CLASSIFYING REPRESENTATIONS WITH

SQUAREFREE TOPS BY WAY OF GRASSMANNIANS

Birge Huisgen-Zimmermann

Dedicated to the memory of Sheila Brenner

Abstract. Let Λ be a finite dimensional algebra. It is proved that existence of a coarse moduli space classifying, up to isomorphism, the representations of Λ with fixed dimension d and fixed squarefree top T is equivalent to existence of a fine moduli space for this classification problem. Accessible criteria are given to characterize this event, and in the positive case

– unexpectedly frequent in light of the stringency of fine classification – the (necessarily projective) classifying variety is explicitly described as a subvariety of a Grassmannian, which can be computed from quiver and relations of Λ. Even when the moduli problem fails to be solvable, this latter variety is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained.

1. Introduction and terminology

While the full representation-theoretic picture of a finite dimensional algebra Λ of wild representation type is beyond the scope of a complete meaningful description, in many wild cases, substantial slices of this representation theory do lie within the reach of adapted classification methods. The slice we address here consists of the representations M of a fixed dimension d with fixed squarefree top T = M/JM, where J is the Jacobson radical of Λ; in other words, we require that the simple (left) Λ-modules occur with multiplicity at most 1 in T. Our primary goal here is to decide when the restricted classification problem for isomorphism classes of d-dimensional representations with top T has a coarse or fine moduli space. In rough terms, this means that one seeks to bijectively parametrize these isomorphism classes by the points of an algebraic variety such that the structure constants ‘evolve smoothly’ as one moves along the parameter space – this idea is made precise in the concept of a family – and so that other smooth parametrizations are induced by the distinguished one – the latter requirement is made precise via universal properties of varying degrees of stringency. (This classification philosophy actually predates the rigorous definition of coarse or fine moduli spaces given by Mumford in the 1960’s; it already underlies Riemann’s classification of nonsingular projective curves in the 1850’s, where the term ‘moduli’ was coined, standing for a ‘structure-determining collection of continuous parameters in ℂ’.)

Typeset by AMS-TEX

This research was partially supported by a grant from the National Science Foundation.

We prove that, in our situation, the two moduli problems are equivalent and give accessible necessary and sufficient conditions for the existence of solutions. In the positive case, we explicitly exhibit the classifying variety for the d-dimensional modules with top T. In doing so, we aim at conditions and constructions which are not only of theoretical interest, but verifiable and implementable, respectively, in case Λ is presented by quiver and relations. As an application, we derive criteria for Λ to have ‘finite local representation type with respect to a simple module T ’, i.e., to have only finitely many representations (of arbitrary dimension) with top T.

The theoretical solution to the moduli problem (Section 4) can be described in the following simple terms: Fixing d and a semisimple module T with projective cover P, we denote by 𝔊𝔯𝔞𝔰𝔰_{d}^{T} that subvariety of the classical Grassmannian of all (dim P − d)-dimensional subspaces of JP which consists of the Λ-submodules of JP having the correct dimension. Clearly, 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is a projective variety endowed with a natural morphic Aut_{Λ}(P)-action, the orbits of which coincide with the fibres of the natural surjection

ϕ : 𝔊𝔯𝔞𝔰𝔰_{d}^{T} ⟶ {isomorphism types of d-dimensional modules with top T }

sending C to the class of P/C.

From now on, we assume that T is squarefree; without this hypothesis, our assertions will fail in general. Adopting the obvious notion of a ‘family of Λ-modules’ (cf. [10] and the last paragraph of Section 1), we then obtain the equivalence of the following statements (see Theorems 4.2 and 4.4 for stronger results and Section 2.A for further terminology):

(1) There exists a coarse moduli space classifying the d-dimensional Λ-modules with top T, up to isomorphism.

(2) There exists a fine moduli space classifying the d-dimensional Λ-modules with top T, up to isomorphism.

(3) No d-dimensional module with top T has a proper degeneration with top T.

(4) The classical quasi-affine variety of d-dimensional Λ-modules with top T has a geometric quotient modulo its GL_{d}-action.

(5) All Aut_{Λ}(P)-orbits of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} are singletons.

(6) Every submodule C ⊆ JP of codimension d in P is fully invariant under endomorphisms of P.

(7) 𝔊𝔯𝔞𝔰𝔰_{d}^{T} \underline{is} the fine moduli space for the d-dimensional Λ-modules with top T.

In fact, the equivalence of (3), (5), (6) holds ‘pointwise’; in particular, a module M with top T, say M = P/C, is devoid of proper top-stable degenerations precisely when C is fully invariant in P. Invariance is a strong requirement, but nonetheless satisfied in a wide range of interesting cases – see Corollary 4.5 for illustration – as is also attested to by the fact that arbitrary irreducible projective varieties arise as irreducible components of varieties 𝔊𝔯𝔞𝔰𝔰_{d}^{T} subject to conditions (1) – (7). (For the sake of contrast: If d ≥ 2, the full collection of d-dimensional modules has a coarse moduli space only when Λ is semisimple.) We note that, in some special cases, the existence of a fine moduli space can alternatively be deduced from work of King [10]; see the comments following Theorem 4.4.

It may be somewhat surprising that, at least when Λ is given through quiver and relations, the variety 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is accessible, in the sense that one can compute polynomials defining it (the algorithm implicit in our proof has polynomial complexity in d). A key feature of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} in this connection – it is the principal bonus afforded by squarefreeness of T – is the existence of a finite affine cover consisting of Aut_{Λ}(P)-stable subvarieties of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}. Even though in general not open, this cover is an isomorphism invariant of Λ, up to permutation and birational equivalence of the components.

In situations where the d-dimensional Λ-modules with fixed top fail to have a moduli space, the natural next step is to subdivide the target class by further discrete invariants. There is an obvious refinement of our primary partition by tops: It amounts to focusing on the modules M having a prescribed sequence of radical layers 𝕊 = 𝕊(M) = (J^{i}M/J^{i+1}M)_{0≤i≤L} with top layer T = M/JM; here L is chosen so that J^{L+1} = 0. In other words, we fix the matrix recording the multiplicities of the simple composition factors of M in a format that keeps track of their ‘layer-locations’. The corresponding locally closed subvariety of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is labeled 𝔊𝔯𝔞𝔰𝔰(𝕊). In contrast to the strong condition that all orbits of the projective variety 𝔊𝔯𝔞𝔰𝔰_{d}^{T} be closed, closedness of the Aut_{Λ}(P)-orbits inside the quasi-projective variety 𝔊𝔯𝔞𝔰𝔰(𝕊) is a comparatively mild requirement and no longer guarantees existence of a moduli space (for the representations with radical layering 𝕊 in this case). On the other hand, it turns out that the Aut_{Λ}(P)-space 𝔊𝔯𝔞𝔰𝔰(𝕊) is hardly less useful than a coarse moduli space in the situation of closed orbits.

In fact, in Sections 3 and 4, we give a first set of reasons for viewing 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, and a fortiori 𝔊𝔯𝔞𝔰𝔰(𝕊), as good approximations to moduli spaces for the corresponding classification problems in general; for further backing of this viewpoint, we refer to [8]. To name one reason, the fibres of the representation map are well-understood in terms of their intrinsic structure: If T is simple, the Aut_{Λ}(P)-orbits are full affine spaces 𝔸^{m}, where m is bounded above by the multiplicity of T in JP. For general squarefree T, each orbit Aut_{Λ}(P).C is a direct product of a full affine space 𝔸^{m} and a torus (K^{∗})^{r}, where dim_{K} T − r is the number of indecomposable summands of P/C. The main benefit of restricting one’s focus to 𝔊𝔯𝔞𝔰𝔰(𝕊) lies in the fact that the distinguished Aut_{Λ}(P)-stable affine cover of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is a union of open covers of the individual subvarieties 𝔊𝔯𝔞𝔰𝔰(𝕊), which in turn forces the Aut_{Λ}(P)-orbits of each 𝔊𝔯𝔞𝔰𝔰(𝕊) to be closed in 𝔊𝔯𝔞𝔰𝔰(𝕊) when T is simple, and ‘almost closed’ in general – see Section 4 for more precision.

Our characterization of the algebras that have finite local representation type with respect to a simple module T (Section 5) lends additional support to the emphasis we place on radical layerings. In fact, finite local representation type makes 𝕊(M) a classifying invariant for the modules M with top T – that is, forces 𝕊(−) to separate isomorphism classes. Moreover, the algorithm implied by the theory permits to effectively list the modules with top T in this event.

In terms of methodology, one can summarize the assets of the Aut_{Λ}(P)-spaces 𝔊𝔯𝔞𝔰𝔰_{d}^{T} as follows: They are complete varieties with readily computable Aut_{Λ}(P)-stable affine covers; the orbit boundaries are accessible by way of these coordinatized covers (a fact which is further elaborated in [8]); the orbits have comparatively low dimensions and simplistic structures; the acting group Aut_{Λ}(P) has a large unipotent radical in all interesting cases. Loosely speaking, this last fact places the Grassmannian varieties of representations with top T on the ‘opposite end’ of the spectrum from their traditional counterparts, which are quasi-affine and endowed with reductive group actions. In view of the bridge between the Grassmannian and classical scenarios, presented in Section 2.C (it was previously established by Bongartz and the author), the different tools available to extract representation-theoretic information from these two settings thus complement each other. An illustration for this interplay is the proof of Theorem 5.2.

Throughout, Λ will be a basic finite dimensional algebra over an algebraically closed field K. So, without loss of generality, we may assume that Λ = KQ/I, where Q is a quiver, and I an admissible ideal in the path algebra KQ. The quiver provides us with a convenient set of primitive idempotents e_{1}, …, e_{n} of Λ, which are in bijective correspondence with the vertices of Q; we will, in fact, not distiguish between the vertices and the e_{i}. As is well-known, the factors S_{i} = Λe_{i}/Je_{i} form a set of representatives of the simple (left) Λ-modules. By L we denote the largest integer for which the power J^{L} of the Jacobson radical does not vanish; in other words, L + 1 is the Loewy length of Λ.

Given any (left) Λ-module M, an element x ∈ M will be called a top element of M if x ∉ JM and x is normed by some e_{i}, that is x = e_{i}x for some i ∈ {1, …, n}. The isomorphism invariant

𝕊(M) = (M/JM, JM/J^{2}M, …, J^{L−1}M/J^{L}M, J^{L}M)

of M will be referred to as the sequence of radical layers of M, or, more briefly, the radical layering of M.

Moreover, we observe the following conventions: The product pq of two paths p and q in KQ stands for ‘first q, then p’ if end(q) = start(p), and zero otherwise (so, in particular, p = pe_{i} means that the path p starts in the vertex e_{i}). In line with this notation, we call a path p_{1} a right subpath of p if p = p_{2}p_{1} for some path p_{2}. We will generally gloss over the distinction between the left Λ-structure of M ∈ Λ-mod and the induced left KQ-structure; in particular, we let paths operate on Λ-modules without using residue notation.

For background on moduli problems, we refer to [12], but recall the definition of a fine moduli space for our specific problem. The appropriate concept of a family of Λ-modules was already introduced by King in [10]: Namely, a family of d-dimensional Λ-modules parametrized by an algebraic variety X is a pair (Δ, δ), where Δ is a (geometric) vector bundle of rank d over X and δ : Λ → End(Δ) a K-algebra homomorphism. Our notion of equivalence of families parametrized by the same variety X, finer than King’s in general, is the coarsest possible to separate isomorphism classes: Namely, (Δ_{1}, δ_{1}) ∼ (Δ_{2}, δ_{2}) precisely when, for each x ∈ X, the fibre of Δ_{1} over x is Λ-isomorphic to the fibre of Δ_{2} over x. As is common, given a family (Δ, δ) parametrized by X and a morphism τ : Y → X of varieties, the induced family τ^{∗}(Δ, δ) over Y is the pullback of (Δ, δ) along τ. In this context, a variety X is a fine moduli space for (families of) d-dimensional modules with top T if there exists a family (Γ, γ) of such modules parametrized by X which has the property that an arbitrary family – parametrized by Y say – is equivalent to a family τ^{∗}(Γ, γ) induced via a unique morphism τ : Y → X. In particular, this entails that every d-dimensional module with top T is isomorphic to precisely one fibre of the bundle Γ. For a definition of a coarse moduli space, see [12, pp. 23,24].

2. The varieties Top_{d}^{T} and 𝔊𝔯𝔞𝔰𝔰_{d}^{T} for general T

We fix d ≥ 1 and a finitely generated semisimple module T, say T = ⨁_{1≤i≤n}(Λe_{i}/Je_{i})^{t_{i}}. Moreover, we let P = ⨁_{1≤i≤n}(Λe_{i})^{t_{i}} be the projective cover of T. In the first three subsections, we discuss the classical and Grassmannian varieties associated to the d-dimensional modules with top T, along with connections between them.

2.A. The classical setup.

If we denote by Q^{∗} the union of the set {e_{1}, …, e_{n}} of vertices with the set of arrows of Q, then Q^{∗} generates Λ as a K-algebra, and the traditional variety of d-dimensional left Λ-modules, Mod_{d}^{Λ}, takes on the form

Mod_{d}^{Λ} = {(x_{α})_{α∈Q^{∗}} ∈ ∏_{α∈Q^{∗}} End_{K}(K^{d}) | the x_{α} satisfy all the rel’ns of the α in Λ}.

This affine algebraic variety carries a morphic GL_{d}-action by conjugation accounting for change-of-basis transformations. Clearly, the fibres of the representation map

R : Mod_{d}^{Λ} ⟶ {isomorphism types of d-dimensional Λ-modules}

are precisely the GL_{d}-orbits of Mod_{d}^{Λ}. The slice of Mod_{d}^{Λ} which will be pivotal for our investigation is the locally closed subvariety

Top_{d}^{T} = {x ∈ Mod_{d}^{Λ} | R(x)/JR(x) ≅ T }.

Clearly, Top_{d}^{T} is stable under the GL_{d}-action.

We start by adapting a criterion for the existence of a coarse moduli space to our situation. On this occasion, we recall two concepts of quotient of an algebraic variety modulo an algebraic group action: Suppose that X is a variety endowed with a morphic action of an algebraic group G. Then a categorical quotient of X by G is a morphism of varieties π : X → Z which is constant on the G-orbits of X and satisfies the following universal property: Every morphism ψ : X → Y that is constant on the orbits factors uniquely through π. Moreover, a morphism π : X → Z having fibres coinciding with the G-orbits is called a geometric quotient of X by G if it is surjective and open, and satisfies the following condition relating the structure sheaf of Z to the rings of invariants for the G-action: for every open subset U of Z, the comorphism π^{∗} induces an isomorphism from the ring O_{Z}(U) of regular functions on U to the ring O_{X}(π^{−1}(U))^{G} of G-invariant regular functions on π^{−1}(U). Every geometric quotient of X by G is a categorical quotient and hence unique in case of existence. It is, moreover, well-known that, in case X is affine and G reductive (e.g., in case X = Mod_{d}^{Λ} and G = GL_{d}), a categorical quotient of X by G always exists; this quotient is geometric precisely when all G-orbits of X are closed. Sufficiency of this latter condition does not extend to quasi-affine varieties, however – for counter-examples based on subvarieties of Mod_{d}^{Λ}, see Section 4.

Criterion 2.1. There exists a coarse moduli space for the (families of isomorphism classes of) d-dimensional modules with top T if and only if the variety Top_{d}^{T} has a categorical quotient π : Top_{d}^{T} → Top_{d}^{T} / GL_{d} which is an orbit map; the latter means that the map π is surjective and that its fibres coincide with the GL_{d}-orbits of Top_{d}^{T}. In the positive case, this quotient Top_{d}^{T} / GL_{d} is the coarse moduli space we are looking for.

Proof. By [12, Proposition 2.13], it suffices to specify a family (Υ, ε) of Λ-modules, parametrized by Top_{d}^{T}, which has the local universal property relative to families of d-dimensional modules with top T. Local universality means that, for an arbitrary family (Δ, δ) – parametrized by some variety Y say – and for any point y ∈ Y, there exists a neighborhood U of y such that Δ|_{U} is induced from (Υ, ε) by way of some (not necessarily unique) morphism U → Top_{d}^{T}. The obvious candidate for (Υ, ε) is as follows: Take Υ to be the trivial bundle of rank d over Top_{d}^{T}, and define ε : Λ → End(Υ) by the requirement that, for α ∈ Q^{∗}, the endomorphism ε(α) of Υ coincides with x_{α} on the fibre above x. To verify the local universal property, it obviously suffices to show that every family (Δ, δ) based on a trivial bundle Δ = U × K^{d} is induced from (Υ, ε). To do so, we let τ : U → Top_{d}^{T} be the morphism which sends any element y in U to the point x ∈ Top_{d}^{T} with the property that, for each α ∈ Q^{∗}, the endomorphism x_{α} equals the restriction of δ(α) to the fibre above y. Then (Δ, δ) is equivalent to τ^{∗}(Υ, ε) as required. □

In fact, the criterion extends to arbitrary GL_{d}-stable, locally closed subsets W of Top_{d}^{T} as follows: Given any such W, the isomorphism classes R(x), x ∈ W, possess a coarse moduli space if and only if the GL_{d}-space W has a categorical quotient which is an orbit map.

A necessary condition for the existence of a coarse moduli space for the d-dimensional representations with top T is thus immediate: Namely, all GL_{d}-orbits of Top_{d}^{T} need to be closed (in standard jargon, this amounts to excluding the ‘jump phenomenon’). In representation-theoretic terms, this condition says that no d-dimensional module M with top T has a proper degeneration sharing that top.

Recall that a degeneration of a d-dimensional module R(x) is a module R(y) with the property that y belongs to the closure of the orbit GL_{d} .x in Mod_{d}^{Λ}. One writes R(x) ≤_{deg} R(y) in that case to reflect the fact that ‘degenerates to’ is a partial order on the isomorphism classes of d-dimensional modules. For background on degenerations, we point to [1, 3, 11, 13, 15]. In view of Criterion 2.1, our present investigation of moduli spaces automatically involves existence questions for degenerations. As by-products, we will thus obtain some preliminary results on what we call ‘top-stable’ and ‘layer-stable’ degenerations. (This thread will be picked up in [8].)

Definition 2.2. A degeneration M′ of a finitely generated left Λ-module M is called top-stable if M/JM ≅ M′/JM′, layer-stable in case J^{l}M/J^{l+1}M ≅ J^{l}M′/J^{l+1}M′ for all l.

Clearly, the layer-stable degenerations of M are a fortiori top-stable. Moreover, it is obvious that the top-stable degenerations of a module M = R(x) with x ∈ Top_{d}^{T} are precisely those R(y) for which y belongs to the relative closure of GL_{d} .x in Top_{d}^{T}. Analogously, if 𝕊 = 𝕊(M), then the layer-stable degenerations of M coincide with the R(y) where y traces the relative closure of GL_{d} .x within the subvariety

Layer(𝕊) = {x ∈ Top_{d}^{T} | R(x) has radical layering 𝕊}.

2.B. The Grassmannian setup.

Keep the notation introduced at the beginning of Section 2, and set m = dim_{K} P − d. We define 𝔊𝔯𝔞𝔰𝔰_{d}^{T} to be the closed subvariety of the Grassmannian of m-dimensional subspaces of JP consisting of those subspaces that are Λ-submodules of JP. In other words, 𝔊𝔯𝔞𝔰𝔰_{d}^{T} contains precisely those Λ-submodules C of JP for which P/C is a d-dimensional module (automatically having top T). Clearly, 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is a closed subvariety of the usual Grassmannian of m-dimensional subspaces of JP and, as such, is in turn projective. Note moreover that the variety 𝔊𝔯𝔞𝔰𝔰_{d}^{T} comes coupled with a surjective map

ϕ = ϕ_{d}^{T} : 𝔊𝔯𝔞𝔰𝔰_{d}^{T} ⟶ {isom. types of d-diml. modules with top T }, C ↦ [P/C].

We will refer to this map as the representation map of the Grassmannian of d-dimensional modules with top T. As already pointed out, the fibres of ϕ coincide with the orbits of the natural morphic action of Aut_{Λ}(P) on 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, denoted by f.C. The fact that the Aut_{Λ}(P)-orbits of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} will turn out to be structurally simpler than the GL_{d}-orbits of Top_{d}^{T} in many cases, hinges on the following elementary fact.

Observation 2.3. Aut_{Λ}(P) ≅ Aut_{Λ}^{u}(P) ⋊ Aut_{Λ}(T), where

Aut_{Λ}^{u}(P) = {id +f_{0} | f_{0} ∈ Hom_{Λ}(P, JP)}

is the unipotent radical of Aut_{Λ}(P). The automorphism group Aut_{Λ}(T) of the top T of P is isomorphic to ∏_{1≤i≤n} M_{t_{i}}(K), a torus if all the t_{i} ≤ 1.

The isomorphism Aut_{Λ}(P) ≅ Aut_{Λ}^{u}(P) ⋊ Aut_{Λ}(T) can be replaced by equality if, for instance, we identify Aut_{Λ}(T) with the subgroup H ≤ Aut_{Λ}(P) consisting of those Λautomorphisms of P which leave the subspace ⨁_{1≤i≤n}(Ke_{i})^{t_{i}} of P = ⨁_{1≤i≤n}(Λe_{i})^{t_{i}} invariant. □

We use hereditary algebras for two simple illustrations: First let Λ = KQ, where Q is

the quiver \xymatrix{2 & 1 \ar[l] \ar[l] \ar[l]} . If T = S_{1}, then 𝔊𝔯𝔞𝔰𝔰_{2}^{T} ≅ 𝔊𝔯𝔞𝔰𝔰_{3}^{T} ≅ ℙ^{2}.

Now suppose that Λ = KQ, where Q is the quiver \xymatrix{2 & 1 \ar[l] \ar[l] \ar[l] \ar[r] \ar[r] \ar[r] & 3} , and again let T = S_{1}. Then 𝔊𝔯𝔞𝔰𝔰_{4}^{T} consists of four irreducible components (which coincide with the connected components), two of them copies of ℙ^{2} × ℙ^{2} and the other two singletons. In all of these instances, the Aut_{Λ}(P)-orbits are reduced to points, that is, the representation map ϕ is a bijection from 𝔊𝔯𝔞𝔰𝔰_{d}^{T} to the isomorphism types of d-dimensional modules with top T.

Yet, in the following example, it is not.

Example 2.4. Let Λ = KQ/⟨ω^{2}⟩, where Q is the quiver

[figure: 1; ω; α; 2]

Let T = S_{1}, and d = 3. Then 𝔊𝔯𝔞𝔰𝔰_{d}^{T} ≅ ℙ^{1}, but there are only two isomorphism classes of d-dimensional modules with top T, namely Λe_{1}/Λ(α − kαω) for k ∈ K and Λe_{1}/Λαω. □

In general, it is difficult to determine the structure of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} ‘at one blow’. This is one of the reasons for subdividing 𝔊𝔯𝔞𝔰𝔰_{d}^{T} into more accessible affine portions, still stable under the Aut_{Λ}(P)-action. Provided that they are nicely located within 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, one can then piece together the representation-theoretic information encoded in 𝔊𝔯𝔞𝔰𝔰_{d}^{T} from the collection of affine frames. Such a subdivision will be introduced in two stages (Sections 2.D and 3.B).

2.C. Connection between Top_{d}^{T} and 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

Geometric information on the orbits and orbit closures of the Aut_{Λ}(P)-action of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} smoothly translates into information about the orbits and orbit closures of the GL_{d}-action of the classical variety Top_{d}^{T}, as spelled out in the following theorem. It is due to Bongartz and the author [5, Proposition C] and was previously applied towards an investigation of uniserial modules, which triggered many of the ideas developed here.

By the term ‘subvariety’ we always mean ‘locally closed subset’.

Proposition 2.5. Consider the map from the set of all Aut_{Λ}(P)-orbits of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} to the set of GL_{d}-orbits of Top_{d}^{T}, sending any Aut_{Λ}(P)-orbit Aut_{Λ}(P).C = ϕ^{−1}(P/C) to the GL_{d}- orbit R^{−1}(P/C). This map extends to a one-to-one inclusion-respecting correspondence between the set of Aut_{Λ}(P)-stable subsets of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} on one hand, and the set of GL_{d}- stable subsets of Top_{d}^{T} on the other, which preserves openness, closures, irreducibility, connectedness, and types of singularities.

Furthermore: Suppose that V is an Aut_{Λ}(P)-stable subvariety of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} and W the corresponding GL_{d}-stable subvariety of Top_{d}^{T}. Then:

(a) V has a geometric quotient by its Aut_{Λ}(P)-action if and only if W has a geometric quotient by its GL_{d}-action, and in the positive case, these quotients coincide.

(b) V has a categorical quotient modulo Aut_{Λ}(P) if and only if W has a categorical quotient modulo GL_{d}. In the positive case, these quotients are isomorphic. The fibres of the former are equal to the Aut_{Λ}(P)-orbits of V if and only if the fibres of the latter are equal to the GL_{d}-orbits of W.

Proof. Only the statements under (b) were not addressed in [5]. They are also immediate consequences of Lemmas 1,2 preceding [5, Proposition C]. Indeed, a variety X endowed with an action of an algebraic group H having the following properties is presented there: H contains Aut_{Λ}(P) and GL_{d} as subgroups meeting only in 1, as well as two closed normal subgroups N_{1} and N_{2} such that H/N_{1} ≅ Aut_{Λ}(P) and H/N_{2} ≅ GL_{d} canonically. Moreover, [5] exhibits morphisms ρ : X → 𝔊𝔯𝔞𝔰𝔰_{d}^{T} and σ : X → Top_{d}^{T} with ϕ ∘ ρ = R ∘ σ having the following properties: ρ is an Aut_{Λ}(P)-invariant geometric quotient of X by N_{1}, and σ a GL_{d}-invariant geometric quotient of X by N_{2}. It is straightforward to deduce (b). □

The final assertion of Proposition 2.5 yields a twin version of Criterion 2.1 which rephrases existence of a coarse moduli space for the d-dimensional modules with top T in terms of the Aut_{Λ}(P)-space 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

Criterion 2.6. Let V be any Aut_{Λ}(P)-stable subvariety of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}. Then there exists a coarse moduli space for the representations P/C with C ∈ V if and only if the Aut_{Λ}(P)- space V has a categorical quotient which is an orbit map. In case of existence, such a quotient V/ Aut_{Λ}(P) is the moduli space we are looking for. □

We also glean an alternate characterization of top-stable degenerations from Proposition 2.5. Observe that, whenever ϕ(C) = P/C ≅ R(x), the correspondence of the theorem pairs the Aut_{Λ}(P)-orbits contained in the closure of Aut_{Λ}(P).C in 𝔊𝔯𝔞𝔰𝔰_{d}^{T} with the GL_{d}-orbits contained in the closure of GL_{d} .x in Top_{d}^{T}. In other words, the top-stable degenerations of P/C are precisely the factors P/C′ with C′ ∈ \overline{Aut_{Λ}(P).C}.

2.D. The partition of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} in terms of radical layers.

In translating geometric features of the variety 𝔊𝔯𝔞𝔰𝔰_{d}^{T} into representation-theoretic data, the partition of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} introduced in this section is pivotal. Again, we let the top T equal ⨁_{1≤i≤n}(Λe_{i}/Je_{i})^{t_{i}} and let P be its projective cover; moreover, we abbreviate the dimension ∑_{i} t_{i} of T by t. Further, we will systematically identify isomorphic semisimple modules in the sequel.

Definitions 2.7. (i) A d-dimensional semisimple sequence with top T is a sequence

𝕊 = (𝕊_{0}, 𝕊_{1}, …, 𝕊_{L})

with the following properties: 𝕊_{0} = T and each 𝕊_{i} is a submodule of J^{i}P/J^{i+1}P, such that ∑_{i} dim 𝕊_{i} = d.

(ii) Given a d-dimensional semisimple sequence 𝕊 = (𝕊_{0}, …, 𝕊_{L}) with top T, we set

𝔊𝔯𝔞𝔰𝔰(𝕊) = {C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T} | 𝕊(P/C) = 𝕊},

and call the (L + 1)-tuple of integers

\underline{dim} 𝕊 = (dim_{K} 𝕊_{0}, …, dim_{K} 𝕊_{L})

the dimension vector of 𝕊.

Since we are identifying isomorphic semisimple modules, there are only finitely many d-dimensional semisimple sequences. The important examples are the radical layerings of the d-dimensional modules with top T; indeed, whenever C ∈ Top_{d}^{T}, the sequence 𝕊(P/C)

– see end of Section 1 – meets our criteria.

We start by recording the ‘layer-stable analogue’ of the final observation of 2.C. Namely – again by Proposition 2.5 – the layer-stable degenerations of a module P/C with C ∈ 𝔊𝔯𝔞𝔰𝔰(𝕊) are precisely the modules represented by the points in the relative closure of Aut_{Λ}(P).C in 𝔊𝔯𝔞𝔰𝔰(𝕊).

It is, moreover, obvious that each of the sets 𝔊𝔯𝔞𝔰𝔰(𝕊) is stable under the Aut_{Λ}(P)- action and that 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is the disjoint union of the 𝔊𝔯𝔞𝔰𝔰(𝕊), where 𝕊 ranges through the d-dimensional semisimple sequences with top T. We point out that, in general, this partition falls slightly short of being a stratification in the technical sense: While the sets 𝔊𝔯𝔞𝔰𝔰(𝕊) are locally closed subvarieties of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} – see part (ii) of the upcoming lemma – their closures are not always unions of 𝔊𝔯𝔞𝔰𝔰(𝕊′)’s. However, this lack is compensated for by the following asset (see part (i) of the lemma): Endow the (L+1)-tuples of non-negative integers with the lexicographic order, and let d ∈ ℕ_{0}^{L+1}. Then the union ⋃_{\underline{dim} 𝕊>d} 𝔊𝔯𝔞𝔰𝔰(𝕊), where 𝕊 runs through the d-dimensional semisimple sequences with top T, is closed in 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

We can do a little better than that by introducing the following partial order on the semisimple sequences themselves. Namely, for two d-dimensional semisimple sequences 𝕊 = (𝕊_{0}, …, 𝕊_{L}) and 𝕊′ = (𝕊′_{0}, …, 𝕊′_{L}), we define

𝕊 ≤ 𝕊′

if and only if ⨁_{0≤i≤L} 𝕊_{i} = ⨁_{0≤i≤L} 𝕊′_{i} and either 𝕊 = 𝕊′ or else 𝕊_{j} is a proper direct summand of 𝕊′_{j} for the smallest index j with 𝕊_{j} ≠ 𝕊′_{j}. Clearly, this partial order is compatible with the order on the dimension vectors, in the sense that 𝕊 ≤ 𝕊′ implies \underline{dim} 𝕊 ≤ \underline{dim} 𝕊′.

Lemma 2.8.

(i) Let 𝕊 be any d-dimensional semisimple sequence with top T. Then ⋃_{𝕊′≤𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′) is open in 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, as is the union ⋃_{𝕊′<𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′). Consequently , the sets ⋃_{\underline{dim} 𝕊≥d} 𝔊𝔯𝔞𝔰𝔰(𝕊) and ⋃_{\underline{dim} 𝕊>d} 𝔊𝔯𝔞𝔰𝔰(𝕊) are closed for any (L + 1)-tuple d of non-negative integers.

(ii) Each of the sets 𝔊𝔯𝔞𝔰𝔰(𝕊) is locally closed in 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

(iii) If M is a d-dimensional module with top T and M′ a top-stable degeneration of M, then \underline{dim} 𝕊(M) ≤ \underline{dim} 𝕊(M′).

Proof. We fix a sequence z_{1}, …, z_{r} of top elements of P, where each z_{r} is normed by a primitive idempotent, e(r) say, such that P = ⨁_{1≤r≤t} Λz_{r}.

(i) Let us call a set B of paths in KQ admissible if |B| = d. Moreover, given a ddimensional semisimple sequence 𝕊, let us call an admissible set B compatible with 𝕊 in case, for each l ∈ {0, …, L} and each i ∈ {1, …, n}, the number of paths of length l in B ending in e_{i} equals the multiplicity of the corresponding simple module S_{i} in 𝕊_{l}. For every admissible set B, finally, we let Aux(B) be the set of those points C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T} which have the property that P/C has basis ⋃_{1≤r≤t}{pz_{r} + C | p ∈ B, p starts in e(r)}.

Each Aux(B) is open in 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, since the condition

C ⊕ (∑_{r≤t} ∑_{\begin{matrix}p∈B \\ length(p)>0 \end{matrix}} Kpz_{r}) = JP

is open in the Grassmannian of m-dimensional subspaces of JP.

To verify openness of ⋃_{𝕊′<𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′), it now suffices to notice that ⋃_{𝕊′<𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′) is the union of the sets Aux(B), with B any admissible set compatible with some 𝕊′ which is strictly smaller than 𝕊. The inclusion of the latter union in the former follows from the fact that, whenever B is compatible with 𝕊′ and C ∈ Aux(B), the module P/C has radical layering 𝕊(P/C) ≤ 𝕊′; the other inclusion is obvious. Analogously, ⋃_{𝕊′≤𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′) is seen to be open.

(ii) This follows from (i) and the fact that 𝔊𝔯𝔞𝔰𝔰(𝕊) is the intersection of ⋃_{𝕊′≤𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′) with ⋃_{𝕊′≮𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′).

(iii) It suffices to observe that the relative closure of 𝔊𝔯𝔞𝔰𝔰(𝕊(M)) in 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is contained in ⋃_{\underline{dim} 𝕊≥\underline{dim} 𝕊(M)} 𝔊𝔯𝔞𝔰𝔰(𝕊). □

2.E. General results on the Aut_{Λ}(P)-orbits.

The following proposition summarizes the information on the Aut_{Λ}(P)-orbits of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} – alias the fibres of the representation map ϕ – available without any restriction on T. Strengthened versions for the squarefree case will come to bear in Section 4.

Again, we let T = ⨁_{1≤i≤n}(Λe_{i}/Je_{i})^{t_{i}}, set t = ∑_{i} t_{i}, and fix a sequence z_{1}, …, z_{r} of top elements of P, each z_{r} being normed by a primitive idempotent e(r), such that P = ⨁_{1≤r≤t} Λz_{r}. If H is the subgroup of Aut_{Λ}(P) consisting of all automorphisms which leave the subspace ⨁_{1≤r≤t} Kz_{i} invariant, and G the unipotent radical of Aut_{Λ}(P), then Observation 2.3 tells us that G is normal in Aut_{Λ}(P) and Aut_{Λ}(P) is the semidirect product G ⋊ H. The incarnation of the maximal torus of H afforded by our choice of top elements of P is T = (K^{∗})^{t}, where we identify (a_{1}, …, a_{t}) ∈ T with the automorphism of P given by z_{i} ↦ a_{i}z_{i}. Finally, given h ∈ H and C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, we denote by h.G.C the set {hg.C | g ∈ G}. Due to the normality of G in Aut_{Λ}(P), the set h.G.C equals the G-orbit of h.C.

Proposition 2.9. Let C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

(1) The orbit 𝔊𝔯𝔞𝔰𝔰_{d}^{T} .C has dimension

dim_{K} End_{Λ}(P) − dim_{K} Hom_{Λ}(P, C) − dim_{K} End_{Λ}(P/C),

and this dimension is generically constant on the irreducible components of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}. (2) The G-orbit G.C in 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is isomorphic to 𝔸^{m} with

m = m(P/C) = dim_{K} Hom_{Λ}(P, JP/C) − dim_{K} Hom_{Λ}(P/C, JP/C).

(3) There exists a point C′ ∈ Aut_{Λ}(P).C such that the T -orbit T .C′ is isomorphic to the torus (K^{∗})^{t−𝔰}, where 𝔰 = 𝔰(P/C) is the number of indecomposable summands of P/C.

(4) The full orbit Aut_{Λ}(P).C is the disjoint union of the subvarieties h.G.C ≅ 𝔸^{m(P/C)}, for h ∈ H, and H acts as a transitive permutation group on these subvarieties.

Before proving the proposition, we single out a lemma for repeated reference.

Lemma 2.10. For any C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, the orbit map Aut_{Λ}(P) → Aut_{Λ}(P).C, f ↦ f.C, is separable, and hence Aut_{Λ}(P).C is isomorphic to the homogeneous Aut_{Λ}(P)-space Aut_{Λ}(P)/ Stab_{Aut_{Λ}(P)} C. Moreover, the dimensions of the stabilizers of C in Aut_{Λ}(P), resp. in G, are:

dim Stab_{Aut_{Λ}(P)} C = dim_{K} Hom_{Λ}(P, C) + dim_{K} End_{Λ}(P/C)  and
dim Stab_{G} C = dim_{K} Hom_{Λ}(P, C) + dim_{K} Hom_{Λ}(P/C, JP/C).

Proof. We first address the separability claim. Combining [6, Proposition 6.7] with [11, AI.5.5, Satz 2], we see that it suffices to check reducedness of the schematic fibre of the orbit map, i.e., reducedness of the stabilizer subgroup Stab_{Aut_{Λ}(P)} C of C. Reducedness of this stabilizer, in turn, can be deduced from the fact that it arises as the solution set of a system of linear equations over K; ascertaining the latter fact is a matter of routine, if notationally cumbersome.

For the first dimension formula, consider the homomorphism ρ : Stab_{Aut_{Λ}(P)} C → Aut_{Λ}(P/C) of algebraic groups sending any map f in the stabilizer of C to the induced automorphism of P/C. Clearly, ρ is onto and the kernel is the subgroup of Aut_{Λ}(P) consisting of the automorphisms of the form id + g_{0} with g_{0} ∈ Hom_{Λ}(P, C). This proves the first equality.

As for the second, consider the restriction of ρ to Stab_{G} C, and notice that the kernel is the same as before, while ρ(Stab_{G} C) equals the unipotent radical of Aut_{Λ}(P/C), namely the subgroup consisting of the automorphisms of the form id + g_{1} with g_{1} ∈ Hom_{Λ}(P/C, JP/C). □

Proof of Proposition 2.9. (1) Since dim Aut_{Λ}(P).C = dim Aut_{Λ}(P) − dim Stab_{Aut_{Λ}(P)} C, the claimed equality can be derived from Lemma 2.10. For the generic behavior of the fibre dimension see, e.g., [11, II.2.6].

(2) In view of Lemma 2.10, G.C is isomorphic to the homogeneous space G/ Stab_{G} C, whence the claim concerning the fibre structure follows from Rosenlicht’s Theorem[14]. The assertion concerning the fibre dimension is a consequence of the equality dim G = dim_{K} Hom_{Λ}(P, JP) and Lemma 2.10.

(3) Clearly, T .C ≅ T / Stab_{T} C is a torus. For the following choice of C′ ∈ Aut_{Λ}(P).C we will show dim Stab_{T} C′ = 𝔰: Decompose P/C into indecomposable summands, say P/C ≅ P_{1}/C′_{1} ⊕ ⋯ ⊕ P_{𝔰}/C′_{𝔰}, where the P_{i} are suitable direct summands of P; it is clearly harmless to assume P_{i} = ⨁_{j∈I(i)} Λz_{j} for some partition ⨆_{1≤i≤𝔰} I(i) of {1, …, t}. Now set C′ = ⨁_{1≤i≤𝔰} C′_{i}. For simplicity of notation, we assume C′ = C. Let T̂ be the K-subspace of End_{Λ}(P) consisting of all endomorphisms of the form (a_{1}, …, a_{t}) ∈ K^{t}, not just the invertible ones; again we identify (a_{1}, …, a_{t}) with the endomorphism of P sending z_{i} to a_{i}z_{i}. Moreover, StabT̂ C will stand for the K-subspace of T̂ consisting of the endomorphisms f with f(C) ⊆ C. Clearly, Stab_{T} C is a dense open subvariety of StabT̂ C, whence the two varieties have the same dimension. As for the dimension of the latter, StabT̂ C obviously contains the span of the vectors 1_{I(i)}, defined as having j-th entry 1 if j ∈ I(i) and 0 otherwise. That these vectors in fact form a basis for StabT̂ C is an immediate consequence of the indecomposability of the P_{i}/C_{i}. We infer that Stab_{T} C has dimension 𝔰, which makes the dimension of T .C equal to t − 𝔰 as asserted.

(4) is obvious in view of the remarks preceding the proposition. □

As part (3) of the proposition suggests, the dimension of the stabilizer subgroup Stab_{T} C′ varies from one point C′ ∈ Aut_{Λ}(P).C to another. In fact, Stab_{T} C′ may be reduced to the scalar multiples of the identity 1 of T , even when 𝔰(P/C) is large.

3. 𝔊𝔯𝔞𝔰𝔰_{d}^{T} for squarefree T

The diagram below is to remind the reader of the relationships among the varieties in which we are interested.

Classical scenario Grassmannian scenario

GL_{d}-action Aut_{Λ}(P)-action

TopT d

R

[figure] isom. types of d-dim’l. modules with top T

{ ; | } ; |

𝔊𝔯𝔞𝔰𝔰T d

ϕ [figure]

∪ ∪ ∪

Layer(𝕊) R [figure]

isom. types of modules with radical layering 𝕊

{ ; | } ; |

𝔊𝔯𝔞𝔰𝔰(𝕊) ϕ [figure]

Throughout this section, T denotes a squarefree semisimple module, say

T = ⨁_{1≤r≤t} Λe_{r}/Je_{r}

and P = ⨁_{1≤r≤t} Λe_{r} its projective cover; in particular, P is now a left ideal of Λ. We again set m = dim_{K} P − d. Our primary goal is to construct finite Aut_{Λ}(P)-stable affine open covers of the subvarieties 𝔊𝔯𝔞𝔰𝔰(𝕊) ⊆ 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

3.A. Local coordinates for 𝔊𝔯𝔞𝔰𝔰_{d}^{T} and a pivotal family of representations parametrized by 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

We could have postponed the restriction to squarefree T until the next section, but adopt it here for its notational pay-off. The coordinatized open cover of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} which we introduce first does not consist of Aut_{Λ}(P)-stable sets in general. It depends on a fixed choice of top elements of P, whereas the cover at which we are ultimately aiming is Aut_{Λ}(P)-stable and invariant under such choices. At the outset, our preferred bases for factor modules P/C may appear more specialized than necessary for the present purpose. They actually are; yet, it is precisely this special format which will make them useful in the sequel.

Definitions 3.1. (i) An abstract d-dimensional skeleton with top T is a set σ consisting of d distinct paths of lengths ≤ L which satisfies the following conditions:

• Each path in σ starts in one of the vertices e_{1}, …, e_{t};

• σ is closed under right subpaths; and

• σ contains {e_{1}, …, e_{t}}.

(ii) Given an abstract d-dimensional skeleton σ with top T, we define

Aux(σ) = {C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T} | P/C has basis {p + C | p ∈ σ}}.

The sets Aux(σ) are merely of temporary importance as auxiliaries. An abstract ddimensional skeleton σ with top T can be visualized as an undirected graph with d nodes, namely the disjoint union of t trees, represented by the sets {p ∈ σ | p = pe_{r}} for 1 ≤ r ≤ t.

Clearly, there are only finitely many abstract d-dimensional skeletons. Moreover, Aux(σ) is empty unless the subset σ of P is linearly independent over K. On the other hand, each point C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T} belongs to at least one of the sets Aux(σ); in other words, the Aux(σ) cover 𝔊𝔯𝔞𝔰𝔰_{d}^{T} – see Lemma 3.9 for a sharper statement.

Since, for any skeleton σ, the condition

C ⊕ ∑_{\begin{matrix}p∈σ \\ length(p)≥1 \end{matrix}} Kp = JP,

defines an open subset in the Grassmannian of m-dimensional subspaces of JP, we obtain:

Lemma 3.2. The Aux(σ) are open subsets of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} □

Next, we will use this cover to construct a family of d-dimensional modules with top T parametrized by 𝔊𝔯𝔞𝔰𝔰_{d}^{T}. The reason why this family is of interest lies in the fact that it melts down to the universal family whenever a fine moduli space for our problem exists.

Lemma 3.3. There exists a family (Γ, γ) of d-dimensional modules parametrized by 𝔊𝔯𝔞𝔰𝔰_{d}^{T} such that, for each C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, the fibre of Γ over C (with the Λ-structure induced by γ) is isomorphic to P/C.

Proof. Let Mod_{d}^{Λ} be the affine variety of Section 2.A and σ an abstract d-dimensional skeleton with top T. We start by pinning down the obvious morphism ρ_{σ} from Aux(σ) to Mod_{d}^{Λ} which matches our coordinatization. To that end, we assume that Aux(σ) ≠ ∅. This implies that the K-subspace V of P generated by σ has dimension d and, given any C ∈ Aux(σ), each element of P has a unique expansion as a K-linear combination of the the elements p ∈ σ modulo C. View V as K^{d}, identifying the distinguished basis σ of V with the canonical basis of K^{d}, where σ is ordered as follows: Start by ordering the finite set P of all paths of length ≤ L in KQ in such a fashion that the paths starting in e_{i} precede those starting in e_{j} whenever i < j. Then let p_{1} be that element of σ which is minimal in P, next let p_{2} ∈ σ be as small as possible in P \ {p_{1}}, etc.

Given C ∈ Aux(σ), we now define

ρ_{σ}(C) = (ρ_{σ}(C)_{α})_{α∈Q^{∗}}

where ρ_{σ}(C)_{α} is the d×d matrix over K recording in its columns the expansion coefficients (relative to σ) of the multiples αp modulo C, for p ∈ σ.

Thus the trivial bundle Aux(σ) × K^{d} becomes a family of Λ-modules via the algebra homomorphism from Λ to the endomorphism ring of Aux(σ) × K^{d}, defined by

α ↦ ((C, v) ↦ (C, ρ_{σ}(C)_{α}v)).

The appropriate choice of morphisms g_{σ,σ′} : Aux(σ)∩Aux(σ′) → GL_{d} to glue the Aux(σ)× K^{d} together to a vector bundle over 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is now automatic: Namely just send each C ∈ Aux(σ) ∩ Aux(σ′) to the transition matrix recording the coefficients of the p′ + C (for p′ ∈ σ′) relative to the p + C, in the prescribed order; then, clearly, the g_{σ,σ′} satisfy the relevant cocycle condition. The resulting vector bundle over 𝔊𝔯𝔞𝔰𝔰_{d}^{T} will be denoted by Γ. Since our algebra homomorphisms Λ → End(Aux(σ) × K^{d}) are compatible with this gluing, they yield an algebra homomorphism γ : Λ → End(Γ) and thus a family of d-dimensional Λ-modules as required. □

3.B. The distinguished Aut_{Λ}(P)-stable affine open cover of 𝔊𝔯𝔞𝔰𝔰(𝕊).

The key to the announced cover comes from the concrete counterparts of the abstract skeletons of section 3.A. By construction, this new cover will depend only on the quiver of Λ and the ideal I of relations. In fact, up to permutation and birational equivalence of the irreducible components, it is an isomorphism invariant of Λ (see Remark 3.17).

Definitions 3.4. (i) Let M be a d-dimensional module with top T. An abstract ddimensional skeleton σ (with top T) is called a skeleton of M provided there exists a sequence x_{1}, …, x_{t} of top elements of M with x_{r} = e_{r}x_{r} such that, for every integer l between 0 and L, the union

⋃_{1≤r≤t}{px_{r} + J^{l+1}M | p = pe_{r} ∈ σ, length(p) = l}

is a K-basis for J^{l}M/J^{l+1}M.

(ii) For any abstract d-dimensional skeleton σ with top T, we set

𝔊𝔯𝔞𝔰𝔰(σ) = {C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T} | σ is a skeleton of P/C}.

(iii) Given an abstract skeleton σ and a semisimple sequence 𝕊, we say that σ is compatible with 𝕊 in case, for each i ∈ {1, …, n}, the number of paths of any prescribed length l in σ which end in the vertex e_{i} equals dim_{K} e_{i}𝕊_{l}.

First we observe that the notion of a skeleton of a module M as in (i) does not depend on a choice of top elements of M. Indeed, as is readily confirmed, if σ is a skeleton of M, then any choice of top elements y_{1} = e_{1}y_{1}, …, y_{t} = e_{t}y_{t} of M yields K-bases ⋃_{1≤r≤t}{py_{r} + J^{l+1}M | p = pe_{r} ∈ σ, length(p) = l} for the radical layers J^{l}M/J^{l+1}M.

It is obvious that isomorphic modules have identical sets of skeletons, which means that the sets 𝔊𝔯𝔞𝔰𝔰(σ) are stable under the action of Aut_{Λ}(P). Moreover, we notice that, whenever C belongs to 𝔊𝔯𝔞𝔰𝔰(σ), the skeleton σ is compatible with the sequence 𝕊(P/C) of radical layers of P/C. Hence 𝔊𝔯𝔞𝔰𝔰(σ) ∩ 𝔊𝔯𝔞𝔰𝔰(𝕊) = ∅ if σ fails to be compatible with 𝕊. On the other hand, 𝔊𝔯𝔞𝔰𝔰(σ) is contained in 𝔊𝔯𝔞𝔰𝔰(𝕊), whenever σ is compatible with 𝕊. Further properties of the sets 𝔊𝔯𝔞𝔰𝔰(σ) are collected in the following theorem. In particular, we glean from it that the 𝔊𝔯𝔞𝔰𝔰(σ) are locally closed affine subvarieties of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

Theorem 3.5. Let 𝕊 be a d-dimensional semisimple sequence with top T.

The sets 𝔊𝔯𝔞𝔰𝔰(σ), where σ runs through the abstract skeletons compatible with 𝕊, form a finite Aut_{Λ}(P)-stable affine open cover of the quasi-projective variety 𝔊𝔯𝔞𝔰𝔰(𝕊). A family of polynomials determining any of the 𝔊𝔯𝔞𝔰𝔰(σ) in a suitable affine coordinate system can be (algorithmically) obtained from the quiver Q and a finite set of relations generating the ideal I as a left ideal of KQ.

The fact that the sets 𝔊𝔯𝔞𝔰𝔰(σ) form an open cover of 𝔊𝔯𝔞𝔰𝔰(𝕊) will be proved in Lemmas 3.8 and 3.9. The remaining claim of Theorem 3.5, namely that each of the sets 𝔊𝔯𝔞𝔰𝔰(σ) is an explicitly computable affine variety, is less obvious. We will tackle it in several steps, providing further detail along the way. Since the statements of most of our later results can be understood without the explicit description of the sets 𝔊𝔯𝔞𝔰𝔰(σ), which we are about to provide, the reader not interested in computational aspects is encouraged to skip ahead to Section 4.

Remark 3.6. The affine cover of Theorem 3.5 permits to resolve the isomorphism problem for d-dimensional top-T modules with with low computational investment: Given any point C in 𝔊𝔯𝔞𝔰𝔰(σ), expressed in terms of an affine coordinate system for 𝔊𝔯𝔞𝔰𝔰(σ), the Aut_{Λ}(P)-orbit of C can be obtained from a system of at most d · (dim_{K} J/J^{2}) linear equations (see [8]).

The following elementary example shows that, if T contains squares of simples, the variety 𝔊𝔯𝔞𝔰𝔰(𝕊) will not have an Aut_{Λ}(P)-stable affine cover in general. In fact, we exhibit an Aut_{Λ}(P)-orbit that fails to be quasi-affine for a situation where T is a square.

Example 3.7. Let Λ be the representation-finite hereditary algebra KQ, where Q is the quiver

\xymatrix{2 & 1 \ar[l]_{α} \ar[r]^{β} & 3}

If 𝕊 = (S_{1}^{2}, S_{2}⊕S_{3}), then 𝔊𝔯𝔞𝔰𝔰(𝕊) is isomorphic to ℙ^{1}×ℙ^{1} and consists of two Aut_{Λ}(P)- orbits, namely the orbits representing the modules Λe_{1}/Λαe_{1} ⊕ Λe_{1}/Λβe_{1} and Λe_{1} ⊕ S_{1}. The orbit of the latter is isomorphic to ℙ^{1}. □

We return to our blanket hypothesis that T be squarefree. In justifying Theorem 3.5, we start with openness of the 𝔊𝔯𝔞𝔰𝔰(σ) in the pertinent 𝔊𝔯𝔞𝔰𝔰(𝕊).

Lemma 3.8. For any skeleton σ compatible with 𝕊, the subset 𝔊𝔯𝔞𝔰𝔰(σ) is open in 𝔊𝔯𝔞𝔰𝔰(𝕊).

Proof. From the fact that any skeleton of a module M with top T satisfies the defining properties of 3.4(i) for arbitrary sequences of top elements, normed by the idempotents e_{r} in the correct order, we infer

𝔊𝔯𝔞𝔰𝔰(σ) = Aux(σ) ∩ 𝔊𝔯𝔞𝔰𝔰(𝕊).

whenever σ is compatible with 𝕊. Now we use Lemma 3.2. □

That the 𝔊𝔯𝔞𝔰𝔰(σ) cover 𝔊𝔯𝔞𝔰𝔰(𝕊), is guaranteed by the following elementary lemma.

Lemma 3.9. Every module M with top T has at least one skeleton.

Proof. First, we set σ_{0} = {e_{1}, …, e_{t}} and choose top elements x_{1} = e_{1}x_{1}, …, x_{t} = e_{t}x_{t} of M giving rise to a basis of M/JM. Next, we pick a set σ_{1} of arrows p_{1}, each starting in one of the vertices in σ_{0}, such that the set {p_{1}x_{r} | p_{1} ∈ σ_{1} ∩ KΓe_{r}, 1 ≤ r ≤ t} yields a K-basis for JM modulo J^{2}M. The following stage is to choose a set σ_{2} of paths p_{2} of length 2, each containing some arrow in σ_{1} as a right subpath, such that the set {p_{2}x_{r} | p_{2} ∈ σ_{2} ∩ KΓe_{r}, 1 ≤ r ≤ t} gives rise to a K-basis of J^{2}M/J^{3}M. We continue in this fashion and set σ = ⋃_{0≤l≤L} σ_{l}. (Keep in mind that J^{L+1} = 0 by our choice of L.) □

From the proof of Lemma 3.9, we can actually glean the following stronger statement: Given any path p of length l with pM ⊈ J^{l+1}M, there exists a skeleton of M which contains p.

We next introduce a convenient affine coordinate grid for each 𝔊𝔯𝔞𝔰𝔰(σ), and then follow with a family of polynomials derived from the relations of Λ. That the vanishing set of these polynomials coincides with 𝔊𝔯𝔞𝔰𝔰(σ), up to an isomorphism which ‘respects’ the representation map, is spelled out under the heading of the more precise Theorem 3.14 below.

3.10. The coordinate system for 𝔊𝔯𝔞𝔰𝔰(σ). Let σ be an abstract d-dimensional skeleton with top T. A pair (α, p), consisting of an arrow α and a path p ∈ σ, is called σ-critical in case αp is a path in KQ which does not belong to σ.

Clearly, the isomorphism type of any module M = P/C with skeleton σ is completely pinned down by the (unique) scalars c_{αp,q} appearing in the following equations in M:

αp + C = ∑_{q∈σ} c_{αp,q}q + C

for the σ-critical pairs (α, p). Moreover, to obtain a more economical coordinatization, we observe that the coefficient c_{αp,q} in the above equation is zero unless q is strictly longer than p and ends in the same vertex as the arrow α. This motivates the following notation: For any σ-critical pair (α, p), we let σ(α, p) be the set of all paths in σ which are longer than p and end in the same vertex as α.

As a consequence, in pinning down M, we only need to keep track of the scalars arising in the equations

αp + C = ∑_{q∈σ(α,p)} c_{αp,q}q + C

for the σ-critical pairs (α, p). In other words, as a Λ-submodule of P, any C ∈ 𝔊𝔯𝔞𝔰𝔰(σ) is generated by the differences

αp − ∑_{q∈σ(α,p)} c_{αp,q}q.

As long as we keep the skeleton σ fixed, it is therefore justified to identify the points C ∈ 𝔊𝔯𝔞𝔰𝔰(σ) with the corresponding families

(c_{αp,q})\begin{matrix}(α,p) σ-critical \\ q∈σ(α,p) \end{matrix}

of scalars in the appropriate affine space 𝔸^{N}, where N is the disjoint union of the sets σ(α, p).

3.11. The congruence relation induced by σ. Keeping σ fixed, we consider the polynomial ring

A = A(σ) := KQ[X_{αp,q} | (α, p) σ-critical, q ∈ σ(α, p)]

over the path algebra KQ. On the ring A, we consider congruence modulo the left ideal

C = C(σ) := ⨁_{t+1≤i≤n} Ae_{i} + ∑_{(α,p) σ-critical} A(αp − ∑_{q∈σ(α,p)} X_{αp,q}q)

and denote this congruence relation by ˆ=.

The following lemma says that A/C is a free left module over the commutative polynomial ring K[X_{αp,q}], with the cosets of the paths in σ serving as a basis. The argument backing the lemma is constructive.

Lemma 3.12. For every z ∈ A, there is a unique family of polynomials τ_{q}(X) = τ_{q}^{z}(X) in K[X_{αp,q}] such that

z ˆ= ∑_{q∈σ} τ_{q}(X) q.

Proof. We only verify existence of the τ_{q}(X) and leave uniqueness to the reader. It is clearly innocuous to assume that z is a path in KQ starting in one of the vertices e_{1}, …, e_{t}. Let p be the longest right subpath of z belonging to σ. We will prove our claim by induction on length(z) − length(p). If this difference is zero, we are done. So suppose it is positive and let α be the (unique) arrow with the property that αp is again a right subpath of z, say z = z′αp for a suitable path z′. Then (α, p) is a σ-critical pair, and

z ˆ= z′( ∑_{q∈σ(α,p)} X_{αp,q}q ) = ∑_{q∈σ(α,p)} X_{αp,q}z′q.

Since all of the differences length(z′q) − length(q) arising from this equation are strictly smaller than length(z) − length(p), our induction hypothesis guarantees that each of the summands X_{αp,q}z′q has the desired form up to ˆ=. Consequently, so does z. □

3.13. Polynomials for 𝔊𝔯𝔞𝔰𝔰(σ). Let R be any finite generating set for the left ideal Ie_{1} + ⋯ + Ie_{t} of KQ – note that such a generating set always exists since all paths of lengths ≥ L + 1 belong to I. For each ρ ∈ R, let τ_{q}^{ρ}(X) be the polynomials in K[X_{αp,q}] with

ρ ˆ= ∑_{q∈σ} τ_{q}^{ρ}(X) q

as guaranteed by the lemma. We will see that the simultaneous vanishing set of these polynomials τ_{q}^{ρ}(X) is an alternate incarnation of 𝔊𝔯𝔞𝔰𝔰(σ).

Solely for the purpose of establishing a suitable isomorphism of varieties, we let Aff(σ) stand for the subvariety of 𝔸^{N} defined by the polynomials τ_{q}^{ρ}(X) for the moment.

It is straightforward that Aff(σ) is independent of our choice of R. (In fact, so is the ideal of K[X_{αp,q}] generated by the τ_{q}^{ρ}(X).) Furthermore, we emphasize that, in order to proceed, we need not preselect ‘eligible’ abstract d-dimensional skeletons, i.e., abstract skeletons which actually occur as skeletons of Λ-modules; Aff(σ) is automatically empty if 𝔊𝔯𝔞𝔰𝔰(σ) is, as is for instance the case when, for some integer l, the set of paths of length l in σ is linearly dependent modulo J^{l+1}P.

Theorem 3.14. Suppose σ is a d-dimensional abstract skeleton. There exists an isomorphism ψ_{σ} : Aff(σ) → 𝔊𝔯𝔞𝔰𝔰(σ) of varieties, together with a surjective map χ_{σ}, such that the following diagram commutes:

\xymatrix{Aff(σ) \ar[dd]^{≅}_{ψ_{σ}} \ar[dr]^{χ_{σ}} \\  & \{isom. types of d-dim’l. modules with skeleton σ\} \\ 𝔊𝔯𝔞𝔰𝔰(σ) \ar[ur]_{ϕ_{σ}}}

Here ϕ_{σ} is the restriction of the representation map ϕ to 𝔊𝔯𝔞𝔰𝔰(σ), and the maps ψ_{σ} and χ_{σ} are as follows: ψ_{σ} sends any point c = (c_{αp,q})\begin{matrix}(α,p) σ-critical \\ q∈σ(α,p) \end{matrix} from Aff(σ) to the submodule

U(c) := ∑_{(α,p) σ-critical} Λ(αp − ∑_{q∈σ(α,p)} c_{αp,q}q)

of P = ⨁_{1≤r≤t} Λe_{r}, and χ_{σ} sends c to the isomorphism type of the factor module P/U(c).

We defer the somewhat technical proof of Theorem 3.14 to Section 6; clearly, Theorem 3.14 covers the remaining claims of Theorem 3.5.

In the sequel, we will identify Aff(σ) with 𝔊𝔯𝔞𝔰𝔰(σ) and χ_{σ} with ϕ_{σ}. This is warranted in light of the commutative diagram linking the pertinent representation maps from 𝔊𝔯𝔞𝔰𝔰(σ) and Aff(σ) to the modules with skeleton σ. We remark that ψ_{σ}^{−1} is a closed immersion of 𝔊𝔯𝔞𝔰𝔰(σ) into the affine space 𝔸^{N}.

An algorithm for finding polynomials that determine 𝔊𝔯𝔞𝔰𝔰(σ) within the affine space 𝔸^{N} is implicit in the preceding discussion. We follow with an observation which reduces the computational effort very significantly in general – the underlying concept of a ‘route’ will reappear in the proof of Theorem 3.14 – and then illustrate the procedure with an example.

Computational remark 3.15. Suppose u is a path of length l which passes through the vertices (e(0), e(1), …, e(l)) in that order; in particular, this means that u starts in e(0) and ends in e(l). Given an abstract d-dimensional skeleton σ with top T, such a path u will be called a route on σ if there exist paths p_{0}, p_{1}, …, p_{l} in σ with length(p_{0}) = 0 and length(p_{i−1}) < length(p_{i}) such that each p_{i} ends in e(i). Note that any path in σ is a route on σ, since skeletons are closed under right subpaths by definition.

Now let C be any point in 𝔊𝔯𝔞𝔰𝔰(σ); in particular, C ⊆ ⨁_{1≤r≤t} Je_{r}. If u is a path in ⨁_{1≤r≤t} KQe_{r} which fails to be a route on σ, then its image in Λ belongs to C; in other words, if we view u as an element of A, we have u ˆ= 0 under the congruence relation of 3.11. Consequently, u can be ignored in the substitution process leading to polynomials defining 𝔊𝔯𝔞𝔰𝔰(σ).

To justify this claim, suppose a path u = α_{s} ⋯ α_{1} for certain arrows α_{i} has the property that u + C ≠ 0 in M = P/C. If, for j ≤ s, the integer l_{j} is such that α_{j} ⋯ α_{1} + C ∈ J^{l_{j}}M \ J^{l_{j}+1}M, then 1 < l_{1} < l_{2} < ⋯ < l_{s}; moreover, for each j, the set σ contains a path of length l_{j} ending in the same vertex as α_{j}. This shows the path u = α_{s} ⋯ α_{1} to be a route on σ, thus backing our remark.

Example 3.16. Consider Λ = KQ/I, where Q is the quiver

[figure: 1; ω_{1}; ω_{2}; α_{1}; α_{2}; 2]

and I ⊆ KQ the ideal generated by the four paths ω_{i}ω_{j} for i, j ∈ {1, 2} and α_{1}ω_{1} − α_{2}ω_{2}. Clearly, the listed relations also generate Ie_{1} as a left ideal of KQ.

For T = Λe_{1}/Je_{1} and d = 4, we consider the skeleton σ = {e_{1}, ω_{1}, α_{1}ω_{1}, α_{2}}, and determine 𝔊𝔯𝔞𝔰𝔰(σ) as follows. First we list the σ-critical pairs (α, p), together with their sets σ(α, p). Obviously, there is no harm in omitting pairs (α, p) with αp ∈ I; the pairs that are left are (ω_{2}, e_{1}) with σ(ω_{2}, e_{1}) = {ω_{1}}, (α_{1}, e_{1}) with σ(α_{1}, e_{1}) = {α_{2}, α_{1}ω_{1}}, and (α_{2}, ω_{1}) with σ(α_{2}, ω_{1}) = {α_{1}ω_{1}}. This leads to the following congruences in KQ[X_{1}, …, X_{4}], where X_{1}, …, X_{4} stand for X_{ω_{2},ω_{1}}, X_{α_{1},α_{2}}, X_{α_{1},α_{1}ω_{1}}, X_{α_{2}ω_{1},α_{1}ω_{1}}. Namely, ω_{2} ˆ= X_{1}ω_{1}, α_{1} ˆ= X_{2}α_{2} + X_{3}α_{1}ω_{1}, and α_{2}ω_{1} ˆ= X_{4}α_{1}ω_{1}. Since all of the four paths ω_{i}ω_{j} in I fail to be routes on σ, Remark 3.15 tells us that ω_{i}ω_{j} ˆ= 0 is automatic, whence the relations ω_{i}ω_{j} = 0 in Λ do not produce any conditions on the X_{k}. On the other hand, inserting the above ‘basic congruences’ into the final relation yields α_{1}ω_{1} − α_{2}ω_{2} ˆ= α_{1}ω_{1} − X_{1}α_{2}ω_{1} ˆ= α_{1}ω_{1} − X_{1}X_{4}α_{1}ω_{1} = (1 − X_{1}X_{4})α_{1}ω_{1}. The last congruence is the expansion of α_{1}ω_{1} − α_{2}ω_{2} ∈ A as in Lemma 3.12, whence

𝔊𝔯𝔞𝔰𝔰(σ) = {(c_{i})_{i≤4} ∈ 𝔸^{4} | 1 − c_{1}c_{4} = 0}. □

Remark 3.17 on uniqueness. Clearly, the locally closed subsets 𝔊𝔯𝔞𝔰𝔰(𝕊) of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} are uniquely determined, up to isomorphism, by the isomorphism type of Λ. The affine open subsets 𝔊𝔯𝔞𝔰𝔰(σ) of any given 𝔊𝔯𝔞𝔰𝔰(𝕊), in their turn, enjoy a somewhat weaker uniqueness property. Namely, they are invariant under algebra isomorphisms of Λ, up to birational equivalence. More precisely: Suppose Λ ≅ Λ′, where Λ′ = KQ′/I′ (it is well-known that Q and Q′ are isomorphic as directed graphs in this situation). Let 𝕊 be a semisimple sequence over Λ, and 𝕊′ the semisimple sequence over Λ′ corresponding to 𝕊 under some algebra isomorphism from Λ to Λ′. Moreover, let Irr(𝕊) be the set of all irreducible components of the affine varieties 𝔊𝔯𝔞𝔰𝔰(σ), where σ runs through the skeletons compatible with 𝕊, and define Irr(𝕊′) analogously. Then there is a bijection Irr(𝕊) → Irr(𝕊′) such that the partners under this pairing are birationally equivalent. One proves this as in the scenario of varieties of uniserial modules in [4]. Finally, one observes that ‘birationally equivalent’ can be strengthened to ‘isomorphic’ in our uniqueness statement, provided that Q is without double arrows; indeed, in that case, both the quiver and the corresponding relations are essentially pinned down by the isomorphism type of Λ.

4. The moduli problem

We continue to assume that T = ⨁_{1≤r≤t} Λe_{r}/Je_{r} is squarefree, and P = ⨁_{1≤r≤t} Λe_{r}. The first subsection contains our main results, while in 4.B we briefly discuss the moduli problem for classes of representations with fixed radical layering.

4.A. Top-stable degenerations and quotients of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} modulo Aut_{Λ}(P).

From Proposition 2.9 and Theorem 3.5, we derive the following consequences concerning top-stable and layer-stable degenerations. The case where T is simple is particularly transparent, since in that situation, the Aut_{Λ}(P)-action coincides with the action of the unipotent radical Aut_{Λ}^{u}(P).

Proposition 4.1.

(1) All Aut_{Λ}(P)-orbits are quasi-affine subvarieties of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

(2) Whenever M is a module with top T, say M = P/C with C ∈ 𝔊𝔯𝔞𝔰𝔰(𝕊), the orbit Aut_{Λ}^{u}(P).C of C under the action of the unipotent radical of Aut_{Λ}(P) is closed in 𝔊𝔯𝔞𝔰𝔰(𝕊). It is a full affine space of dimension

m(M) = μ_{T} (M) − (t + dim_{K} Hom_{Λ}(M, JM)),

where μ_{T} (M) is the sum of the multiplicities of the simple summands of T as composition factors of M.

(3) In case T is simple and M as in part (2), the orbit Aut_{Λ}(P).C is closed in 𝔊𝔯𝔞𝔰𝔰(𝕊) and isomorphic to 𝔸^{m} with

m = μ_{T} (M) − dim_{K} End_{Λ}(M),

where μ_{T} (M) is the multiplicity of T in M.

Proof. The first assertion is an immediate consequence of the existence of an Aut_{Λ}(P)- stable affine cover of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}. As for the second, let C ∈ 𝔊𝔯𝔞𝔰𝔰(𝕊). To see that the Aut_{Λ}^{u}(P)-orbit of C is closed in 𝔊𝔯𝔞𝔰𝔰(𝕊), we need only make sure that its intersections with the open subvarieties 𝔊𝔯𝔞𝔰𝔰(σ), where σ runs through the skeletons compatible with 𝕊, are relatively closed. But in light of Theorem 3.5, this follows from a result of Kostant’s which guarantees that the orbits of a unipotent group morphically acting on an affine variety are closed (cf. [9, p.115, Exercise 8]). The claimed dimension of Aut_{Λ}^{u}(P).C can be deduced from Proposition 2.9. For part (3) finally, observe that Aut_{Λ}(P) ≅ K^{∗} ×Aut_{Λ}^{u}(P), with K^{∗} acting trivially, and apply part (2). □

As is common, we call a module local in case it has simple top; thus the local modules are characterized by having precisely one maximal submodule.

Theorem 4.2. Suppose that M is a d-dimensional module with top T, say M ≅ P/C with C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T}. Then the following conditions are equivalent:

(1) M has no proper top-stable degenerations.

(2) The orbit Aut_{Λ}(P).C is closed in 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

(3) The orbit Aut_{Λ}(P).C is a singleton.

(4) C ⊆ P is fully invariant under Λ-endomorphisms of P, i.e., Cu ⊆ C for any path u starting in one of the vertices e_{r} with r ≤ t.

(5) M is a direct sum of local modules and

μ_{T} (M) = t + dim_{K} Hom_{Λ}(M, JM),

where μ_{T} (M) is the sum of the multiplicites of the simple summands of T as composition factors of M.

Proof. The equivalence of (1) and (2) is immediate from Proposition 2.5, and the implications (4) ⟹ (3) ⟹ (2) are trivial.

To verify (2) ⟹ (3), suppose (2) holds. Using the fact that 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is a projective variety, we infer completeness of Aut_{Λ}(P).C. On the other hand, this orbit is quasiaffine by Proposition 4.1, and therefore must be zero dimensional. Since Aut_{Λ}(P).C is irreducible, (3) follows.

For (3) ⟹ (4), assume (3), which means that C is invariant under automorphisms of P. Given any f ∈ End_{Λ}(P), pick a nonzero scalar a such that −a is not an eigenvalue of f. Then a · id +f belongs to Aut_{Λ}(P), and hence leaves C invariant; but this means f(C) ⊆ C.

(3) ⟹ (5). By Proposition 2.9(3), the number of summands in a decomposition of M into indecomposables equals t. Since t = dim_{K} T, this means that M splits into local summands. The numerical equality follows from Proposition 4.1(2).

(5) ⟹ (3). Without loss of generality, we may assume that C equals the point C′ ∈ Aut_{Λ}(P).C specified in Proposition 2.9(3). Thus T .C = {C}. Moreover, (5) guarantees that Aut_{Λ}^{u}(P).C ≅ 𝔸^{0}, whence Aut_{Λ}(P).C is a singleton as well. □

In case T contains squares, the statements of the theorem fail in general. For instance, in Example 3.7, let M = Λe_{1} ⊕ S_{1}. Then M = P/C with C = Λ(0, α) + Λ(0, β) has no proper top-stable degeneration, while Aut_{Λ}(P).C is the projective line.

The simplest instance of a (local) module M having a proper top-stable degeneration can be found in Example 2.4. In this example, 𝔊𝔯𝔞𝔰𝔰_{d}^{T} consists of two Aut_{Λ}(P)-orbits, those of the modules M = Λe_{1}/Λα and M′ = Λe_{1}/Λαω. The Aut_{Λ}(P)-orbit of C = Λα is isomorphic to 𝔸^{1}, while that of C′ = Λαω is a singleton. Thus M degenerates top-stably to M′. However, M′ is not a layer-stable degeneration of M, as is predicted by the following consequence of Proposition 4.1(3).

Corollary 4.3. If M is a Λ-module with simple top, then M has no proper layer-stable degenerations. □

In contrast, modules with non-simple tops do have proper layer-stable degenerations in general. Yet, the layer-stable degenerations of any module with squarefree top are finite in number and can be pinned down explicitly from a presentation of the module under consideration (see [8]).

From Corollary 4.3, it is intuitively clear that absence of proper layer-stable degenerations of the modules represented by 𝔊𝔯𝔞𝔰𝔰(𝕊) does not suffice to guarantee existence of a moduli space for the representations with radical layering 𝕊; a confirmation will follow in 4.B. In light of this fact, the next theorem distinguishes Top_{d}^{T} and its closed subvarieties among the quasi-affine subvarieties of Mod_{d}^{Λ}.

Theorem 4.4. Suppose that T is squarefree and W a closed GL_{d}-stable subvariety of the classical variety Top_{d}^{T} of d-dimensional modules with top T. Let V be the corresponding closed Aut_{Λ}(P)-stable subvariety of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} (under the bijection of Proposition 2.5 – in particular, W = Top_{d}^{T} implies V = 𝔊𝔯𝔞𝔰𝔰_{d}^{T}). Then the following statements are equivalent:

(1) There exists a fine moduli space for the isomorphism classes of modules represented by W, i.e., for the isomorphism classes R(x), x ∈ W.

(2) There exists a coarse moduli space for the isomorphism classes of modules represented by W.

(3) V \underline{is} the fine moduli space for the modules represented by W.

(4) W has a geometric quotient modulo GL_{d}.

(5) All R(x) with x ∈ W are devoid of top-stable degenerations.

(6) All C ∈ V are fully invariant in P. In case the modules represented by W possess a moduli space, they all split into local direct summands. Moreover, restriction to V of the family (Γ, γ) over 𝔊𝔯𝔞𝔰𝔰_{d}^{T} constructed in Lemma 3.3 yields the universal family in this situation.

Proof. That (3) ⟹ (1) ⟹ (2) is trivial, as is (4) ⟹ (5). Moreover, (6) ⟹ (4) is due to the fact that (6) makes V its own geometric quotient modulo Aut_{Λ}(P); by Proposition 2.5, (4) follows. The equivalence of (5) and (6) can be obtained from Theorem 4.2. That (2) implies (5) can be deduced from Criterion 2.1 (just restrict the locally universal bundle over Top_{d}^{T} to W).

(6) ⟹ (3). Adopt (6), which means that all Aut_{Λ}(P)-orbits of V are reduced to points. Then V is its own geometric quotient modulo Aut_{Λ}(P) and, using Criterion 2.6, we infer that V is a coarse moduli space for the families of modules isomorphic to P/C with C ∈ V. To see that V is even a fine moduli space, we restrict the family (Γ, γ) parametrized by 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, constructed in Lemma 3.3, to the subvariety V and show that this latter family is universal. So let (Δ, δ) be any family of d-dimensional modules recruited from the P/C with C ∈ V, where Δ is a vector bundle over a variety X say. Define a map τ : X → V ⊆ 𝔊𝔯𝔞𝔰𝔰_{d}^{T} by sending any point x in X to the unique point C ∈ 𝔊𝔯𝔞𝔰𝔰_{d}^{T} with the property that the fibre of Δ above x is Λ-isomorphic to P/C. In view of the fact that V is already known to be a coarse moduli space, meaning that the bijection α = ϕ|_{V}^{−1} from the set of isomorphism classes of modules P/C with C ∈ V to the variety V satisfies the conditions of 1.6′ in [12, p.24], it is now routine to check that τ is a morphism and (Δ, δ) is equivalent to τ^{∗}((Γ, γ)|_{V}). Obviously, τ is unique with this property.

The final statements follow from Theorem 4.2 and the previous paragraph, respectively. □

For an illustration of the fact that squarefreeness cannot be discarded from the hypothesis of Theorem 4.4 either, we again refer to Example 3.7. Namely, let W ⊆ Top_{d}^{T} be the GL_{d}-orbit representing the module M = Λe_{1} ⊕ S_{1}. Then W is closed in Top_{d}^{T}, but, while W has a trivial geometric quotient modulo GL_{d}, condition (6) of the theorem is violated.

Theorem 4.4 shows, in particular, that the isomorphism classes of d-dimensional modules with top T have a fine moduli space provided that the simple summands of T do not recur as composition factors of JP/(soc JP). As was pointed out to the author by Crawley-Boevey, the following special case of Theorem 4.4 can also be derived from a result of King [10]: Namely, if T = Λe/Je is simple and eJe = 0, then, for any d ∈ ℕ, the modules with top T and fixed class in K_{0}(Λ) – that is, with fixed dimension vector – possess a fine moduli space. (Indeed, if the total dimension is d and Θ : K_{0}(Λ) → ℝ is the ℤ-linear map which sends the class of T to −(d − 1) and all other simples to 1, then the modules specified are Θ-stable.)

The situation where, for all d, the d-dimensional modules with fixed simple top T have a fine moduli space, can actually be completely characterized in terms of quiver and relations of Λ, a justification being immediate from Theorem 4.4.

Corollary 4.5. Given a simple module T = Λe/Je, the following statements are equivalent:

(a) For all d, the d-dimensional modules with top T have a coarse moduli space.

(b) For all d, the variety 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is a fine moduli space for the d-dimensional modules with top T.

(c) For every element λ ∈ Je and every oriented cycle ω ∈ eJe, the product λω belongs to Λλ. □

To illustrate the richness of the representation theory in situations where the modules with top T are classifiable, we show that arbitrary irreducible projective varieties occur, up to birational equivalence, as irreducible components of moduli spaces for families of modules with fixed dimension and fixed top. The algebras underlying these examples were previously constructed in a study of uniserial modules in [7]. We omit arguments that can be adapted from [7, Theorem G].

Examples 4.6. Given any irreducible projective variety V, there exists an algebra Λ, together with a positive integer d and a simple module T such that

• 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is the fine moduli space for the d-dimensional modules with top T, and

• V is birationally equivalent to an irreducible component of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}.

Proof. Given the irreducible projective variety V, let U be a non-empty affine open subset of V. Without loss of generality, we may assume that U ⊆ 𝔸^{l} is the vanishing set of a family of polynomials f_{1}, …, f_{N} in K[X_{1}, …, X_{l}] with the property that any monomial appearing in any of the f_{j} is squarefree, i.e., of the form X_{i_{1}} ⋯ X_{i_{s}} for distinct indices i_{1}, …, i_{s}. We can thus write the f_{j} in the form

f_{j} = ∑_{A∈P(l)} c_{j}(A) ∏_{i∈A} X_{i},

where P(l) is the power set of {1, …, l} and c_{j}(A) ∈ K. Let Λ = KQ/I, where Q is the quiver

[figure: 1; α_{1}; γ_{1}; 2; β_{1}; 3; α_{2}; γ_{2}; 4; β_{2}; 5; ⋯; 2l−1; α_{l}; γ_{l}; 2l; β_{l}; 2l+1]

and I ⊆ KQ the ideal generated by the relations R_{1}, …, R_{s} with

R_{j} = ∑_{A∈P(l)} c_{j}(A)p_{l}(A) ⋯ p_{1}(A),

where p_{i}(A) = γ_{i} for i ∈ A and p_{i}(A) = β_{i}α_{i} for i ∉ A. Moreover, let T = S_{1}, d = 2l + 1, and 𝕊 = (S_{1}, …, S_{d}). Then 𝔊𝔯𝔞𝔰𝔰(𝕊) ≅ U, and 𝔊𝔯𝔞𝔰𝔰_{d}^{T} clearly satisfies conditions (1)–(6) of Theorem 4.4. Moreover, we note that the irreducible variety 𝔊𝔯𝔞𝔰𝔰(𝕊) is not contained in the closure of the union ⋃ 𝔊𝔯𝔞𝔰𝔰(𝕊′), where 𝕊′ runs through the d-dimensional top-T semisimple sequences different from 𝕊, since this union is closed by Lemma 2.8; indeed all factors P/C with C ∈ ⋃_{𝕊′≠𝕊} 𝔊𝔯𝔞𝔰𝔰(𝕊′) have strictly larger dimension vectors (with respect to the lexicographic order) than the d-dimensional uniserial modules with top T. This shows that the closure of 𝔊𝔯𝔞𝔰𝔰(𝕊) in 𝔊𝔯𝔞𝔰𝔰_{d}^{T} is an irreducible component of 𝔊𝔯𝔞𝔰𝔰_{d}^{T}. By construction, this component is birationally equivalent to the initial projective variety. □

4.B. Remarks on quotients of 𝔊𝔯𝔞𝔰𝔰(𝕊).

As one might suspect, there is a plethora of cases where the d-dimensional modules with top T do not possess a moduli space, whereas, for each d-dimensional semisimple sequence with top T, the modules with radical layering 𝕊 do. The simplest example illustrating this fact is Example 2.4: If 𝕊 = (S_{1}, S_{1}, S_{2}) and 𝕊′ = (S_{1}, S_{1} ⊕ S_{2}, 0), each of the two subvarieties 𝔊𝔯𝔞𝔰𝔰(𝕊) and 𝔊𝔯𝔞𝔰𝔰(𝕊′) trivially has a moduli space (a singleton in each case), whereas 𝔊𝔯𝔞𝔰𝔰_{3}^{S_{1}} = 𝔊𝔯𝔞𝔰𝔰(𝕊) ∪ 𝔊𝔯𝔞𝔰𝔰(𝕊′) does not (by Theorem 4.4).

From Criterion 2.6, we know that the representations with radical layering 𝕊 have a coarse moduli space precisely when the Aut_{Λ}(P)-space 𝔊𝔯𝔞𝔰𝔰(𝕊) has a categorical quotient which is an orbit map. Obviously, closedness of the Aut_{Λ}(P)-orbits in 𝔊𝔯𝔞𝔰𝔰(𝕊)

(equivalently, absence of proper layer-stable degenerations among the modules with radical layering 𝕊) is a necessary condition for this event. In view of Corollary 4.3, it is automatically satisfied when T is simple (but not so for more general T; see [8]). Yet, even for a sequence 𝕊 with simple top T, the modules with radical layering 𝕊 frequently fail to possess a coarse moduli space. Indeed, another obstacle is as follows: Assume, for the moment that 𝔊𝔯𝔞𝔰𝔰(𝕊) is irreducible and π : 𝔊𝔯𝔞𝔰𝔰(𝕊) → 𝔊𝔯𝔞𝔰𝔰(𝕊)/ Aut_{Λ}(P) a categorical quotient which is an orbit map. Then π is dominant, and hence, by [6, AG.10.1], all Aut_{Λ}(P)-orbits of π have the same dimension. For an example where this condition is violated, let l ≥ 2 and consider the algebra

Λ = KQ/⟨all paths of length l + 1⟩,

where Q is the quiver

[figure: •; ω_{1}; ω_{2}]

If S is the unique simple left Λ-module and 𝕊 the (l + 1)-dimensional semisimple sequence (S, S, …, S), then 𝔊𝔯𝔞𝔰𝔰(𝕊) = (ℙ^{1})^{l} × 𝔸^{l(l−1)/2} is irreducible, but all numbers between 0 and l − 1 arise as dimensions of Aut_{Λ}(P)-orbits of 𝔊𝔯𝔞𝔰𝔰(𝕊); for details, see [4, Example].

On the other hand, we believe such skips in the orbit dimension to be the only impediment (beyond nonclosedness of the orbits for nonsimple T) in the way of a moduli space. In fact, a variety of examples led us to the following sharper conjecture.

Conjecture 4.7. The following conditions are equivalent for a semisimple sequence 𝕊 with squarefree top T:

(1) All Aut_{Λ}(P)-orbits of 𝔊𝔯𝔞𝔰𝔰(𝕊) are closed and the orbit dimension is constant on each irreducible component of 𝔊𝔯𝔞𝔰𝔰(𝕊).

(2) There exists a coarse moduli space for the representations with radical layering 𝕊.

(3) There exists a fine moduli space for the representations with radical layering 𝕊.

(4) 𝔊𝔯𝔞𝔰𝔰(𝕊) has a geometric quotient modulo Aut_{Λ}(P).

If T is simple, the first requirement under (1) is automatic, and the second is equivalent to constant dimension of the endomorphism rings End_{Λ}(P/C), where C traces an irreducible component of 𝔊𝔯𝔞𝔰𝔰(𝕊).

5. Algebras of finite local representation type

Our main objective in this section is to characterize the algebras which, up to isomorphism, permit only finitely many modules with a fixed simple top T. We start by considering an invariant that measures the size of the category Add_{Λ}^{T} consisting of the direct sums of left Λ-modules with top T. This invariant fails to be left-right symmetric in general.

Definition 5.1. Given a simple left Λ-module T with projective cover P, the local dimension of Λ-mod at T, denoted

loc dim^{T} (Λ-mod),

is the maximum of the following differences

dim C − generic fibre dimension of ϕ on C, where C runs through the irreducible components of the varieties 𝔊𝔯𝔞𝔰𝔰_{d}^{T} for 1 ≤ d ≤ dim_{K} P. (Recall that ϕ stands for the pertinent representation map.)

This invariant of Λ-mod can be explicitly obtained from the Aut_{Λ}(P)-stable affine covers of the Grassmannians 𝔊𝔯𝔞𝔰𝔰_{d}^{T} for d ≥ 1. Indeed, loc dim^{T} (Λ-mod) clearly equals the maximum of the differences ‘dim D − generic Aut_{Λ}(P)-orbit dimension on D’, where D traces the irreducible components of the 𝔊𝔯𝔞𝔰𝔰(σ) and σ ranges through all abstract skeletons with top T. For a computation of the generic fibre dimensions on the irreducible components D via a (fairly small) system of linear equations, we refer to [8]. Yet, in checking whether loc dim^{T} (Λ-mod) = 0 for an algebra Λ presented in terms of quiver and relations, it is usually more convenient to use one of the last two of the equivalent conditions presented in Theorem 5.2 below and to combine the procedures of Section 3.B with Gröbner methods; for examples see [8]. In particular, we will find that, in case Λ-mod has vanishing local dimension at T, all local modules with top T can be explicitly computed from quiver and relations of Λ, their sequences of radical layers providing a complete classification. On the side, we point out that we know of no coarser invariant separating isomorphism types in that situation; dimension vectors do not distinguish between non-isomorphic representations.

Theorem 5.2. For any simple left Λ-module T with projective cover P, the following statements are equivalent:

(1) There are only finitely many left Λ-modules with top T, up to isomorphism.

(2) loc dim^{T} (Λ-mod) = 0.

(3) For every semisimple sequence 𝕊 with top T, there exists at most one module M with 𝕊(M) = 𝕊, up to isomorphism.

(4) For every semisimple sequence 𝕊 with top T, the variety 𝔊𝔯𝔞𝔰𝔰(𝕊) is either empty or irreducible of dimension

μ_{T} (M) − dim_{K} End_{Λ}(M)

for some (equivalently, for all) M with radical layering 𝕊.

(5) For every skeleton σ with top T, the variety 𝔊𝔯𝔞𝔰𝔰(σ) is either empty or consists of a single Aut_{Λ}(P)-orbit.

If these equivalent conditions are satisfied, the varieties 𝔊𝔯𝔞𝔰𝔰(𝕊), where 𝕊 ranges through the semisimple sequences with top T, are either empty or isomorphic to full affine spaces.

Proof. (1) ⟹ (2). Adopt (1) and assume 𝕊 to be a semisimple sequence with top T such that 𝔊𝔯𝔞𝔰𝔰(𝕊) ≠ ∅. If C is an irreducible component of 𝔊𝔯𝔞𝔰𝔰(𝕊), then C consists of only finitely many fibres of ϕ by hypothesis. All of them are closed in 𝔊𝔯𝔞𝔰𝔰(𝕊) by Corollary 4.3, and a fortiori closed in C. Therefore C consists of a single fibre. In particular, dim C equals the generic fibre dimension of ϕ on C. Since, for each d, the irreducible components of 𝔊𝔯𝔞𝔰𝔰_{d}^{T} are among the closures, in 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, of the irreducible components of the 𝔊𝔯𝔞𝔰𝔰(𝕊)’s covering 𝔊𝔯𝔞𝔰𝔰_{d}^{T}, we conclude that loc dim^{T} (Λ-mod) = 0.

(2) ⟹ (1). Given (2), it suffices to prove that, for any semisimple sequence 𝕊 with top T, there are only finitely many modules M with 𝕊(M) = 𝕊 up to isomorphism; indeed, there are just finitely many such sequences 𝕊, since their total dimensions are all bounded by dim_{K} P. Assume 𝔊𝔯𝔞𝔰𝔰(𝕊) ≠ ∅ and let C be an irreducible component of 𝔊𝔯𝔞𝔰𝔰(𝕊).

If C ∈ C has an Aut_{Λ}(P)-orbit of maximal dimension, then dim Aut_{Λ}(P).C = dim C by hypothesis, and since Aut_{Λ}(P).C is irreducible and closed in 𝔊𝔯𝔞𝔰𝔰(𝕊) by Proposition 4.1, we infer that C = Aut_{Λ}(P).C, i.e., ϕ(C) is a singleton.

(1) ⟹ (3). We assume (1), let 𝕊 = (𝕊_{0}, …, 𝕊_{L}) be any semisimple sequence with top T such that 𝔊𝔯𝔞𝔰𝔰(𝕊) is nonempty, and C ∈ 𝔊𝔯𝔞𝔰𝔰(𝕊). We show by induction on the dimension d of 𝕊 that M = P/C is the only local module with semisimple sequence 𝕊, up to isomorphism. Let k be the largest integer with 𝕊_{k} ≠ 0. Without loss of generality, d ≥ 2, which means that k ≥ 2, because dim_{K} 𝕊_{0} = dim_{K} T = 1. Then M′ = M/J^{k}M is local with semisimple sequence 𝕊(M′) = (𝕊_{0}, …, 𝕊_{k−1}, 0, …, 0), and by the induction hypothesis, M′ is unique with this property. This focuses our attention on the following subset of Mod_{d}^{Λ}, namely on

Ext = {x ∈ Mod_{d}^{Λ} | ∃ an exact sequence 0 → 𝕊_{k} → R(x) → M′ → 0}.

By [1, 6.3], Ext is an irreducible subset of Mod_{d}^{Λ}. Therefore Ext ∩ Top_{d}^{T} is irreducible as well; indeed, due to the upper semicontinuity of the maps dim_{K} Hom_{Λ}(−, S_{r}), this intersection is open in Ext. Moreover, our construction guarantees that

Layer(𝕊) = Ext ∩ Layer(𝕊) = (Ext ∩ Top_{d}^{T} ) \ ( ⋃_{\underline{dim} 𝕊′>\underline{dim} 𝕊} Layer(𝕊′)).

Since the union ⋃_{\underline{dim} 𝕊′>\underline{dim} 𝕊} Layer(𝕊′) is closed in Top_{d}^{T} by Lemma 2.8(i) and Proposition 2.5, we infer that Layer(𝕊) is again irreducible. Hence so is 𝔊𝔯𝔞𝔰𝔰(𝕊) by Proposition 2.5. In light of our hypothesis, 𝔊𝔯𝔞𝔰𝔰(𝕊) consists of finitely many orbits, one of which is Aut_{Λ}(P).C, and the fact that all of these orbits are closed in 𝔊𝔯𝔞𝔰𝔰(𝕊) by part (3) of Proposition 4.1 thus yields 𝔊𝔯𝔞𝔰𝔰(𝕊) = Aut_{Λ}(P).C. This finishes the induction and establishes (3).

(3) ⟺ (4) is immediate from Proposition 4.1(3), as is the fact that (3) implies the supplementary statement. The implications (3) ⟹ (5) ⟹ (2) are obvious. □

As a special case we recover the following result of Bongartz in [2]:

Corollary 5.3. Let 𝕊 = (𝕊_{0}, …, 𝕊_{L}) be a semisimple sequence with the property that each nonzero 𝕊_{i} is simple. If Λ has only finitely many uniserial modules, up to isomorphism, there is at most one uniserial left Λ-module with sequence 𝕊 of consecutive composition factors. □

The following extension of Example 2.4 shows that, even in case loc dim^{T} (Λ-mod) = 0, the varieties 𝔊𝔯𝔞𝔰𝔰_{d}^{T} may have arbitrarily large dimension.

Example 5.4. For any m ≥ 1, we present a finite dimensional algebra Λ, together with a simple left Λ-module T, such that loc dim^{T} (Λ-mod) = 0 while 𝔊𝔯𝔞𝔰𝔰_{m+2}^{T} contains a copy of ℙ^{m}. Let Λ = Q/I, where Q is the quiver

[figure: 1; ω; α; 2]

and I the ideal generated by ω^{m+1}. Moreover, let T = Λe_{1}/Je_{1}. Using Theorem 5.2, one readily verifies that loc dim^{T} (Λ-mod) = 0. Via the methods of [8], one moreover checks that 𝔊𝔯𝔞𝔰𝔰_{m+2}^{T} contains precisely m + 1 distinct orbits representing modules with dimension vector (m + 1, 1); these are isomorphic to 𝔸^{0}, 𝔸^{1}, …, 𝔸^{m} and correspond to the submodules C_{j} = ∑_{0≤i≤m, i≠j} Λαω^{i} of Λe_{1}, respectively. From Lemma 2.8 and Theorem 4.2 one thus derives by induction that the orbit closures are stacked into one another as follows:

\overline{Aut_{Λ}(P).C_{j}} = ⋃_{i≤j} Aut_{Λ}(P).C_{i}.

Here is a sketch: Given that Aut_{Λ}(P).C_{1} ≅ 𝔸^{1}, this orbit fails to be closed, and hence its closure is Aut_{Λ}(P).C_{1} ∪ Aut_{Λ}(P).C_{0}. Similarly, the closure of Aut_{Λ}(P).C_{2} is seen to contain Aut_{Λ}(P).C_{1}, and so on. In particular, we conclude that the closure of Aut_{Λ}(P).C_{m} is isomorphic to ℙ^{m}. □

The stacking phenomenon, observed in Example 5.4, is typical of finite local representation type at T. Namely, in that case the modules represented by the closure of an orbit in any 𝔊𝔯𝔞𝔰𝔰_{d}^{T} are linearly ordered by the degeneration order (see [8]).

6. Proof of Theorem 3.14

We start by showing that χ_{σ} is well-defined and surjective and then check that ψ_{σ} is a bijective set map. That ψ_{σ} is actually an isomorphism of varieties can be verified by an argument modeled on that of Bongartz and the author in [4, Theorem A]; we leave it to the reader to make the obvious modifications. That the triangle of maps commutes, is obvious.

It will be convenient to write σ^{(r)} for the set of paths in σ which start in the vertex e_{r} and to let σ_{l}, resp. σ_{l}^{(r)}, be the subset of σ, resp. of σ^{(r)}, consisting of the paths of length l; accordingly, σ_{l} = ⋃_{1≤r≤t} σ_{l}^{(r)} and σ = ⋃_{0≤l≤L} σ_{l}.

For well-definedness of χ_{σ}, we need to show that, for any point c ∈ Aff(σ), the factor module M(c) = P/U(c) has skeleton σ. Set x_{r} = e_{r} + U(c) ∈ M(c) for 1 ≤ r ≤ t. Clearly, M(c) is generated by the elements px_{r} for r ≤ t and p ∈ σ^{(r)}. Indeed, with the argument backing Lemma 3.12, one checks that, given any path q and r ≤ t, the element qx_{r} of M(c) is a K-linear combination of terms px_{s} with p ∈ σ^{(s)}, s ≤ t. So it suffices to verify that, for each l ∈ {0, …, L}, the subset ⋃_{1≤r≤t}{px_{r} | p ∈ σ_{l}^{(r)}} of M(c) is linearly independent modulo J^{l+1}M(c). On the assumption that this fails, let l_{0} be minimal with respect to failure for some factor algebra of Λ and some point c in a suitable Aff(σ). It is harmless to assume that l_{0} = L. For otherwise we can enlarge the ideal I ⊆ KQ so as to include all paths of length l_{0} + 1, replace σ by (⋃_{l≤l_{0}} σ_{l}), and let d be the cardinality of this union; in this modified setup l_{0} is still minimal with respect to failure of our independence condition. So we only need to refute the assumption that the set σ_{L}^{(1)}x_{1} ∪ ⋯ ∪ σ_{L}^{(t)}x_{t} is linearly dependent in M(c).

To obtain a convenient framework for a comparison of coefficients, consider the projective left ideal P̂ := KQe_{1} ⊕ ⋯ ⊕ KQe_{t} of KQ and note that P̂/IP̂ ≅ P. Further, we let V (c) be the submodule of P̂ generated by the differences αp − ∑_{q∈σ(α,p)} c_{αp,q}q, where (α, p) ranges through the σ-critical pairs. Viewed as a KQ-module, M(c) is then isomorphic to the quotient P̂/(V (c) + IP̂).

Next, we note that, for 1 ≤ r ≤ t, the set of paths in KQe_{r} is the disjoint union of the following two sets: The set σ^{(r)}, and that of all paths of the form uαp, where (α, p) is a σ-critical pair with p ∈ σ^{(r)} and u a path of length ≥ 0. This permits us to define a K-linear transformation

F_{c} : P̂ ⟶ P̂

as follows: Suppose that p is any path starting in one of the vertices e_{r} for r ≤ t. We set F_{c}(p) = p if p ∈ σ^{(r)}; moreover, we define F_{c}(uαp) = ∑_{q∈σ(α,p)} c_{αp,q}uq in the second case. Our choice of c in Aff(σ) guarantees that F_{c}^{L}(IP̂) = 0. Indeed, if v is any path, of length l say, Lemma 3.12 yields a congruence

v ˆ= ∑_{q∈σ} τ_{q}^{v}(X)q

in the polynomial ring A over KQ, where the τ_{q}^{v}(X) are suitable polynomials in K[X_{αp,q}]. Along the induction backing Lemma 3.12, we then find F_{c}^{l}(v) = ∑_{q∈σ} τ_{q}^{v}(c)q. Hence, if ρ ∈ IP̂ = Ie_{1} + ⋯ + Ie_{t}, then F_{c}^{L}(ρ) = ∑_{q∈σ} τ_{q}^{ρ}(c)q = 0 because the τ_{q}^{ρ}(X) annihilate all points in Aff(σ) by construction. It follows that F_{c}^{L}(IP̂) = 0. Next we observe that, for each element z ∈ P̂, the difference z − F_{c}(z) belongs to V (c), which makes V (c) invariant under F_{c}.

By assumption, there is a finite list p_{1}, …, p_{s} of distinct paths in σ, say p_{i} ∈ σ^{(r_{i})}, together with nonzero scalars k_{1}, …, k_{s}, such that the linear combination ∑_{1≤i≤s} k_{i}p_{i}x_{r_{i}} vanishes in M(c). On moving to the above presentation of M(c) as a left KQ-module, we infer that the element ∑_{1≤i≤s} k_{i}p_{i} of P̂ is a sum z + a with z ∈ V (c) and a ∈ IP̂. Now F_{c}^{L} annihilates a and maps z back to V (c), while leaving the sum ∑_{1≤i≤s} k_{i}p_{i} fixed, thus yielding ∑_{1≤i≤s} k_{i}p_{i} ∈ V (c). Since any path including a non-route on σ as a right subpath is again a non-route, this yields an equality

(†)  ∑_{1≤i≤s} k_{i}p_{i} = ∑_{(α,p) σ-critical} ∑_{k≥1} b_{αp,k}u_{αp,k}(αp − ∑_{q∈σ(α,p)} c_{αp,q}q),

for suitable scalars b_{αp,k} and paths u_{αp,k} of lengths ≥ 0, starting in the endpoint of α, respectively; clearly, we may assume that u_{αp,j} ≠ u_{αp,k} for j ≠ k. This being an equality in P̂, we are now in a position to compare coefficients.

In doing this, the concept of a route on σ, introduced in Remark 3.15, will come in handy. Since none of the paths u_{αp,k}αp on the right-hand side of (†) belongs to σ, each p_{i} on the left-hand side must equal one of of the paths u_{αp,k}q. Moreover, one observes that, whenever we have an equality p_{i} = u_{αp,k}q for some q ∈ σ(α, p), the path u_{αp,k}αp is a route on σ. We can therefore find a σ-critical pair (α_{0}, p_{0}) such that p_{0} ∈ σ has minimal length with respect to the following property: There exists an index, say k = 1, with

• b_{α_{0}p_{0},1} ≠ 0 and such that

• u_{α_{0}p_{0},1}α_{0}p_{0} is a route on σ. Set w = u_{α_{0}p_{0},1}α_{0}p_{0} and note that w ∉ σ. The left-hand side of (†) being a K-

linear combination of paths in σ, the path w must cancel out of the right-hand side.

Observe that w does not equal any path of the form u_{αp,k}αp with (α, p) ≠ (α_{0}, p_{0}), for p_{0} is the longest right subpath of w which belongs to σ; nor does w coincide with one of the paths u_{α_{0}p_{0},k}α_{0}p_{0} for k ≠ 1. Consequently, w must be one of the u_{αp,k}q for some σ-critical pair (α, p), some choice of q ∈ σ(α, p), and some k with b_{αp,k} ≠ 0. In particular, this makes u_{αp,k}q a route on σ, which entails that u_{αp,k}αp is also a route on σ. We infer length(p) ≥ length(p_{0}) by the minimality of length(p_{0}), and further deduce length(q) > length(p_{0}) because q belongs to σ(α, p). On the other hand, the equality

w = u_{α_{0}p_{0},1}α_{0}p_{0} = u_{αp,k}q

implies length(p_{0}) ≥ length(q), once again due to the maximal length of p_{0} as a right subpath of w that belongs to σ. This contradiction shows our original assumption ∑_{i≤s} k_{i}p_{i}x_{r_{i}} = 0 to be absurd. Hence σ is a skeleton of M(c) as claimed, which proves well-definedness of χ_{σ}.

To see that χ_{σ} is a surjection, let M be any Λ-module with skeleton σ; in particular, this ensures that M is d-dimensional with top T. Choose a sequence y_{1}, …, y_{t} of top elements of M with y_{r} = e_{r}y_{r}. For each l, the products py_{r} for p ∈ σ_{l}^{(r)} and 1 ≤ r ≤ t then form a basis for J^{l}M/J^{l+1}M. Moreover, for any σ-critical pair (α, p), with p ∈ σ^{(r)} say, let c_{αp,q} be the unique scalars such that αp y_{r} = ∑_{s≤t} ∑_{q∈σ(α,p)∩σ^{(s)}} c_{αp,q}q y_{s}. It clearly suffices to show that the corresponding point c = (c_{αp,q}) ∈ 𝔸^{N} belongs to the variety Aff(σ), for it is then clear that M ≅ P/U(c); just send y_{r} to e_{r} + U(c). So let us check c ∈ Aff(σ). For that purpose, suppose v ∈ Ie_{r} for some r between 1 and t. Viewing v as an element of A, consider the expansion v ˆ= ∑_{q∈σ} τ_{q}^{v}(X) q with τ_{q}^{v}(X) ∈ K[X] supplied by Lemma 3.12. In tandem with the inductive procedure justifying the lemma, one obtains the following equalities in M:

v y_{r} = ∑_{s≤t} ∑_{q∈σ^{(s)}} τ_{q}^{v}(c) q y_{s};

here we refer to the induced KQ-structure of M. Consequently, the linear independence of the elements q y_{s}, for q ∈ σ^{(s)} and s ≤ t, yields vanishing of all the values τ_{q}^{v}(c). This means that c ∈ Aff(σ) as asserted and shows that χ_{σ} is indeed a surjection.

That the map ψ_{σ} is injective, follows from the first part of the proof; indeed, it suffices to observe that, for any c ∈ Aff(σ), the paths in σ are linearly independent modulo U(c). To verify surjectivity, let C ∈ 𝔊𝔯𝔞𝔰𝔰(σ). Then P/C has skeleton σ and top elements e_{r} +C. Therefore, as explained in the surjectivity argument for ψ_{σ}, there exists a point c ∈ Aff(σ) with the property that, for any σ-critical pair (α, p) with p ∈ σ^{(r)},

αp(e_{r} + C) = ∑_{s≤t} ∑_{q∈σ(α,p)∩σ^{(s)}} c_{αp,q}q(e_{s} + C).

In other words, U(c) ⊆ C. But we already know that U(c) has codimension d in P as does C, and hence ψ_{σ}(c) = U(c) = C. This completes the proof of Theorem 3.14. □

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Department of Mathematics, University of California, Santa Barbara, CA 93106

E-mail address: birge@math.ucsb.edu