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Abstract
We study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. This class of moduli spaces unifies Grassmannians of subrepresentations of rigid representations and moduli spaces of representations of generalized Kronecker quivers. With homological methods, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel’fand MacPherson correspondence, and derive a formula for the Poincaré polynomial in singular cohomology of these moduli spaces.
1 Introduction
In this paper we construct and study moduli spaces parameterizing isomorphism classes of representations of so-called one-point extensions of path algebras of quivers. This constitutes a class of algebras of global dimension two, for which many of the favorable properties of moduli spaces of representations of quivers still hold. Namely, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel’fand MacPherson correspondence, and derive a formula for the Poincaré polynomial in singular cohomology of these moduli spaces. We explicitly apply the developed theory in several examples.
To do this, we fix a path algebra A = kQ of a finite quiver, which we extend by a representation T of A to the one-point extension algebra . We construct standard projective resolutions for representations of
. One of the most important consequences of this is an explicit description of the space
of representations, which allows us to conclude its vanishing on so-called full representations (under the assumption of T being rigid). See Theorems 3.1, 3.3, 3.4, and 3.6 for precise formulations. Moreover, the standard resolutions allow us to calculate the Euler form of
in Theorem 3.7, Corollaries 3.8 and 3.10. After these preparations, we consider the representation varieties of
, interpret the found homological properties in this geometric setting, and rediscover some results by Schofield and Crawley-Boevey (with different methods) in Theorem 4.3, Corollaries 4.4 and 4.5. Moreover, in this way we can determine the Zariski tangent space of the representation variety in each point (Theorem 4.1) and conclude that the open subset of full representations is smooth and irreducible (Theorem 4.8).
We follow the GIT approach of King in the construction of moduli spaces. For this, we choose a canonical stability condition, such that the resulting spaces unify quiver Grassmannians of subrepresentations of rigid representations (Theorem 7.1) and moduli spaces of representations of generalized Kronecker quivers. We find a numerical criterion for semistability in Theorem 5.2, which allows us to conclude that semi-stable representations are full representations (Corollary 5.7). In this way, we can apply the above geometric properties of representation varieties to prove that the resulting moduli spaces are smooth, irreducible and of expected dimension in Theorem 5.8.
After this, we prove a relative version of a recursive numerical criterion [Citation9] for non-emptiness of the semi-stable locus in Theorems 5.9 and 5.10. A set of generators for the ring of semi-invariants, closely following Schofield and Van den Bergh ([Citation13]) is given in section 6.
Moreover, we find a form of Gel’fand MacPherson correspondence in terms of these moduli spaces in Theorem 7.1. We finish the paper by deriving a recursive formula to determine the Poincaré polynomial of the moduli spaces (Theorem 8.2).
2 One-point extensions and their representations
We fix an algebraically closed field k of characteristic zero. Let A = kQ be the path algebra of a finite quiver Q, and let T be a finite dimensional A-module. We consider the one-point extension of A by T
Recall that the multiplication in (the k-algebra) is given by the formal matrix multiplication
for
and componentwise addition.
We define a category as follows. Its objects M are tuples (M1, M2), where M1 is a left A-module and M2 is a k-vector space, together with a map of A-modules
. A morphism
is a tuple
, where
is a map of A-modules and
a k-linear map, such that the diagram
commutes. Composition of morphisms is defined componentwise.
Lemma 2.1.
The category is equivalent to the category of left
-modules.
Proof.
See for example [Citation11]. □
We can easily describe a quiver and relations
in this situation such that
. By extending Q the following way
we obtain the quiver
. We obtain the relations
by using the transformation matrices induced by the fixed representation T: For each vertex
we choose a k-basis
of Ti
and write
for all and each arrow
. The coefficients of these linear combinations induce the relations
(1)
(1)
Obviously this construction of relations does not depend (up to isomorphism of k-algebras) on the basis we choose in Ti
for each vertex .
Example 2.1.
Let .
By extending the path algebra of Q with we get
3 Homological properties
Since A is the path algebra of a quiver Q, we can and will identify A with the tensor algebra , where R is the semisimple k-algebra generated by the vertices of Q and X is the R-R-bimodule generated (as a k-vector space) by the arrows of Q.
3.1 The standard resolution
Let be the complete set of primitive orthogonal idempotents given by the length 0 paths in A. Then
is a complete set of primitive orthogonal idempotents of
. Let
be the k-subalgebra of
given by
The multiplication of A resp. induces the Eilenberg sequence [Citation2, Proposition 2.7.3]
resp.
with
resp.
.
This sequence of A-bimodules resp. -bimodules splits in the category of right A-modules resp.
-modules.
Theorem 3.1.
Let be an
-module. Then there is a short exact sequence
where
Proof.
We obtain the short exact sequence by tensoring the split short exact sequence of right -modules
over
with
.
The first equation follows immediately using the definition of . Namely, for
we have
and
.
The second equation follows from the following commutative diagram with exact rows:
where
and g is defined as
for
. □
Remark 3.2.
To simplify the notation, set and
. Combining Theorem 3.1 with the standard resolution
of A-modules, we obtain the following long exact sequence:
Using this construction, we obtain:
Theorem 3.3.
For -modules
we have the standard projective resolution:
Here g and h are defined as
for
and
.
In particular, we have .
3.2 Characterizing ![](//:0)
![](//:0)
Let be
-modules.
By using the Eilenberg sequence we obtain the following commutative diagram with exact rows and columns
Note that, since R is a semisimple k-algebra, and
are projective modules for all A-modules L.
Applying , we obtain the following commutative diagram
Now we consider the standard projective resolution of of Theorem 3.3
and the induced cochain complex
:
Then for i = 0, 1, 2.
We have
Moreover, we have the following commutative diagrams:
Thus we obtain:
(2)
(2)
We sum up and obtain finally:
We thus arrive at the following description of :
Theorem 3.4.
For -modules
there is an isomorphism:
In particular, if T is projective, so is , thus
vanishes identically. In other words,
is again hereditary in this case.
Definition 3.5.
We call an -module
full, if f is a surjective map.
Theorem 3.6.
Assume . Then, for full
-modules
, the vanishing property
holds.
Proof.
Write and
. We consider
and apply
and obtain the long exact sequence
Now we will show that holds by showing that
holds. We also have:
Applying to this, we obtain the long exact sequence:
Since , in particular
holds.
So and therefore
. Using Theorem 3.4, we conclude the proof. □
3.3 Derivations and the Euler form of ![](//:0)
![](//:0)
Let be
-modules. To shorten notation, we define
. In this section we determine the dimension of a space of derivations
To do this, we consider the canonical exact sequence
where c is defined as
We obtain the equality:
On the other hand, we have the following description:
We consider the standard projective resolution of (Theorem 3.3)
and the induced cochain complex
:
Then .
This way we obtain the equality:
We easily determine:
By using the characterization of of Theorem 3.4, we end up in:
Theorem 3.7.
For finite-dimensional -modules
and
we have
where
denotes the homological Euler-form of A.
Corollary 3.8.
If we assume , then for full
-modules
and
we have
Proof.
The claim follows immediately using Theorems 3.7 and 3.6. □
We notate a dimension vector as a tuple (s, d), where
and
.
Definition 3.9.
For dimension vectors we set
Corollary 3.10.
For the homological Euler-form of the following identity holds:
where
are (finite-dimensional)
-modules and
denotes the Euler-form of Q.
Proof.
The identity follows from the above discussion. □
4 Varieties of representations of one-point extensions
For all standard notions on varieties of representations of algebras, we refer to [Citation5]. Let (s, d) be a dimension vector of . Using the isomorphism
, we can realize the variety of representations of
with dimension (s, d) as a (Zariski-)closed subvariety of the variety of representations of
with dimension (s, d) denoted by
:
4.1 The Zariski-tangent space
For we have (see [Citation5, Example 3.10.]):
We set and get by using Theorem 3.7:
Theorem 4.1.
For we have
4.2 A well-behaved subvariety in the rigid case
We consider
where
denotes the general rank of homomorphism from T s
to a representation of dimension d (see [Citation12, Section 5]).
Theorem 4.2.
If we assume and
, then
is smooth and every component has dimension
i.e.
is a local complete intersection.
Proof.
By counting the explicit defining polynomial equations
induced by (1), we find that every (nonempty) component of
has dimension
.
On the other hand, for each we have
The claim follows immediately by using the dimension formula in Theorem 4.1. □
4.3 On homomorphisms from a fixed representation
We denote the open dense subset
of
by
. Here,
is the dimension of the space of homomorphisms from the fixed representation T to a general representation of dimension vector d (see [Citation3]). Moreover,
is the unique maximal rank of homomorphisms from T to M.
We consider the regular map
which induces a regular map
where
is the open preimage
.
Obviously is irreducible and the fibers of π are irreducible of dimension
. So there is a unique irreducible component
of
of maximal dimension which dominates π, that is, we have
(Proposition 4.7)and
Since the regular points in form an open dense subset, there is regular point
in the irreducible component
, and we have:
Using the description of the tangent space by derivations, we thus find:
Theorem 4.3.
We have
for a general A-module homomorphism
.
As a side remark, we rediscover the following result of Schofield [Citation12] and Crawley-Boevey [Citation3]:
Corollary 4.4.
We have
for a general A-module homomorphism
.
Proof.
Use the Euler form of Q and the identity
□
Corollary 4.5.
If we assume and
, then
.
4.4 An irreducible component in the rigid case
We need the following facts from algebraic geometry:
Proposition 4.6.
Let be a regular map of affine varieties. Then there is an open dense subset
such that for all
Proof.
See for example [Citation4]. □
Proposition 4.7.
Let be a regular map of quasi-projective varieties.
If is irreducible and all fibers of f are irreducible and of same dimension d (in particular f is surjective), then:
There is a unique irreducible component
of
that dominates
, i.e.
.
Each irreducible component
of
is a union of fibers of f. Its dimension is equal to
.
In particular, we can conclude is irreducible if either of the following holds:
is equidimensional.
f is closed.
Proof.
See [Citation6]. □
Theorem 4.8.
If we assume and
, then
is irreducible and smooth of dimension
Proof.
By Theorem 4.2, it remains to prove that is irreducible. We consider the dominant map
For each the regular map
induces a split mono for the differential
in
i.e. is surjective. For each
we have
There is an open dense subset such that for each
we have
(Theorem 4.6). Since is surjective, by using Theorem 4.5 we obtain:
for each
. So
Since is dense,
is dense, too.
The map induces a surjective regular map
where
is dense (and thus irreducible). All fibers of π are irreducible of equal dimension and
is equidimensional, thus
has to be irreducible by Theorem 4.7. Using
, we can conclude the claim. □
5 Semistability
For all notions concerning stability and moduli spaces of representations we refer to [Citation9]. For an -module
we define its slope
Definition 5.1.
An -module
is called semi-stable (resp. stable) if
(resp.
) for all proper subrepresentation
.
5.1 A first criterion
Theorem 5.2.
Let be an
-module with
. Then
is (semi-)stable iff for all subspaces
the following inequality is fulfilled:
Proof.
For all subobjects , we have:
We can easily determine the total space of the smallest subobject of
containing a given
. Namely, we have
for each
. □
5.2 An observation on the Harder-Narasimhan filtration
At first we recall the notion of Harder-Narasimhan filtration.
Definition 5.3.
A dimension vector
is called (semi-)stable if there is a (semi-)stable representation of
with dimension vector (s, d).
A tuple
of dimension vectors is of HN-type if each
(
) is semi-stable and
A filtration
of a representation
(of
) is called Harder-Narasimhan (HN) if each quotient
(
) is semi-stable and
Proposition 5.4.
Every representation of
admits a unique Harder-Narasimhan filtration.
Proof.
See [Citation9]. □
Definition 5.5.
For a HN type we denote by
the subset of representations whose HN filtration is of type
.
is called HN stratrum for the HN type
. More generally, we denote by
the subset of representations
possessing a filtration of type
, i.e. there is a chain of subrepresentations
with
for
.
Theorem 5.6.
Let be a representation of
with
. Then the following are equivalent:
is surjective.
For the HN filtration of
we have
Proof.
a) b): We denote
, which we write as
Since is a subrepresentation, we have the following commutative diagram:
Assume , i.e.
. So we obtain W = V, and since f is surjective we can conclude from the above commutative diagram that U already equals M. In other words,
, contradicting our assumption.
b)
a): Assume the structure morphism
is not surjective. Then we can consider the proper subobject
induced by the commutative diagram
Obviously then we have , and since f is not surjective, we neither have
. So
is a semi-stable representation.
Now look at the HN filtration of . Since the structure morphism of
is surjective we can conclude from the first part of this proof that for the HN filtration of
we have
Since by definition the slope is always we can conclude
Using the uniqueness of the HN filtration we finally obtain that
must be the HN filtration of
. But this contradicts our assumption about the HN filtration of
. So f has to be surjective. □
Consequences of Theorem 5.6 are:
Corollary 5.7.
For we have the following connection between the semi-stable representations and full representations:
5.3 Geometric consequences for the moduli space
The linear algebraic group
acts on via the base change action
is stable under this
-action. By definition, the
-orbits in
correspond bijectively to isomorphism classes
of representations of
of dimension vector (s, d). We consider the stability function
for
given by
and
for
. The associated slope function on representations of
coincides with the slope function μ on
-modules. This allows us to define moduli spaces
resp.
as the algebraic quotient of
, resp. the geometric quotient of
, by
.
Theorem 5.8.
Assume and
.
If , then both
and
are irreducible and smooth of dimension
Proof.
We calculate fiber dimensions for the geometric quotient
and use Corollary 5.7, Theorem 3.6 and the fact that the endomorphism rings of stable representations are trivial. □
Next, we introduce the Harder-Narasimhan stratification. Note that the term stratification is used in a weak sense, meaning a finite decomposition of a variety into locally closed subsets.
5.4 Harder-Narasimhan stratification
In this section we write for
to simplify notation.
Theorem 5.9.
Let assume and
.
The HN-strata for the HN-types
with weight (s, d), i.e.
, and
define a stratification of
.
The codimension of in
is given by:
Proof.
Let be a flag of type
in the
-graded vector space
, i.e.
for
, and denote by
the i-component of F l
.
Denote by the closed subvariety of
of representations
which are compatible with
, i.e.
for
and for all arrows
in
. We have the regular map
given by the projection
mapping
to the sequence of subquotients with respect to
. The map
induces a regular map
where
A minute reflection shows that
and it is a locally closed subset. By Theorem 4.8,
We set
We obtain the regular map
with fibers at
given by
Thus all fibers are irreducible and of equal dimension (Corollaries 5.7 and 3.8). By Theorem 4.7 we have
Now we set
Therefore we obtain the regular map
Since
the map induces a regular map
As above, we see that this regular map has irreducible fibers of equal dimension, that is, for , we have:
Inductively, we obtain in this way a regular map
with irreducible fibers of equal dimension, where
Summing up, for this yields:
Together with the formula in Corollary 3.8, we find:
To simplify the notation in the following, we write (resp. p) for
(resp.
). The preimage of
under p gives us an open subvariety of
and
. Since the varieties
are irreducible, we see in a similar manner to
that
holds.
The action of on
induces actions of the parabolic subgroup
of
, consisting of elements fixing the flag
, on
and
. The image of the associated fiber bundle
under the action morphism m equals
, which is thus a closed subvariety of
. The image of
under m equals
, and
is the full preimage. By the uniqueness of the HN filtration, the morphism m is bijective over
, which therefore is a locally closed subvariety of
.
The canonical map is (Zariski) locally trivial. Therefore
The codimension of in
is now easily computed as
using the identity
and the above description of
. □
From this description, we can derive a recursive criterion for the existence of semi-stable representations:
Theorem 5.10.
Let us assume . A dimension type
is semi-stable if and only if
and there exists no HN type
with weight (s, d) and
such that
Proof.
Let
Obviously H is a finite set. Using Theorem 5.6 we get
If is of equal dimension like
. So
for
implies
.
Let n < l. Since resp.
are semi-stable, we find semi-stable representations
resp.
of
with
resp.
and by using Corollaries 3.6 and 5.7 we get the relation
From we deduce
, in particular
So the equation
is fulfilled if and only if
for all
. □
Example 5.1.
We carry on with Example 2.1 here. Obviously the -orbit of
is dense in
. Therefore we have
.
Applying the recursive criterion we derive that (2, 4, 1) is semi-stable. In fact, by using the first criterion in Theorem 5.2, we see that in this case the semi-stability notion equals to the stability notion. From the geometry of the moduli (Theorem 5.8) we can conclude
.
Analogous statements as in 1) hold for the dimension vector (3, 6, 2). And we can deduce
.
6 Generating semi-invariants
In this section we assume that Q is an acyclic quiver. We determine a set of functions generating the ring of semi-invariants for given dimension vector on
under the
base change action.
We start with a general observation:
Lemma 6.1.
Let be a linear reductive group and
an affine
-variety. Let
be a closed and
-stable subset. Then, for every semi-invariant function
there is a semi-invariant
such that
holds.
Proof.
Let χ be a character of . We consider the action of
on
given by
where
.
Since is closed and
-stable, the categorical quotient
is closed, i.e.
is surjective. □
In the following, to simplify the notation we denote by the path algebra of the one-point extended quiver
.
Let be a representation of
with projective resolution
For we apply the functor
to this resolution and get
The condition is equivalent to
that is, in this case we end up with a linear map between vector spaces of equal dimension. Schofield and Van den Bergh proved [Citation13] that all semi-invariant functions arise as linear combination of functions
induced by representations
of
with
.
Using the relation we can conclude with Lemma 6.1:
Lemma 6.2.
The functions for representations
of
such that
generate the ring of invariants on
.
We can improve this description further by using the canonical exact sequence for one-point extensions and the explicit description of the standard projective resolution in Theorem 3.3: Let be a representation of
. Then we have the canonical exact sequence
which in more detail reads
Obviously can be interpreted as a representation of Q, with standard resolution of length 1. Thus, we arrive at the following commutative diagram with exact columns and rows:
Thus, if and
hold, we can conclude from the diagram that the determinants
and
can be formed. This discussion shows:
Theorem 6.3.
The ring of semi-invariant functions on is generated by the functions
induced by representations L of Q such that
and full representations
of
such that
.
From the homological properties we can further conclude:
Theorem 6.4.
For a character χ of , a representation
is χ-semi-stable iff there is a non-trivial finite-dimensional representation
of
such that
, and
.
Proof.
Take the standard resolution as described as in Theorem 3.3
and consider the cochain complex
:
Then you have .
In this way, we achieve the relation:
Both conditions in the theorem are equivalent to being an isomorphism. □
Example 6.1.
We carry on with Example 5.1 here. Long calculations yields to following generating and algebraic independent semi-invariant functions:
The regular maps
defined by
,
, and
give rise to the geometric quotient
by
in the following way
Thus, we have
.
In the case of the dimension vector (3, 6, 2) the regular maps
defined by
,
,
, and
give rise to the geometric quotient
by
in the following way
This shows that
.
7 Higher Gel’fand MacPherson correspondence
We first recall the definition of quiver Grassmannians (see for example [Citation1]). For a quiver Q, a representation X of Q of dimension vector d and another dimension vector , we define
as the set of subrepresentations U of X of dimension vector d – e. This carries a natural scheme structure as the geometric quotient by the base change group
of the set
of surjections (that is, rank e maps) from X to a representation of dimension vector e.
From now on, we assume and
.
In this case, the regular map
which is always a locally trivial fiber bundle [Citation3, Lemma 1.2.], has single element fibers, and thus is an isomorphism of varieties. Moreover, we have the
-bundle
All in all, we achieve a -bundle
where
is open.
We thus have an induced map
The linear reductive group acts naturally on
and for
clearly we have
We thus find:
Theorem 7.1
(Higher Gel’fand MacPherson correspondence). There is an isomorphism of varieties:
Proof.
By Theorem 5.8 is smooth, in particularly normal. Theorem 4.8 shows that the quiver Grassmanian of a representation without self-extensions is irreducible. Since π is surjective the claim follows from [Citation7, Theorem 4.2]. □
8 Motive of the moduli space
In this section, we assume and
.
As an application of the arguments in Theorem 5.9 and the explicit recursive formula given there, we derive a formula for the motive of the (smooth) moduli space (over
).
To achieve this, we will follow closely the strategy of [Citation8, Section 6], but replace counts of rational points over finite fields by motives as in the proof of [Citation10, Theorem 3.5]. We denote by the free abelian group generated by representatives
of isomorphism classes of complex varieties X, modulo the relation
if C is isomorphic to a closed subvariety of X with open complement isomorphic to U. Multiplication in
is given by
. We denote by
the class of the affine line; the following calculations will be performed in the localization
At this point, we recall a notation from the Section 5.4. Let
In this section, we write for
and
for
.
8.1 Motive
Obviously H is finite and by Theorem 5.6 we have
and thus
in
. We then find
Fix . As in the proof of Theorem 5.9 we have
and we arrive at
This provides us with the motivic HN-recursion:
Theorem 8.1.
Using the arguments of [Citation8, Theorem 6.7.], we see that the following relation
holds and we obtain:
Theorem 8.2.
Let (s, d) be a dimension vector such that semi-stability and stability coincide. Then, with we have:
Proof.
As in [Citation8, Proposition 6.6.] we have
and the claim follows from the previous theorem. □
8.2 Applications and examples
Lemma 8.3.
Let Q be of Dynkin type, let be the set theoretic quotient of
by the structure group
via the base change action, and for
let
Then, we have
Proof.
We consider the map
and note that
is constant along orbits. □
Overall, we find:
Theorem 8.4.
Let Q be of Dynkin type, and let (s, d) be a dimension vector such that semi-stability and stability coincide. Then we have:
Example 8.1.
Let
and
. Then H consists of
We calculate all components in the formula for each element in H:
We continue to calculate and get
In total, we end up with:
This result was to be expected if one remembers our explicit calculations of the moduli space.
Acknowledgments
The authors are supported by the DFG SFB/Transregio 191 “Symplektische Strukturen in Geometrie, Algebra und Dynamik.”
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