Euler characteristics and geometric properties of quiver Grassmannians [Elektronische Ressource] / Nicolas Haupt. Mathematisch-Naturwissenschaftliche Fakultät

Nicolas Haupt

Sujets

• Mathematics

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2011
• Nombre de lectures: 23
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Euler characteristics
and geometric properties
of quiver Grassmannians
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakult at
der
Rheinischen Friedrich-Wilhelms-Universit at Bonn
vorgelegt von
Nicolas Haupt
aus
Georgsmarienhutte, Deutschland
Bonn, Mai 2011Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult at der
Rheinischen Friedrich-Wilhelms-Universit at Bonn
Erstgutachter: Prof. Dr. Jan Schr oer
Zweitgutachter: Priv.-Doz. Dr. Igor Burban
Tag der Promotion: 27. September 2011
Erscheinungsjahr: 2011
34Abstract
Let k be an algebraically closed eld, Q a nite quiver and M a nite-dimensional
Q-representation. The quiver Grassmannian Gr (M) is the projective variety of sub-d
representations of M with dimension vector d.
Quiver Grassmannians occur naturally in di erent contexts. Fomin and Zelevinsky in-
troduced cluster algebras in 2000. Caldero and Keller used Euler characteristics of quiver
Grassmannians for the categori cation of acyclic cluster algebras. This was generalized
to arbitrary antisymmetric cluster algebras by Derksen, Weyman and Zelevinsky. The
quiver Grassmannians play a crucial role in the construction of Ringel-Hall algebras.
Moreover, they arise in the study of general representations of quivers by Scho eld and
in the theory of local models of Shimura varieties. Motivated by this, we study the ge-
ometric properties of quiver Grassmannians, their Euler characteristics and Ringel-Hall
algebras. This work is divided into three parts.
In the rst part of this thesis, we study geometric properties of the quiver Grass-
mannian Gr (M). In some cases we compute the dimension of this variety, we detectd
smooth points and we prove semicontinuity of the rank functions and of the dimensions
of homomorphism spaces. Moreover, we compare the geometry of the quiver Grassman-
nian Gr (M) with the geometry of the module variety rep (Q) and we develop tools tod d
decompose Gr (M) into irreducible components.d
In the following we consider some special classes of quiver representations, called
string, tree and band modules. There is an important family of nite-dimensional k-
algebras, called string algebras, such that each indecomposable module is either a string
or a band module.
In the second part, for k = we compute the Euler characteristics of quiver Grass-
mannians Gr (M) and of quiver ag varieties (M) in the case that M is aF (1) (r)d d ;:::;d
direct sum of string, tree and band modules. We prove that these Euler characteristics
are positive if the corresponding variety is non-empty. This generalizes some results of
Cerulli Irelli.
In the third part, we consider the Ringel-Hall algebraH(A) of a string algebraA over
. We give a complete combinatorial description of the product of the subalgebraC(A)
of the Ringel-Hall algebraH(A).
In covering theory we obtain the following results, which resemble the results of the
^last two parts. LetQ be a locally nite quiver with a free action of a free or free abelian
^group and : Q!Q the corresponding projection on the orbit space Q. Thus for each
^ nite-dimensional Q-representation V we get a Q-representation (V ) and induces
^Q Q0 0a map : ! of dimension vectors. We show that the Euler characteristic of
the quiver Grassmannian Gr ( (V )) is the sum of the Euler characteristics of Gr (V ),d t
1 ^where t runs over all dimension vectors in (d). Moreover, the morphism : Q!Q
^ ^of quivers induces a morphismC():C( Q)!C( Q) of the Ringel-Hall algebras.
5
NCNCCCContents
1 Introduction 9
1.1 Geometric properties of quiver Grassmannians . . . . . . . . . . . . . . . 9
1.2 Euler characteristics of quiver Grassmannians . . . . . . . . . . . . . . . . 11
1.3 Ringel-Hall algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Preliminaries 15
2.1 Quivers and quiver representations . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Tree and band modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Ringel-Hall algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Geometric properties of quiver Grassmannians 33
3.1 Isomorphism classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Morphisms induced by homomorphisms . . . . . . . . . . . . . . . . . . . 39
3.3 Semicontinuity and group action . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Connections to degenerations of representations . . . . . . . . . . . . . . . 47
3.5 Representation nite case . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Euler characteristics of quiver Grassmannians 83
4.1 Gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Tree and band modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Quiver ag varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.6 Coverings of quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.7 Proof of the main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Ringel-Hall algebras 103
5.1 Morphisms of Ringel-Hall algebras . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Liftings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3 Gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Product in the Ringel-Hall algebra . . . . . . . . . . . . . . . . . . . . . . 108
5.5 String algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
71 Introduction
Let k be an algebraically closed eld, Q = (Q ;Q ) a locally nite quiver, M =0 1
(M;M ) a nite-dimensional Q-representation and d = (d ) a dimensioni i2Q ; 2Q i i2Q0 1 0
vector. A subrepresentation of M with dimension vector d is a tuple (U ) of d -i i2Q i0
dimensional subspaces U of the k-vector space M such that (U;M j ) isi i i U i2Q ; 2Qs() 0 1
again a Q-representation. The quiver Grassmannian Gr (M) is the projective varietyd
over k of all these subrepresentations of M with dimension vector d. This is a closed
subvariety of a product of classical Grassmannians (see Lemma 2.3.7).
Following [46] quiver Grassmannians appear in the study of general representations of
quivers (see Crawley-Boevey [19] and Scho eld [48]) and their Euler characteristics in
the theory of cluster algebras (see Caldero and Chapoton [9], Caldero and Keller [11] and
Derksen, Weyman and Zelevinsky [21]). Cluster algebras were introduced by Fomin and
Zelevinsky [24, 25, 26] in 2000. For instance, Caldero and Keller [10, 11] showed that the
Euler characteristic plays a central role for the categori cation of cluster algebras. In this
context the positivity of these Euler characteristics is essential. The Euler characteristic
of such a projective variety is a much studied, but very rough invariant (see Caldero
and Zelevinsky [13] and Cerulli Irelli [14]). The representation theoretic properties of
these quiver Grassmannians are studied for instance by Fedotov [23], Lusztig [39] and
Reineke [42]. Moreover, G ortz [30, Section 4] showed that they appear in the theory of
local models of Shimura varieties.
It is easy to see that an ideal I of a quiver Q does not a ect our results. Let M be
a (Q;I)-representation. SoM is also aQ-representation. Each subrepresentation of the
Q-representation M is also a subrepresentation of the (Q;I)-representation M. Thus
the variety Gr (M) for a nite-dimensional ( Q;I)-representation M equals the varietyd
Gr (M) for the Q-representation M.d
This thesis is organized as follows: After this introduction we state the necessary
basic notions in Chapter 2. Most of these de nitions and results are well-known. In
the remaining three chapters we present our own results. In Chapter 3 we study the
geometry of the quiver Grassmannian Gr (M) as a scheme. In Chapter 4 we computed
the Euler characteristics of some quiver Grassmannians. These results are applied to
Ringel-Hall algebras in Chapter 5. Some results of the last two chapters are already
published in [32].
1.1 Geometric properties of quiver Grassmannians
We study basic geometric properties of quiver Gr (M) building on workd
of Caldero and Reineke [12] (see also Cerulli Irelli and Esposito [15], Scho eld [48] and
9