (for X finite dimensional). For any vectors ρ, σ ∈ ℕ^{I} we have hom(X, ρ + σ) ≤ hom(X, ρ) + hom(X, σ). Now if \frac{1}{d} hom(X, dβ) = λ and we write r = ad + b with a, b ∈ ℕ and b < d, then

\frac{1}{r} hom(X, rβ) ≤ \frac{1}{ad} hom(X, (ad + b)β)

≤ \frac{1}{ad}(a hom(X, dβ) + b hom(X, β))

≤ λ + \frac{1}{a} hom(X, β)

and this tends to λ as r → ∞. It follows that the sequence \frac{1}{r} hom(X, rβ) converges to its infimum μ.

If Y ⊆ X then hom(X, rβ) = dim Hom(X, M) for some M of dimension rβ, so

\frac{1}{r} hom(X, rβ) = \frac{1}{r} dim Hom(X, M)

≥ \frac{1}{r} dim Hom(X/Y, M)

≥ \frac{1}{r} dim Hom(X/Y, M) − \frac{1}{r} dim Ext(X/Y, M)

= \frac{1}{r}⟨\underline{dim} X/Y, rβ⟩ = ⟨\underline{dim} X/Y, β⟩ = β(X/Y).

Thus μ ≥ max{β(X/Y) | Y is a submodule of X}. As r increases the vector γ_{X,rβ} is bounded by the dimension vector of X, so some dimension vector α arises infinitely often; say for r = r_{1}, r_{2}, …. By assumption there are maps from X to representations of dimension r_{i}β of rank α, so there must be a subrepresentation Y of X of codimension α. By the theorem we have

\frac{1}{r_{i}} hom(X, r_{i}β) ≤ ⟨α, β⟩ + \frac{1}{r_{i}} dim Gr_{KQ}(X, α).

Thus μ ≤ ⟨α, β⟩ = ⟨\underline{dim} X/Y, β⟩ = β(X/Y). This proves the corollary.

Proof of Corollary 2 (for X finite dimensional). Any β-semistable representation has a filtration with β-stable quotients, so we may assume that X is β-stable. By Corollary 1 we have lim_{r→∞} \frac{1}{r} hom(X, rβ) = 0.

As in the previous corollary let α = γ_{X,rβ} for r = r_{1}, r_{2}, … and let Y be a submodule of X of codimension α. By the theorem

\frac{1}{r_{i}} hom(X, r_{i}β) = ⟨α, β⟩ + \frac{1}{r_{i}} dim U_{i}

with U_{i} open in Gr_{KQ}(X, α). Letting r_{i} → ∞ we must have ⟨α, β⟩ = 0. Thus β(Y) = 0, which contradicts the β-stability of X unless α = 0 or α = \underline{dim} X. In either case Gr_{KQ}(X, α) is reduced to a point, so that dim U_{i} = 0, and hence hom(X, r_{i}β) = 0, as required.

3. Passage to finitely presented modules

In this section we extend the corollaries from the case when X is finite dimensional to X being finitely presented (in case the quiver has oriented cycles, so that there are infinite dimensional finitely presented modules). The idea of using universal localization here was suggested to the author by A. Schofield.

We define a new quiver Q′ as follows. If I = {1, …, n} then Q′ has vertex set I′ = {1′, …, n′, 1′′, …, n′′}. For each vertex i ∈ I there is an arrow v_{i} : i′ → i′′ in Q′, and for each arrow a : i → j in Q there is an arrow a′ : i′ → j′′ in Q′. Observe that the vertices 1′, …, n′ are sources while 1′′, …, n′′ are sinks. Thus KQ′ is finite dimensional.

Let A = KQ′ and B = M_{2}(KQ). There is an equivalence KQ-Mod → B-Mod sending a KQ-module M to the vector space X = M^{2} with B acting as a matrix on a column vector. In what follows we usually make no distinction between a KQ-module and the corresponding B-module. (This is perfectly acceptable for categorical notions like Hom and Ext, but care is needed for traditional notions like the elements of a module.)

Let ϕ : A → B be the homomorphism defined by

ϕ(e_{i′}) = (\begin{matrix}e_{i} & 0 \\ 0 & 0 \end{matrix}) , ϕ(e_{i′′}) = (\begin{matrix}0 & 0 \\ 0 & e_{i} \end{matrix}) , ϕ(v_{i}) = (\begin{matrix}0 & 0 \\ e_{i} & 0 \end{matrix}) , ϕ(a′) = (\begin{matrix}0 & 0 \\ a & 0 \end{matrix}) .

We write B ⊗ − for B ⊗_{A} −, and we write _{A}X for the restriction of a B-module via ϕ. Restriction by ϕ sends a representation M of Q to the representation Z of Q′ with Z_{i′} = Z_{i′′} = M_{i}, with v_{i} the identity map Z_{i′} → Z_{i′′}, and with the linear map corresponding to an arrow a′ the same as the linear map for M corresponding to a. Thus restriction by ϕ induces an equivalence from B-Mod to the full subcategory of KQ′-Mod consisting of those representations in which the v_{i} are isomorphisms. This observation implies that ϕ : A → B is the universal localization of KQ′ with respect to the maps KQ′e_{i′′} → KQ′e_{i′} corresponding to the v_{i}. The next lemma collects some properties of universal localization, see [S1, Theorems 4.5, 4.7, 4.8].

Lemma 3.1. (1) Any finitely presented B-module is induced from a finitely presented (hence finite dimensional) A-module.

(2) Hom_{B}(X, Y) ≅ Hom_{A}(X, Y) and Ext_{B}^{1}(X, Y) ≅ Ext_{A}^{1}(X, Y) for any Bmodules X and Y.

(3) B ⊗_{A} X ≅ X and Tor_{1}^{A}(B, X) = 0 for any B-module X.

If β ∈ ℤ^{I} we define β′ ∈ ℤ^{I′} by β′(i′) = β′(i′′) = β(i). Clearly if X is a B-module of dimension vector β then _{A}X has dimension vector β′.

Lemma 3.2. If Z is a finite dimensional A-module and β ∈ ℕ^{I}, then (1) hom(B ⊗ Z, β) = hom(Z, β′), and

(2) β(B ⊗_{A} Z) ≥ β′(Z), with equality if Tor_{1}^{A}(B, Z) = 0.

Proof. (1) Since Hom_{A}(Z,_{A} Y) ≅ Hom_{B}(B ⊗ Z, Y) we certainly have the relation hom(Z, β′) ≤ hom(B ⊗ Z, β). On the other hand, in a general representation of Q′ of dimension β′ the arrows v_{i} are isomorphisms, so the representation is the restriction of a B-module. Thus there is a B-module Y of dimension β with

hom(Z, β′) = dim Hom_{A}(Z,_{A} Y)
= dim Hom_{B}(B ⊗ Z, Y) ≥ hom(B ⊗ Z, β).

(2) If Z is projective we have equality β(B ⊗_{A} Z) = β′(Z) since

B ⊗ Ae_{i′} ≅ B ⊗ Ae_{i′′}

corresponds to the KQ′-module KQ′e_{i}. In general if 0 → P′ → P → Z → 0 is a projective resolution then there is an exact sequence

0 → Tor_{1}^{A}(B, Z) → B ⊗ P′ → B ⊗ P → B ⊗ Z → 0,

and the kernel Tor_{1}^{A}(B, Z) must be a projective B-module since it embeds in B⊗P′. Thus β(Tor_{1}^{A}(B, Z)) ≥ 0, and the result follows by additivity.

Lemma 3.3. If Z is an A-module and \overline{Z} is the image of Z under the natural map Z → B ⊗ Z, then B ⊗ Z ≅ B ⊗ \overline{Z} and Tor_{1}^{A}(B, \overline{Z}) = 0.

Proof. Tensoring the maps Z ↠ \overline{Z} ↪ B ⊗ Z with B we have

B ⊗ Z ↠ B ⊗ \overline{Z} → B ⊗ (B ⊗ Z) ≅ B ⊗ Z

so B ⊗ Z ≅ B ⊗ \overline{Z}. Also Tor_{1}^{A}(B, \overline{Z}) ↪ Tor_{1}^{A}(B, B ⊗ Z) since A is hereditary, and this last space is zero by Lemma 3.1.

Proof of Corollary 1 (for X finitely presented). As in the finite dimensional case \frac{1}{r} hom(X, rβ) converges to its infimum μ and

μ ≥ max{β(X/Y) | Y a finitely presented submodule of X}.

It only remains to find a finitely presented submodule Y of X with β(X/Y) ≥ μ. Now X is induced from a finite dimensional A-module W. By the finite dimensional case of Corollary 1 we have

inf\{\frac{1}{r} hom(W, rβ′) | r ≥ 1\} = β′(Z)

for some quotient Z of W. By Lemma 3.2 this becomes

inf\{\frac{1}{r} hom(X, rβ) | r ≥ 1\} ≤ β(B ⊗ Z)

and since B⊗Z is isomorphic to the quotient of X by a finitely presented submodule (using that path algebras are left coherent), we have the result.

Proof of Corollary 2 (for X finitely presented). By Lemma 3.1, the module X is induced from a finite dimensional A-module Z, and by Lemma 3.3 we may suppose that Tor_{1}^{A}(B, Z) = 0. By assumption X is β-semistable. Now β′(Z) = β(X) = 0 by Lemma 3.2, and if Z has a quotient W with β′(W) > 0 then X has a quotient B ⊗ W with β(B ⊗ W) > 0, contradicting the fact that X is β-semistable. Thus Z is β′-semistable. By the finite dimensional case of Corollary 2 there is some r > 0 with hom(Z, rβ′) = 0, and then hom(X, rβ) = 0, as required.

4. Extension of the theorem to finitely generated modules

In this section we extend the theorem to the case when X is a finitely generated module (assuming that Q has oriented cycles, so that not all finitely generated modules are finite dimensional). In order even to state the theorem, we need to define a variety Gr_{KQ}(X, α) of submodules of X of codimension α. It is convenient to use the functorial language of Demazure and Gabriel [DG] in which a scheme (over K) is a certain type of functor from commutative K-algebras to sets. Recall that a module for a commutative ring has rank r if it is finitely generated and at each prime it is free of rank r. It is then automatically projective by [B, II, §5, no. 2]. If X is a finitely generated KQ-module and α is a dimension vector, we write Gr_{KQ}(X, α) for the functor sending a commutative K-algebra R to the set of R⊗_{K} KQ-submodules Y of R⊗_{K} X with the property that for each vertex i the Rmodule e_{i}((R ⊗_{K} X)/Y) has rank α(i). We prove that Gr_{KQ}(X, α) is a separated algebraic scheme. Its reduced induced structure is then the variety Gr_{KQ}(X, α).

Theorem. If X is a finitely generated KQ-module and β ∈ ℕ^{I} then

hom(X, β) = ⟨γ_{X,β}, β⟩ + dim U

for some non-empty open subset U of Gr_{KQ}(X, γ_{X,β}).

In order to prove the theorem (and indeed to prove that Gr_{KQ}(X, α) is an algebraic scheme), we construct schemes RepHom_{KQ}(X, β)_{α} and Hom_{KQ}(X, β)_{α} and others corresponding to constructions in §1. The theorem then follows by the same argument as §1. We skip the details. (Note that an algebraic scheme has the same dimension as its reduced induced structure.)

We work in slightly greater generality. Let K be any field, let V be a vector space over K, possibly infinite dimensional, let A be a finitely generated K-algebra, and let X be a finitely generated A-module. Fix a complete set of (not necessarily primitive) orthogonal idempotents e_{i} (i ∈ I, a finite set), so e_{i}e_{j} = δ_{ij}e_{i} and ∑_{i∈I} e_{i} = 1. Any A-module M has a decomposition M = ⨁_{i} M_{i} where M_{i} = e_{i}M. We define functors as follows:

• Gr(V, r) is the functor which sends a commutative K-algebra R to the set of R-submodules W of R ⊗_{K} V with (R ⊗_{K} V)/W of rank r.

• Hom(V, m) is the functor sending R to Hom_{R}(R ⊗_{K} V, R^{m}).

• Hom(V, m)_{r} is the subfunctor of Hom(V, m) sending R to the set of θ with Im(θ) a direct summand of R^{m} of rank r.

• Gr_{A}(X, α) is the subfunctor of ∏_{i} Gr(X_{i}, α(i)) consisting of those tuples Y_{i} ⊆ R ⊗_{K} X_{i} with ⨁_{i} Y_{i} an R ⊗_{K} A-submodule Y of R ⊗_{K} X.

• Hom_{A}(X, β)_{α} is the subfunctor of ∏_{i} Hom(X_{i}, β(i))_{α(i)} sending R to the set of tuples (θ_{i}) with ⨁_{i}(Ker θ_{i}) an R ⊗_{K} A-submodule of R ⊗_{K} X.

• Rep_{A}(β) sends R to the set of R⊗_{K} A-module structures M on the R-module ⨁_{i} R^{β(i)}, with e_{i} acting as the projection onto R^{β(i)}.

• RepHom_{A}(X, β) is the subfunctor of Rep_{A}(β) × ∏_{i} Hom(X_{i}, β(i)) sending R to the pairs (M, (θ_{i})) with the θ_{i} defining an R⊗_{K}A-module homomorphism R ⊗_{K} X → M.

• RepHom_{A}(X, β)_{α} is the subfunctor of RepHom_{A}(X, β) defined by also demanding that θ_{i} ∈ Hom(X_{i}, β(i))_{α(i)}(R).

The first lemma is presumably already known, but I couldn’t find a suitable reference. Of course it is all standard in case V is finite dimensional.

Lemma 4.1. (1) Hom(V, m) is an affine scheme.

(2) Gr(V, r) is a separated scheme.

(3) Hom(V, m)_{r} is a locally closed subscheme of Hom(V, m).

(4) The natural transformation κ : Hom(V, m)_{r} → Gr(V, r), which sends a homomorphism R ⊗_{K} V → R^{m} to its kernel, is a locally trivial bundle with fibre Hom(K^{r}, m)_{r}.

Proof. (1) If V has basis v_{λ} (λ ∈ Λ) then

Hom_{R}(R ⊗_{K} V, R^{m}) ≅ Hom_{K-alg}(K[x_{λ,i} | λ ∈ Λ, 1 ≤ i ≤ m], R).

(2) We show that G = Gr(V, r) is a local functor by modifying the argument of [DG, I, §1, 3.13]. Suppose that R is a commutative K-algebra, (f_{i}, x_{i}) is a partition of unity, and Y_{i} ∈ G(R_{f_{i}}) are compatible in the sense that Y_{i} and Y_{j} have the same image in G(R_{f_{i}f_{j}}) for all i and j. We need to show that there is some Y ∈ G(R) inducing Y_{i} in each G(R_{f_{i}}). Let B = ∏_{i} R_{f_{i}}. By the argument of [DG], the Bmodule ∏_{i} Y_{i} is induced from an R-submodule Y of R ⊗_{K} V. It remains to show that the R-module M = (R⊗_{K} V)/Y has rank r. Now the induced module B ⊗_{R} M is isomorphic to ∏_{i}(R_{f_{i}} ⊗_{K} V)/Y_{i}, so it has rank r, and it follows that M is finitely generated projective by [B, I, §3, no. 6, Proposition 12]. By faithful flatness any prime 𝔮 of R comes from a prime 𝔭 of B, and B_{𝔭} ⊗_{R_{𝔮}} M_{𝔮} ≅ (B ⊗_{R} M)_{𝔭} is a free B_{𝔭}-module of rank r, so M_{𝔮} is free of rank r. Thus M does have rank r.

Next we find a covering of G by affine open subfunctors. Let Q be an rdimensional subspace of V, and define

U_{Q}(R) = {Y ∈ G(R) | Y ⊕ (R ⊗_{K} Q) = R ⊗_{K} V }.

The argument of [DG, I, §1, Example 3.9] shows that U_{Q} is an open subfunctor of G. Let P be a complement to Q in V, and let p_{λ} (λ ∈ Λ) be a basis of P. The submodules Y of R ⊗_{K} V complementary to R ⊗_{K} Q are in 1–1 correspondence with homomorphisms from R ⊗_{K} P to R ⊗_{K} Q, so

U_{Q}(R) ≅ Hom_{R}(R ⊗_{K} P, R ⊗_{K} Q)
≅ Hom_{K}(P, R ⊗_{K} Q)
≅ Hom_{K-alg}(K[x_{λ,i} | λ ∈ Λ, 1 ≤ i ≤ r], R).

Thus U_{Q} is an affine scheme.

For any extension field F of K and any Y ∈ G(F) there is some Q such that Y ∈ U_{Q}(F). Thus the U_{Q} cover G. It follows that G is a scheme.

Suppose R is a commutative K-algebra and Y, Y′ ∈ G(R). If ϕ : R → S is a homomorphism of commutative K-algebras then S ⊗_{R} Y ⊆ S ⊗_{R} Y′ if and only if ϕ(ψ(Y′+y)) = 0 for all y ∈ Y and ψ : (R⊗_{K} V)/Y′ → R. Thus S⊗_{R}Y = S⊗_{R}Y′ if and only if ϕ(I) = 0 where I is the ideal of R generated by the elements ψ(Y′+y) and ψ′(Y + y′) with y ∈ Y, y′ ∈ Y′, ψ : (R ⊗_{K} V)/Y′ → R and ψ′ : (R ⊗_{K} V)/Y → R. This implies that G is separated.

(3) Let θ : R ⊗_{K} V → R^{m} be a homomorphism and write I_{d} for the ideal in R generated by all d × d minors of θ (after choosing a basis of V). If ϕ : R → S is a homomorphism of commutative K-algebras then S ⊗_{R} Im(θ) is a summand of S^{m} of rank r if and only if S ⊗_{R} Coker(θ) has rank m − r. By the theory of Fitting ideals (see for example [E, §20.2], but this only deals with noetherian rings) this holds if and only if Sϕ(I_{r}) = S and ϕ(I_{r−1}) = 0. The claim follows.

(4) κ^{−1}(U_{Q})(R) = {θ ∈ Hom(V, m)_{r}(R) | Ker(θ) ⊕ (R ⊗_{K} Q) = R ⊗_{K} V } and the map to Hom(Q, m)_{r}(R) × Gr(V, r)(R) sending θ to (θ|_{R⊗Q}, Ker θ) is a bijection. Thus κ^{−1}(U_{Q}) → U_{Q} is a trivial bundle with fibre Hom(Q, m)_{r}.

Lemma 4.2. Rep_{A}(β) and RepHom_{A}(X, β) are affine algebraic schemes.

Proof. For Rep_{A}(β) this is fairly standard, see for example [P, Chapter 4]. To study RepHom_{A}(X, β), write X as a quotient of A^{k}, for then RepHom_{A}(X, β) is a closed subfunctor of RepHom_{A}(A^{k}, β). Now a homomorphism from R ⊗_{K} A^{k} to an R⊗_{K} A-module M is given by a k-tuple of elements of M, so RepHom_{A}(A^{k}, β) is isomorphic to the product of Rep_{A}(β) with k ×∑_{i} β(i)-dimensional affine space. Thus it is an affine algebraic scheme.

Lemma 4.3. Gr_{A}(X, α), Hom_{A}(X, β)_{α} and RepHom_{A}(X, β)_{α} are separated algebraic schemes and the natural transformation Hom_{A}(X, β)_{α} → Gr_{A}(X, α) is a locally trivial bundle with fibre ∏_{i} Hom(K^{α(i)}, β(i))_{α(i)}.

Proof. RepHom_{A}(X, β)_{α} is a locally closed subscheme of RepHom_{A}(X, β), so it is separated and algebraic.

Let R be a commutative K-algebra and let Y_{i} ∈ Gr(X_{i}, α(i))(R). Write Y = ⨁_{i} Y_{i}. If ϕ : R → S is a homomorphism of commutative K-algebras, then S ⊗_{R} Y is an S ⊗_{K} A-submodule of S ⊗_{K} X if and only if ϕ(ψ(Y + ay)) = 0 for all y ∈ Y, all a ∈ A, and all ψ : (R ⊗_{K} X)/Y → R. This implies that Gr_{A}(X, α) is a closed subscheme of ∏_{i} Gr(X_{i}, α(i)).

Now Hom_{A}(X, β)_{α} is the pullback of Gr_{A}(X, α) along

∏_{i} Hom(X_{i}, β(i))_{α(i)} ⟶ ∏_{i} Gr(X_{i}, α(i)),

so it is a closed subscheme of the product ∏_{i} Hom(X_{i}, β(i))_{α(i)} and also the natural transformation Hom_{A}(X, β)_{α} → Gr_{A}(X, α) is a locally trivial bundle with fibre ∏_{i} Hom(K^{α(i)}, β(i))_{α(i)}.

Now we have RepHom_{A}(X, α)_{α} ≅ Hom_{A}(X, α)_{α} since if θ_{i} : R ⊗_{K} X_{i} → R^{α(i)} are surjective maps and ⨁_{i} Ker θ_{i} is an R ⊗_{K} A-submodule of R ⊗_{K} X, then there is a unique R ⊗_{K} A-module structure on ⨁_{i} R^{α(i)} such that (θ_{i}) is an R ⊗_{K} Amodule homomorphism R ⊗_{K} X → ⨁_{i} R^{α(i)}. Thus Hom_{A}(X, α)_{α} is algebraic, so the bundle Hom_{A}(X, α)_{α} → Gr_{A}(X, α) implies that Gr_{A}(X, α) is algebraic. Now for general β the bundle Hom_{A}(X, β)_{α} → Gr_{A}(X, α) implies that Hom_{A}(X, β)_{α} is algebraic.

References

[B] N. Bourbaki, Algèbre Commutative, Hermann, Paris, 1961–83.

[DG] M. Demazure and P. Gabriel, Groupes Algébriques, Tome 1, Masson, Paris, 1970. MR 46:1800

[E] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer-Verlag, New York, 1995. CMP 95:10

[P] C. Procesi, Rings with polynomial identities, Marcel Dekker, New York, 1973. MR 51:3214 [S1] A. Schofield, Representations of rings over skew fields, London Math. Soc. Lec. Note Ser.

92, Cambridge Univ. Press, 1985. MR 87c:16001

[S2] A. Schofield, General representations of quivers, Proc. London Math. Soc. 65 (1992), 46–64. MR 93d:16014

Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England

   E-mail address: w.crawley-boevey@leeds.ac.uk