THE GEOMETRY OF UNISERIAL REPRESENTATIONS OF

ALGEBRAS II. ALTERNATE VIEWPOINTS AND UNIQUENESS

Klaus Bongartz and Birge Huisgen-Zimmermann

Abstract. We provide two alternate settings for the family of varieties, introduced in [3], which model the uniserial representations of a finite dimensional algebra. The first is a quasiprojective subvariety of a Grassmannian containing the members of the mentioned family as a principal affine open cover; among other benefits, one derives invariance from this intrinsic description of the ‘uniserial varieties’. The second viewpoint re-interprets these varieties as locally closed subsets of the traditional module varieties; in particular, it exhibits closedness of the fibres of the canonical maps from the uniserial varieties to the uniserial representations.

1. Introduction, notation, and further prerequisites

Let Λ = KΓ/I be a finite dimensional path algebra modulo relations, based on a quiver Γ and an algebraically closed field K of coefficients. In [3], a family of affine algebraic varieties was introduced which not only provides a crisp fit for the interlaced parameters governing the structure of the uniserial Λ-modules, but which is concretely accessible and manageable. In [4, 2] these ‘uniserial varieties’ were used to derive representation-theoretic information from quiver and relations.

Originally, the present note was solely aimed at giving a proof for the invariance of the uniserial varieties under a change of coordinatization of Λ – invariance up to birational equivalence, more precisely. It turned out that this goal is most smoothly achieved via a shift of viewpoint, exhibiting the mentioned family of varieties as an affine open cover of a quasi-projective subvariety of a Grassmannian. This realization somewhat widened the scope of the note.

In fact, we will present two alternate incarnations of the varieties of [3] and provide an application for each. In particular, we will build a bridge to the classical varieties of representations and use it towards an easy proof for the fact that the natural maps from the uniserial varieties to the set of isomorphism types of uniserial Λ-modules always have closed fibres. Our argument is based on the interesting observation that there are no nontrivial uniserial degenerations of uniserial modules (Section 3). On the side, we mention a third re-interpretation of our varieties in terms of Hesselink stratifications of nullcones, given by L. Le Bruyn [5], which provides yet another angle on the subject.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16G10, 16G20, 16G60, 16P10.

The research of the second author was partially supported by a National Science Foundation grant.

For more detail on Section 2, we fix a sequence

𝕊 = (S(0), …, S(l))

of simple left Λ-modules S(i) = Λe(i)/Je(i), where the e(i) are vertices of Γ, identified with the corresponding primitive idempotents of Λ. Note that for each uniserial left Λ-module U with composition length l+1 and sequence 𝕊 of consecutive composition factors, there is at least one path p of length l passing through the vertices e(0), e(1), …, e(l) in this sequence such that pU ≠ 0. Each such path will be referred to as a mast of U; for short, we will moreover say that U has composition series 𝕊 and that p is a path passing through 𝕊. We briefly recall the definition of the varieties V_{p} of [3], where p traces the paths through 𝕊. If p_{0} = e(0), p_{1}, …, p_{l} are the right subpaths of a mast p of a given uniserial module U = Λx, ordered by length (i.e., p = q_{i}p_{i}, meaning q_{i} after p_{i}, and length(p_{i}) = i), we describe U by recording the effect which multiplication by arrows has on the basis (p_{0}x, …, p_{l}x). Not surprisingly, just ‘a few’ of the arising coefficients need to be recorded to pin down the isomorphism class of U: Given m ≤ l and an arrow α in Γ, we call the pair (α, p_{m}) a detour on p – and write (α, p_{m}) ≀≀ p – in case αp_{m} ∈ KΓ \ I is different from p_{m+1}, but there exists an index s > m such that p_{s} ends in the same vertex as α. Letting I(α, p_{m}) be the set of all such indices s, we then obtain αp_{m}x = ∑_{i∈I(α,p_{m})} k_{i}(α, p_{m})p_{i}x for unique scalars k_{i}(α, p_{m}). It is straightforward to check that the family (k_{i}(α, u))_{i∈I(α,u),(α,u)≀≀p} determines U. In [3], it was shown that the set of all such coordinate tuples, as U runs through the uniserials with mast p, is an affine algebraic variety, a defining set of polynomials for which can easily be derived from the quiver Γ and a set of generators for I. The variety V_{p} comes equipped with an obvious map Φ_{p} : V_{p} → {iso types of uniserials with mast p} given by

k = (k_{i}(α, u)) ↦ Λe(0) / ( ∑_{(α,u)≀≀p} Λ(αu − ∑_{i∈I(α,u)} k_{i}(α, u)p_{i}) + ∑_{q not a route on p} Λqe(0)),

where a path q = qe(0) is called a route on p if the sequence of vertices through which it consecutively passes is a subsequence of (e(0), …, e(l)). One of the virtues of this setup is the snug fit of the V_{p}’s; ‘often’, Φ_{p} is a bijection, this being for example the case when p does not pass through an oriented cycle (see [3]).

Now V_{𝕊} denotes the finite family of irreducible components of the varieties V_{p}, where p traces the paths through 𝕊. In case there is more than one such path, this juxtaposition of partial frames is somewhat unsatisfactory; it reflects the shortcoming that, up to now, there was no obvious gluing of these affine patches to a variety incorporating the complete picture. This deficiency is removed in Section 2, where we give an interpretation of the varieties V_{p} as subvarieties G -Uni(p) of a Grassmannian. While being computationally inaccessible (unless one takes the detour through the V_{p}’s), the varieties G -Uni(p) have the advantage of embedding – by dint of their very construction – into a quasiprojective variety G -Uni(𝕊) inside the Grassmannian in which we are operating. The family of varieties G -Uni(p), where p traces the paths through 𝕊, turns out to be an affine cover of G -Uni(𝕊) consisting of principal open sets and very natural in the following sense: First, there is a canonical map ϕ_{𝕊} from G -Uni(𝕊) to the set of isomorphism types of uniserials with composition series 𝕊, the restrictions of which to the patches G -Uni(p) are nothing else but the reincarnations of the maps Φ_{p} in the new setting. In other words, if we denote the restriction of ϕ_{𝕊} to G -Uni(p) by ϕ_{p}, our isomorphism G -Uni(p) → V_{p} renders the following triangles commutative:

\xymatrix{V_{p} \ar[dr]^{Φ_{p}} \ar[dd]_{≅} \\  & \{iso types of uniserials in Λ-mod with mast p\} \\ G -Uni(p) \ar@{-->}[ur]_{ϕ_{p}}}

Second, all of the subvarieties G -Uni(p) of G -Uni(𝕊) are unions of fibres of ϕ_{𝕊}. Thus the variety G -Uni(𝕊) provides a joint geometric framework for the collection of the V_{p}’s which makes the uniqueness statements of Theorems E and F of [3] quite obvious, in that the definition of G -Uni(𝕊) is intrinsic. Moreover, this setting will turn out to be very helpful in the description of the fibres of the maps Φ_{p} in [2].

2. Uniserial varieties as subvarieties of Grassmannians. Uniqueness

Let 𝕊 = (S(0), …, S(l)) be a sequence of simple left Λ-modules, say S(i) = Λe(i)/Je(i). We abbreviate e(0) to e, and let p ∈ KΓ \ I be a path through 𝕊. As before, p_{0}, …, p_{l} will be the right subpaths of p with length(p_{i}) = i, and by A_{p} we will denote the K-subspace of Λe spanned by the p_{i}. Moreover, we set m = dim_{K} Λe − dim_{K} A_{p} = dim_{K} Λe − (l + 1). Finally, G_{m}(Λe) will stand for the Grassmannian of all m-dimensional subspaces of Λe. One readily checks that the subset G_{m}^{Λ}(Λe) of G_{m}(Λe) consisting of all Λ-submodules of Λe having K-dimension m is a closed subvariety. It is inside this closed subset that the varieties G -Uni(p) and G -Uni(𝕊) will be located. We start by observing that, given any module C ∈ G_{m}^{Λ}(Λe) with Λe = A_{p} ⊕ C, we have J^{i}(Λe/C) = Λ(p_{i} + C), and hence Λe/C is uniserial with mast p.

Definition. Let G -Uni(p) be the set of those objects in G_{m}^{Λ}(Λe) which complement the subspace A_{p} in Λe. We define G -Uni(𝕊) as the union of the sets G -Uni(q), where q traces the paths through 𝕊.

Moreover, we label by ϕ_{𝕊} the map

G -Uni(𝕊) ⟶ {iso types of uniserials with composition series 𝕊}

given by C ↦ Λe/C, and by ϕ_{p} we will denote its restriction to G -Uni(p).

By construction, the image of the map ϕ_{p} is just the set of isomorphism classes of uniserial modules with mast p, and hence G -Uni(𝕊) equals the set of all Λ-submodules C of Λe with the property that Λe/C is uniserial with composition series 𝕊. In particular, the fibres of the restrictions ϕ_{p} of ϕ_{𝕊} coincide with fibres of ϕ_{𝕊}, and consequently all intersections of the patches G -Uni(p) are unions of fibres. We have an obvious map ψ_{p} : V_{p} → G -Uni(p) which makes the triangle

\xymatrix{V_{p} \ar[dr]^{Φ_{p}} \ar[dd]_{ψ_{p}} \\  & \{iso types of uniserials in Λ-mod with mast p\} \\ G -Uni(p) \ar@{-->}[ur]_{ϕ_{p}}}

commutative. Namely,

ψ_{p}((k_{i}(α, u)_{(α,u)≀≀p}) = ∑_{(α,u)≀≀p} Λ(αu − ∑_{i∈I(α,u)} k_{i}(α, u)p_{i}) + ∑_{qe(0) not a route on p} Λq

satisfies this requirement (see [3]).

Examples. (1) Let 𝕊 = (S(0), S(1)). If there are s ≥ 1 arrows from e(0) to e(1), say γ_{1}, …, γ_{s}, then clearly each Λ-submodule C of Λe(0) with the property that Λe(0)/C is 2-dimensional with composition series 𝕊 contains B = J^{2}e(0) +∑_{e′≠e(1)} e′Je(0). We infer that the points of G -Uni(𝕊) are in one-to-one correspondence with the (s − 1)-dimensional K-subspaces of Je(0)/B. Consequently, G -Uni(𝕊) is a projective variety, a copy of ℙ^{s−1} in fact. Moreover, the masts p run through the γ_{i}, and the family (G -Uni(p)) provides a canonical affine cover.

(2) In case Λ = KΓ is a hereditary algebra, we are often presented with similarly symmetric situations. For instance, if Γ is the quiver

[figure: 0; ⋮; 1; ⋮; 2; ⋮; ···; ⋮; l]

containing precisely s(i) distinct arrows from the vertex e(i − 1) to the vertex e(i), and if 𝕊 is the sequence (Λe_{0}/Je_{0}, …, Λe_{l}/Je_{l}), the subset G -Uni(𝕊) of G_{m}(Λe(0)) is a multiprojective subvariety; in fact, it is a copy of ℙ^{s(1)−1} × ⋯ × ℙ^{s(l)−1}. Again the varieties V_{p}, where p traces the paths through 𝕊, constitute the standard affine cover in the most convenient coordinatization.

(3) Now let Γ be the quiver consisting of a single vertex and two loops labeled α and β. Fixing a natural number l, let Λ = KΓ/I, where I ⊆ KΓ is the ideal generated by all paths of length l + 1. Moreover, let 𝕊 be the sequence lining up l + 1 copies of the unique simple left Λ-module. An extension of the considerations under (2) yields that G -Uni(𝕊) ≅ ℙ^{1} × ⋯ × ℙ^{1} × 𝔸^{l(l+1)/2} with l factors ℙ^{1}. Once more, the 2^{l}-member family of affine open subvarieties G -Uni(p), where p traces the paths through 𝕊, coincides with the canonical affine cover of G -Uni(𝕊).

This illustration suggests the following general connection between the varieties V_{p} and the Grassmannians. The first part is essentially obvious. The main point, namely that the ψ_{p}’s are isomorphisms, is equally elementary; the slightly technical appearance of our argument is due to the fact that there is no general machinery available for dealing with the V_{p}’s.

Theorem A. Each of the sets G -Uni(p), where p traces the paths through the sequence 𝕊, is an open affine subvariety of G_{m}^{Λ}(Λe); in fact, G -Uni(p) is the intersection of a principal affine subvariety of G_{m}(Λe) with the set of m-dimensional submodules of Λe. Moreover, the maps ψ_{p} are isomorphisms V_{p} → G -Uni(p).

Proof. On all counts, it is harmless to assume that there exists a uniserial module with mast p; indeed, this is equivalent to G -Uni(p) being nonempty, and also to V_{p} being nonempty. In particular, this implies that p_{0}, p_{1}, …, p_{l} ∈ Λe are K-linearly independent in Λe (throughout this proof, we identify paths with their canonical images in Λ). Thus we can supplement the p_{i} to a K-basis p_{0}, p_{1}, … p_{l}, b_{1}, …, b_{m} of Λe. Let B be an ordered basis of ⋀^{m} Λe recruited from the subsets of cardinality m of {p_{i}, b_{j}}, with first vector b_{1}∧⋯∧b_{m}. Then the requirement that a space D in G_{m}(Λe) meet the subspace A_{p} = ∑_{0≤i≤l} Kp_{i} of Λe in zero only is tantamount to asking that the first of the homogeneous coordinates of ⋀^{m} D ∈ ℙ(⋀^{m} Λe) relative to B be nonzero.

For a proof of the final assertion, start by observing that ψ_{p} is onto. Indeed, each Λsubmodule C of Λe which complements the subspace A_{p} contains all those paths starting in e which are non-routes on p; actually, C is generated by the non-routes and some elements of the form αu − ∑_{i>length(u)} k_{i}p_{i} over Λ, where (α, u) is a detour on p and the k_{i} are scalars. It is thus sufficient to invoke the definition of V_{p} to conclude that each object in G -Uni(p) is hit by ψ_{p}. Injectivity of ψ_{p} is obvious in view of the linear independence of the elements p_{i} + ψ_{p}(k) ∈ Λe/ψ_{p}(k), 0 ≤ i ≤ l.

To verify that the map ψ_{p} and its inverse are morphisms, it will turn out convenient to choose vectors b_{1}, …, b_{m} along the line of the first paragraph, as follows: First supplement p_{0}, …, p_{l} to a basis of A_{p} + ∑_{(α,u)≀≀p} Kαu by picking suitable elements b_{1}, …, b_{m_{1}} from {αu | (α, u) ≀≀ p}. Write b_{i} = α_{i}u_{i} for i = 1, …, m_{1}, and impose the side condition that the sum ∑_{1≤i≤m_{1}} length(u_{i}) be maximal. Next supplement p_{0}, …, p_{l}, b_{1}, …, b_{m_{1}} to a basis of

A_{p} + ∑_{(α,u)≀≀p} Kαu + ∑_{q∈Λe a non-route on p} Kq

by picking certain non-routes b_{m_{1}+1}, …, b_{m_{2}} in Λe, and finally supplement p_{0}, …, p_{l}, b_{1}, …, b_{m_{2}} to a basis of Λe by recruiting suitable routes b_{m_{2}+1}, …, b_{m} on p. We claim that each module C ∈ G -Uni(p) then has a unique vectorspace basis of the form

{b_{i} − ∑_{length(u_{i})<j≤l} k_{j}(α_{i}, u_{i})p_{j} | 1 ≤ i ≤ m_{1}} ∪ {b_{m_{1}+1}, …, b_{m_{2}}}

∪ {b_{i} − ∑_{1≤j≤l} l_{ij}p_{j} | m_{2} + 1 ≤ i ≤ m},

where the k_{j}(α_{i}, u_{i}) are scalars and the l_{ij} are polynomial expressions in the k_{j}(α_{i}, u_{i}), with polynomials not depending on the choice of C.

Uniqueness is obvious in view of the fact that C ∩ A_{p} = 0, whereas existence follows from the following considerations. First note that, apart from the claim concerning the l_{ij}, existence is once more an immediate consequence of the fact that Λe = A_{p} ⊕ C; just keep in mind that all non-routes on p are zero modulo C and that αu + C belongs to J^{length(u)+1}(Λe/C) for any detour (α, u), whereas each residue class p_{i} + C lies outside J^{i+1}(Λe/C). So our task is reduced to showing that, given any route v on p, there exist polynomials 𝔮_{r} such that, for any choice of C in G -Uni(p), the element v ∈ Λe is congruent to ∑_{1≤r≤l} 𝔮_{r}(k_{j}(α_{i}, u_{i}))p_{r} modulo C; here the scalars k_{j}(α_{i}, u_{i}) depend of course on C. We will do this by induction on l − d, where d is maximal with the property that p_{d} is a right subpath of v. In case l − d = 0, we have v = p. So suppose this difference is positive. It is clearly harmless to assume that v is not a right subpath of p, i.e., v = v′βp_{d}, where (β, p_{d}) is a detour on p and length(v′) ≥ 0. By our length-maximal choice of b_{1}, …, b_{m_{1}}, there are scalars h_{r}, k_{s} such that βp_{d} = ∑_{1≤r≤l} h_{r}p_{r} + ∑_{1≤s≤m_{1}} k_{s}α_{s}u_{s} with length(u_{s}) ≥ d whenever k_{s} ≠ 0. Again, since p_{r} + C ∉ J^{r+1}(Λe/C), we see that, for all r with h_{r} ≠ 0, we have length(p_{r}) ≥ d + 1. To compute v + C = v′βp_{d} + C, let us multiply the equation for βp_{d} by v′ from the left and individually focus on the residue classes v′p_{r} +C and v′α_{s}u_{s} +C with r ≥ d+1 and length(u_{s}) ≥ d, respectively. If the path v′p_{r} is not a route on p, its residue class modulo C is always zero. In the remaining case, our induction hyposthesis yields polynomials 𝔭_{t}, 1 ≤ t ≤ l, depending only on v′ and r such that v′p_{r} +C = ∑_{1≤t≤l} 𝔭_{t}(k_{j}(α_{i}, u_{i}))p_{t} +C. As for the other terms: By the definition of the scalars k_{z}(α_{s}, u_{s}), we have v′α_{s}u_{s} + C = ∑_{length(u_{s})<z≤l} k_{z}(α_{s}, u_{s})v′p_{z} + C, and in view of length(u_{s}) ≥ d, we obtain v′p_{z} + C = ∑_{1≤t≤l} 𝔭_{t}(k_{j}(α_{i}, u_{i}))v′p_{t} + C for the terms of the latter sum. This completes the proof of our intermediate claim.

Note that the argument of the preceding paragraph holds for arbitrary routes v on p so, in particular, for all routes αu, where (α, u) is a detour. It follows that V_{p} is isomorphic to its projection onto the components (k_{j}(α_{i}, u_{i}))_{j∈I(α_{i},u_{i}),1≤i≤m_{1}}. Again we use our special basis for Λe to form an ordered basis B for ⋀^{m} Λe consisting of m-element subsets and containing b_{1} ∧ ⋯ ∧ b_{m} as first vector. The first paragraph of the proof permits us to identify the points of G -Uni(p) with points of ⋀^{m} Λe having first coordinate 1, which identifies ψ_{p} with the map sending a point k = (k_{i}(α, u)) to c_{1} ∧ ⋯ ∧ c_{m}, where c_{1}, …, c_{m} is the basis of the space ψ_{p}(k) specified in the preceding claim; in particular, c_{i} = b_{i} − ∑_{length(u_{i})<j≤l} k_{j}(α_{i}, u_{i})p_{j} for 1 ≤ i ≤ m_{1}. The claim clearly guarantees that ψ_{p} is a morphism. As for ψ_{p}^{−1}, it will be enough to show that, up to signs, all the scalars k_{j}(α_{i}, u_{i}) show up as coefficients in the expansion of c_{1} ∧ ⋯ ∧ c_{m} relative to B. But since, up to a factor ±1, the elements d_{ij} = d_{ij1} ∧ ⋯ ∧ d_{ijm}, 1 ≤ i ≤ m_{1}, with d_{ijr} = b_{r} for r ≠ i and d_{iji} = p_{j} belong to B, this is clear. □

Corollary B. G -Uni(𝕊) is a quasiprojective variety, and the family (G -Uni(p)), where p traces the paths through 𝕊, is an affine cover consisting of principal open subsets of G -Uni(𝕊). □

The next consequence of our observation proves Theorems E and F of [3].

Corollary C. If Λ ≅ Λ′ and 𝕊′ is the sequence of simple left Λ′-modules corresponding to 𝕊 under an algebra isomorphism, the varieties G -Uni(𝕊) and G -Uni(𝕊′) are isomorphic.

In particular, given a coordinatization KΓ′/I′ of Λ′, there is a one-to-one correspondence between the collection of all irreducible components of the V_{p′}, where p′ traces the paths in Γ′ through 𝕊′, and the irreducible components of the V_{p}, where p traces the paths in Γ through 𝕊, such that the partners under this pairing are birationally equivalent.

Proof. Let f : Λ → Λ′ be an algebra isomorphism and e′ = f(e), then clearly the induced isomorphism G_{m}(Λe) ≅ G_{m}(Λ′e′) restricts to an isomorphism G -Uni(𝕊) ≅ G -Uni(𝕊′). Since the affine patches G -Uni(p′) of G -Uni(𝕊′) relative to any coordinatization of Λ′ form an open cover of G -Uni(𝕊′), the second claim is a consequence of the first. □

3. The uniserial varieties as subvarieties of the classical ones. Closedness of the fibres of the canonical maps.

Let e_{1}, …, e_{n} be the distinct vertices of Γ, and let Γ∗ be the union of {e_{1}, …, e_{n}} with the set of arrows of Γ. Recall that, given d ∈ ℕ, the variety Mod_{Λ}^{d} of (bounden) representations of Λ = KΓ/I with K-dimension d consists of those points

x = (α(x)) ∈ ∏_{α∈Γ∗} Hom_{K}(K^{d}, K^{d}),

the components of which satisfy the relations in I. By R we denote the canonical map from Mod_{Λ}^{d} to the set of isomorphism classes of d-dimensional left Λ-modules. If 𝕊 = (S(0), …, S(l)) is a sequence of simple modules as before and d = l + 1, the subset Uni(𝕊) ⊆ Mod_{Λ}^{d} of those points which correspond to uniserial modules with composition series 𝕊 is an open subvariety of Mod_{Λ}^{d} which is closed under the standard action of GL_{d}; indeed, if p traces the paths through 𝕊, then Uni(𝕊) is the union of the open sets Uni(p) consisting of the points in Mod_{Λ}^{d} which correspond to the uniserials with mast p (i.e., the points x with γ_{l}(x) ⋯ γ_{1}(x) ≠ 0, provided that p = γ_{l} ⋯ γ_{1}). It is within these open sets that we will find closed subvarieties Rep(p) as desired, namely again canonically isomorphic to the V_{p}.

In the sequel, we will write the points of x ∈ Mod_{Λ}^{d} as sequences of d × d-matrices, representing the maps α(x) relative to the canonical basis of K^{d}.

Definition. Given a path p = γ_{l} ⋯ γ_{1} through 𝕊, where γ_{i} is an arrow e(i − 1) → e(i), we define Rep(p) to be the closed subset of Uni(p) consisting of those points x which are represented by tuples of lower triangular matrices (α(x))_{α∈Γ∗} having the additional property that the i-th column of the matrix γ_{i}(x) equals the (i + 1)-st canonical basis vector of K^{d} for 1 ≤ i ≤ l.

It is clear that the image R(Rep(p)) is the full set of isomorphism classes of uniserial modules with mast p; in other words, the image of R restricted to Rep(p) coincides with the image of Φ_{p}.

Theorem D. There is a canonical isomorphism ρ_{p} : Rep(p) → V_{p} of varieties which makes the following diagram commutative:

\xymatrix{V_{p} \ar[dr]^{Φ_{p}} \\  & \{iso types of uniserials in Λ-mod with mast p\} \\ Rep(p) \ar[ur]_{R} \ar[uu]^{ρ_{p}}_{≅}}

Proof. Define ρ_{p}(x) to be (k_{i}^{x}(α, p_{m}))_{i∈I(α,p_{m}),(α,p_{m})≀≀p}, where k_{i}^{x}(α, p_{m}) is the the entry in position (i, m) of the matrix α(x). In other words, ρ_{p} amounts to a projection onto a certain subset of coordinates of Mod_{Λ}^{d} and is thus a morphism. All other entries of x are either 0 or 1, depending solely on p, which shows that ρ_{p} is ‘essentially’ equal to the identity: Indeed, a point x ∈ Rep(p), describing a uniserial module U say, essentially just consists of matrix representations of the multiplications by arrows, relative to a suitable basis of the form (p_{0}z, …, p_{l}z), z a top element of U. Focusing on a matrix α(x) for some arrow α, we note that, for a pair (α, p_{m}) which fails to be a detour, the entry in position (i, m) is 1 or 0 depending on whether or not i = m + 1 and αp_{m} = p_{m+1}; in case of a detour, on the other hand, the corresponding entry is zero for all i ∉ I(α, p_{m}). This clearly guarantees that ρ_{p} is an isomorphism.

Commutativity of the diagram follows from the obvious fact that x and ρ_{p}(x) describe the same uniserial module with mast p, up to isomorphism. □

Remark. Again let d = l + 1, where l = length(p). As will become clear in [2], the variety Rep(p) is – via fibre bundles and geometric quotients – intrinsically related to the open subvariety of Mod_{Λ}^{d} consisting of all those representations which are not annihilated by p (note that such representations are automatically uniserial). Before growing aware of the interpretation in terms of Grassmannians described in Section 2, we made use of this relationship to prove Corollary C. Our present, more comprehensive, viewpoint requires a bit more notational ballast, but has a twofold edge over the one focusing on two incarnations of the V_{p} only. First, it very naturally yields an action of a unipotent group on Rep(p) such that the orbits coincide with the isomorphism classes of uniserials (see [2]); second, it provides us with a global quasi-projective variety which glues the various V_{p}’s together to a variety representing all uniserial modules with a fixed sequence of composition factors.

As an application, we give an elementary proof for the fact that the fibres of the canonical maps Φ_{p} are closed. In view of Theorems A and D, this is clearly tantamount to closedness of the fibres of the sibling maps ϕ_{p}, as well as to closedness – inside Uni(𝕊) – of the intersections of the GL_{d}-orbits of Mod_{Λ}^{d} with Uni(𝕊).

Proposition E. Suppose that U, U′ ∈ Λ-mod are non-isomorphic uniserial modules of the same K-dimension. Then U does not degenerate to U′.

Proof. The beginning of our induction on dim_{K} U is trivial since, in the variety of representations, the points corresponding to semisimple modules have closed orbits. Now suppose that dim_{K} U > 1, and assume that U′ is a degeneration of U which fails to be isomorphic to U. As is well-known, we then have dim_{K} Hom_{Λ}(U, X) ≤ dim_{K} Hom_{Λ}(U′, X) and dim_{K} Hom_{Λ}(X, U) ≤ dim_{K} Hom_{Λ}(X, U′) for all X ∈ Λ-mod (see, e.g., [1]). If S denotes the socle of U, then plugging S = X into the second inequality, we see that S is the socle of U′ as well. Theorem 2.4 of [1] now tells us that Q = U/S degenerates to Q′ = U′/S, and hence the induction hypothesis entails Q ≅ Q′. Since U and U′ are non-isomorphic but have the same K-dimension, we therefore obtain an isomorphism Hom_{Λ}(U′, U) ≅ Hom_{Λ}(Q, U), which yields dim_{K} Hom_{Λ}(U, U) > dim_{K} Hom_{Λ}(Q, U) = dim_{K} Hom_{Λ}(U′, U). This contradicts the first of the cited inequalities. □

Corollary F. The fibres of Φ_{p} : V_{p} → {iso types of uniserials with mast p} are closed subvarieties of V_{p}. □

References

1. K. Bongartz, On degenerations and extensions of finite dimensional modules, Advances in Math. 121 (1996), 245-287.

2. K. Bongartz and B. Huisgen-Zimmermann, Varieties of uniserial representations IV. Kinship to geometric quotients, preprint.

3. B. Huisgen-Zimmermann, The geometry of uniserial representations of finite dimensional algebras I, J. Pure Appl. Algebra 127 (1998), 39-72.

4. B. Huisgen-Zimmermann, The geometry of uniserial representations of finite dimensional algebras. III: Finite uniserial type, Trans. Amer. Math. Soc. 348 (1996), 4775-4812.

5. L. Le Bruyn, Optimal filtrations on representations of finite dimensional algebras, Trans. Amer. Math. Soc., (to appear); see http://win-www.uia.ac.be/u/lebruyn/PAPERS/optimalnew.dvi.

FB Mathematik, Universität Gesamthochschule Wuppertal, 42119 Wuppertal, Germany

  E-mail address: bongartz@math.uni-wuppertal.de

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

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