Strong exceptional sequences provided by quivers

Klaus Altmann Lutz Hille

Abstract

Let Q be a finite quiver without oriented cycles. Denote by U → M(Q) the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext_{M(Q)}^{l}(U, U) = 0 for all l > 0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra of the quiver Q. If so, then the bounded derived categories of finitely generated right kQ-modules and that of coherent sheaves on M(Q) are related via the full and faithful functor − ⊗_{kQ}^{IL} U.

1 Introduction

(1.1) Let Q be a quiver (i.e. an oriented graph) without oriented cycles; denote by Q_{0} the vertices and by Q_{1} the arrows of Q. For a fixed dimension vector d, that is a map d : Q_{0} → ZZ_{≥0}, we denote by IH(d) := {θ : Q_{0} → IR | ∑_{q∈Q_{0}} θ_{q}d_{q} = 0} the vector space of the so-called weights with respect to d. We fix an algebraically closed field k. To each θ ∈ IH(d) there exists the moduli space M^{θ}(Q, d) of θ-semistable k-representations of Q with dimension vector d (cf. [Ki]). This space is known to be projective and, in case θ is in general position and d is indivisible, also smooth. Moreover, if we restrict ourselves to thin sincere representations, that is d_{q} = 1 for all q ∈ Q_{0}, then M^{θ}(Q) is also toric (cf. [Hi2]). In any case, each integral weight θ induces an ample line bundle L(θ) on M^{θ}(Q, d). If θ is in general position and d indivisible, then M^{θ}(Q, d) is, in addition, a fine moduli space admitting a universal bundle U. The universal bundle splits into a direct sum of vector bundles U = ⊕_{q∈Q_{0}}U_{q}, and the summands U_{q} have rank d_{q} (cf. [Ki]). All known examples suggest that the universal bundles on those moduli spaces have no self-extensions, i.e. Ext _{M^{θ}(Q,d)}^{l}(U, U) = 0 for all l > 0. The issue of this paper is to prove this formula in special cases. The meaning of this property and its relation to tilting theory will be discussed in (1.4). In this paper we restrict ourselves to thin sincere representations; the corresponding moduli spaces are called toric quiver varieties. Because d = (1, …, 1) is fixed, we will omit it in all notation introduced above. The direct summands of the universal bundle are line bundles, and they are characterized, up to a common twist, by the following property: For any arrow α ∈ Q_{1} pointing from p to q (p, q ∈ Q_{0}) the invertible sheaf U_{p}^{−1}⊗U_{q} corresponds to the divisor of all representations assigning the zero map to α. Furthermore, there exists a distinguished weight θ^{c} (see (1.2) and (2.7) for a definition and first properties). We denote the corresponding moduli space by M(Q).

(1.2) Polarized projective toric varieties may be constructed from lattice polytopes. If one wants to forget about the polarization, simply consider the inner normal fan of the polytope. In §2 we give a detailed description of the moduli space M^{θ}(Q) of thin sincere representations via its “defining polytope” Δ(θ). The easiest way to obtain Δ(θ) from the quiver is to imagine Q as a one-way pipe system carrying liquid; a weight θ ∈ IH describes the input (possibly negative) into the system at each knot. Using this language, Δ(θ) is simply given as the set of all possible flows respecting both the direction of the pipes and the given input θ (see (2.3)). Considering the opposite viewpoint, each flow through our pipe system requires a certain input, i.e. a weight. In particular, from the special flow that is constant 1 at each pipe we obtain a special, so-called canonical, weight θ^{c}. The corresponding Δ(θ^{c}) is a reflexive polytope (in the sense of Batyrev, [Bat]), i.e. the moduli space M(Q) is Fano (Proposition (2.7)). Fixing a weight θ in general position, i.e. M^{θ}(Q) is smooth, flows and weights have still another meaning. Each flow defines an equivariant, with respect to the defining torus, effective divisor, and each weight θ′ defines an element L(θ′) in the Picard group of M^{θ}(Q). Assigning a flow its input weight corresponds to assigning a divisor its class in the Picard group (see (3.1)).

Example: 1) In the special case θ′ := θ this recovers our ample line bundle introduced before.

2) The line bundle U_{p}^{−1} ⊗ U_{q} corresponds to the weight with values 1 at p, −1 at q, and zero at all other points.

(1.3) Our first main result is Theorem (3.6) stating the lack of self-extensions of U on the moduli space M(Q) with respect to the canonical weight, i.e. Ext _{M(Q)}^{l}(U, U) = 0 for all l > 0. This is proved by using a slightly generalized Kodaira vanishing argument which works for toric varieties, cf. Theorem (3.5). As a corollary of Theorem (3.6) we conclude that we obtain a full and faithful functor from the bounded derived category of finitely generated right modules over the endomorphism algebra A of U into the bounded derived category of coherent sheaves on the moduli space M(Q) (Theorem (4.4)). Moreover, in Theorem (4.3) we provide a criterion for A = End_{M(Q)}(U, U) to be isomorphic to the path algebra kQ of the quiver Q. Combining both results we obtain the following relation between the derived categories of right kQ-modules and of coherent sheaves on M(Q), respectively.

Theorem: Assume Q is a quiver lacking (1, 0)- and (t, t)-walls (see (2.2) for an explanation). Then,

− ⊗_{kQ}^{IL} U : D^{b}(mod–kQ) ⟶ D^{b}(Coh(M(Q)))

is a full and faithful functor from the bounded derived category of finitely generated right kQ-modules into the bounded derived category of coherent sheaves on M(Q).

(1.4) The result above is closely related to tilting theory. Since the fundamental paper [Be], tilting theory has become a major tool in classifying vector bundles; a tilting sheaf induces an equivalence of bounded derived categories, as in the previous Theorem. To be precise we recall the definition of a tilting sheaf ([Bae]). A sheaf T on a smooth projective variety is called a tilting sheaf if

1) it has no higher self-extensions, that is Ext^{l}(T , T ) = 0 for all l > 0,

2) the direct summands generate the bounded derived category, and

3) the endomorphism algebra A of T has finite global (homological) dimension.

Then, the functors IRHom(T , −) and − ⊗_{A}^{IL} T define mutually inverse equivalences of the bounded derived categories of coherent sheaves on the underlying variety of T and of the finitely generated right A-modules, respectively. For constructions of tilting bundles and their relations to derived categories we refer to the following papers [Ka], [Be], [Ru], [Bo], and [Or]. For the similar notion of a tilting module we refer to [HR]. For our purpose, the notion of an exceptional sequence is more useful. Let C be any of the categories introduced above: the category of finitely generated right modules over a finite dimensional algebra, the category of coherent sheaves on a smooth projective variety, or one of its derived categories. Thus, C is either an abelian or a triangulated k-category. Each object in C has a unique, up to isomorphism and reordering, decomposition into indecomposable direct summands, i.e. C is a Krull-Schmidt category. Moreover, the extension groups are defined and globally bounded; they are finite-dimensional k-vector spaces. An object in C is called exceptional if it has no self-extensions and its endomorphism ring is k. A sequence (E_{0}, …, E_{n}) of objects in C is called exceptional if 1) each object E_{i} for i = 0, …, n is exceptional and

2) Ext^{l}(E_{j}, E_{i}) = 0 for all l ≥ 0, and j > i.

Such a sequence is called strong exceptional if, additionally, 3) Ext^{l}(E_{i}, E_{j}) = 0 for all l > 0 and all i, j = 0, …, n.

Finally, it is called full if in addition to 1), 2) and 3)

4) the objects E_{i} for i = 0, …, n generate the bounded derived category.

Thus, each full strong exceptional sequence defines a tilting bundle ⊕_{i=0}^{n}E_{i}, because the endomorphism algebra of ⊕_{i=0}^{n}E_{i} has global dimension at most n. Vice versa, each tilting bundle whose direct summands are line bundles gives rise to a strong exceptional sequence. Using this language, our vanishing result Theorem (3.6) means that the direct summands of U form a strong exceptional sequence.

(1.5) In general, this sequence cannot be full. Assume the contrary; then the bounded derived categories in the previous theorem are equivalent. The first one is a derived category of a hereditary abelian category, whose structure is well-known ([Ri2]). In particular, the Serre functor (see [BK] for the definition) coincides with the Auslander-Reiten translation and fixes objects up to translation only in case the category is tame or just semi-simple (cf. [Hap] §1.4/5). On the other hand, the Serre functor in the bounded derived category of coherent sheaves fixes all skyscraper sheaves up to a translation. Consequently, an equivalence implies that M is a point or a projective line in case the algebra kQ is semi-simple or tame, respectively. It follows that Q is a point or the Kronecker quiver; the remaining tame cases may not appear (see [Ri1] Theorem p. 158). Nevertheless, there is some hope that one may find a complement \overline{U} such that U ⊕ \overline{U} is a tilting bundle. At least a class of very particular examples of tilting bundles on toric quiver varieties is known ([Hi2], Theorem 3.9).

(1.6) For an introduction to quivers and path algebras we refer the reader to [Ri1] and [ARS]; the theory of localizations may be found in [S]. For an introduction to moduli spaces we mention [N] and for moduli of representations of quivers we refer to the work of King [Ki]. For results on triangulated categories we refer to [Hap] and [Har]. Our standard reference for toric geometry is [Ke]; for a short introduction to this area we also mention [F]. We would like to thank G. Hein, A. King, and A. Schofield for helpful discussions.

2 Moduli spaces of thin sincere representations

(2.1) Let Q be a connected quiver without oriented cycles; it consists of a set Q_{0} of vertices, a set Q_{1} of arrows, and two functions s, t : Q_{1} → Q_{0} assigning to each arrow α ∈ Q_{1} its source s(α) and its target t(α). A representation of Q is a collection of finite dimensional k-vector spaces x(q) for each vertex q together with a collection of linear maps x(α) : x(s(α)) → x(t(α)) for each arrow α ∈ Q_{1}. The dimension vector d = (d_{q} | q ∈ Q_{0}) of a representation x is defined by d_{q} = dim x(q). A representation is called thin if dim x(q) ≤ 1 for all q ∈ Q_{0} and sincere if dim x(q) ≥ 1 for all q ∈ Q_{0}. In this paper we consider only thin sincere representations. We denote by R = ⊕_{α∈Q_{1}}k the space of all thin sincere representations, that is x(q) = k. By G = ×_{q∈Q_{0}}k^{∗} we denote the torus acting via conjugation on R. The orbits of this action are exactly the isomorphism classes of thin sincere representations, i.e. their moduli space may be obtained via GIT. Doing so, we have to deal with the notion of stability with respect to a given weight (cf. [Ki]).

Definition: The elements of the real vector space IH := {θ : Q_{0} → IR | ∑_{q∈Q_{0}} θ_{q} = 0} are called weights of the quiver Q.

Let θ ∈ IH. A thin sincere representation x of Q is θ-stable (-semistable) if for each proper nontrivial subrepresentation y ⊂ x we have ∑_{q∈Q_{0}|y(q)≠0} θ_{q} < 0 (≤ 0 respectively). Two semistable representations x and y are called S-equivalent with respect to θ if the factors of the stable Jordan-Hölder filtration coincide. A subquiver Q′ ⊆ Q with Q′_{0} = Q_{0} is θ-stable (θ-semistable) if it has a θ-stable (θ-semistable) representation. Two quivers are S-equivalent with respect to θ if they admit θ-semistable representations of the same S-equivalence classes. Finally, we denote by T (θ) the set of all θ-semistable subtrees T ⊆ Q with T_{0} = Q_{0} and by Q_{1}(θ) the set of all arrows α such that Q \ {α} is a θ-stable subquiver.

In other words, a representation x is θ-stable precisely when the subquiver Q_{0}∪{α ∈ Q_{1} | x(α) ≠ 0} is θ-stable. Moreover, a subquiver Q′ is θ-stable if and only if for all non-trivial proper subsets S ⊂ Q_{0} which are closed under successors in Q′ we have ∑_{q∈S} θ_{q} < 0 (see [Hi1] §2 and [Hi2] Lemma 1.4). We also note that each S-equivalence class contains a unique minimal θ-semistable subquiver – just take the disjoint union of the support of the Jordan-Hölder factors.

(2.2) As already mentioned in the beginning, for any given weight θ the moduli space M^{θ}(Q) exists; however, different weights θ may cause different moduli spaces. According to [Hi1] there is a chamber system in IH, and the type of M^{θ}(Q) can only flip if θ crosses walls of the following type:

Definition: W ⊆ IH is called a (t^{+}, t^{−})-wall if

W = {θ ∈ IH | ∑_{q∈Q_{0}^{+}} θ_{q} = − ∑_{q∈Q_{0}^{−}} θ_{q} = 0}

for some decomposition Q_{0} = Q_{0}^{+}⊔Q_{0}^{−} such that the full subquivers Q^{+} and Q^{−} are both connected and such there are exactly t^{+} arrows pointing from Q_{0}^{+} to Q_{0}^{−} and t^{−} arrows the other way around. We say that θ is in general position if θ does not lie on any wall and if the moduli space is not empty.

Assume θ is in general position, then M^{θ}(Q) is smooth and has the (maximal) dimension d = #Q_{1} − #Q_{0} + 1, see [Hi1]. Moreover, for those weights, every semistable thin sincere representation is stable.

(2.3) To describe the toric structure of M^{θ}(Q) we introduce the real vector space of flows defined as IF := {r : Q_{1} → IR} = IR^{Q_{1}}. A flow is called regular if it has only non-negative values, i.e. if it respects the direction of the pipes. For any α ∈ Q_{1} we denote by f^{α} ∈ IF the characteristic flow mapping α to 1 and keeping the remaining pipes dry. More generally, for each walk w without cycles in Q we define the characteristic flow f^{w} mapping an arrow α ∈ w to 1, an arrow β with β^{−1} ∈ w to −1, and the remaining arrows to 0. This flow is regular if and only if w is a path, i.e. respects the orientation in Q. There is a canonical linear map π : IF → IH describing the input of flows; if α ∈ Q_{1} points from p to q, then π(f^{α}) sends p and q onto 1 and −1, respectively. Thus

(π(r))_{q} = ∑_{s(α)=q} r_{α} − ∑_{t(α)=q} r_{α}.

This leads to the following definition:

Definition: The convex polytope of flows Δ(θ) assigned to a weight θ is defined as the intersection

Δ(θ) := π^{−1}(θ) ∩ IR_{≥0}^{Q_{1}} .

This means that Δ(θ) consists of exactly those regular flows respecting the prescribed input θ. Moreover, Δ(θ) is compact since Q has no oriented cycles. The vector spaces IF and IH contain the lattices IF_{Z Z} and IH_{Z Z} of integral flows and weights, respectively. For any integral weight θ : Q_{0} → ZZ we define the affine lattice M^{θ} ⊆ IF as the fiber IF_{Z Z} ∩ π^{−1}(θ), i.e.

M^{θ} = {r ∈ ZZ^{Q_{1}} | ∑_{s(α)=q} r_{α} − ∑_{t(α)=q} r_{α} = θ_{q} for all q ∈ Q_{0}} .

Any element r ∈ M^{θ} provides obviously an isomorphism (+r) : M := M^{0} ⟶^{∼} M^{θ}.

(2.4) The following lemma will be crucial for the understanding of our flow polytope as well as for proving the upcoming vanishing theorem in §3; it explicitly provides points of the lattices M^{θ}. Let T ⊆ Q be an arbitrary maximal tree. Each arrow α ∈ T_{1} divides Q_{0} into two disjoint subsets, the source S_{T} (α) and the target T_{T} (α).

[figure: T_{T} (α); α; S_{T} (α)]

Lemma: Fix a maximal tree T and let θ ∈ IH. For any flow ε there is a unique element r = r^{T} ∈ π^{−1}(θ) ⊆ IF satisfying r_{α} := ε(α) for α ∉ T_{1}. Its remaining coordinates (i.e. for α ∈ T_{1}) are given by

r_{α} − ε(α) = ∑_{q∈S_{T} (α)}θ_{q} + ∑_{T_{T} (α)→^{β}S_{T} (α)}ε(β) − ∑_{S_{T} (α)→^{β}T_{T} (α)}ε(β) .

Proof: First, we should note that both ε and r are flows – the different notation for their coordinates (ε(α) and r_{α}, respectively) was chosen for psychological reasons only. Now, let r be some element of IF. If π(r) = θ, then for any subdivision Q_{0} = Q_{0}^{+} ⊔ Q_{0}^{−} we obtain

∑_{s(β)∈Q_{0}^{+},t(β)∈Q_{0}^{−}} r_{β} − ∑_{s(β)∈Q_{0}^{−},t(β)∈Q_{0}^{+}} r_{β} = ∑_{q∈Q_{0}^{+}} θ_{q} = − ∑_{q∈Q_{0}^{−}} θ_{q}

just by summing up the M^{θ}-equations with q ∈ Q_{0}^{+}. The reverse implication is also true, even if we restrict ourselves to the special subdivisions provided by arrows α ∈ T_{1} via Q_{0}^{+} := S_{T} (α) and Q_{1}^{+} := T_{T} (α). On the other hand, these subdivisions have the important property that α is the only arrow that belongs to both the tree T and to one of the index sets {β | s(β) ∈ Q_{0}^{+}, t(β) ∈ Q_{0}^{−}} or {β | s(β) ∈ Q_{0}^{−}, t(β) ∈ Q_{0}^{+}}. In particular, by just taking care of this single exception, in the above equations we may always replace r_{β} by ε(β). □

If the weight θ and the flow are integral, then so is r, i.e. r ∈ M^{θ}.

(2.5) Proposition: Let θ ∈ IH be an integral weight, then the polytope of flows Δ(θ) ⊆ M^{θ} is always a lattice polytope. The associated projective toric variety equals M^{θ}(Q). Moreover, Δ(θ) provides an ample, equivariant line bundle L(θ) on M^{θ}(Q).

Proof: First, we establish a one-to-one correspondence between vertices of Δ(θ) and S-equivalence classes of θ-semistable trees. Faces of Δ(θ) in any dimension are obtained by forcing certain coordinates of IF to be zero. In particular, vertices are points with a maximal set of vanishing coordinates. Let r ∈ Δ(θ) be a vertex and denote its support by

supp r := {α ∈ Q_{1} | r_{α} > 0} .

If supp r contained any cycle of Q, then we could replace r by a different regular flow with the same weight and a smaller support. Hence, supp r is contained in maximal trees of Q. Moreover, we obtain that

• every maximal tree T containing supp r is θ-semistable, and

• those trees are stable if and only if supp r = T_{1} (which determines the tree uniquely).

To prove these facts, take a proper subset Q_{0}^{+} ⊂ Q_{0} that is closed under successors in T; denoting Q_{0}^{−} := Q_{0} \ Q_{0}^{+} this means that there are no arrows pointing from Q_{0}^{+} to Q_{0}^{−} in T. Hence, using the formula mentioned in the previous proof,

∑_{q∈Q_{0}^{+}} θ_{q} = − ∑_{[Q_{0}^{−}→^{β}Q_{0}^{+}]} r_{β} ≤ 0.

Conversely, let T be any maximal tree. The previous lemma tells us that there is exactly one r ∈ π^{−1}(θ) ⊆ IF such that supp r ⊆ T_{1}; it has integer coordinates r_{α} = ∑_{S_{T} (α)} θ_{q} for α ∈ T_{1}. Moreover, if T is θ-semistable, then these numbers are non-negative, meaning that r ∈ Δ(θ). Thus, it must be a vertex. Moreover, two different trees define the same vertex if and only if they are S-equivalent.

What does the inner normal fan Σ(θ) look like? Denote by C the matrix describing the incidences of our quiver; C consists of #Q_{0} rows and #Q_{1} columns, and for q ∈ Q_{0}, α ∈ Q_{1} we have

C_{qα} := \{ \begin{matrix}+1 & if q = s(α) \\ −1 & if q = t(α) \\ 0 & otherwise . \end{matrix}

The free abelian group N := ZZ^{Q_{1}}/(C–rows) is dual to the lattice M := M^{0} = (C–rows)^{⊥} ⊆ IF. Hence, writing a^{α} ∈ N for the images of the canonical vectors e^{α} ∈ ZZ^{Q_{1}}, we obtain

Σ(θ) = \{⟨a^{α_{1}}, …, a^{α_{k}}⟩ | \begin{matrix}Q \ \{α_{1}, …, α_{k}\} is the minimal element in some S-equivalence class \\ of θ-semistable subquivers \}. \end{matrix}

According to [Hi2], Theorem 1.7 and (3.3), this is exactly the fan defining the moduli space M^{θ}(Q).

□

Example: In the quiver below the canonical weight is θ^{c} = (2, 1, −1, −2).

[figure: 4; 1; 2; 3]  ^{C = ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ −1 & 0 & 1 & 1 & 0 \\ 0 & −1 & −1 & 0 & 1 \\ 0 & 0 & 0 & −1 & −1 \end{matrix} )}

The corresponding fan Σ(θ), and the polytope Δ(θ^{c}) look like the following:

[figure]

It is known that the toric variety of this fan is the blow up of the projective plane in two points, which is isomorphic to the blow up of the two-dimensional smooth quadric in one point.

(2.6) Equivariant (with respect to the torus action), invertible sheaves L on a toric variety X(Σ) are completely determined by their order function ord L : Σ^{(1)} → ZZ or its piecewise linear continuation ord L : N_{IR} → IR. If r(σ) ∈ M provides a local generator x^{r(σ)} of L on U(σ) ⊆ X(Σ), then ord L(a) is defined as ⟨a, r(σ)⟩ if a ∈ σ (cf. [Od]). Moreover, if L is an ample (or at least globally generated) invertible sheaf given by a lattice polytope Δ ⊆ M_{IR}, then the local generators of L correspond to the vertices of Δ. In particular, ord L(a) = min ⟨a, Δ⟩. Shifting the polytope Δ by a vector r ∈ M means to replace L by x^{r} · L. The corresponding order functions differ by the globally linear function ⟨•, r⟩.

Lemma: Let r ∈ M^{θ} be an arbitrary element. Then, the mapping a^{α} ↦ −r_{α} gives the order function of L(θ) on M^{θ}(Q). A different choice r′ ∈ M^{θ} just changes the order function by the linear summand ⟨•, r′ − r⟩.

Proof: We may use r ∈ M^{θ} to carry Δ(θ) into the “right” lattice M (see the end of (2.3)). Then, the order function applied to a^{α} ∈ Σ(θ)^{(1)} ⊆ N_{IR} is

ord L(θ)(a^{α}) = min ⟨a^{α}, Δ(θ) − r⟩ = min ⟨e^{α}, Δ(θ)⟩ − r_{α} = −r_{α} . □

(2.7) Given the quiver Q, the canonical weight θ^{c} announced in the introduction is defined as the weight of the flow r^{c} that is constant 1 on every arrow. Explicitly, this means

θ^{c}(q) := #{arrows with source q} − #{arrows with target q} .

The advantage of θ^{c} is the existence of a unique interior lattice point in the polytope Δ(θ^{c}): it is again the flow r^{c} = [1, …, 1].

Proposition: The polytope Δ(θ^{c}) is reflexive (in the sense of [Bat]), its order function is −1 on the generators a^{α} ∈ Σ(θ^{c})^{(1)}, and the ample divisor L(θ^{c}) is anti-canonical.

Proof: The three claims are synonymous, i.e. we just have to look at the order function of L(θ^{c}). Applying Lemma (2.6) on r^{c} = \underline{1} yields the result. □

3 The cohomology of the universal bundle

(3.1) From now on we assume that θ is an integral weight in general position, i.e. M^{θ}(Q) is a smooth variety. To each integral flow we associate a divisor as follows:

f^{α} ∈ IF_{Z Z} ↦ D_{α} := \{x ∈ M^{θ}(Q) | x_{α} = 0\} = \{ \begin{matrix}\overline{orb (α)} & if a^{α} ∈ Σ^{(1)} = Q_{1}(θ) \\ ∅ & otherwise \end{matrix}

with \overline{orb (α)} denoting the closed orbit corresponding to the one-dimensional cone α. One obtains surjective maps IF_{Z Z} ⟶ ⟶ Div M^{θ}(Q) from the space of integral flows onto the equivariant divisors and, as a consequence, IH_{Z Z} ⟶ ⟶ Pic M^{θ}(Q), θ′ ↦ L(θ′) from the integral weights to the Picard group (see also [Hi2], Theorem 2.3). Applying the map π : IF_{Z Z} ⟶ ⟶ IH_{Z Z} means to assign a divisor its class in the Picard group. Copying the definition of (2.3), every weight θ′ gives rise to

\overline{Δ}(θ′) := \{r ∈ π^{−1}(θ′) | r_{α} ≥ 0 for α ∈ Q_{1}(θ)\} .

Even if L(θ′) is not ample on M^{θ}(Q), the polytope \overline{Δ}(θ′) may still be used to describe the global sections:

Proposition: The lattice points of \overline{Δ}(θ′) provide a basis of the global sections of L(θ′). Moreover, if Q_{1}(θ′) ⊆ Q_{1}(θ), then both polytopes Δ(θ′) and \overline{Δ}(θ′) coincide.

Proof: Given θ′, we choose a flow r ∈ M^{θ′} providing the order function of a divisor in the class defined by θ′. The corresponding polytope of global sections is contained in M = M^{0}; via the isomorphism (+r) : M^{0} → M^{θ′} it is mapped onto \overline{Δ}(θ′). If Q_{1}(θ′) ⊆ Q_{1}(θ), then \overline{Δ}(θ′) sits between Δ(θ′) and \{r ∈ π^{−1}(θ′) | r_{α} ≥ 0 for α ∈ Q_{1}(θ′)\}. On the other hand, Proposition (2.5) implies that the latter two polytopes are equal; its proof shows quite directly that the inequalities parametrized by Q_{1} \ Q_{1}(θ′) are redundant for the definition of Δ(θ′). □

(3.2) Since θ is in general position, there is a universal bundle U on M^{θ}(Q); it splits into a direct sum U = ⊕_{q∈Q_{0}}U_{q} of line bundles. The direct summands U_{p,q} := U_{p}^{−1} ⊗ U_{q} of \underline{End} (U) have the following shape: Choose a walk from p to q along (possibly reversed) arrows α_{1}^{ε(1)}, …, α_{m}^{ε(m)}, i.e. α_{1}, …, α_{m} ∈ Q_{1} and ε(i) ∈ {±1}. Then, denoting by O(α) := O(D_{α}) the sheaf corresponding to the prime divisor D_{α},

U_{p,q} = U_{p}^{−1} ⊗ U_{q} = ⨂_{i=1}^{m} O(α_{i})^{ε(i)} .

In the Picard group of M^{θ}(Q) this sheaf does not depend on the particular choice of the walk from p to q: Using the language of (3.1), the sheaves ⊗_{i} O(α_{i})^{ε(i)} are induced from the flows ∑_{i} ε(i)·f^{α_{i}}, which all have the same weight.

Notation: Setting ε(α^{i}) := ε(i) and ε(α) := 0 for α ∉ {α^{1}, …, α^{m}} provides a function ε : Q_{1} → {1, −1, 0} for every walk. This is the characteristic flow introduced in (2.3). Then, the sheaf U_{p,q} may be written as U_{p,q} = U(ε) = ⊗_{α∈Q_{1}}O(α)^{ε(α)}; the corresponding weight θ_{p,q} := π(ε) has value 1 in p, −1 in q, and 0 in all other vertices.

(3.3) Proposition:

(1) Let θ ∈ IH_{Z Z} be an integral weight in general position. Then, the sheaves U_{p,q} ⊗ L(θ) and U_{p,q}^{−1} ⊗ L(θ) are generated by their global sections.

(2) If, additionally, θ = θ^{c}, then the polytopes \overline{Δ}(θ^{c} ± θ_{p,q}) (describing the global sections) have the same dimension as Δ(θ^{c}).

Proof: Since U_{q,p} = U_{p,q}^{−1}, it is sufficient to consider the latter sheaf. The corresponding polytope \overline{Δ}(θ − θ_{p,q}) may be studied in different level sets:

\begin{aligned}\overline{Δ}(θ − θ_{p,q}) & = & \{r ∈ π^{−1}(θ − θ_{p,q}) | r_{α} ≥ 0 for α ∈ Q_{1}(θ)\} \\ & ≅ & \{r ∈ π^{−1}(θ) | r_{α} ≥ ε(α) for α ∈ Q_{1}(θ)\} . \end{aligned}

We will use the second description.

(1) The vertices of Δ(θ) and thus also the top-dimensional cones of Σ(θ) are in a one-to-one correspondence with the θ-stable trees in Q. Let T ∈ T (θ); the corresponding vertex Δ^{T} of Δ(θ) provides a local generator of L(θ). Since the Δ^{T} are characterized by the property Δ_{α}^{T} = 0 for α ∉ T, we obtain the local generators of U_{p,q}^{−1} ⊗ L(θ) from the lattice points r^{T} ∈ M^{θ} assigned via Lemma (2.4) to the map ε : Q_{1} → ZZ describing a walk from p to q. We have to show that these local generators r^{T} are regular on any open, affine subset U_{T′} ⊆ M^{θ}(Q) corresponding to some possibly different tree T′ ∈ T (θ). That means, it remains to check that r^{T} ∈ r^{T′} + (σ_{T′})^{∨} where σ_{T′} denotes the cone corresponding to T′, i.e. σ_{T′} is spanned by those arrows not contained in T′.

Claim: Let T ∈ T (θ) be a θ-stable tree, and let α be any arrow in Q. Then r_{α}^{T} ≥ ε(α) is true. Before we prove that claim, we remark that it solves our problem, as, for any tree, we know for T′ that r_{α}^{T′} = ε(α) for α ∉ T′. Hence, the claim implies r_{α}^{T} ≥ r_{α}^{T′} for α ∉ T′. On the other hand, (σ_{T′})^{∨} is just given by the inequalities r_{α} ≥ 0 for those α.

The claim is trivial for α ∉ T; if α ∈ T, we use the formula for r_{α} − ε(α) presented in Lemma (2.4). First, as already used in (2.5), stability of T implies ∑_{q∈S_{T} (α)} θ_{q} ≥ 1. Now, the point is to

interpret the two remaining sums well: together they just count the number of arrows α_{i}^{ε(i)} in the walk from p to q pointing from T_{T} (α) to S_{T} (α) minus those from S_{T} (α) to T_{T} (α). In particular,

∑_{T_{T} (α)→^{β}S_{T} (α)}ε(β) − ∑_{S_{T} (α)→^{β}T_{T} (α)}ε(β) = \{ \begin{matrix}−1 & if p ∈ S_{T} (α) and q ∈ T_{T} (α) \\ 1 & if q ∈ S_{T} (α) and p ∈ T_{T} (α) \\ 0 & if p, q ∈ S_{T} (α) or p, q ∈ T_{T} (α) . \end{matrix}

In any case, r_{α} − ε(α) remains non-negative.

(2) For the second part, we consider θ = θ^{c}. Since ε has only −1, 0, or 1 as values, the canonical flow r^{c} = \underline{1} ∈ Δ(θ^{c}) is also contained in \overline{Δ}(θ^{c}−θ_{p,q}) ⊆ π^{−1}(θ^{c}). Assuming dim \overline{Δ}(θ^{c}−θ_{p,q}) < dim Δ(θ^{c}), this means that there exist arrows α^{1}, …, α^{k} ∈ Q_{1}(θ^{c}) having the following two properties:

(i) The flow r^{c} satisfies the α^{v}-inequalities of \overline{Δ}(θ^{c} − θ_{p,q}) sharp, i.e. 1 = ε(α^{v}) for v = 1, …, k.

(ii) It is possible to represent 0 ∈ N_{IR} as a positive linear combination of the vectors a^{α^{v}} ∈ N, v = 1, …, k. (Recall that a^{α} is the normal vector of the supporting hyperplane corresponding to the inequality “r_{α} ≥ ε(α)”.)

The first property means that, along the chosen walk ε from p to q, the arrows α^{1}, …, α^{k} have all the same direction. On the other hand, the second property implies that there is a decomposition Q_{0} = Q_{0}^{+} ⊔ Q_{0}^{−} with α^{1}, …, α^{k} pointing from Q_{0}^{+} to Q_{0}^{−} and being the only arrows connecting these two parts. This yields a contradiction. □

(3.4) Let Σ be a complete fan in some d-dimensional vector space N_{IR} with lattice N. Denote by M the dual lattice. By [Ke], I/§3 we know that the cohomology groups of equivariant, invertible sheaves L are M-graded and how to calculate their summands as reduced topological cohomology groups of certain subsets of N_{IR}. With r ∈ M and A_{r} := {a ∈ N_{IR} | ⟨a, r⟩ < ord L(a)} it follows that H^{l}(X_{Σ}, L)_{r} = H̃^{l−1}(A_{r}, k) for l ≥ 1. We would like to use this method to prove a generalization of Kodaira-vanishing which holds for toric varieties. We restrict the subsets A_{r} in question to the (d − 1)-dimensional unit sphere S^{d−1} ⊂ N_{IR}:

Lemma: Let ϕ : N_{IR} → IR be a continuous function which is linear on the cones of Σ. For B_{r} := {a ∈ S^{d−1} ⊂ N_{IR} | ⟨a, r⟩ < ϕ(a)} we denote by Σ_{r} ⊆ Σ the subfan consisting of all cones σ ∈ Σ such that σ ∩ S^{d−1} ⊆ B_{r}. Then, the sets B_{r} and |Σ_{r}| ∩ S^{d−1} are homotopy equivalent. Moreover, the assertion remains true if we replace “<” by “≤” in the definition of B_{r}. (|Σ_{r}| denotes the union of all cones contained in Σ_{r}.)

Proof: If the fan Σ is simplicial (for instance for smooth varieties X_{Σ}), then it is possible to prove in one strike that |Σ_{r}| ∩ S^{d−1} is a deformation retract of B_{r}. In the general case, however, it seems to be necessary to project B_{r} successively down dimension by dimension. Using the following fact, each step can be worked out in the cones of Σ separately: Let P be a compact convex polytope and H^{+} an open or closed half space. Then ∂P ∩ H^{+} is a deformation retract of P ∩ H^{+}. We leave the proof of this obvious fact to the reader. □

(3.5) Proposition: (Kodaira-vanishing) Let X_{Σ} be a complete toric variety with at most Gorenstein singularities. Assume that L is an equivariant line bundle which is generated by its global sections. Then, if the lattice polytope Δ ⊆ M_{IR} describing H^{0}(X_{Σ}, L) (as explained in (2.6)) is full-dimensional, we have H^{l}(X_{Σ}, L ⊗ ω_{X}) = 0 for l ≥ 1.

Proof: Denote by ψ the order function of L ⊗ ω_{X}. The order function φ_{K} of the canonical divisor equals 1 on the skeleton Σ^{(1)}; “Gorenstein” means that φ_{K} is linear on the cones. Thus, we obtain by the previous Lemma that the sets B_{r} := {a ∈ S^{d−1} ⊂ N_{IR} | ⟨a, r⟩ < ψ(a)} and C_{r} := {a ∈ S^{d−1} ⊂ N_{IR} | ⟨a, r⟩ ≤ ψ(a) − φ_{K}(a)} are homotopy equivalent. The first one computes the r-th graded piece of the desired cohomology, and the latter is contractible since ψ − φ_{K} is the order function of L, i.e. (ψ − φ_{K})(a) = min⟨a, Δ⟩. □

Remark: The assumption about the dimension of Δ means that L may not be obtained via pull back from some lower-dimensional variety.

(3.6) Theorem: Assume θ^{c} is in general position. Then Ext _{M(Q)}^{l}(U, U) = 0 for all l > 0.

Proof: Recall that Ext _{M(Q)}^{l}(U, U) = H^{l}(M(Q), \underline{End} U) = ⨁_{p,q∈Q_{0}} H^{l}(M(Q), U_{p,q}). Then, since U_{p,q} ⊗ ω_{M}^{−1} ≃ U_{p,q} ⊗ L(θ^{c}) is globally generated with dim \overline{Δ}(θ_{p,q} + θ^{c}) = dim Δ(θ^{c}) (cf. Propositions (2.7) and (3.3)), the result follows from Kodaira vanishing. □

(3.7) The previous theorem asks for the canonical weight to be in general position. We would like to close this section with a criterion for this fact to hold. Moreover, we present a criterion for Q_{1} = Q_{1}(θ^{c}).

Proposition: The canonical weight θ^{c} is in general position if and only if there does not exist any (t, t)-wall. Moreover, Q_{1}(θ^{c}) = Q_{1} if and only if there are no (1, 0)- or (1, 1)-walls.

Proof: Assume a (t^{+}, t^{−})-wall is given. Then, ∑_{q∈Q_{0}^{+}} θ_{q}^{c} = t^{+} − t^{−} by adding the M^{θ}-equations. Consequently, θ^{c} lies on this wall precisely when t^{+} − t^{−} = 0. This proves the first claim. For the second claim, we include the case of a (1, 1)-wall in brackets. Let Q_{0} = Q_{0}^{+} ⊔ Q_{0}^{−} be the subdivision defining the wall (see (2.2)), and denote by α the unique arrow with s(α) ∈ Q_{0}^{+} and t(α) ∈ Q_{0}^{−} [by β the unique arrow with s(β) ∈ Q_{0}^{−} and t(β) ∈ Q_{0}^{+}]. We show that α is not in Q_{1}(θ^{c}) [α and β are not both in Q_{1}(θ^{c})]. The set Q_{0}^{+} is closed under successors in Q \ {α}, but ∑_{q∈Q_{0}^{+}} θ_{q}^{c} = 1 [= 0]. Thus, Q \ {α} is not stable [both Q \ {α} and Q \ {β} are not stable]. It remains to show the converse. Assume we have no (1, 0)-wall and no (1, 1)-wall. Thus, t^{+} ≥ 2 for each wall W. Assume further that W is a (t^{+}, t^{−})-wall with t^{+} > t^{−}. We define the open halfspace W^{+} := {θ ∈ IH | ∑_{q∈Q_{0}^{+}} θ_{q} > 0}, i.e. θ^{c} ∈ W^{+}. Using the wall crossing formula from [Hi2], Lemma 3.4, we obtain Q_{1}(θ^{c}) = ∪_{θ∈IH}Q_{1}(θ). It remains to show that Q_{1} = ∪_{θ∈IH}Q_{1}(θ). By assumption, for each α ∈ Q_{1} there exists a tree in Q \ {α}. But for each tree T there exists a weight θ such that T is θ-stable ([Hi1], Proposition 2.5). This finishes the proof. □

4 The endomorphism algebra of the universal bundle

(4.1) In this section we always assume that θ is in general position, i.e. the universal bundle on M^{θ}(Q) exists. We start this section with a result about the endomorphism algebra A of the universal bundle. This algebra is non-commutative and finite-dimensional; in order to formulate the statements in this section, we need some basic results about those algebras. We denote by rad(A) the radical of A. It consists of all strongly nilpotent elements a of A, that is (aA)^{n} = 0 for n sufficiently large. Thus A is isomorphic to the quotient of the tensor algebra of the A/rad(A)- bimodule rad(A)/rad^{2}(A) by some admissible ideal I

A ≃ T_{(A/rad(A))}(rad(A)/rad^{2}(A))/I.

Recall that an ideal I is called admissible if

T_{(A/rad(A))}^{n}(rad(A)/rad^{2}(A)) ⊂ I ⊂ T_{(A/rad(A))}^{2}(rad(A)/rad^{2}(A))

for some n. In case A is the path algebra of a quiver, the radical of A is the ideal generated by paths of length at least one.

A finite-dimensional algebra A is called basic if the semisimple quotient A/rad(A) is a product of fields. Because we deal with basic algebras over an algebraically closed field, this semisimple quotient is a product of copies of the ground field. It turns out that each basic finite-dimensional algebra is isomorphic to the quotient of a path algebra of a finite quiver by some admissible ideal. Moreover, each finite-dimensional algebra is Morita equivalent to a basic finite-dimensional algebra, that is the module categories of both algebras are isomorphic. Thus, if we are interested in module categories, we may restrict ourselves to modules over basic algebras. The endomorphism ring of U is basic precisely when U contains only pairwise non-isomorphic direct summands. Consequently, for End(U) to be isomorphic to the path algebra of Q, it is necessary that the direct summands U_{q} are pairwise non-isomorphic. In fact, in the theorem below we will see that the converse is also true.

(4.2) In any case it would be desirable to know End(U) and its Morita equivalent basic algebra. This leads to the following definitions (cf. [S]): Let Q be a quiver without oriented cycles. Thus, the path algebra kQ is finite-dimensional. For an arrow α ∈ Q_{1} we define the localization kQ[α^{−1}] by formally adjoining the inverse α^{−1} of the arrow α, i.e. s(α^{−1}) = t(α), t(α^{−1}) = s(α), and α^{−1}α = e_{t(α)}, αα^{−1} = e_{s(α)} where e_{q} is the idempotent in kQ corresponding to the vertex q. In particular, α and α^{−1} ∈ kQ[α^{−1}] are not in the radical. Consequently, kQ[α^{−1}] is not basic, because kQ[α^{−1}]/rad(kQ[α^{−1}]) contains the two-by-two full matrix ring with basis e_{s(α)}, e_{t(α)}, α, α^{−1}. We consider the quotient quiver \overline{Q} := Q/(Q_{1} \ Q_{1}(θ)) defined by killing the arrows from Q_{1} \ Q_{1}(θ) while identifying their sources and targets, respectively. Each weight of Q provides in a canonical way a weight of \overline{Q} – just add the values of the identified vertices. The corresponding moduli spaces are isomorphic. The localization Q[α^{−1} | α ∉ Q_{1}(θ)] is Morita equivalent to k\overline{Q}. Moreover, this localization is finite-dimensional if and only if the quotient quiver \overline{Q} contains no oriented cycle.

Lemma: The line bundle U_{q} is isomorphic to the line bundle U_{p} if and only if there exists a walk from p to q consisting of arrows not in Q_{1}(θ).

Proof: The class of the bundle U_{p}^{−1}⊗U_{q} is trivial if and only if the divisor ∑_{α∈w} D_{α}−∑_{α^{−1}∈w} D_{α} is linearly equivalent to zero for a walk w from p to q in Q. Moreover, this divisor is also equivalent to ∑_{α∈w∩Q_{1}(θ)} D_{α} − ∑_{α^{−1}∈w∩Q_{1}(θ)} D_{α}, because D_{α} ∼ 0 for α ∉ Q_{1}(θ). This divisor is linearly equivalent to 0 if and only if w ∩ Q_{1}(θ) is a cycle in Q/(Q_{1} \ Q_{1}(θ)). This is true if and only if w ∩ Q_{1}(θ) = ∅. □

In particular, the indecomposable direct summands of the universal bundles on M^{θ}(Q) and M\overline{θ}(\overline{Q}) are isomorphic; in U(\overline{Q}) we have just cancelled multiple summands.

(4.3) Theorem: The endomorphism algebra A of the universal bundle U on M^{θ}(Q) is isomorphic to the localization of the path algebra of the quiver by all arrows not in Q_{1}(θ). If Q_{1}(θ) = Q_{1}, then A is isomorphic to the path algebra of the quiver Q.

Proof: As explained in the previous remarks, we may assume that Q_{1}(θ) = Q_{1}. By Proposition (3.1) we know that

Hom_{M^{θ}(Q)}(U_{p}, U_{q}) = H^{0}(M^{θ}(Q), U_{p}^{−1} ⊗ U_{q}) = k · {lattice points of Δ(θ_{p,q})}

where θ_{p,q} is the weight introduced in (3.2). Since an integral flow in Δ(θ_{p,q}) has values in {0, 1}, we obtain a bijection between the lattice points of Δ(θ_{p,q}) and the paths from p to q in Q; hence End_{M^{θ}(Q)}(U) ≃ kQ. □

(4.4) Let U be a vector bundle on a smooth projective algebraic variety M. Let A be the endomorphism algebra of U, which is finite-dimensional. Moreover, let U = ⊕_{q∈Q_{0}}U_{q} be a decomposition into indecomposable direct summands. Then A = ⊕_{q∈Q_{0}}e_{q}A is a decomposition of A into indecomposable projective right A-modules. We denote by K^{b}(U_{q} | q ∈ Q_{0}) and K^{b}(e_{q}A | q ∈ Q_{0}) the homotopy category of bounded complexes {C^{i}}, where each C^{i} is a direct sum of copies of U_{q} or e_{q}A, respectively. The functor induced by the map U_{q} ↦ e_{q}A is an equivalence between K^{b}(U_{q} | q ∈ Q_{0}) and K^{b}(e_{q}A | q ∈ Q_{0}) because the endomorphism algebra of A viewed as right A-module is A.

Theorem: Assume Q is a quiver without any (t, t)-wall. Then the equivalence above induces a full and faithful functor

D^{b}(mod–A) ⟶ D^{b}(Coh(M(Q))).

Proof: We define p ≤ q if Hom(U_{p}, U_{q}) ≠ 0. This is a partial order on Q_{0} because U_{q} is a line bundle for all q ∈ Q_{0}. Consequently, End(U) is a directed algebra (there is an order on Q_{0} such that Hom_{A}(e_{p}A, e_{q}A) = 0 for p > q). A directed algebra is of finite global dimension, thus the bounded derived category D^{b}(mod–A) of finitely generated right A-modules is equivalent to the bounded homotopy category K^{b}(e_{q}A | q ∈ Q_{0}) (cf. [Hap] §1, 3.3). Since U has no self-extension (Theorem (3.6)), the natural functor K^{b}(U_{q} | q ∈ Q_{0}) → D^{b}(Coh(M(Q))) is full and faithful. □

Proof of Theorem (1.3): If there exists no (1, 0)-wall and no (1, 1)-wall we have A ≃ kQ by Proposition (3.7) and Theorem (4.3). Thus, the result follows from Theorem (4.4) □

References

[ARS] Auslander, M., Reiten, I., Smalo, S. O.: Representation Theory of Artin Algebras. Cambridge studies in advanced mathematics 36, Cambridge University Press, Cambridge 1995.

[Bae] Baer, D.: Tilting Sheaves in Representation Theory of Algebras. Manuscripta mathematica 60 (1988), 323-347.

[Bat] Batyrev, V. V.: Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties. J. Algebraic Geometry 3 (1994), 493-535.

[Be] Beilinson, A. A.: Coherent Sheaves on IP^{n} and Problems of Linear Algebra. Funkt. Anal. Appl. 12 (1979), 214-216.

[Bo] Bondal, A. I.: Representations of associative algebras and coherent sheaves (Russ.). Sbornik Akad. Nauk SSSR 53 (1988). (An English version is contained in [Ru].)

[BK] Bondal, A. I., Kapranov M. M.: Representable Functors, Serre Functor, and Mutations. Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1183-1205; engl. transl. in Math. USSR Izv. 35 (1990).

[F] Fulton, W.: Introduction to Toric Varieties. Princeton University Press, Princeton 1993.

[HR] Happel, D., Ringel, C. M.: Tilted Algebras. Transactions of the AMS, vol. 274, 2 (1982), 399-443.

[Hap] Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Mathematical Society, Lecture Note Series 119, Cambridge University Press, Cambridge 1988.

[Har] Hartshorne, R.: Residues and Duality. LNM 20, Springer Verlag, Berlin-Heidelberg-New York 1966.

[Hi1] Hille, L.: Moduli Spaces of Thin Sincere Representations of Quivers. Preprint Chemnitz 1995.

[Hi2] Hille, L.: Toric Quiver Varieties. Preprint Chemnitz 1996.

[Ka] Kapranov, M. M.: On the Derived Category of Coherent Sheaves on some Homogeneous Spaces. Invent. Math. 92i (1988), 479-508.

[Ke] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. LNM 339, Springer-Verlag, Berlin-Heidelberg-New York 1973.

[Ki] King, A.: Moduli of Representations of Finite Dimensional Algebras. Quarterly J. Math. Oxford 45 (1994), 515-530.

[N] Newstead, P. E.: Introduction to Moduli Problems and Orbit Spaces. Tata Institute of Fund. Research, Springer Verlag, Berlin-Heidelberg-New York 1978.

[Od] Oda, T.: Convex Bodies and Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3/15), Springer Verlag, Berlin-Heidelberg 1988.

[Or] Orlov, D. O.: Projective Bundles, Monoidal Transformations and the Derived Category of Coherent Sheaves. Math USSR Izv. 38 (1993), 133-141.

[Ri1] Ringel, C. M.: Tame Algebras and Integral Quadratic Forms. LNM 1099, Springer Verlag, Berlin-Heidelberg-New York-Tokio 1984.

[Ri2] Ringel, C. M.: Finite Dimensional Hereditary Algebras of Wild Representation Type. Math. Z. 161 (1978), 235-255.

[Ru] Seminar Rudakov, Helices and Vector Bundles. London Mathematical Society, Lecture Note Series 148, Cambridge University Press, Cambridge 1990.

[S] Schofield, A. H.: Representations of Rings over Skew Fields. London Mathematical Society, Lecture Note Series 92, Cambridge University Press, Cambridge 1985.

Klaus Altmann Institut für reine Mathematik der Humboldt-Universität zu Berlin Ziegelstr. 13A D-10099 Berlin, Germany E-mail: altmann@mathematik.hu-berlin.de Lutz Hille Mathematisches Seminar der Universität Hamburg Bundesstr. 55 D-20146 Hamburg, Germany E-mail: hille@math.uni-hamburg.de