THE GEOMETRY OF UNISERIAL REPRESENTATIONS

OF FINITE DIMENSIONAL ALGEBRAS I

Birge Huisgen-Zimmermann

1. Introduction

This is the first part of a trilogy which is to lay the foundations for a geometric approach to the uniserial representations of finite dimensional algebras. The need for a solid understanding of the full class of uniserials arose within a program aimed at approximating finitely generated modules by modules of a simpler structure, the basic building blocks of ‘helpful’ approximations being uniserial modules (cf. [6]). Of course, a uniserial module viewed by itself does not hold much interest, as is the case for a point on a curve when considered outside the context of the curve. In classifying families of uniserial modules, the interest lies in the number of and interplay among the parameters which offer themselves for the description of these modules.

For good focus, let us start by recording two problems which have been propagated by Maurice Auslander since the 1970’s and have now appeared among the eleven open problems stated in [2, pp. 411-412]:

(1) Give a method for deciding when two uniserial modules over an artin algebra are isomorphic.

(2) Which artin algebras of infinite representation type have only a finite number of pairwise nonisomorphic uniserial modules?

Specializing to finite dimensional algebras over algebraically closed fields, we present a solution to problem 1 and supply the tools to tackle question 2 in [8]. More generally, our results address split basic finite dimensional algebras Λ over an arbitrary field K, i.e., algebras of the form KΓ/I, where Γ is a quiver and I an admissible ideal in the path algebra KΓ. Given such a ‘coordinatization’ of Λ, we introduce irreducible affine algebraic varieties over K, the points of which parametrize the isomorphism types of uniserial Λ-modules, and the geometry of which reflects the constraints met in constructing such modules. While being a priori defined in terms of quiver and relations of Λ, this list of varieties will turn out to be uniquely determined by the isomorphism type of Λ, up to order and birational equivalence [5].

There are several ‘classical’ ways of viewing representations as points of algebraic varieties; all of these varieties contain classes of uniserial representations as open subvarieties. The most time-honored procedure is to fix a K-basis for Λ, say λ_{1}, …, λ_{m}, and to define the variety 𝕄od_{d}(Λ) of d-dimensional Λ-modules as a closed subvariety of M_{d}(K)^{m} where the matrix in the i-th slot represents module multiplication by λ_{i}. The sets of points

Typeset by AMS-TEX

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

corresponding to the isomorphism classes of modules arise as orbits under the obvious GL_{d}(K)-action (c.f. [9, §§12.16, 12.17]). It is not difficult to see that the d-dimensional uniserial modules with a fixed sequence of consecutive composition factors form an open subvariety of 𝕄od_{d}(Λ), which is invariant under the canonical GL_{d}(K)-action. However, these varieties are very large, and the information they contain is encoded in the GLactions. In contrast, the ones which we will introduce and study here provide a close fit for the uniserial modules. In many cases, they are isomorphic to the geometric quotient of the above-mentioned uniserial subvarieties of 𝕄od_{d}(Λ) modulo the GL-action; in particular, there is then a 1–1 correspondence between points and isomorphism types. When there is not, the ‘slack’ occurring in the varieties turns out to be fairly harmless.

For more detail, let us concentrate on the class of uniserial representations of length l+1 having a fixed sequence 𝕊 = (S(1), …, S(l + 1)) of simple composition factors. It turns out that there is a natural subdivision of this class, possibly with overlaps, so that each of the segments is described by an irreducible affine variety. A primary, somewhat rougher, subdivision is in terms of ‘masts’, as follows. Clearly, for each uniserial Λ-module U with sequence 𝕊 of consecutive composition factors, there exists at least one path p of length l in KΓ such that pU ≠ 0; necessarily p passes in order through the sequence (e(1), …, e(l+1)) of those vertices in Γ which represent the simple modules S(i). (Here we consider U as a KΓ-module in the obvious fashion.) Each such path will be called a mast for U, and for each mast p we will construct an affine variety V_{p} over K – not necessarily irreducible – and a canonical surjection Φ_{p} from V_{p} onto the set of isomorphism types of uniserial left Λ-modules with mast p (Section 3). If Γ does not have any double arrows, there is clearly at most one path p passing through the sequence (e(1), …, e(l +1)) of vertices, and the variety V_{p} parametrizes the full set of isomorphism classes of uniserial modules with composition sequence 𝕊. In case Γ does contain double arrows, several varieties of the form V_{p} are needed to account for the uniserial modules with composition sequence 𝕊 in general. In this case, the irreducible components of all the pertinent V_{p}’s are combined into a family V_{𝕊} of varieties, as discussed below. Nonetheless, the ‘packaging’ of irreducible components in terms of masts remains the most accessible for both proofs and computations and hence will play a crucial role in our work.

Loosely speaking, the points of V_{p} are – as in the classical approach – strings of coordinate vectors determining module multiplication. But the bases for the (l + 1)-dimensional K-space underlying the representations in the image of Φ_{p} used here are being shifted from one point of V_{p} to the next, in a fashion that is tied to the multiplication of the uniserial modules labeled by these points. While we lose the natural GL_{l+1}-action coming with 𝕄od_{l+1}(Λ) in the process, we gain, among other things, a particularly transparent connection between the points of V_{p} and the graphs of the uniserial modules they represent, as well as a geometric picture which cleanly shows how the relations in the ideal I impinge on the interplay of the parameters of the uniserials. As another bonus of restricting our focus to uniserial representations, we can make do with a small selection of paths from KΓ as representative ‘multipliers’ and with bases for small K-subspaces of the modules considered in order to pin down the effect of multiplication. The main price we pay for tight fit and manageability, on the other hand, lies in the fact that independence of the chosen coordinate system for Λ is not at all self-evident in this setting. For instance, certain irreducible components of V_{p} may shift to a different variety V_{q} under a change of coordinatization. However, if we denote by V_{𝕊} the full collection of all the irreducible components of the varieties V_{q}, where q runs through the paths of length l that pass through the sequence (e(1), …, e(l + 1)), the isomorphism type of Λ uniquely determines V_{𝕊} up to birational equivalence. In case Γ has no double arrows, the irreducible components in V_{𝕊} are even unique up to isomorphism. Uniqueness will be proved in a subsequent joint note with K. Bongartz [5]. It turns out that translating the varieties considered here into certain closed subvarieties of 𝕄od_{l+1}(Λ) provides the most convenient setting for this purpose.

To understand the emerging picture, we further require an in-depth study of intersections of the form Φ_{p}(V_{p})∩Φ_{q}(V_{q}), where p and q are two paths of length l running through the vertices (e(1), …, e(l + 1)) in that order. We prove that, if X and Y are irreducible components of V_{p} and V_{q} respectively, such that Φ_{p}(X) ∩ Φ_{q}(Y) ≠ ∅, then X and Y are birationally equivalent, and consequently one of them can be deleted from V_{𝕊} as ‘a double’ without loss of information (Section 5).

In case the path p ∈ KΓ does not start with an oriented cycle, the surjection

Φ_{p} : V_{p} → {isomorphism types of uniserials in Λ -mod with mast p}

is a bijection; as a consequence, the points of V_{p} serve as complete isomorphism invariants for the uniserial modules with mast p in this situation. On the other hand, Φ_{p} may fail to be bijective in the presence of certain types of oriented cycles within p. In Section 4, we characterize the fibres of the maps Φ_{p}, thus providing a general solution to the isomorphism problem. It turns out that there is a system of equations with the following property: on insertion of arbitrary points P and Q of V_{p}, it specializes to a linear system in the remaining variables, the consistency of which is equivalent to ‘Φ_{p}(P) ≅ Φ_{p}(Q)’. This classification is quite gratifying since the varieties V_{p}, as well as the pertinent systems of equations, are very accessible – they can be readily computed on the basis of quiver and relations for Λ – and the points P of V_{p} store information on the modules Φ_{p}(P) in an easily decodable form. We note that K. Bongartz proved in [4] that the isomorphism problem for any pair of finite dimensional modules can be resolved in a finite number of steps through a Gaussian elimination process. In contrast, our procedure, being tailored specifically for uniserial modules, avoids the complications involved in handling arbitrary finite dimensional modules.

In a nutshell, Section 6 is devoted to showing that every affine variety over K arises as the geometric quotient, modulo the GL-action, of the open subvariety of 𝕄od_{l+1}(Λ) consisting of the points representing uniserial modules with fixed sequence 𝕊 of composition factors. In our present terminology: Given any affine variety V over K, the family of irreducible components of V can be realized as a V_{𝕊} for a suitable finite dimensional algebra Λ = KΓ/I – with Γ acyclic – and a suitable sequence 𝕊 of simple Λ-modules. We can even assume that Γ is without double arrows, which implies that the family V_{𝕊} consists precisely of the irreducible components of a single variety V_{p} which is isomorphic to V on one hand, and uniquely determined up to isomorphism by Λ on the other. Recall, moreover, that acyclicity of Γ entails bijectivity of the corresponding natural map Φ_{p} as discussed above.

The author would like to thank Axel Boldt for numerous helpful comments on a first draft of this paper.

2. Preliminaries

Throughout, K will stand for an arbitrary field, and Λ ≅ KΓ/I will be a finite dimensional path algebra modulo relations over K; here Γ is a quiver and I is an admissible ideal in the path algebra KΓ. Our convention for the composition of paths p, q ∈ KΓ is as in [2], namely, qp stands for ‘q after p’ whenever the concatenation is defined. Moreover, J will denote the Jacobson radical of Λ. As is well known, the ideal I factored out of KΓ is, in general, not an isomorphism invariant of Λ, but depends on the choice of a complete set e_{1}, …, e_{n} of primitive idempotents and on the choice of ‘arrows’ from e_{i} to e_{j}, that is, of elements from e_{j}Je_{i} which give rise to a K-basis for e_{j}Je_{i}/e_{j}J^{2}e_{i}. Any choice of an ideal I in KΓ, together with an isomorphism from KΓ/I onto Λ, will be called a coordinatization of Λ. For simplicity, we will assume that Λ is equal to KΓ/I, unless otherwise specified, and identify the vertices of Γ with the corresponding primitive idempotents e_{1}, …, e_{n} of Λ. Our Λ-modules will be left modules throughout.

Definition 1. Given a uniserial Λ-module U of length l + 1, any path p of length l in KΓ with pU ≠ 0 is called a mast of U.

Elementary Observations. 1. If the quiver Γ has no double arrows, then each uniserial Λ-module has a unique mast. Conversely, the uniqueness of masts in all uniserial modules implies absence of double arrows.

2. Not every path in KΓ with nonzero image in Λ needs to occur as a mast of a uniserial module. For example, if Λ = KΓ/I, where Γ is the quiver

[figure: 2; β; 1; α; γ; 3; 4; δ]

and I = ⟨βα − δγ⟩, then neither βα nor δγ is a mast of a uniserial Λ-module.

3. Of course, this concept of mast depends on a given coordinatization of Λ. Our choice of coordinates will therefore impinge on the varieties of uniserials with given mast, to be introduced in the next section. As a consequence, the effect of a coordinate change needs to be discussed (see [5]).

Let M be a Λ-module. A top element of M is an element x ∈ M \ JM with e_{i}x = x for some i ∈ {1, …, n}; in that case, x will also be called a top element of type e_{i}. Clearly, given a uniserial Λ-module U of length l + 1 with top element x, a path p ∈ KΓ of length l is a mast of U if and only if px ≠ 0. A useful tool in visualizing and communicating uniserial modules will be their labeled and layered graphs. Suppose that p = α_{l} ⋯ α_{1} where each arrow α_{i} has starting point e(i) and endpoint e(i + 1). The labeled and layered graph of U with respect to a top element x and a mast p consists of (a) the mast p, drawn vertically with e(1) at the top and edges labeled by the arrows α_{i}, together with (b) an edge labeled ω from e(i) to e(j) whenever i < j and ω is an arrow from e(i) to e(j) such that Λωα_{i−1} ⋯ α_{1}x = J^{j−1}U.

For example, let Γ be the quiver

[figure: 1; α; β; ϵ; 2; δ; γ; 3]

and Λ = KΓ/⟨α^{2}, γ^{2}⟩. That the (layered and labeled) graph of a uniserial module U with top element x be

1_{α}  _{β}  _{ϵ} 1_{β}2_{γ}2_{δ}3

means that U has mast p = δγβα, that Λβx = J^{2}U, and Λϵx = J^{4}U; in other words, βx is congruent to a nonzero scalar multiple of βαx modulo J^{3}U, and ϵx is a nonzero scalar multiple of px. Observe that these layered and labeled graphs are, in general, not completely determined by the isomorphism types of the modules they represent, but may depend on the choice of top element. For instance, if Λ is as above and U = Λe_{1}/(Λγβα + Λ(β −βα)+Λϵ), then the graphs of U relative to the top elements x = \overline{e}_{1} and y = \overline{e}_{1} −α\overline{e}_{1} are

1_{α}  _{β}  1_{α} 1_{β}  and  1_{β} 2  2

respectively. Conversely, layered and labeled graphs of uniserial modules do not pin these modules down up to isomorphism. Indeed, if Λ = KΓ, where Γ is the Kronecker quiver

[figure: 1; α; β; 2]

then any uniserial module U_{k} = Λe_{1}/Λ(β − kα) for k ∈ K \ {0} has graph

1_{α} _{β} 2

while obviously U_{k} ≅ U̸_{l} for k ≠ l.

Finally, we will speak of subpaths of a path p ∈ KΓ: A path q ∈ KΓ is a right subpath (respectively, left subpath) of p if there exists a path r with p = rq (respectively, p = qr). In particular, if p is a path from a vertex e(1) to a vertex e(2), then e(1) is the unique right subpath of length zero of p, and e(2) is the unique left subpath of length zero. A proper right subpath of p is a right subpath which is strictly shorter than p. It will be convenient to communicate the statement ‘q is a right subpath of p’ in the form ‘p = •q’ to arrive at compact formulas.

For the geometric terms we use, the reader should consult the introductory texts by Hartshorne and Mumford ([7],[10]).

3. Description of the varieties of uniserials with fixed mast

Throughout this section, let p be a path of length l from a vertex e(1) to a vertex e(l + 1). We will use p to denote both the element in KΓ and its residue class in Λ, unless there is a danger of ambiguity.

Roughly speaking, the affine K-variety V_{p} corresponding to the uniserial Λ-modules with mast p consists of points that are families of coordinate vectors of the following type: Given a uniserial left Λ-module U with mast p and top element x, we equip U with the Kbasis p_{i}x, for p = •p_{i}, and string up the coordinate vectors of the elements qx, where q runs through the paths in KΓ not vanishing in Λ. Of course, these coordinate strings pin down the corresponding uniserials. The first point to be addressed is the fact that these strings of coordinate vectors actually form an affine algebraic variety V_{p} which, in fact, is readily accessible on the basis of a coordinatization of Λ. A crude outline of the procedure for assembling a finite set of polynomials which defines V_{p} is as follows: Replace the scalars k_{i,q} arising in the linear dependence relations qx = ∑_{p=•p_{i}} k_{i,q}p_{i}x inside an arbitrary uniserial module with mast p and top element x by generic indeterminates X_{i,q}, and expand a representative set of relations from I by means of substitutions q = ∑_{p=•p_{i}} X_{i,q}p_{i} inside the polynomial ring KΓ[X_{i,q}]. A repetition of this substitution process will eventually reduce the relations to equations of the form ∑_{p=•p_{i}} τ_{i}p_{i} = 0, where the τ_{i} are polynomials in K[X_{i,q}]. Reflecting the fact that the elements p_{i}x are K-linearly independent, our interest will be in the simultaneous vanishing set of these polynomials τ_{i}. What makes the resulting affine variety fairly manageable is the fact that we can significantly reduce the set of variables without renouncing information.

Next we give a formal description of the variety V_{p} and the canonical map Φ_{p} from V_{p} onto the set of isomorphism classes of uniserial Λ-modules with mast p. Instead of considering all the indeterminates X_{i,q} mentioned above, we will restrict our attention to those of the form X_{i,αu}, where (α, u) is a ‘detour’ as defined below.

Definitions 2.

1. A detour on the path p is a pair (α, u), where α is an arrow and u is a right subpath of p (length 0 being allowed) such that

(i) αu ≠ 0 in KΓ,

(ii) αu is not a right subpath of p in KΓ, but

(iii) there exists a right subpath v of p with length(v) ≥ length(u) + 1 such that the endpoint of v coincides with the endpoint of α.

(See Figure (*) below.) For ease of notation, we will often abbreviate the statement ‘(α, u) is a detour on p’ by ‘(α, u) ≀≀ p’.

2. Suppose that p is a path of length l that passes consecutively through the vertices e(1), …, e(l + 1). A route on p is any path in KΓ which starts in e(1) and passes through a subsequence of the sequence (e(1), …, e(l + 1)). (We include e(1) and p in the set of routes on p.)

Remarks. (a) In particular, each right subpath of p is a route on p. More generally, any right subpath of a route is again a route.

(b) In our work with routes, the following factorization property will be crucial: A path r ∈ KΓ is a route on p if and only if it can be written in the form

r = r′α_{m}u_{m} ⋯ α_{1}u_{1}

for some m ≥ 0 such that there exists a corresponding factorization

p = p′ · w_{m} ⋯ w_{1}

of p with the property that (α_{i}, u_{i}) is a detour on w_{i} with endpoint(α_{i}) = endpoint(w_{i}) for each i ≤ m, and r′ is a right subpath of p′. (See Figure (**) below.) Note that such a factorization of p corresponding to the given route r need not be unique.

(c) The length of any route r on p is bounded above by length(p).

(d) Whenever rw is a route on pw, then r is a route on p.

[figure: u; u; v; p; w_{1}; u_{1}; p; α_{1}; α; w_{2}; u_{2}; α_{2}; p′; r′; Figure (*); Figure (**)]

We start by describing the polynomial ring in which we will be working. Given any detour (α, u) on p, let

V (α, u) = {v_{i}(α, u) | i ∈ I(α, u)}

be the family of right subpaths of p in KΓ which are longer than u and have the same endpoint as α. In most cases, it will be more convenient to refer to the index set I(α, u) for V (α, u) than to the set V (α, u) itself. The polynomial ring of our choice will then be

KΓ[X] = KΓ[X_{i}(α, u) | i ∈ I(α, u), (α, u) ≀≀ p]

with coefficients in the path algebra KΓ and independent variables X_{i}(α, u). Next we introduce an equivalence relation on KΓ[X] as follows: Let L(p) be the left ideal of KΓ[X] generated by all the paths q in KΓ which fail to be routes on p together with all the differences

αu − ∑_{i∈I(α,u)} X_{i}(α, u) · v_{i}(α, u)

for detours (α, u) on p. Then, clearly, the relation ‘σ ˆ= τ ⟺ σ − τ ∈ L(p)’ for σ, τ ∈ KΓ[X] defines a congruence relation relative to addition and left multiplication. The proof of the following easy observation is left to the reader.

Observation 3. Each element of the path algebra KΓ is ˆ=-congruent to a unique element of the form ∑_{p=•p′} τ_{p′}(X)p′, where the τ_{p′}(X) are polynomials in

K[X] = K[X_{i}(α, u) | i ∈ I(α, u), (α, u) ≀≀ p]. □

To obtain these polynomials τ_{p′}(X) for a given element z ∈ KΓ algorithmically, consider the following substitution equations for p: First, q ˆ= 0 for any path q in KΓ which fails to be a route on p, and second,

αu ˆ= ∑_{i∈I(α,u)} X_{i}(α, u) · v_{i}(α, u)

for all detours (α, u) on p. We use the phrase ‘inserting the substitution equations from the right’ for the following steps:

q′′q′ ˆ= 0  if q′ is a non-route on p;
q′′q′ ˆ= ∑_{i∈I(α,u)} X_{i}(α, u) · q′′v_{i}(α, u)  if q′ = αu with (α, u) ≀≀ p.

Note that if q′ is not a route on p, then neither is q′′q′. Inserting the substitution equations from the right into the paths occurring in the element z ∈ KΓ and repeating this procedure clearly leads to the equivalence z ˆ= ∑ τ_{p′}(X)p′ with τ_{p′}(X) ∈ K[X] after at most d steps, where d is an upper bound on the lengths of the paths involved in z.

Now let L be the Loewy length of Λ – i.e., L is minimal with respect to J^{L} = 0 – and denote by I^{(L)} the K-subspace of I consisting of all elements which can be written as Klinear combinations of paths of lengths at most L. Moreover, choose a finite K-generating set t_{1}, …, t_{s} for the space I^{(L)}. By the above remarks, there are unique polynomials τ_{i,p′}(X) ∈ K[X] with the property that t_{i} ˆ= ∑_{p=•p′} τ_{i,p′}(X)p′ for 1 ≤ i ≤ s. We now give a definition of the variety V_{p} depending on this choice of relations t_{1}, …, t_{s}; this is the most convenient description for purposes of computation. However, as we will immediately observe, this definition of V_{p} is independent of the choice of the t_{i}.

Definition 4. Let V_{p} = V (τ_{i,p′}(X) | 1 ≤ i ≤ s and p = •p′) be the simultaneous vanishing locus of the polynomials τ_{i,p′}(X) in affine N-space 𝔸^{N} = 𝔸^{N}(K), where N = ∑_{(α,u)≀≀p} |I(α, u)|.

Remarks. 1. The affine algebraic set V_{p} is independent of the choice of a K-generating set for the K-space I^{(L)}; in fact, V_{p} is the vanishing locus of all polynomials τ_{p′}(X) arising in congruences z ˆ= ∑_{p=•p′} τ_{p′}(X)p′ for elements z from the ideal I. The former assertion is an immediate consequence of our construction, while the latter can easily be checked as follows: If the length of p exceeds the Loewy length L of Λ, then V_{p} is the empty set, because the right subpath p_{L} of p of length L belongs to I^{(L)}, and hence the congruence p_{L} ˆ= 1 · p_{L} places the constant 1 into the ideal of V_{p}. If, on the other hand, p has length ≤ L, all paths in KΓ of length greater than L are non-routes on p and are hence reduced to zero under our equivalence relation.

2. Our next remark along this line often saves a considerable amount of computational effort. Namely, observe that, whenever r_{1}, …, r_{m} ∈ I generate I as a left ideal of KΓ, then V_{p} = V (ρ_{i,q}(X) | 1 ≤ i ≤ m, p = •q), where r_{i} ˆ= ∑_{p=•q} ρ_{i,q}(X)q with ρ_{i,q}(X) ∈ K[X].

3. A priori, the family of algebraic varieties V_{p}, where p runs through the set of paths in KΓ which do not vanish in Λ, clearly depends on the chosen coordinatization of the split algebra Λ, that is, on a fixed set e_{1}, …, e_{n} of primitive idempotents and K-bases for the spaces e_{i}Je_{j}/e_{i}J^{2}e_{j}. In fact, both the labels of the varieties considered and their realization in affine space depend on the choice of coordinates. This leads us to the following uniqueness problem: If KΓ/I ≅ KΓ/I′, are the varieties V_{p} and V′_{p} formed relative to the two ideals I and I′ isomorphic? While in general there is uniqueness only up to birational equivalence (see Section 5), the answer to this isomorphism question is positive for large classes of algebras (see [5]).

It is easy to compute the varieties V_{p} from a given coordinatization, i.e., from a presentation of Λ in terms of quiver and relations. These varieties will permit us to classify the uniserial modules in terms of the correspondence described in the main theorem of this section (Theorem A). Before we state this theorem, we illustrate the construction of the varieties V_{p}.

Example 5. Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α_{1}; β_{1}; 2; α_{2}; β_{2}; γ; 3; α_{3}; β_{3}; 4; α_{4}; β_{4}; 5]

and I is the ideal of KΓ generated by the following relations:

γ^{3},  β_{2}α_{1},  β_{2}γα_{1},  α_{4}α_{3}β_{2}β_{1},  α_{4}α_{3}β_{2}γ^{2}α_{1} − α_{4}α_{3}α_{2}γ^{2}α_{1},
β_{4}β_{3}α_{2}γα_{1} − α_{4}α_{3}α_{2}β_{1},  α_{4}α_{3}α_{2}γ^{2}β_{1} − α_{4}α_{3}α_{2}α_{1},  α_{4}α_{3}α_{2}γα_{1} + α_{4}α_{3}β_{2}γβ_{1},
α_{4}α_{3}α_{2}γα_{1} − α_{4}β_{3}α_{2}γ^{2}α_{1}, α_{4}α_{3}α_{2}γα_{1} − β_{4}α_{3}α_{2}γ^{2}α_{1}.

Moreover, consider the path p = α_{4}α_{3}α_{2}γ^{2}α_{1}. To compute the variety V_{p}, observe that the detours on p and the corresponding substitution equations are as follows:

(β_{1}, e_{1})  β_{1} ˆ= X_{1}α_{1} + X_{2}γα_{1} + X_{3}γ^{2}α_{1}
(α_{2}, α_{1})  α_{2}α_{1} ˆ= X_{4}α_{2}γ^{2}α_{1}
(β_{2}, α_{1})  β_{2}α_{1} ˆ= X_{5}α_{2}γ^{2}α_{1}
(α_{2}, γα_{1})  α_{2}γα_{1} ˆ= X_{6}α_{2}γ^{2}α_{1}
(β_{2}, γα_{1})  β_{2}γα_{1} ˆ= X_{7}α_{2}γ^{2}α_{1}
(β_{2}, γ^{2}α_{1})  β_{2}γ^{2}α_{1} ˆ= X_{8}α_{2}γ^{2}α_{1}
(β_{3}, α_{2}γ^{2}α_{1})  β_{3}α_{2}γ^{2}α_{1} ˆ= X_{9}α_{3}α_{2}γ^{2}α_{1}
(β_{4}, α_{3}α_{2}γ^{2}α_{1})  β_{4}α_{3}α_{2}γ^{2}α_{1} ˆ= X_{10}p

Next observe that the relations listed above, together with the paths γ^{3}α_{1} and γ^{3}β_{1}, generate I as a left ideal of KΓ; consequently, Remark 2 following Definition 4 tells us that we need only consider these elements of I in determining a generating set of polynomials for the ideal of V_{p}. Since these last two paths, as well as γ^{3}, are non-routes on p, they are ˆ=-equivalent to 0 and hence do not lead to conditions on the indeterminates X_{i}. We now insert the substitution equations into the remaining relations.

First β_{2}α_{1} ˆ= X_{5}α_{2}γ^{2}α_{1}, which, in view of β_{2}α_{1} ∈ I, gives us the equation X_{5} = 0 for V_{p}. Analogously, the combination of β_{2}γα_{1} ˆ= X_{7}α_{2}γ^{2}α_{1} and β_{2}γα_{1} ∈ I implies X_{7} = 0 for the points of V_{p}. Moreover, α_{4}α_{3}β_{2}β_{1} ˆ= α_{4}α_{3}β_{2}(X_{1}α_{1} +X_{2}γα_{1} +X_{3}γ^{2}α_{1}) ˆ= X_{1}X_{5}p+ X_{2}X_{7}p+X_{3}X_{8}p yields X_{1}X_{5}+X_{2}X_{7}+X_{3}X_{8} = 0 on V_{p}. In view of X_{5} = X_{7} = 0, we obtain X_{3}X_{8} = 0. We further compute α_{4}α_{3}β_{2}γ^{2}α_{1} − p ˆ= (X_{8} − 1)p to conclude that X_{8} − 1 = 0, and consequently also X_{3} = 0, on V_{p}. Next, β_{4}β_{3}α_{2}γα_{1} − α_{4}α_{3}α_{2}β_{1} ˆ= X_{6}β_{4}β_{3}α_{2}γ^{2}α_{1} − X_{1}α_{4}α_{3}α_{2}α_{1} − X_{2}α_{4}α_{3}α_{2}γα_{1} − X_{3}α_{4}α_{3}α_{2}γ^{2}α_{1} ˆ= X_{6}X_{9}X_{10}p − X_{1}X_{4}p − X_{2}X_{6}p − X_{3}p, which gives us X_{6}X_{9}X_{10} − X_{1}X_{4} − X_{2}X_{6} − X_{3} = 0, and X_{6}X_{9}X_{10} − X_{1}X_{4} − X_{2}X_{6} = 0 in view of the preceding equations. Analogously, the last four relations yield X_{1} − X_{4} = 0, X_{6} + X_{1}X_{7} + X_{2}X_{8} = X_{6} + X_{2} = 0, X_{6} − X_{9} = 0, and X_{6} − X_{10} = 0, respectively. We deduce that

V_{p} = V (X_{3}, X_{5}, X_{7}, X_{8}−1, X_{6}X_{9}X_{10}−X_{1}X_{4}−X_{2}X_{6}, X_{1}−X_{4}, X_{2}+X_{6}, X_{6}−X_{9}, X_{6}−X_{10}).

Clearly, V_{p} is isomorphic to the variety V (Y^{2} − X^{3} − X^{2}) in affine 2-space. Over the real numbers, we thus obtain the standard α-curve:

In this example, the correspondence between the points of V_{p} and the isomorphism types of uniserial Λ-modules with mast p, as described in Theorem A below, is bijective, even though our path p does include an oriented cycle. □

The following theorem contains basic information about the ‘back and forth’ between points on the variety V_{p} on one hand and uniserial Λ-modules with mast p on the other.

Theorem A. Suppose that p is a path in KΓ starting in the vertex e(1).

(I) There is a surjective map Φ_{p} from the variety V_{p} to the set of isomorphism types of uniserial left Λ-modules with mast p. It assigns to each point k = (k_{i}(α, u))_{i∈I(α,u), (α,u)≀≀p} in V_{p} the isomorphism type of the module Λe(1)/U_{k}, where

U_{k} = ( ∑_{(α,u)≀≀p} Λ(αu − ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u))) + ( ∑_{q not a route on p} Λqe(1)).

Alternately, the uniserial module Λe(1)/U_{k} representing Φ_{p}(k) can be described as the unique uniserial factor module with mast p of the module

Λe(1)/( ∑_{(α,u)≀≀p} Λ(αu − ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u))).

(II) The variety V_{p} is nonempty if and only if there exists a uniserial left Λ-module with mast p.

(III) Provided that p does not have a proper right subpath which is an oriented cycle of positive length, the map Φ_{p} is bijective.

Proof. The following maps will repeatedly prove useful. For any family of scalars k = (k_{i}(α, u))_{i∈I(α,u), (α,u)≀≀p} in 𝔸^{N}, we will consider a K-linear transformation F_{k} : KΓ → KΓ depending on k: Namely, F_{k} sends all the paths which fail to be routes on p to zero and acts on routes q as follows: If q is a right subpath of p, then F_{k}(q) = q, and if q = q′αu, where (α, u) is a detour on p, then F_{k}(q) = ∑_{i∈I(α,u)} k_{i}(α, u)q′v_{i}(α, u). Then, given any element z ∈ KΓ which is a K-linear combination of paths of length at most m, the image of z under (F_{k})^{m} equals ∑_{p=•p′} τ_{p′}(k)p′, where ∑_{p=•p′} τ_{p′}(X)p′ is the unique element in ∑_{p=•p′} K[X]p′ equivalent to z under ‘ˆ=’; here we are working inside the polynomial ring KΓ[X], where X stands for the family of variables X_{i}(α, u). In particular, if z belongs to I, the polynomials τ_{p′}(X) fall into the ideal of V_{p} (by the first of the above remarks) and consequently vanish at k whenever k ∈ V_{p}; in other words, (F_{k})^{m}(z) = 0 in that case.

To any family k = (k_{i}(α, u)) ∈ 𝔸^{N} as above, we moreover assign the left ideal W_{k} of the path algebra KΓ, generated by the paths q which are non-routes on p and by the elements αu − ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u), where (α, u) runs through the detours on p. Observe that, for any element z ∈ KΓ and any choice of k, the difference z − F_{k}(z) belongs to W_{k}; in particular, the subspace W_{k} of KΓ is invariant under F_{k}.

Let Φ = Φ_{p} be defined as in the statement of the theorem and, for k ∈ V_{p}, identify Φ(k) with the module Λe(1)/U_{k}.

Claim 1. For each k ∈ V_{p}, the module Φ(k) is uniserial with mast p.

Indeed, write p = β_{l} ⋯ β_{1}, where the β_{i} are arrows, and set x = e(1) + U_{k} ∈ Φ(k). It clearly suffices to show that Φ(k) has K-basis β_{i} ⋯ β_{1}x, 0 ≤ i ≤ l. Indeed, once this is established, we have J^{l}Φ(k) ≠ 0, and, for reasons of dimension, the radical powers J^{i}Φ(k), 0 ≤ i ≤ l, constitute a composition series for Φ(k).

To see that the elements β_{i} ⋯ β_{1}x, 0 ≤ i ≤ l, form a K-generating set for Φ(k), it suffices to check that, given any route r on p, the element rx ∈ Φ(k) is a K-linear combination of the β_{i} ⋯ β_{1}x. We prove this by induction on d = l − length(u_{1}), where u_{1} is the right subpath of r of maximal length occurring also as a right subpath of p. If d = 0, then r = p, and there is nothing to prove. So suppose d ≥ 1, and let r = r′α_{s}u_{s} ⋯ α_{1}u_{1}. If s = 0, then r is a (proper) right subpath of p, and we are again done. If, on the other hand, s ≥ 1, the definition of Φ(k) yields

rx = ∑_{i∈I(α_{1},u_{1})} k_{i}(α_{1}, u_{1})r′α_{s}u_{s} ⋯ α_{2}u_{2}v_{i}(α_{1}, u_{1})x.

If the path r′α_{s}u_{s} ⋯ α_{2}u_{2}v_{i}(α_{1}, u_{1}) is not a route on p, then r′α_{s}u_{s} ⋯ α_{2}u_{2}v_{i}(α_{1}, u_{1})x = 0 by construction. The induction hypothesis applies to each of the remaining terms r′α_{s}u_{s} ⋯ α_{2}u_{2}v_{i}(α_{1}, u_{1})x, since the lengths of the right subpaths v_{i}(α_{1}, u_{1}) of p exceed the length of u_{1}.

To prove that the elements β_{i} ⋯ β_{1}x for 0 ≤ i ≤ l are linearly independent, it suffices to verify that px = β_{l} ⋯ β_{1}x ≠ 0. Assume the contrary, and consider the following presentation of Φ(k) as a KΓ-module: Φ(k) = KΓe(1)/(Ie(1) + W_{k}e(1)), where W_{k} is as defined above. So px = 0 is equivalent to the existence of an element a ∈ Ie(1) such that p + a ∈ W_{k}. To reach a contradiction, we will first derive that p ∈ W_{k}. Suppose that a is a K-linear combination of paths of lengths at most m. In view of the fact that a ∈ I, the first paragraph of the proof shows that F_{k}^{m}(a) = 0. Moreover, due to the F_{k}-invariance of W_{k}, the fact that p+a ∈ W_{k} yields p = F_{k}^{m}(p+a) ∈ W_{k} as required. But the containment p ∈ W_{k} is in turn impossible, as we now prove.

Assume that p is a K-linear combination of terms q′(αu − ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u)) and terms q′′q, where q is not a route on p and q′, q′′ are paths in KΓ. Since none of the paths of the form αu going with a detour (α, u) on p is a right subpath of p, the above equality forces p to be equal to one of the terms q′v_{i}(α, u), say p = q′_{0}v_{i}(α_{0}, u_{0}). This clearly implies that q′_{0}α_{0}u_{0} is a route on p. Observe that, more generally, q′αu is a route on p, whenever (α, u) is a detour with the property that q′v_{j}(α, u) is a route on p for some j ∈ I(α, u). Let u_{1} be a right subpath of p that has minimal length with respect to the following properties:

(a) for some arrow α_{1}, the pair (α_{1}, u_{1}) is a detour on p, and, for a suitable path q′_{1}, the element q′_{1}(α_{1}u_{1} − ∑ k_{i}(α_{1}, u_{1})v_{i}(α_{1}, u_{1})) occurs nontrivially in the above representation of p as an element of W_{k};

(b) q′_{1}α_{1}u_{1} is a route on p.

Since α_{1}u_{1} is not a right subpath of p and, a fortiori, q′_{1}α_{1}u_{1} ≠ p, the term q′_{1}α_{1}u_{1} must cancel out of our representation of p as an element of W_{k}. Clearly q′_{1}α_{1}u_{1} ≠ q′′q whenever q fails to be a route on p, and q′_{1}α_{1}u_{1} ≠ q′_{2}α_{2}u_{2} whenever (α_{2}, u_{2}) is a detour different from (α_{1}, u_{1}). Thus q′_{1}α_{1}u_{1} = q′_{2}v_{j}(α_{2}, u_{2}) for some j and some detour (α_{2}, u_{2}) of p such that q′_{2}(α_{2}u_{2} − ∑_{i∈I(α_{2},u_{2})} k_{i}(α_{2}, u_{2})v_{i}(α_{2}, u_{2})) occurs non-trivially in our representation of p. On one hand, this forces q′_{2}v_{j}(α_{2}, u_{2}), and consequently also q′_{2}α_{2}u_{2}, to be a route on p, whence length(u_{2}) ≥ length(u_{1}) by our choice of u_{1}. On the other hand, the equality ‘q′_{1}α_{1}u_{1} = q′_{2}v_{j}(α_{2}, u_{2})’ is only possible when length v_{j}(α_{2}, u_{2}) ≤ length(u_{1}), for u_{1} is the longest right subpath of p which is also a right subpath of q′_{1}α_{1}u_{1}. But this, in turn, entails length(u_{2}) < length(u_{1}). We have reached a contradiction, which proves that p ∉ W_{k}, and which thus completes the proof of Claim 1.

Claim 2. Whenever U in Λ-mod is a uniserial module with mast p, there exists a point k ∈ V_{p} such that U ≅ Φ(k).

To find a suitable point k ∈ V_{p}, let x = e(1)x be a top element of U; then the map f : Λe(1) → U = Λx which sends e(1) to x is a projective cover of U. Again write p = β_{l} ⋯ β_{1}, where the β_{i} are arrows. Then, clearly, J^{i}U = Λβ_{i} ⋯ β_{1}x for i ≤ l, and the elements β_{i} ⋯ β_{1}x, 0 ≤ i ≤ l, form a K-basis for U.

If (α, u) is a detour on p, then Λαux = J^{m}x for some m ≥ length(u) + 1. Consequently, letting e be the primitive idempotent with eα = α, we obtain that αux is a K-linear combination of those elements β_{i} ⋯ β_{1}x of U for which β_{i} ⋯ β_{1} is a path longer than u ending in e. But these latter paths are precisely the ones that belong to the family (v_{i}(α, u))_{i∈I(α,u)}, which shows that αux = ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u)x for suitable scalars k_{i}(α, u) ∈ K. (Note on the side: Once we have fixed a top element x of U, the scalars k_{i}(α, u) are actually uniquely determined by U, since the elements v_{i}(α, u)x, i ∈ I(α, u), are K-linearly independent.) In other words, via the map f, the uniserial U is an epimorphic image of the module M := Λe(1)/ ∑_{(α,u)≀≀p} Λ(αu − ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u)). Set k = (k_{i}(α, u))_{i∈I(α,u), (α,u)≀≀p}. In order to prove Claim 2, it is thus enough to show that (a) qx = 0, whenever q ∈ KΓ is a path which fails to be a route on p, and (b) k ∈ V_{p}. For reasons of dimension, we will then obtain that the epimorphism Φ(k) → U induced by f is an isomorphism. In particular, statement (a) will guarantee that, up to isomorphism, U is the only factor module of M which is uniserial with mast p.

For (a), suppose that the path q is a non-route on p. It is clearly harmless to assume that q = qe(1). Write q = q′u, where u is the longest right subpath of q (possibly of length zero) which is a route on p. Moreover, let p′ be the shortest right subpath of p such that u is a route on p′, say p = p′′p′. Then length q′ ≥ 1, that is, q′ = q′′γ for some arrow γ, and the fact that γu is not a route on p is equivalent to the non-existence of a right subpath v of p which is strictly longer than p′ and ends in the same vertex as γ. If ux ≠ 0, then the minimal choice of p′ implies that Λux = Λw′x where w′ is a right subpath of p with length(w′) ≥ length(p′). If p = w′′w′, then w′′ is a left subpath of p′′ such that Λux = Λw′x is a uniserial module with mast w′′. Consequently, the fact that p′′, and hence also w′′, is devoid of right subpaths of positive length ending in the same vertex as γ forces γux to be zero. Thus qx = 0 as required.

To verify that k is a point in V_{p}, let z be any element in I; say z is a linear combination of paths of lengths bounded above by some integer m. Moreover, let ∑_{p=•p′} τ_{p′}(X)p′ be the unique element in ∑_{p=•p′} K[X]p′ which is equivalent to z under ˆ=. We want to show that all of the polynomials τ_{p′}(X) vanish at k. For that purpose, view U again as a KΓ-module under the action induced by that of Λ, and note that zU = 0, as well as W_{k}x = 0. Since F_{k}(y)−y ∈ W_{k} for all y ∈ KΓ, we infer that F_{k}^{r}(z)x = 0 for all r ≥ 1. But as we pointed out in the first paragraph, F_{k}^{m}(z) = ∑_{p=•p′} τ_{p′}(k)p′, and so, in particular, ∑_{p=•p′} τ_{p′}(k)p′x = 0. Now use the fact that the elements p′x, where p′ runs through the right subpaths of p, are K-linearly independent, to conclude τ_{p′}(k) = 0 for all p′ as required. This completes the proof of part (I). Part (II) is an obvious consequence of Part (I).

(III) In proving the surjectivity of Φ, we saw that, given a uniserial module U with mast p and a fixed top element x, there is a unique point k = (k_{i}(α, u)) ∈ Φ^{−1}(U) such that αux = ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u)x for each detour (α, u) on p. In other words, each subfamily (k_{i}(α, u))_{i∈I(α,u)} of k is the coordinate vector of αux relative to the K-

basis v_{i}(α, u)x, i ∈ I(α, u), of eJ^{length(u)+1}U; here e is the primitive idempotent in which α terminates. Since this coordinate vector will not change if x is replaced by ax for some nonzero scalar a, we see that, if U has a unique top element, up to scalar factors – equivalently, if p is not of the form p = p′c for a cycle c of positive length from e(1) to e(1) – the set Φ^{−1}(U) is a singleton, i.e., Φ is injective. This leaves us to deal with the case, where p is a cycle from e(1) to e(1) such that no proper right subpath of p is a cycle e(1) → e(1). In that case, an arbitrary top element y of U is of the form y = ax + bpx for a, b ∈ K and a nonzero, and since αuy = αuax and v_{i}(α, u)y = v_{i}(α, u)ax for each detour (α, u) on p, the coordinate vector of αuy with respect to the new basis is the same as that of αux with respect to the old basis. Again we deduce that Φ is injective. □

Note that, in case K is algebraically closed, Theorem A(II) provides us with an algorithmic procedure to decide whether a given path p : e → e′ in KΓ occurs as mast of a uniserial left Λ-module. Indeed, when combined with Hilbert’s Nullstellensatz, Theorem A yields the following: There is a uniserial Λ-module with mast p if and only if, for some K-generating set t_{1}, …, t_{s} for I^{(L)}, the resulting polynomials τ_{ij}(X), as in the definition of the variety V_{p}, generate a proper ideal of K[X]. The ensuing question, whether or not 1 belongs to the ideal generated by the τ_{ij}(X) is well-known to be decidable by way of the Groebner method.

We will extract an observation from the preceding argument which will be useful on several occasions. For a smooth formulation, we require the following notational convention: Given a uniserial module U with mast p and a detour (α, u) on p such that α ends in the vertex e, we denote by U(α, u) the K-subspace eJ^{length(u)+1}U of U. Note that, for any top element x of U, the set {v_{i}(α, u)x | i ∈ I(α, u)} forms a basis for this subspace.

Corollary to the proof of Theorem A. Let N be the disjoint union of the index sets I(α, u), where (α, u) runs through the detours on p, and let k be a point in 𝔸^{N}(K), say k = (k_{i}(α, u))_{i∈I(α,u), (α,u)≀≀p}. Then k belongs to V_{p} precisely when there exists a uniserial module U with mast p and top element x such that, for each detour (α, u) on p, the projection (k_{i}(α, u))_{i∈I(α,u)} of k onto 𝔸^{I(α,u)} is the coordinate vector of the element αux ∈ U(α, u) with respect to the K-basis {v_{i}(α, u)x | i ∈ I(α, u)}.

In the positive case, any such uniserial module U belongs to Φ_{p}(k), and the top element x = e(1) + U_{k} of Λe(1)/U_{k} has the property that αux = ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u)x for all (α, u) ≀≀ p; here U_{k} is as in the statement of Theorem A. □

Let L again denote the Loewy length of Λ, and I^{(L)} the K-subspace of I consisting of all elements of I which can be written as K-linear combinations of paths of lengths ≤ L. In the definition of V_{p}, we picked a K-generating set for I^{(L)} to arrive – via the substitution equations for p – at a set of polynomials that determines V_{p}. This set may be vastly redundant, as we already pointed out after Definition 4. In fact, it suffices to consider elements t_{1}, …, t_{r} ∈ I^{(L)} such that I^{(L)} ⊆ ∑_{i=1}^{r} KΓt_{i}; if again τ_{ij}(X) ∈ K[X] are such that t_{i} ˆ= ∑_{j=0}^{l} τ_{ij}(X)p_{j} for 1 ≤ i ≤ s, then

V_{p} = V (τ_{ij}(X) | 1 ≤ i ≤ r, 0 ≤ j ≤ l).

On the other hand, it does not suffice to consider a set of relations which generates I as an ideal, as the following example demonstrates.

Example 6. Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α; 2; β; γ; 3; δ; ϵ; 4]

and I ⊆ KΓ is the ideal generated by the relations δβ − ϵγ, ϵβ, and δγ. Moreover, let p = δβα. The detours on p are (γ, α) and (ϵ, βα), and hence the substitution equations for p are γα ˆ= X_{1}βα, ϵβα ˆ= X_{2}p, as well as q ˆ= 0 whenever q fails to be a route on p. Note that none of the paths occurring in the above relations is a route on p, and so the polynomials resulting from these relations via the substitution equations are all zero. On the other hand, the relations δβα − ϵγα and ϵβα yield 1 − X_{1}X_{2} = 0 and X_{2} = 0, whence V_{p} = ∅. □

The following instance of failure of bijectivity of Φ_{p} is prototypical.

Examples 7. Let Λ = KΓ/⟨α^{2}⟩, where Γ is the quiver

[figure: 1; α; β; 2]

If p = βα, then V_{p} = 𝔸^{1}, while Φ_{p}(V_{p}) is a singleton; indeed, for any scalar k ∈ K, the uniserial modules Λe_{1}/Λ(β − kβα) and Λe_{1}/Λβ are isomorphic.

However, the sufficient condition for bijectivity of Φ_{p} given in Theorem A(III) is not necessary. Indeed, if Γ is the quiver

[figure: 3; γ; 1; β; α; 2]

and Λ = KΓ/⟨αγα⟩, the path p = βγα again consists of a cycle followed by a nontrivial left subpath. Again we have V_{p} = 𝔸^{1}, but this time the map Φ_{p} is bijective. □

Definition 8. Again, let p be a path in KΓ and Λ = KΓ/I. We will refer to V_{p} as the uniserial variety of Λ -mod at p. The irreducible components of the uniserial variety V_{p} will be called the uniserial components of Λ -mod at p. Moreover, we will say that a uniserial component W of V_{p} intersects V_{q} if Φ_{p}(W) ∩ Im(Φ_{q}) ≠ ∅.

Let us start by looking at two trivial cases: If p = e is a primitive idempotent, then V_{p} is a singleton, represented by the simple module centered at e, and if p = α is an arrow e → e′, then V_{p} = 𝔸^{n}, where n is the number of arrows e → e′ different from α. In each of these cases, the uniserial variety at p consists of a single component.

Observe that, in case the quiver Γ of Λ has no double arrows between any two vertices, the images Φ_{p}(V_{p}) of the uniserial varieties, where p runs through the paths in KΓ, yield a disjoint partitioning of the set of isomorphism classes of uniserial objects in Λ -mod. In general, however, the uniserial varieties at p and q, where p and q are distinct paths of the same length passing through the same sequence of vertices, may intersect. We will see in Section 5 that the uniserial components at p which intersect V_{q} and those components at q which intersect V_{p} coincide in number and can be arranged into birationally equivalent pairs.

Two further problems impose themselves as follow-ups to the preceding theorem. One concerns functoriality of the assignment Λ = KΓ/I ↦ (V_{p})_{p a path in KΓ} and, in particular, the behavior of the family of varieties V_{p} under algebra isomorphism. The other is the isomorphism problem for uniserial modules. We defer the former to [5], and proceed by tackling the latter.

4. The isomorphism problem for uniserial modules

Since an obvious necessary condition for isomorphism of two uniserial modules is a joint mast, we wish to explicitly describe the equivalence relation on V_{p} which partitions V_{p} into the fibres Φ_{p}^{−1}(U), where U runs through the uniserial modules in the image Φ_{p}(V_{p}). Theorem A(III) provides us with a partial answer: If the path p does not start with an oriented cycle, i.e., if p is not of the form p′c where c is an oriented cycle of positive length, the map Φ_{p} is a bijection; in other words, the points on the variety V_{p} form a complete system of isomorphism invariants for the uniserials with mast p in that case. In general, injectivity of Φ_{p} may fail, as we know from Example 7. To fill the resulting gap in the information on the uniserial modules with mast p stored in V_{p}, we will construct a system of equations S_{p}(X, Y, Z) in the variables X_{i}(α, u), Y_{i}(α, u) (for (α, u) ≀≀ p and i ∈ I(α, u)) and finitely many variables Z_{j}, which is linear in the Z_{j} over K[X, Y] such that, for any pair of points k, k′ ∈ V_{p}, the linear system S_{p}(k, k′, Z) is consistent if and only if Φ_{p}(k) ≅ Φ_{p}(k′). So, loosely speaking, the family of uniserial modules with mast p can be identified with the variety V_{p} modulo a certain system of linear equations with coefficients in K. While easy to establish, this system will provide a handy decision process for the isomorphism problem on the basis of the varieties V_{p}.

To describe the system S_{p}(X, Y, Z), suppose that the path p : e(1) → e(l + 1) has precisely t right subpaths of positive length ending in the starting vertex e(1) of p, say w_{1}, …, w_{t}. Then our system will have the t linear variables Z_{1}, …, Z_{t}. Start by considering the following equations E(α, u) in KΓ[X, Y, Z], one for each detour (α, u) on p:

(E(α, u))  αu(e(1) + ∑_{j=1}^{t} Z_{j}w_{j}) = ∑_{i∈I(α,u)} X_{i}(α, u)v_{i}(α, u)(e(1) + ∑_{j=1}^{t} Z_{j}w_{j});

here the v_{i}(α, u) are as in the definition of the substitution equations for p. Now expand both sides of these equations by successively inserting from the right the substitution equations βv ˆ= ∑_{i∈I(β,v)} Y_{i}(β, v)v_{i}(β, v) for detours (β, v) on p, and the equivalences q ˆ= 0 for those paths q ∈ KΓ which fail to be routes on p. As pointed out in Observation 3 of Section 3, the equation E(α, u) will eventually take on the form

∑_{i∈I(α,u)} a_{i}(X, Y, Z)v_{i}(α, u) = ∑_{i∈I(α,u)} b_{i}(X, Y, Z)v_{i}(α, u)

for suitable polynomials a_{i}(X, Y, Z), b_{i}(X, Y, Z) ∈ K[X, Y, Z] which are uniquely determined by the left-hand and right-hand sides of equation E(α, u). Indeed, this is always the terminal stage of the substitution process, since each substitution step replaces a path q by a linear combination of paths of lengths ≥ length(q), all of which have the same endpoint as q. Now collect all of the equations of the form a_{i}(X, Y, Z) = b_{i}(X, Y, Z), i ∈ I(α, u), arising in this way for arbitrary detours (α, u) on p, and label the resulting system S_{p}(X, Y, Z). Observe that this system is polynomial in the X_{j} and Y_{j}, linear in the Z_{j}.

Theorem B. For k, k′ ∈ V_{p}, the linear system S_{p}(k, k′, Z) in Z = (Z_{1}, …, Z_{t}) is consistent if and only if Φ_{p}(k) ≅ Φ_{p}(k′).

Proof. First suppose that there exists an isomorphism f : Φ_{p}(k) → Φ_{p}(k′) of Λ-modules, and let x and y be top elements of Φ_{p}(k) and Φ_{p}(k′), with the property that αux = ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u)x and αuy = ∑_{i∈I(α,u)} k′_{i}(α, u)v_{i}(α, u)y, respectively, for any detour (α, u) on p (here we identify the isomorphism classes Φ_{p}(k) and Φ_{p}(k′) with the distinguished representatives described in Theorem A). The equation f(x) = e(1)f(x) clearly yields f(x) = c_{0}y + ∑_{j=0}^{t} c_{j}w_{j}y with c_{j} ∈ K and c_{0} ≠ 0; without loss of generality, we may assume c_{0} = 1. Let (α, u) be a detour on p such that α ends in the primitive idempotent e, and consider the following equality inside the K-vectorspace eΦ_{p}(k′) which results from our isomorphism f:

(†)  αu(y + ∑_{j=1}^{t} c_{j}w_{j}y) = ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u)(y + ∑_{j=1}^{t} c_{j}w_{j}y).

Clearly, the left-hand side of equality (†) reduces to the form

∑_{i∈I(α,u)} a_{i}(k, k′, c)v_{i}(α, u)y ∈ eΦ_{p}(k′),

where c = (c_{1}, …, c_{t}) and a_{i}(X, Y, Z) is the polynomial occurring in the definition of the system S_{p}(X, Y, Z); this is clear from the discussion prior to Theorem B. Analogously, the right-hand side of (†) is equal to the element

∑_{i∈I(α,u)} b_{i}(k, k′, c)v_{i}(α, u)y ∈ eΦ_{p}(k′)

with b_{i}(X, Y, Z) ∈ K[X, Y, Z] as above, and consequently equality (†) reduces to

(‡)  ∑_{i∈I(α,u)} a_{i}(k, k′, c)v_{i}(α, u)y = ∑_{i∈I(α,u)} b_{i}(k, k′, c)v_{i}(α, u)y.

Since the vectors v_{i}(α, u)y, i ∈ I(α, u), are K-linearly independent, this shows that the scalars c_{1}, …, c_{t} satisfy the system S_{p}(k, k′, Z).

Conversely, suppose that the system S_{p}(k, k′, Z) is consistent, and let c = (c_{1}, …, c_{t}) ∈ K^{t} be a solution. Moreover, denote the residue class of the vertex e(1) in Φ_{p}(k) by x, that in Φ_{p}(k′) by y (again we identify Φ_{p}(k) and Φ_{p}(k′) with the representatives described in Theorem A). We wish to show that the assignment x ↦ y + ∑_{j=1}^{t} c_{j}w_{j}y extends to a welldefined Λ-isomorphism Φ_{p}(k) → Φ_{p}(k′). It clearly suffices to show that the annihilator of x in Λ is contained in the annihilator of y + ∑_{j=1}^{t} c_{j}w_{j}y, since these two elements are top elements of the uniserials Φ_{p}(k) and Φ_{p}(k′), respectively, and the uniserial modules Φ_{p}(k) and Φ_{p}(k′) have the same length. Recall that, by the definiton of Φ_{p}(k), the left annihilator of x is, as a left ideal of Λ, generated by the (residue classes in Λ of) the elements αu − ∑_{i∈I(α,u)} k_{i}(α, u)v_{i}(α, u), where (α, u) runs through the detours on p, and by all the (residue classes of) paths q which fail to be routes on p. That each of the former elements annihilates y + ∑_{j=1}^{t} c_{j}w_{j}y is guaranteed by the fact that c = (c_{1}, …, c_{t}) satisfies the system S_{p}(k, k′, Z). Indeed, from the fact that c satisfies (‡), we infer that c also satisfies equality (†) above. So let q be a non-route on p. By construction, q also annihilates y; moreover, given any of the subpaths w_{j} of p ending in e(1), the composition qw_{j} is not a route on p either by Remark (a) following Definitions 2. This implies that also qw_{j}y = 0 for all j, and hence that q(y + ∑_{j=1}^{t} c_{j}w_{j}y) = 0 as required. □

We illustrate our method for solving the isomorphism problem with two examples. In the first, V_{p} ≅ 𝔸^{3} and each of the fibres of Φ_{p} is a subvariety isomorphic to 𝔸^{2}; in the second, V_{p} ≅ 𝔸^{1} consists of a single fibre.

Example 9. Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α; β; 2; γ]

and I is the ideal in KΓ generated by α^{2}, γβγ, γβαγ. Consider the path p = βαγβα. The detours on p are (β, e_{1}), (β, γβα), and (α, α), yielding the substitution equations

β ˆ= X_{1}βα + X_{2}p,  βγβα ˆ= X_{3}p,  and  α^{2} ˆ= X_{4}γβα + X_{5}αγβα.

Inserting the substitution equations from the right into a K-generating set for I^{(6)} – note that 6 is the Loewy length of Λ – gives us X_{4} = X_{5} = 0, while imposing no conditions on X_{1}, X_{2}, X_{3}. Thus V_{p} ≅ 𝔸^{3}.

To determine the system S_{p}(X, Y, Z), we observe that there are precisely three right subpaths of p of positive length which end in e_{1}, namely α, γβα, and αγβα; thus Z = (Z_{1}, Z_{2}, Z_{3}). Setting z = e_{1} + Z_{1}α + Z_{2}γβα + Z_{3}αγβα and inserting the substitution equations β ˆ= Y_{1}βα + Y_{2}p, βγβα ˆ= Y_{3}p and α^{2} ˆ= 0 repeatedly into the three starting equations E(β, e_{1}) : βz = X_{1}βαz + X_{2}pz; E(β, γβα) : βγβαz = X_{3}pz and E(α, α) : α^{2}z = X_{4}γβαz + X_{5}αγβαz as described ahead of Theorem B, we obtain

X_{1}βα + Z_{2}X_{1}p + X_{2}p = Y_{1}βα + Y_{2}p + Z_{1}βα + Z_{2}βγβα + Z_{3}p
= Y_{1}βα + Y_{2}p + Z_{1}βα + Z_{2}Y_{3}p + Z_{3}p

from equation E(β, e_{1}), X_{3}p = Y_{3}p from the second of the starting equations, and 0 = 0 from the third. Thus the system S_{p}(X, Y, Z) is

X_{1} = Y_{1} + Z_{1},  Z_{2}X_{1} + X_{2} = Y_{2} + Z_{2}Y_{3} + Z_{3},  X_{3} = Y_{3}

in this example. In particular, given any two points k = (k_{1}, k_{2}, k_{3}) and k′ = (k′_{1}, k′_{2}, k′_{3}) in V_{p} ≅ 𝔸^{3}, the linear system S_{p}(k, k′, Z) in Z_{1}, Z_{2}, Z_{3} is consistent if and only if k_{3} = k′_{3}. This shows that, up to isomorphism, there is only a one-parameter family of uniserial modules with mast p, the parameter being k_{3} ∈ K.

Let us translate this information into the graphs of the uniserial modules with mast p. Each of the uniserial modules corresponding to a nonzero value of k_{3} all has three different graphs, depending on the choice of top element. These graphs are:

1_{α}  _{β}  1_{α}  _{β}  1_{α} 1_{β}  1_{β}  1_{β} 2_{γ}  or  2_{γ}  or  2_{γ} 1_{α}  _{β}  1_{α} _{β}  1_{α}  _{β} 1_{β}  1_{β}  1_{β} 2  2  2

The uniserial module corresponding to the value k_{3} = 0 has graphs

1_{α}  _{β}  1_{α}  _{β}  1_{α} 1_{β}  1_{β}  1_{β} 2_{γ}  or  2_{γ}  or  2_{γ} 1_{α}  1_{α}  1_{α} 1_{β}  1_{β}  1_{β} 2  2  2

again depending on the choice of a top element. □

Example 10. Let Λ = KΓ/I, where Γ is the quiver

[figure: 1; α; 2; β; ϵ; δ; 3; γ; 4]

and I is the ideal in KΓ generated by the relations

ϵα − ϵγβα, δαδα, αδαδ, βγ, δγ.

Observe that these relations, together with ϵαδ − ϵγβαδ and ϵαδα − ϵγβαδα, as well as eight monomial relations of the forms βγq and δγq, generate I as a left ideal. Moreover, consider the mast p = ϵγβαδα. The detours on p are (β, α), (ϵ, α), and (ϵ, αδα). Inserting the corresponding substitution equations

βα ˆ= X_{1}βαδα,  ϵα ˆ= X_{2}p,  ϵαδα ˆ= X_{3}p

into the relations yields V_{p} = V (X_{2} − X_{1}, X_{3} − 1) = {(k_{1}, k_{1}, 1) | k_{1} ∈ K} ≅ 𝔸^{1}.

Note that w = δα is the only right subpath of positive length of p which ends in e_{1}, whence the family Z of variables in the system S_{p}(X, Y, Z) is reduced to a single one. Using the above method for determining the system S_{p}(X, Y, Z), we obtain, for any two points k = (k_{1}, k_{1}, 1) and k′ = (k′_{1}, k′_{1}, 1) in V_{p}:

(S_{p}(k, k′, Z))  k′_{1} + Z = k_{1},  k′_{1} + Z = k_{1},  1 = 1.

Since this system is consistent for arbitrary choice of k, k′ ∈ V_{p}, there is, up to isomorphism, precisely one uniserial left Λ-module with mast p. The following are all its graphs relative to suitable top elements:

1_{α}  1_{α} 2_{δ}  2_{δ}  _{β}  _{ϵ} 1_{α}  1_{α} 2_{β}_{ϵ}  and  2_{β}_{ϵ} 3_{γ}  3_{γ} 2_{ϵ}  2_{ϵ} 4  4  □

We conclude the section with a look at hereditary algebras. As is readily seen, in that case, all of the varieties V_{p} are full affine spaces of ‘maximum’ dimension. We will see that the split hereditary algebras are actually characterized by their varieties of uniserial modules, which answers a question of K. R. Fuller.

Proposition C. The algebra Λ is hereditary if and only if, for each path p in KΓ, the variety V_{p} is isomorphic to the full affine space 𝔸^{N(p)}, where

N(p) = ∑{|I(α, u)| : (α, u) ≀≀ p}

and 𝔸^{0} stands for a singleton.

In case these conditions are satisfied, all the maps

Φ_{p} : V_{p} ⟶ {isomorphism types of uniserials in Λ -mod with mast p}

are bijections, i.e., Φ_{p}(k) ≅ Φ_{p}(k′) precisely when k = k′. Moreover, for a hereditary algebra Λ over an infinite field K, the following statements are equivalent:

(i) There are only finitely many isomorphism types of uniserial Λ-modules.

(ii) Given any pair of vertices e and e′ in Γ and an arrow α : e ⟶ e′, the arrow α is the only \underline{path} from e to e′ in KΓ.

(iii) For any finite sequence of simple Λ-modules there is either no or precisely one uniserial module having this composition series, depending on whether or not there is a path in KΓ which passes through the corresponding sequence of vertices.

Proof. To prove the first equivalence, observe that, for any path p, the coordinate ring of the variety V_{p} is K[X_{i}(α, u) | i ∈ I(α, u), (α, u) ≀≀ p] modulo the ideal generated by all the polynomials arising from an insertion of the substitution equations into relations involving routes on p. In particular, this ideal will be nonzero whenever p makes a non-trivial appearance in a relation of Λ, in which case dim V_{p} < N(p). Consequently, isomorphism of V_{p} with 𝔸^{N(p)} for all paths p does not allow for any non-trivial relations. Conversely, it is clear that V_{p} ≅ 𝔸^{N(p)} for all p when Λ is hereditary.

Now suppose that Λ is hereditary. Since the quiver Γ is acyclic in this case, bijectivity of the maps Φ_{p} follows from part (III) of Theorem A. In view of the first part of the theorem, condition (i) is therefore equivalent to ‘N(p) = 0’ for all p, whence (i) implies (iii), the converse of this implication being trivial. But condition (ii) also translates into the nonexistence of detours on any path p in KΓ, that is, into the equality ‘N(p) = 0’ for all p. This completes the proof. □

5. Varieties of uniserials with fixed sequence of composition factors

A more natural subdivision of the varieties of uniserial modules than that in terms of masts – the latter depending a priori on the given coordinatization of Λ – is in terms of sequences of consecutive composition factors. In order to understand the uniserials of composition length l + 1 with a fixed sequence (S(1), …, S(l + 1)) of simple composition factors, we need to first study the correlation among the varieties V_{p}, where p runs through all paths of length l which pass precisely through the vertices e(1), …, e(l+1), in that order. In particular, we need to explore the intersections Φ_{p}(V_{p}) ∩ Φ_{q}(V_{q}), where p and q are two such paths. In a first easy step, we will observe that there is a 1−1 correspondence between the detours on p and those on q which preserves the cardinalities of the corresponding index sets I(α, u). Hence, the varieties V_{p} and V_{q} live in the same affine space 𝔸^{N}. Let D = Φ_{p}(V_{p}) ∩ Φ_{q}(V_{q}) = Im(Φ_{p}) ∩ Im(Φ_{q}). It turns out that the preimages Φ_{p}^{−1}(D) and Φ_{q}^{−1}(D) are Zariski-open in V_{p} and V_{q}, respectively, and isomorphic. (We do not require varieties to be irreducible and correspondingly mean by an isomorphism between two varieties a homeomorphism which has regular coordinate functions in both directions.) In particular, the irreducible components of V_{p} which intersect V_{q} and those of V_{q} which intersect V_{p} can be paired off into pairs of birationally equivalent partners. (Recall that, by Definition 8, an irreducible component W of V_{p} intersects V_{q} if Φ_{p}(W)∩Im(Φ_{q}) ≠ ∅.) The gist of this is that, if we are looking for a set of representatives of the birational equivalence classes of all the uniserial varieties of Λ -mod at the paths p running through the above sequence of vertices, we will not lose information in proceeding as follows: Let q_{1}, …, q_{t} be the distinct paths of length l passing through the sequence (e(1), …, e(l+1)). We start by determining the irreducible components of V_{q_{1}}. Then we find the irreducible components W of V_{q_{2}} such that Φ_{q_{2}}(W) ∩ Im(Φ_{q_{1}}) = ∅, next the components W of V_{q_{3}} with the property that Φ_{q_{3}}(W)∩(Im(Φ_{q_{1}})∪Im(Φ_{q_{2}})) = ∅, and so forth. Eventually, this procedure will lead us to a family of irreducible affine varieties which contains a representative of each birational equivalence class occurring among the irreducible components of the V_{q}, q = q_{1}, …, q_{t}.

Lemma 11. Suppose that p and q are paths of length l passing through the same sequence of vertices (e(1), …, e(l + 1)) in the given order. Then there is a bijection ρ from the set of detours on p to the set of detours on q such that |I(α, u)| = |I(ρ(α, u))| for all detours (α, u) ≀≀ p. In particular, if

N = ∑_{(α,u)≀≀p} |I(α, u)|,

then V_{p} and V_{q} are both subvarieties of affine N-space 𝔸^{N} over K.

Proof. It clearly suffices to focus on the case where p and q differ in precisely one arrow; an obvious induction will then complete the proof. Say p = α_{l} ⋯ α_{1} and q = α_{l} ⋯ α_{r+1}β_{r}α_{r−1} ⋯ α_{1}. Let ρ be the identity on those detours (γ, u) on p for which either length u < r − 1, or else length u = r − 1 and γ ≠ β_{r}, and set ρ(β_{r}, α_{r−1} ⋯ α_{1}) = (α_{r}, α_{r−1} ⋯ α_{1}) if γ = β_{r}. Any detour of the form (γ, α_{s} ⋯ α_{1}) on p with s ≥ r, finally, we match up with the detour (γ, α_{s} ⋯ α_{r+1}β_{r}α_{r−1} ⋯ α_{1}) on q. It is easy to check that this bijection ρ preserves the cardinalities of the corresponding index sets as claimed. □

The proof of the following theorem meets with a few technical hurdles which will, however, cease to play a role in the further development of the subject.

Theorem D. Let p and q be paths of length l passing through the same sequence of vertices, and let N be as in the preceding lemma. Moreover, consider the intersection D = Φ_{p}(V_{p}) ∩ Φ_{q}(V_{q}). Then the preimages Φ_{p}^{−1}(D) and Φ_{q}^{−1}(D) in 𝔸^{N} are isomorphic Zariski open subsets of V_{p} and V_{q}, respectively.

More precisely, if p and q differ in exactly s arrows, then there exist Zariski open subsets Z_{1} and Z_{2} of the form Z_{i} = 𝔸^{N} \ V (X_{i1} ⋯ X_{is}) in 𝔸^{N} such that Z_{1} ∩ V_{p} = Φ_{p}^{−1}(D) and Z_{2} ∩ V_{q} = Φ_{q}^{−1}(D), together with an isomorphism ψ : Z_{1} ∩ V_{p} → Z_{2} ∩ V_{q} which makes the following diagram commutative:

[figure: Φ_{p}^{−1}(D); ψ; Φ_{p}; Φ_{q}^{−1}(D); Φ_{q}; D]

Proof. To construct an isomorphism ψ : Φ_{p}^{−1}(D) → Φ_{q}^{−1}(D), we assume that D ≠ ∅. We start with a point k ∈ Φ_{p}^{−1}(D), and set U = Φ_{p}(k). Moreover, we let x ∈ U be a top element such that, for each detour (γ, u) on p, we have γux = ∑_{i} k_{i}(γ, u)v_{i}(γ, u)x. Then k is the family of coordinate vectors (k_{i}(γ, u))_{i∈I(γ,u)} of the elements γux with respect to the basis (v_{i}(γ, u)x), i ∈ I(γ, u), for the K-space U(γ, u) = eJ^{length(u)+1}U, where e is the endpoint of γ. Let ρ(γ, u) = (γ′, u′) where ρ is as in Lemma 11; the lemma then allows us to assume that I(γ′, u′) = I(γ, u). As we will see, the coordinate vector (k′_{i}(γ′, u′))_{i∈I(γ′,u′)} of the element γ′u′x relative to the basis v′_{i}(γ′, u′)x, i ∈ I(γ′, u′), for U(γ, u) = U(γ′, u′) does not depend on the choice of the element x as above. The assignment k ↦ k′ = (k′_{i}(γ′, u′))_{i∈I(γ′,u′),(γ′,u′)≀≀q} thus yields a well-defined map ψ : Φ_{p}^{−1}(D) → Φ_{q}^{−1}(D) by the Corollary to Theorem A. Furthermore, we will find that the coordinates of ψ are rational functions in the k_{i}(γ, u) which are defined on the whole domain and which depend only on p and q. In particular, this will show that the coordinate functions of ψ are regular maps. Once this is established, it will readily follow that ψ is an isomorphism from Φ_{p}^{−1}(D) to Φ_{q}^{−1}(D). Indeed, if ψ′ : Φ_{q}^{−1}(D) → Φ_{p}^{−1}(D) is the map constructed in complete analogy to ψ, then ψ′ has regular coordinate functions by symmetry, and it is straightforward to check that ψ′ψ and ψψ′ are the identities on Φ_{p}^{−1}(D) and Φ_{q}^{−1}(D), respectively.

As in the proof of Lemma 11, we will assume that p and q differ in precisely one arrow and leave the general case to the reader. Say

p = α_{l} ⋯ α_{1}  and  q = α_{l} ⋯ α_{r+1}β_{r}α_{r−1} ⋯ α_{1}

where each α_{i} is an arrow e(i) → e(i + 1) and β_{r} is an arrow e(r) → e(r + 1). Set u_{0} = α_{r−1} ⋯ α_{1}. Clearly, we then have Φ_{p}^{−1}(D) = {k ∈ V_{p} | k_{1}(β_{r}, u_{0}) ≠ 0} and Φ_{q}^{−1}(D) = {k′ ∈ V_{q} | k′_{1}(α_{r}, u_{0}) ≠ 0}, where we have chosen our indices so that v_{1}(β_{r}, u_{0}) = α_{r}u_{0} and v′_{1}(α_{r}, u_{0}) = β_{r}u_{0}. In other words, if Z_{1} = 𝔸^{N} \ V (X_{1}(β_{r}, u_{0})) and Z_{2} = 𝔸^{N} \ V (X_{1}(α_{r}, u_{0})), then Φ_{p}^{−1}(D) = Z_{1} ∩ V_{p} and Φ_{q}^{−1}(D) = Z_{2} ∩ V_{q}.

To follow the strategy outlined above, we let (γ, u) be a detour on p, and treat the following cases separately.

1. u = u_{0} and γ = β_{r}. Then (γ′, u′) = (α_{r}, u_{0}).

2. u = u_{0} and γ ≠ β_{r}. Then (γ′, u′) = (γ, u).

3. length(u) < length(u_{0}) = r − 1. Then (γ′, u′) = (γ, u).

4. length(u) > length(u_{0}). If u = α_{s} ⋯ α_{1}, then (γ′, u′) = (γ, α_{s} ⋯ α_{r+1}β_{r}u_{0}).

Throughout, assume that I(γ, u) is a set of natural numbers and that the paths v_{i}(γ, u) and v′_{i}(γ′, u′) are ordered by length, that is,

length v_{i}(γ, u) < length v_{i+1}(γ, u)

for all i, and similarly for (γ′, u′); then, clearly, length v_{i}(γ, u) = length v′_{i}(γ′, u′) for all i ∈ I(γ, u) = I(γ′, u′). Again, let k ∈ Φ_{p}^{−1}(D) = Z_{1}∩V_{p}, and identify Φ_{p}(k) with a uniserial module U having top element x such that δvx = ∑ k_{i}(δ, v)v_{i}(δ, v)x for each detour (δ, v) on p.

In case 1, we write each path v′_{i}(α_{r}, u_{0}) in the form u_{i}β_{r}u_{0}, where u_{i} is a path of length ≥ 0; in particular, since v′_{1}(α_{r}, u_{0}) = β_{r}u_{0}, we have u_{1} = e(r + 1). We infer that v_{i}(β_{r}, u_{0}) = u_{i}α_{r}u_{0} = u_{i}v_{1}(β_{r}, u_{0}) and compute

α_{r}u_{0}x = ∑_{i∈I(α_{r},u_{0})} k′_{i}(α_{r}, u_{0})v′_{i}(α_{r}, u_{0})x

= ∑_{i∈I(α_{r},u_{0})} k′_{i}(α_{r}, u_{0})u_{i}( ∑_{j∈I(β_{r},u_{0})} k_{j}(β_{r}, u_{0})v_{j}(β_{r}, u_{0})x)

= ∑_{i≥1} k′_{i}(α_{r}, u_{0})k_{1}(β_{r}, u_{0})v_{i}(β_{r}, u_{0})x + ∑_{{i≥1 \atop j≥2}} k′_{i}(α_{r}, u_{0})k_{j}(β_{r}, u_{0})u_{i}v_{j}(β_{r}, u_{0})x.

Observing that, for i ≥ 1 and j ≥ 2, we have length(u_{i}v_{j}(β_{r}, u_{0})) > length(u_{i}v_{1}(β_{r}, u_{0})) = length(v_{i}(β_{r}, u_{0})) and length(u_{i}v_{j}(β_{r}, u_{0})) ≥ length(v_{j}(β_{r}, u_{0})), we obtain

u_{i}v_{j}(β_{r}, u_{0})x = ∑_{s≥max(i+1,j)} σ_{ijs}v_{s}(β_{r}, u_{0})x

for all i ≥ 1 and j ≥ 2, where the σ_{ijs} are polynomials in the k_{t}(δ, v), for detours (δ, v) on p, which – aside from dependence on the indices – depend only on p, q and the detour considered.

Inserting the second equality into the first yields

α_{r}u_{0}x = ∑_{i≥1}(k′_{i}(α_{r}, u_{0})k_{1}(β_{r}, u_{0}) + ∑_{{1≤s≤i−1 \atop 2≤j≤i}} k′_{s}(α_{r}, u_{0})k_{j}(β_{r}, u_{0})σ_{sji})v_{i}(β_{r}, u_{0})x.

On the other hand, α_{r}u_{0}x = 1 · v_{1}(β_{r}, u_{0})x, and since the elements v_{i}(β_{r}, u_{0})x of U are K-linearly independent, a comparison of coefficients leads us to the system of equations k′_{1}(α_{r}, u_{0})k_{1}(β_{r}, u_{0}) = 1 and

k′_{i}(α_{r}, u_{0})k_{1}(β_{r}, u_{0}) + ∑_{1≤s≤i−1} k′_{s}(α_{r}, u_{0})( ∑_{2≤j≤i} k_{j}(β_{r}, u_{0})σ_{sji}) = 0

for i ≥ 2. This system for the ‘unknowns’ k′_{i}(α_{r}, u_{0}) has size |I(β_{r}, u_{0})| × |I(β_{r}, u_{0})| and is lower triangular, the scalar k_{1}(β_{r}, u_{0}) holding all positions along the main diagonal of the coefficient matrix; this scalar is nonzero, because we chose k ∈ Z_{1}. Consequently, the system is uniquely solvable for the k′_{i}(α_{r}, u_{0}), and the solution is of the form k′_{i}(α_{r}, u_{0}) = τ_{i}/(k_{1}(β_{r}, u_{0}))^{i}, where the τ_{i} are polynomials in the k_{t}(δ, v) depending only on i, p, q and on the detour (α_{r}, u_{0}) on q.

We will leave the details of cases 2,3 to the reader, but will carry out case 4. In that case, length v′_{i}(γ′, u′) = length v_{i}(γ, u) > length(u′) = length(u) > length(u_{0}), and hence v′_{i}(γ′, u′) = u_{i}β_{r}u_{0} for some path u_{i} of length ≥ 1, while v_{i}(γ, u) = u_{i}α_{r}u_{0} for all i ∈ I(γ′, u′) = I(γ, u).

Again we compute γ′u′x in two ways. On one hand,

γ′u′x = ∑_{i∈I(γ′,u′)} k′_{i}(γ′, u′)u_{i}β_{r}u_{0}x

= ∑_{i∈I(γ,u)} k′_{i}(γ′, u′)u_{i}( ∑_{j∈I(β_{r},u_{0})} k_{j}(β_{r}, u_{0})v_{j}(β_{r}, u_{0})x)

= ∑_{i∈I(γ,u)} k′_{i}(γ′, u′)k_{1}(β_{r}, u_{0})v_{i}(γ, u)x + ∑_{{i∈I(γ,u) \atop j∈I(βr ,u_{0}),j≥2}} k′_{i}(γ′, u′)k_{j}(β_{r}, u_{0})u_{i}v_{j}(β_{r}, u_{0})x.

Here we use the facts that I(γ′, u′) = I(γ, u) and that u_{i}v_{1}(β_{r}, u_{0}) = u_{i}α_{r}u_{0} = v_{i}(γ, u). To evaluate the terms u_{i}v_{j}(β_{r}, u_{0})x for j ≥ 2, we observe that

length(u_{i}v_{j}(β_{r}, u_{0})) > length(u_{i}v_{1}(β_{r}, u_{0})) = length(u_{i}α_{r}u_{0}) = length(v_{i}(γ, u))

for all j ≥ 2. Thus, for j ≥ 2, we have u_{i}v_{j}(β_{r}, u_{0})x = ∑_{s≥i+1} σ_{ijs}v_{s}(γ, u)x, where the σ_{ijs} are again polynomials in the k_{t}(δ, v) depending solely on the indices, the paths p, q, and the detour (γ, u). Inserting this information into the above equality yields

(I) γ′u′x =

∑_{i∈I(γ,u)}(k′_{i}(γ′, u′)k_{1}(β_{r}, u_{0}) + ∑_{{1≤s≤i−1 \atop s∈I(γ,u)}} k′_{s}(γ′, u′) ∑_{{j∈I(βr ,u_{0}) \atop j≥2}} k_{j}(β_{r}, u_{0})σ_{sji})v_{i}(γ, u)x.

On the other hand, we can write u = wα_{r}u_{0} for a suitable path w of length ≥ 0. Then u′ = wβ_{r}u_{0}, and we obtain

γ′u′x = γwβ_{r}u_{0}x = ∑_{j∈I(β_{r},u_{0})} k_{j}(β_{r}, u_{0})γwv_{j}(β_{r}, u_{0})x.

Now each of the paths γwv_{j}(β_{r}, u_{0}) is longer than u and has the same endpoint as γ. Therefore

γwv_{j}(β_{r}, u_{0})x = ∑_{i∈I(γ,u)} τ_{ji}v_{i}(γ, u)x,

where the v_{i}(γ, u) are suitable polynomials in the k_{t}(δ, v). This gives us

(II)  γ′u′x = ∑_{i∈I(γ,u)}( ∑_{j∈I(β_{r},u_{0})} k_{j}(β_{r}, u_{0})τ_{ji})v_{i}(γ, u)x,

and a comparison of coefficients of the K-linearly independent elements v_{i}(γ, u)x ∈ U in equations (I), (II) once more yields a square system of equations for the k′_{i}(γ′, u′). Again the coefficient matrix is lower triangular, and all diagonal positions are occupied by k_{1}(β_{r}, u_{0}). Since k_{1}(β_{r}, u_{0}) ≠ 0 by the choice of k, the system has a unique solution, expressing the k′_{i}(γ′, u′) as polynomials in the k_{t}(δ, v) divided by powers of k_{1}(β_{r}, u_{0}). □

Let W_{1}, …, W_{m} be a full set of representatives for the birational equivalence classes of all the irreducible components occurring in the varieties V_{p}, where p runs through the sequence of vertices (e(1), …, e(l + 1)). Since, in particular, W_{i} is not birationally equivalent to W_{j} for i ≠ j, Theorem D tells us that, for any choice of i ≠ j and p ≠ q with W_{i} ⊆ V_{p} and W_{j} ⊆ V_{q}, the intersection Φ_{p}(W_{i}) ∩ Φ_{q}(W_{j}) is empty; in other words, among the irreducible varieties W_{i} selected above, any two corresponding to different masts are disjoint. We can therefore determine the set {W_{1}, …, W_{m}} in the manner described in the beginning of the section. In particular, Theorem D guarantees that the result will not depend on the ordering of the paths p_{1}, …, p_{t}; in other words, the set of birational equivalence classes obtained will be invariant under permutation of the p_{i}.

We interrupt the theory to give two examples. In each case, p and q are two paths of the same length passing through the same sequence of vertices. In the first example, V_{p} and V_{q} are irreducible with Φ_{p}(V_{p}) ∩ Φ_{q}(V_{q}) ≠ ∅, but V_{p} ≅ V̸_{q}; Theorem D guarantees that V_{p} and V_{q} are birationally equivalent in that case. In the second, again Im(V_{p}) ∩ Im(V_{q}) ≠ ∅, but this time V_{p} is irreducible while V_{q} is not; in this situation, Theorem D guarantees that V_{p} is birationally equivalent to an irreducible component of V_{q}.

Example 12. Let Γ be the quiver

[figure: 1; α′; α; 2; β′; β; 3]

and Λ = KΓ/I, where I is generated by β′α′ − βα. If p is equal to βα or β′α′ and q to β′α or βα′, then V_{p} = V (X_{1}X_{2} − 1) is isomorphic to the punctured line, while V_{q} = V (X_{1} − X_{2}) is isomorphic to the full affine line over K. Thus V_{p} ≅ V̸_{q}. Note, however, that Φ_{p}(V_{p}) ∩ Φ_{q}(V_{q}) ≠ ∅. □

Example 13. Let Γ be the quiver

[figure: 1; α′; α; 2; β′; β; 3; γ′; γ; 4; δ′; δ; 5; ϵ′; ϵ; 6]

and Λ = KΓ/I, where I is the ideal generated by the relations

ϵ′δ′γβα − ϵδγ′β′α′,  ϵδγβ′α − ϵδγβα′,  ϵδγ′βα − ϵδγβα′,  ϵδ′γβα − ϵ′δγβα.

If p = ϵδγβα and q = ϵ′δγβ′α, then

V_{p} = V (X_{4}X_{5} − X_{1}X_{2}X_{3}, X_{2} − X_{1}, X_{3} − X_{1}, X_{4} − X_{5}) ≅ V (Y^{2} − X^{3})

is the standard cusp:

In particular, V_{p} is irreducible. On the other hand,

V_{q} = V (X_{2}, X_{5}) ∪ V (X_{4}X_{2} − X_{5}X_{3}X_{1}, X_{1}X_{2} − 1, X_{3} − X_{1}, X_{5}X_{4} − 1)
≅ 𝔸^{3} ∪ V (X_{4}^{2} − X_{1}^{3}, X_{1}X_{2} − 1, X_{5}X_{4} − 1),

the latter component being isomorphic to the cusp from which the singular point has been deleted, or, in other words, isomorphic to the punctured line. Note that, in this case, still Φ_{p}(V_{p}) ∩ Φ_{q}(V_{q}) ≠ ∅, but Φ_{p}(V_{p}) ∩ Φ_{q}(V (X_{2}, X_{5})) = ∅. □

As repeatedly announced, the birational equivalence classes of the irreducible components of the varieties V_{p}, where p runs through the paths of length l through a fixed sequence (e(1), …, e(l + 1)) of vertices, form an isomorphism invariant of the algebra Λ. To see this, we will require a technical theorem which describes the behavior of the varieties V_{p} under change of coordinatization. The proof of this theorem constitutes a rather lengthy detour from our main train of thought; in particular, it is based on an extension of our conceptual framework. Therefore, it will be carried out separately in [5]. The statement of the theorem is as follows:

Theorem E. Given are two coordinatizations of Λ, namely Λ = KΓ/I ≅ KΓ̂/Î, where the quiver Γ is based on the primitive idempotents e_{1}, …, e_{n} as before, and Γ̂ (necessarily isomorphic to Γ as a directed graph) is based on primitive idempotents ê_{1}, …, ê_{n} of Λ, ordered in such a way that each e_{i} is congruent, modulo J, to the image of ê_{i} under our isomorphism.

If p̂ is a path of length l in KΓ̂ passing through the sequence of vertices (ê(1), …, ê(l + 1)), then each irreducible component of Vp̂ is birationally equivalent to a component of some variety V_{p}, where p is a path of length l in KΓ passing through the sequence (e(1), …, e(l + 1)). Here Vp̂ is the variety of p̂ relative to the coordinates Γ̂ and Î, and V_{p} the variety of p relative to Γ and I. □

Theorem F and Definition. Fix a sequence 𝕊 = (S(1), …, S(l + 1)) of simple left Λmodules and let (e(1), …, e(l + 1)) be the corresponding canonical sequence of primitive idempotents in a fixed coordinatization Λ = KΓ/I. Then the set of birational equivalence classes of the irreducible components of the non-empty varieties V_{p}, where p runs through the paths of length l in KΓ passing through (e(1), …, e(l + 1)) in that order, is independent of the coordinatization. In other words, the set V_{𝕊} consisting of the irreducible components of the non-empty varieties V_{p} as above is uniquely determined by the isomorphism type of Λ, up to birational equivalence. We denote by V_{𝕊} the set of \underline{irreducible uniserial varieties of Λ at 𝕊}.

Moreover, if V_{𝕊} = \{W_{1}, …, W_{m}\}, the \underline{uniserial genus of Λ at 𝕊} is defined to be the maximum of the numbers genus(W_{1}), …, genus(W_{m}), and is denoted by genus V_{𝕊}. By the preceding paragraph, genus V_{𝕊} is uniquely determined by the isomorphism class of Λ. □

An example of non-trivial uniserial genus can be found in Section 6 (Example 14).

6. Realization of arbitrary varieties as varieties of uniserials

The aim of this section is to show that each affine variety V is isomorphic to a variety of uniserial modules over a suitable finite dimensional algebra Λ = KΓ/I, under the additional restrictions that Γ be acyclic and without double arrows. More precisely, we will realize our given variety V as a variety V_{p} for some path p in Γ. Let 𝕊 be the sequence of simple modules corresponding to the consecutive vertices along p. Observe that, due to the absence of double arrows, V_{𝕊} then consists precisely of the irreducible components of V_{p}. In other words, the set of irreducible components of an arbitrary variety V can be realized as a V_{𝕊}.

By Theorem A(III), the first of the above restrictions on Γ guarantees that Φ_{p} is bijective, meaning that there is a 1–1 correspondence between the points of V_{p} and the uniserial left Λ-modules with mast p. The second condition on Γ ensures that all of the varieties V_{q} where q runs through the paths in KΓ, are actually isomorphism invariants of Λ [5]. In particular, our theorem thus provides a taste of the opulence and complexity of uniserial representations of finite dimensional algebras.

Theorem G. Given a field K and any affine algebraic variety V over K, there exists a finite dimensional path algebra modulo relations, Λ = KΓ/I, together with a path p in KΓ, such that V ≅ V_{p}. We can, moreover, choose the quiver Γ to be acyclic and without double arrows.

Proof. Say V = V (f_{1}, …, f_{M}), where the f_{i} are polynomials in K[X_{1}, …, X_{m}] for some m ≥ 1. In a preliminary step, we construct an affine variety V′ isomorphic to V such that V′ is the vanishing set of polynomials f′ in a certain polynomial ring K[X_{ij}] which have the property that none of the variables X_{ij} occurs in a power higher than 1 in any monomial.

To that end, let d_{i} = max_{1≤j≤M} deg_{X_{i}}(f_{j}) for 1 ≤ i ≤ m, and introduce new variables X_{11}, X_{12}, …, X_{1d_{1}}, X_{21}, X_{22}, …, X_{2d_{2}}, …, X_{m1}, …, X_{md_{m}}. For each s ∈ {1, …, M}, construct a polynomial f′_{s} ∈ K[X_{ij} | 1 ≤ i ≤ m, 1 ≤ j ≤ d_{i}] as follows: Replace any monomial X_{1}^{r_{1}} ⋯ X_{m}^{r_{m}} occurring nontrivially in f_{s} by

(X_{11} ⋯ X_{1r_{1}})(X_{21} ⋯ X_{2r_{2}}) ⋯ (X_{m1} ⋯ X_{mr_{m}})

in K[X_{ij}]; this is possible, since r_{i} ≤ d_{i} for 1 ≤ i ≤ m by construction. Clearly the total degree of this new monomial in the X_{ij} is the same as that of X_{1}^{r_{1}} ⋯ X_{m}^{r_{m}}. Define

V′ = V (f′_{1}, …, f′_{M}, X_{is} − X_{it} | 1 ≤ i ≤ m, 1 ≤ s, t ≤ d_{i}).

That V ≅ V′ is then an obvious consequence of our construction, and hence we may assume, without loss of generality, that V = V′; in other words, we may assume that d_{i} ≤ 1 for all i ∈ {1, …, m}. This concludes the preliminary step.

We now let Γ be the quiver

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

and set p = β_{m}α_{m} ⋯ β_{2}α_{2}β_{1}α_{1}. To define suitable relations in KΓ, we write each f_{j} in the form f_{j} = ∑_{A∈P} c_{j}(A) ∏_{i∈A} X_{i}, where P is the power set of {1, …, m} and c_{j}(A) ∈ K; this is possible by the preceding paragraph. For each A ∈ P, define a path p(A) in KΓ as p(A) = p_{m}(A) ⋯ p_{1}(A), where the p_{i}(A) are defined by

p_{i}(A) = \{ \begin{matrix}γ_{i} & if i ∈ A \\ β_{i}α_{i} & if i ∉ A \end{matrix} .

Note that each p(A) is a path from 1 to 2m + 1. Let r_{j} = ∑_{A∈P} c_{j}(A)p(A) for 1 ≤ j ≤ M, and observe that each r_{j} is a relation in KΓ. Finally, define Λ = KΓ/I, where I ⊆ KΓ is the ideal generated by r_{1}, …, r_{M}, and note that I^{(L)} = I^{(2m+1)} = I is generated by the r_{j} as a K-space. We now verify that V ≅ V_{p} as follows.

Clearly, there are no paths in KΓ starting in 1 which fail to be routes on p, and the detours on p are precisely the pairs (γ_{i}, β_{i−1}α_{i−1} ⋯ β_{1}α_{1}) for 1 ≤ i ≤ m; here β_{0}α_{0} stands for the primitive idempotent e_{1} identified with the vertex 1. Moreover, for each detour (γ, u) on p, there exists precisely one right subpath v(γ, u) = v_{1}(γ, u) of p longer than u and ending in the same vertex as γ, namely v(γ_{i}, β_{i−1}α_{i−1} ⋯ β_{1}α_{1}) = β_{i}α_{i} ⋯ β_{1}α_{1}. Writing X_{i} for the variable X_{1}(γ_{i}, β_{i−1}α_{i−1} ⋯ β_{1}α_{1}), we thus obtain the substitution equations

γ_{1}e_{1} ˆ= X_{1}β_{1}α_{1}
γ_{2}β_{1}α_{1} ˆ= X_{2}β_{2}α_{2}β_{1}α_{1}
⋮
γ_{m}β_{m−1}α_{m−1} ⋯ β_{1}α_{1} ˆ= X_{m}β_{m}α_{m} ⋯ β_{1}α_{1} = X_{m}p