A HOMOLOGICAL INTERPRETATION OF THE TRANSVERSE QUIVER GRASSMANNIANS

GIOVANNI CERULLI IRELLI, GRÉGOIRE DUPONT, AND FRANCESCO ESPOSITO

Abstract. In recent articles, the investigation of “canonically positive bases” of cluster algebras associated to affine quivers led the second–named author to introduce a variety called transverse quiver Grassmannian and the first–named and third–named authors to consider the smooth part of quiver Grassmannians. In this paper, we prove that, for any affine quiver Q, the transverse quiver Grassmannian of an indecomposable representation M is the set of points N in the quiver Grassmannian of M such that Ext^{1}(N, M/N) = 0. As a corollary we obtain that when the transverse quiver Grassmannian is non–empty, it coincides with the smooth locus of the quiver Grassmannian.

1. Introduction

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in order to design a combinatorial model for understanding positivity in algebraic groups and canonical bases in quantum groups [15, 16, 2, 17].

To any quiver Q with vertex set Q_{0}, without loops and 2–cycles, is associated a (coefficient–free) cluster algebra A_{Q} which is a certain commutative ℤ–subalgebra of the field F = ℚ(x_{1}, ⋯, x_{n}) of rational functions in n = |Q_{0}| variables. The ring A_{Q} is equipped with a distinguished set of generators called cluster variables which are grouped into possibly overlapping free generating sets of the field F called clusters. In particular every element of A_{Q} can be expressed as a rational function in every cluster of A_{Q}. By the Laurent phenomenon every element of A_{Q} can actually be expressed as a Laurent polynomial in every cluster of A_{Q} [15].

In [3, 4, 12], the authors provide closed formulas for the computation of the Laurent expansion of every cluster variable in every cluster. Let us recall this result in the particular case when the cluster is associated to an acyclic seed (we address the reader to the survey [18]). Let Q be a quiver without oriented cycles and let M be a finite-dimensional complex representation of Q with dimension vector d = (d_{i})_{i∈Q_{0}}. Following the notations of [7] we associate to M the following Laurent polynomial in the variables {x_{i} : i ∈ Q_{0}}

(1)  CC(M) = \frac{∑_{e∈ℕ^{Q_{0}}} χ(Gr_{e}(M)) ∏_{i,j∈Q_{0}} (x_{i}^{d_{j}−e_{j}}x_{j}^{e_{i}})^{b_{ij}}}{∏_{i∈Q_{0}} x_{i}^{d_{i}}}

where b_{ij} denotes the number of arrows in Q from the vertex i to the vertex j, Gr_{e}(M) denotes the complex projective variety of e-dimensional sub–representations of M (see section 2 for details) and χ(Gr_{e}(M)) denotes its Euler-Poincaré characteristic. The following result, due to B. Keller and P. Caldero, provides the promised description of the cluster variables of A_{Q}. Recall that a representation M of Q is called rigid if Ext^{1}(M, M) = 0.

Theorem 1.1. [4] Let A_{Q} be the (coefficient-free) cluster algebra with initial seed (Q, {x_{i} : i ∈ Q_{0}}). Then the correspondence M ↦ CC(M) is a bijection between the set of isomorphism classes of indecomposable rigid representations of the quiver Q, and the set of all cluster variables in A_{Q} not belonging to {x_{i} : i ∈ Q_{0}}.

In the theory of cluster algebras, it is natural to consider the cone of positive elements, i.e. elements whose expansion in every cluster has positive coefficients. The study of this cone leads to look for a ℤ–basis B of A_{Q} whose positive linear combinations coincide with this cone. The existence of such a basis was proved only in a few cases :

Theorem 1.2. [21][8] Let Q be a quiver of type Ã_{1,1} or of type Ã_{2,1} and let A_{Q} be the associated cluster algebra. Then there exists a ℤ–basis B of A_{Q} such that the cone of positive elements of A_{Q} coincides with the cone of positive linear combinations of elements of B.

In the same spirit as theorem 1.1, the authors of the present article provided independently in [14] and [9] two geometric realizations of the basis of theorem 1.2. Let us briefly recall these two constructions.

In [9] the first and third–named authors study the geometry of quiver Grassmannians associated to indecomposable representations of the Kronecker quiver. As a consequence of their results it is natural to consider the map which associates to a representation M of Q the Laurent polynomial CC^{(0)}(M) defined by slightly modifying (1) as follows:

(2)  CC^{(0)}(M) = \frac{∑_{e} χ(Gr_{e}(M)^{(0)}) ∏_{i,j∈Q_{0}} (x_{i}^{d_{j}−e_{j}}x_{j}^{e_{i}})^{b_{ij}}}{∏_{i∈Q_{0}} x_{i}^{d_{i}}} .

where Gr_{e}(M)^{(0)} is the subvariety of Gr_{e}(M) constituted of sub-representations N of M such that Ext^{1}(N, M/N) = 0.

In [14] the second–named author defines a subvariety Tr_{e}(M) of Gr_{e}(M) called transverse quiver Grassmannian for any indecomposable representation M of an arbitrary affine quiver Q (see section 3). To any such representation, let θ(M) be the following slight modification of the right–hand side of (1):

(3)  θ(M) = \frac{∑_{e} χ(Tr_{e}(M)) ∏_{i,j∈Q_{0}} (x_{i}^{d_{j}−e_{j}}x_{j}^{e_{i}})^{b_{ij}}}{∏_{i∈Q_{0}} x_{i}^{d_{i}}} .

We define θ on direct sums by θ(M ⊕ N) = θ(M)θ(N).

In [9, theorems 3.1 and 3.2] and [14, theorem 5.1] we provide a geometric realization of the basis B of theorem 1.2 as follows:

Theorem 1.3. [9][14] Let Q be a quiver either of type Ã_{1,1} or of type Ã_{2,1}. Then for every element b in the basis B of theorem 1.2 there exists a representation M_{b} of Q and a monomial m_{b} in the initial cluster variables such that

b = m_{b}.CC^{(0)}(M_{b}) = m_{b}.θ(M_{b}).

Theorem 1.3 shows that the maps M ↦ CC^{(0)}(M) and M ↦ θ(M) coincide when evaluated on particular representations of particular affine quivers. It hence suggests a relation between the two subvarieties Gr_{e}(M)^{(0)} and Tr_{e}(M) of Gr_{e}(M). The following theorem, which is our main result, states that they actually coincide :

Theorem 1.4. Let Q be an affine quiver and let M be an indecomposable representation of Q. Then

(4)  Tr(M) = {N ∈ Gr(M) : Ext^{1}(N, M/N) = 0}.

The following corollary of theorem 1.4 provides a geometric interpretation of the transverse quiver Grassmannian where ⟨·, ·⟩ denotes the Euler form of Q.

Corollary 1.5. Let Q be an affine quiver and let M be an indecomposable representation of Q with dimension vector d. For any dimension vector e, the variety Tr_{e}(M) is non–empty if and only if Gr_{e}(M) is reduced and has dimension ⟨e, d−e⟩. In this case Tr_{e}(M) coincides with the smooth part of Gr_{e}(M).

Theorem 1.4 is of double interest. On one hand, it provides a natural interpretation of the transverse quiver Grassmannian whose definition given in [14] was ad hoc. On the other hand, together with corollary 1.5, the definition of Tr(M) makes it possible to compute explicitly smooth parts of quiver Grassmannians associated to indecomposable representations.

The paper is organized as follows: in section 2 we recall some facts about representations of affine quivers and about the geometry of quiver Grassmannians. In section 3 we recall the definition of transverse quiver Grassmannians. In section 4 we prove theorem 1.4. In section 5 we prove corollary 1.5.

2. Quiver Grassmannians

In this section we recall some well-known facts about quiver Grassmannians. Such varieties are considered in several places in the literature; our references are [20, 11, 10, 6, 12, 3, 5, 4].

2.1. Representations of quivers. A quiver Q = (Q_{0}, Q_{1}) is an oriented graph where Q_{0} denotes the set of vertices and Q_{1} the set of arrows. We always assume that Q_{0} and Q_{1} are finite sets and that the underlying unoriented graph of Q is connected. A quiver is called acyclic if it does not contain any oriented cycle.

Let ℂ denote the field of complex numbers. A (complex) representation M of Q consists of a collection of (finite dimensional complex) vector spaces M(i), one for each vertex i in Q_{0}, together with a collection of ℂ-linear maps M(a) : M(i)⟶ M(j), one for each arrow a : i⟶ j in Q_{1}. A sub–representation N of M consists of vector subspaces N(i) of M(i) such that M(a)(N(i)) ⊂ N(j) for every arrow a : i → j in Q_{1}.

Given two representations M and M′ of Q, a morphism of Q–representations g : M → M′ is a Q_{0}–tuple (g(i) : i ∈ Q_{0}) of ℂ-linear maps g(i) : M(i) → M′(i) such that M(a)∘g(i) = g(j)∘M′(a) for every arrow a : i → j in Q_{1}. We denote by rep(Q) the category of Q–representations. As usual, the category rep(Q) is identified with the category ℂQ-mod of finitely generated left-modules over the path algebra ℂQ of Q. Thus, a Q–representation will sometimes be called a module. Note that throughout the article, unadorned Hom and Ext are taken in the category rep(Q).

By definition, the dimension vector of a Q-representation M is dim (M) = (dim M(i) : i ∈ Q_{0}) ∈ ℕ^{Q_{0}}. The Euler form on rep(Q) is the bilinear form given by

⟨M, N⟩ = dim Hom(M, N) − dim Ext^{1}(M, N)

which only depends on dim M and dim N so that it induces a ℤ-bilinear form on ℤ^{Q_{0}} which is given by

⟨d, d′⟩ = ∑_{i∈Q_{0}} d_{i}d′_{i} − ∑_{a:i→j∈Q_{1}} d_{i}d′_{j}

for any d, d′ ∈ ℤ^{Q_{0}}.

We denote by τ the Auslander-Reiten translation on ℂQ-mod and by τ^{−1} its inverse (see e.g. [1]). An indecomposable representation M is called preprojective if it is in the τ^{−1}-orbit of an indecomposable projective module. An indecomposable representation M is called preinjective if it is in the τ-orbit of an indecomposable injective module. An indecomposable Q–representation M is called regular if it is neither preprojective nor preinjective. Direct sums of indecomposable preprojective (resp. regular, preinjective) representations are called preprojective (resp. regular, preinjective) representations. Any representation M can be uniquely decomposed into M = M_{P} ⊕ M_{R} ⊕ M_{I} where M_{P} is preprojective, M_{R} is regular and M_{I} is preinjective.

Any two representations M and N satisfy the Auslander-Reiten formula :

Ext^{1}(M, N) ≃ DHom(N, τM)

where D = Hom_{ℂ}(−, ℂ) denotes the standard duality.

The indecomposable preprojective (resp. preinjective) representations form a connected component in the Auslander-Reiten quiver of Γ_{Q} of rep(Q). This component is called preprojective (resp. preinjective). All the other connected components are called regular.

From now on, we assume that Q is an affine quiver, that is, an acyclic orientation of an euclidean diagram of type Ã, D̃ or Ẽ. Note that these quivers are also called extended Dynkin quiver in the literature. The representation-theory of such quivers is well-known, we refer the reader to [19, 22] for details.

Regular components in Γ_{Q} form a ℙ^{1}ℂ-family and each component is a (stable) tube, that is of the form ℤ𝔸_{∞}/(τ^{p}) for some integer p ≥ 1 called the rank of the tube. A tube is called homogeneous if p = 1 and exceptional otherwise. Slightly abusing notations, the full subcategory of rep(Q) formed by objects in a given tube T is still denoted by T . Each tube T in Γ_{Q} is abelian, uniserial and standard, that is, isomorphic to the mesh category of the quiver T . A module at the mouth of a tube is called quasi-simple. Let R be a quasi-simple module in a tube T , then there exists a unique sequence of irreducible monomorphisms

R = R^{(1)}⟶ R^{(2)}⟶ ⋯ ⟶ R^{(n)}⟶ ⋯

called ray starting at R and a unique sequence of irreducible epimorphisms

⋯ ⟶ τ^{n−1}R^{(n)}⟶ ⋯ ⟶ τR^{(2)}⟶ R^{(1)} = R

called coray ending at R.

An indecomposable module M has a self-extension if and only if it is regular in a tube T of rank p ≥ 1 and if M ≃ R^{(n)} for some quasi-simple R in T and some integer n ≥ p.

If P is any preprojective module, I is any preinjective module and R, R′ are any regular modules taken in two distinct tubes, then

Hom(R, P) = 0, Hom(I, R) = 0, Hom(I, P) = 0 and Hom(R, R′) = 0,

or equivalently, using the Auslander-Reiten formula,

(5) Ext^{1}(P, R) = 0, Ext^{1}(R, I) = 0, Ext^{1}(P, I) = 0 and Ext^{1}(R′, R) = 0.

2.2. Quiver Grassmannians. Let Q be any acyclic quiver and M be any representation of Q. To any e = (e_{i})_{i∈Q_{0}} ∈ ℕ^{Q_{0}}, the quiver Grassmannian of e– dimensional subrepresentations is the set

Gr_{e}(M) = {N sub–representation of M : dim N = e}.

This is a closed subset of the product ∏_{i∈Q_{0}} Gr_{e_{i}}(M(i)) of usual Grassmannians of vector subspaces and it is hence a complex projective variety. The quiver Grassmannian is the (finite) union

Gr(M) = ⨆_{e∈ℕ^{Q_{0}}} Gr_{e}(M).

In [20] (see also [6]) it is shown that the tangent space T_{N}(Gr_{e}(M)) at a point N of Gr_{e}(M) equals:

(6)  T_{N}(Gr_{e}(M)) = Hom(N, M/N).

Moreover the following inequalities hold :

(7) ⟨e, d − e⟩ ≤ dim Gr_{e}(M) ≤ dim T_{N}(Gr_{e}(M)) ≤ ⟨e, d − e⟩ + dim Ext^{1}(M, M)

where d = dim M so that dim M/N = d − e. We recall that a point N in Gr_{e}(M) is called smooth if dim T_{N}(Gr_{e}(M)) = dim Gr_{e}(M) and Gr_{e}(M) is called smooth if every point in Gr_{e}(M) is smooth. It follows in particular that when M is rigid, Gr_{e}(M) is smooth for any e ∈ ℕ^{Q_{0}}.

Note that the inequalities (7) can be strict as shown in the following example.

Example 2.1. Consider the acyclic quiver Q of type Ã_{2,1} and the indecomposable regular representation of Q :

\xymatrix{ &  & k^{3} \ar[dr]^{J_{3}(0)} \\ M := & k^{3} \ar[ur]^{=} \ar[rr]_{=} &  & k^{3}}

of dimension (3, 3, 3) (here J_{3}(0) denotes the 3 × 3 indecomposable nilpotent Jordan block). In the notation of section 2.1, M lies in a tube of rank two and has the form M = R_{0}^{(6)} where R_{0} is the quasi-simple (which is actually simple in this case) of dimension (0, 1, 0). Moreover τM = R_{1}^{(6)} where R_{1} is the unique indecomposable representation of dimension (1, 0, 1). It follows that dim Ext^{1}(M, M) = dim Hom(R_{0}^{(6)}, R_{1}^{(6)}) = 3.

Let e = (0, 2, 1). Then ⟨e, d − e⟩ = 0. It is easy to verify that Gr_{e}(M) is isomorphic to ℙ^{1} and it is hence smooth of dimension 1 (see also remark 3.1). It thus follows that the inequalities in (7) are all strict except the middle one which is an equality.

3. Transverse quiver Grassmannians

Assume now that Q is an affine quiver. We recall the definition of the transverse quiver Grassmannian provided in [14]. Let M be an indecomposable ℂQ-module. If M is rigid, we set Tr(M) = Gr(M).

If M is non-rigid, it is regular and it is thus contained in a tube T of rank p ≥ 1. We denote by R_{i}, i ∈ ℤ/pℤ the quasi-simple modules in T ordered such that τR_{i} ≃ R_{i−1} for any i ∈ ℤ/pℤ. For any n ≥ 0 and any i ∈ ℤ/pℤ, we denote by R_{i}^{(n)} the unique indecomposable ℂQ-module in T with quasi-socle R_{i} and quasi-length n with the convention that R_{i}^{(0)} = 0. Up to relabeling the R_{i}’s, there exist l ≥ 1 and 0 ≤ k ≤ p − 1 such that M ≃ R_{0}^{(lp+k)}.

For any m ≥ n, Hom(R_{i}^{(n)}, R_{i}^{(m)}) = ℂι for some monomorphism ι so that we can identify Gr(R_{i}^{(n)}) with a closed subset of Gr(R_{i}^{(m)}). By definition, the transverse quiver Grassmannian of M is thus :

Tr(R_{0}^{(lp+k)}) = Gr(R_{0}^{(lp+k)}) \ Gr^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)})

where Gr^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)}) is the set of subrepresentations in Gr(R_{0}^{(lp−1)}) containing R_{0}^{(k+1)} as a subrepresentation. In other words,

Tr(M) = Gr(M) \ {N ∈ Gr(M) : R_{0}^{(k+1)} ⊂ N ⊂ R_{0}^{(lp−1)}} .

Note that

Gr^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)}) ≃ Gr_{dimR_{0}^{(lp−1)}−dimR_{0}^{(k+1)}}(R_{0}^{(lp−1)}/R_{0}^{(k+1)}).

For any indecomposable ℂQ-module M and any dimension vector e, we set

Tr_{e}(M) = Tr(M) ∩ Gr_{e}(M).

As mentioned in the introduction, the motivation for the definition of the transverse quiver Grassmannian comes from the study of [13, Conjecture 7.10] concerning canonically positive bases in cluster algebras associated to affine quivers and more specifically from higher difference properties introduced and studied in [14].

Remark 3.1. Let Q be an affine quiver and M be an indecomposable representation of Q. According to the corollary 1.5, for any dimension vector e ∈ ℕ^{Q_{0}}, when Tr_{e}(M) is not empty it coincides with the set of smooth points in Gr_{e}(M). Nevertheless, Tr_{e}(M) does not coincide with the set of smooth points in Gr_{e}(M) in general. Indeed, in example 2.1 it is shown a quiver Grassmannian Gr_{e}(M) which is smooth of dimension 1. Then one can check directly that for every N ∈ Gr_{e}(M), dim Ext^{1}(N, M/N) = 1 (this follows from the fact that N ≃ R_{0}^{(1)} ⊕ P_{2} where P_{2} is the projective of dimension (0, 1, 1), from (5) and from the combinatorics of the tube of rank two containing M). But in view of theorem 1.4 this computation implies that Tr_{e}(M) is empty. So Gr_{e}(M)^{(0)} = Gr_{e}(M) but T r_{e}(M) is empty.

4. Proof of Theorem 1.4

In this section we prove the main theorem of the article. We start by proving the following lemma :

Lemma 4.1. For every subrepresentations N_{1} and N_{2} of M we have

dim Ext^{1}(N_{1}, M/N_{2}) ≤ dim Ext^{1}(M, M).

Proof. Applying Ext^{1}(N_{1}, −) to the exact sequence M⟶ M/N_{2}⟶ 0, we get an epimorphism \xymatrix{Ext^{1}(N_{1}, M) \ar@{->>}[r] & Ext^{1}(N_{1}, M/N_{2})} since ℂQ is hereditary. Similarly, Ext^{1}(−, M) applied to the exact sequence 0⟶ N_{1}⟶ M, we get an epimorphism \xymatrix{Ext^{1}(M, M) \ar@{->>}[r] & Ext^{1}(N_{1}, M)}. Composing the two epimorphisms one gets an epimorphism \xymatrix{Ext^{1}(M, M) \ar@{->>}[r] & Ext^{1}(N_{1}, M/N_{2})}. This finishes the proof. □

Fix now an affine quiver Q and an indecomposable representation M of Q. Then we need to prove that

(8)  Tr(M) = {N ∈ Gr(M) : Ext^{1}(N, M/N) = 0}.

If M is rigid, i.e. Ext^{1}(M, M) = 0, then Tr(M) = Gr(M) by definition and it follows from Lemma 4.1 that Ext^{1}(N, M/N) = 0 for any N ∈ Gr(M) and thus (8) holds.

We thus assume that M is not rigid. With notations of section 3, we write M ≃ R_{0}^{(lp+k)} for some l ≥ 1 and 0 ≤ k ≤ p − 1 where p ≥ 1 is the rank of the tube containing M.

In order to prove (8) we prove the following two facts:

(9) if N ∈ Gr_{e}^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)}) then Ext^{1}(N, M/N) ≠ 0 ;

(10) if N ∈ Tr_{e}(M) then Ext^{1}(N, M/N) = 0.

Since M is regular, any submodule N of M is of the form N = N_{R} ⊕ N_{P} where N_{R} is regular indecomposable (or zero) and N_{P} is preprojective. Moreover, the quotient M/N is of the form M/N = (M/N)_{R} ⊕(M/N)_{I} where (M/N)_{R} is regular indecomposable (or zero) and (M/N)_{I} is preinjective. It follows that

Ext^{1}(N, M/N) = Ext^{1}(N_{P} ⊕ N_{R}, (M/N)_{R} ⊕ (M/N)_{I}) ≃ Ext^{1}(N_{R}, (M/N)_{R}).

Let us prove (9). Let N ∈ Gr^{R_{0}^{(k+1)}}(R_{0}^{(lp−1)}). Since Hom(R_{0}^{(k+1)}, N_{P}) = 0, it follows that R_{0}^{(k+1)} is a submodule of N_{R} and thus

N_{R} = R_{0}^{(t)} for some k + 1 ≤ t ≤ lp − 1.

\xymatrix{R_{0} \ar@{^{(}->}[dr] &  &  &  &  &  &  &  &  &  &  &  & R_{k−1} \\  & ⋯ \ar@{^{(}->}[dr] &  &  &  &  &  &  &  &  &  & ⋯ \ar@{->>}[ur] \\  &  & R_{0}^{(k+1)} \ar@{^{(}->}[dr] &  &  &  &  &  &  &  & R_{p−1}^{(k+1)} = τR_{0}^{(k+1)} \ar@{^{(}->}[dr] \ar@{->>}[ur] \\  &  &  & ⋯ \ar@{^{(}->}[dr] &  &  &  &  &  & ⋯ \ar@{->>}[ur] &  & ⋯ \ar@{^{(}->}[dr] \\  &  &  &  & R_{0}^{(lp−1)} \ar@{^{(}->}[dr] &  &  &  & R_{k+1}^{(lp−1)} \ar@{->>}[ur] &  &  &  & τR_{0}^{(lp−1)} \\  &  &  &  &  & ⋯ \ar@{^{(}->}[dr] &  & ⋯ \ar@{->>}[ur] \\  &  &  &  &  &  & R_{0}^{(lp+k)} \ar@{->>}[ur]}

Figure 1. The ray and the coray passing through M = R_{0}^{(lp+k)}

Thus, in the tube containing M, the module N_{R} is contained between R_{0}^{(k+1)} and R_{0}^{(lp−1)} in the ray of the tube passing through M.

Moreover there are surjective morphisms:

\xymatrix{R_{k+1}^{(lp−1)} \ar@{->>}[r] & (M/N)_{R} \ar@{->>}[r] & R_{p−1}^{(k+1)}}

i.e. (M/N)_{R} is contained between R_{k+1}^{(lp−1)} and R_{p−1}^{(k+1)} in the coray passing through M. Indeed there are surjective morphisms:

\xymatrix{M/R_{0}^{(k+1)} \ar@{->>}[r] & (M/N)_{R} \ar@{->>}[r] & M/R_{0}^{(lp−1)}}

and clearly

(11)  M/R_{0}^{(k+1)} ≃ R_{k+1}^{(lp+k−(k+1))} = R_{k+1}^{(lp−1)}

and

(12)  M/R_{0}^{(lp−1)} ≃ R_{lp−1}^{(lp+k−(lp−1))} = R_{p−1}^{(k+1)}

We claim that Ext^{1}(R, R′) ≠ 0 if R is an indecomposable regular module contained between R_{0}^{(k+1)} and R_{0}^{(lp−1)} in the ray of the tube passing through M and R′ is an indecomposable regular module contained between R_{k+1}^{(lp−1)} and R_{p−1}^{(k+1)} in the coray passing through M. According to the Auslander-Reiten formula, it is equivalent to prove that Hom(R′, τR) ≠ 0. Since Hom(R_{k+1}^{(lp−1)}, R_{p−1}^{(k+1)}) = ℂπ where π is an epimorphism, this follows easily from the combinatorics of the tubes (see figure 1). This proves the claim and thus (9) holds.

We now prove (10). By definition N ∈ Tr(M) if either R_{0}^{(k+1)} is not a subrepresentation of N or N is not a subrepresentation of R_{0}^{(lp−1)}. We claim that N ∈ Tr(M) if either

(13)  N_{R} ≃ R_{0}^{(t)} for some 0 ≤ t ≤ k

or there exists a surjective morphism

(14)  \xymatrix{R_{0}^{(lp+k)}/R_{0}^{(lp)} ≃ R_{lp}^{(k)} ≃ R_{0}^{(k)} \ar@{->>}[r] & (M/N)_{R}} .

(13) is clearly equivalent to the fact that R_{0}^{(k+1)} is not a subrepresentation of N. We now prove that (14) is equivalent to the fact that N ⊂ R_{0}^{(lp+k)} is not a subrepresentation of R_{0}^{(lp−1)}. Let a be the minimal integer such that N ⊂ R_{0}^{(lp+a)} so that 0 ≤ a ≤ k. Consider the composition f ∈ Hom(M, (M/N)_{R}) given by

M → M/N → (M/N)_{R}

which is surjective. Since M and (M/N)_{R} are regular modules and that each tube is closed under kernels, ker f is a regular module. Since it is a submodule of M and N ⊂ ker f, we get ker f ≃ R_{0}^{(lp+b)} with a ≤ b ≤ k. Thus, there exists an epimorphisms

R_{0}^{(lp+k)}/R_{0}^{(lp)} → R_{0}^{(lp+k)}/R_{0}^{(lp+b)} = R_{0}^{(lp+k)}/ ker(f) ≃ (M/N)_{R}.

We now prove that if N satisfies either (13) or (14) then Ext^{1}(N, M/N) = 0. If (13) holds, N_{R} ≃ R_{0}^{(t)} for some 0 ≤ t ≤ k and then it is easy to see that Ext^{1}(N_{R}, R′) = 0 for any regular quotient R′ of R_{0}^{(lp+k)}. In particular, we get Ext^{1}(N_{R}, (M/N)_{R}) = 0. Similarly, if (14) holds, then (M/N)_{R} ≃ R_{j}^{k−j} for some 0 ≤ j ≤ k and thus, it follows that Ext^{1}(R_{0}^{(s)}, (M/N)_{R}) = 0 for any s ≥ 0. In particular, Ext^{1}(N_{R}, (M/N)_{R}) = 0. This finishes the proof of theorem 1.4. □

5. Proof of Corollary 1.5

We conclude the article by proving corollary 1.5 which points out a geometric interpretation of the transverse quiver Grassmannians. We thus fix an affine quiver Q and an indecomposable representation of Q with dimension vector d. Let e be a dimension vector. Then we need to prove that Tr_{e}(M) is non–empty if and only if Gr_{e}(M) is reduced and has dimension ⟨e, d − e⟩ and that in this case Tr_{e}(M) coincides with the smooth part of Gr_{e}(M).

Suppose that Tr_{e}(M) is non empty and let N be an element of it. Then, in view of theorem 1.4, Ext^{1}(N, M/N) = 0 and hence dim T_{N}(Gr_{e}(M)) = ⟨e, d − e⟩, in view of (6). Then, by (7), dim Gr_{e}(M) = ⟨e, d − e⟩ and hence Gr_{e}(M) is reduced because dim T_{N}(Gr_{e}(M)) = dim Gr_{e}(M) and N is smooth.

On the other hand if Gr_{e}(M) is smooth then there exists a smooth point N ∈ Gr_{e}(M), i.e. dim T_{N}(Gr_{e}(M)) = dim Gr_{e}(M). By hypothesis dim T_{N}(Gr_{e}(M)) = ⟨e, d − e⟩ = dim Gr_{e}(M) and hence, by (7), Ext^{1}(N, M/N) = 0 and N ∈ Tr_{e}(M) in view of theorem 1.4.

In this case the smooth part of Gr_{e}(M) consists of subrepresentations N of M such that dim T_{N}(Gr_{e})(M)) = dim Gr_{e}(M)) which is equivalent to the equality dim T_{N}(Gr_{e})(M)) = ⟨e, d − e⟩ which is equivalent to Ext^{1}(N, M/N) = 0 which is equivalent to N ∈ Tr_{e}(M). This finishes the proof of corollary 1.5. □

Acknowledgments

This paper was written while the second–named author was at the university of Sherbrooke as a CRM-ISM postdoctoral fellow under the supervision of Ibrahim Assem, Thomas Brüstle and Virginie Charette. The first–named author thanks C. De Concini for the financial support for this project, as part of a post–doctoral scholarship at the Department of Mathematics of “Sapienza Università di Roma”.

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Sapienza Università di Roma, Dipartimento di Matematica. Piazzale Aldo Moro 2, 00185, Roma (ITALY)

E-mail address: cerulli@mat.uniroma1.it

Université de Sherbrooke, Département de Mathématiques. 2500, Boul. de l’Université, J1K 2R1, Sherbrooke QC (CANADA).

E-mail address: gregoire.dupont@usherbrooke.ca

Università degli studi di Padova, Dipartimento di Matematica Pura ed Applicata. Via Trieste 63, 35121, Padova (ITALY)

E-mail address: esposito@math.unipd.it