Moduli of representations of one-point extensions

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 Poincare polynomial in singular cohomology of these moduli spaces.


Introduction
In this paper we construct and study moduli spaces parametrizing 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 favourable 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 A[T ].We construct standard projective resolutions for representations of A[T ].One of the most important consequences of this is an explicit description of the space Ext 2 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, 3.6 for precise formulations.Moreover, the standard resolutions allow us to calculate the Euler form of A[T ] in Theorem 3.7, Corollary 3.8, 3.10.After these preparations, we consider the representation varieties of A[T ], 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, Corollary 4.4, 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 ( [9]) for non-emptiness of the semi-stable locus in Theorems 5.9, 5.10.A set of generators for the ring of semi-invariants, closely following Schofield and Van den Bergh ( [13]) 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).

One-point extensions and their representations
We fix an algebraically closed field k.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) A[T ] is given by the formal matrix multiplication for a, a ′ ∈ A, λ, λ ′ ∈ k, t, t ′ ∈ T and componentwise addition.We define a category Rep A (T ⊗ k ? 2 →? 1 ) as follows.Its objects M are tuples (M 1 , M 2 ), where M 1 is a left A-module and M 2 is a k-vector space, together with a map of A-modules f M : Proof.See for example [11].
We can easily describe a quiver Q and relations R in this situation such that we obtain the quiver Q.We obtain the relations R by using the transformation matrices induced by the fixed representation T : For each vertex i ∈ Q 0 we choose a k-basis ρ 1,(i) , . . ., ρ dim Ti,(i) of T i and write for all l = 1, . . ., dim T i and each arrow (α : i → j) ∈ Q 1 .The coefficients of these linear combinations induce the relations R s,l ρ s,(j) .
Obviously this construction of relations does not depend (up to isomorphism of k-algebras) on the basis we choose in By extending the path algebra of with mβ = 0, mγ = 0.

Homological properties
Since A is the path algebra of a quiver Q, we can and will identify A with the tensor algebra T R X, 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 {e 1 , . . ., e n } be the complete set of primitive orthogonal idempotents given by the length 0 paths in A. Then ).This sequence of A-bimodules resp.A[T ]-bimodules splits in the category of right A-modules resp.A[T ]-modules. where Proof.We obtain the short exact sequence by tensoring the split short exact sequence of right The first equation follows immediately using the definition of R. Namely, for M 0 := V we have The second equation follows from the following commutative diagram with exact rows: and g is defined as Remark 3.2.To simplify the notation, set Ω A = Ω(A) and N = Ω A ⊗ A M .Combining Theorem 3.1 with the standard resolution of A-modules, we obtain the following long exact sequence: Using this construction, we obtain: we have the standard projective resolution: Here g and h are defined as In particular, we have gldim 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, Ω(A) and Ω A ⊗ A L are projective modules for all A-modules L.
Applying Hom A (?, N ), we obtain the following commutative diagram Now we consider the standard projective resolution of M of Theorem 3.3 and the induced cochain complex C := Hom B P • ( M ), Ñ : Moreover, we have the following commutative diagrams: HomA(?,N ) Thus we obtain: We sum up and obtain finally: We thus arrive at the folllowing description of Ext 2 A[T ] : there is an isomorphism: In particular, if T is projective, so is ker(f ), thus Ext 2 A[T ] vanishes identically.In other words, A[T ] is again hereditary in this case.
and apply Hom A (ker(f ), ?) and obtain the long exact sequence

Now we will show that Ext
holds.We also have: Applying Hom A (?, T ⊗ k W ) to this, we obtain the long exact sequence: 0 and therefore Ext A (ker(f ), N ) = 0. Using Theorem 3.4, we conclude the proof.

Derivations and the Euler form of
To shorten notation, we define B = A[T ].In this section we determine the dimension of a space of derivations dim Der R B, Hom R( Ñ , M ) .
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 Ñ (Theorem 3.3) and the induced cochain complex C := Hom B P • ( Ñ ), M : ).This way we obtain the equality: We easily determine: By using the characterization of Ext 2 A[T ] of Theorem 3.4, we end up in: where ., .A denotes the homological Euler-form of A.
If we assume Ext A (T, T ) = 0, then for full A[T ]-modules M and Ñ we have Proof.The claim follows immediately using Theorem 3.7 and 3.6.
For the homological Euler-form of A[T ] the following identity holds: where M , Ñ are (finite-dimensional) A[T ]-modules and ., .Q denotes the Eulerform of Q.
Proof.The identity follows from the above discussion.

Varieties of representations of one-point extensions
For all standard notions on varieties of representations of algebras, we refer to [5].Let (s, d) be a dimension vector of Q.Using the isomorphism A[T ] ≃ k Q/(R), we can realize the variety of representations of A[T ] with dimension (s, d) as a (Zariski-)closed subvariety of the variety of representations of Q with dimension (s, d) denoted by Rep (s,d) ( Q): We set t := dim T and get by using Theorem 3.7:

4.2.
A well-behaved subvariety in the rigid case.We consider where γ T s ,d ∈ N Q0 denotes the general rank of homomorphism from T s to a representation of dimension d (see [12,Section 5]).
induced by (1), we find that every (nonempty ).The claim follows immediately by using the dimension formula in Theorem 4.1.

On homomorphisms from a fixed representation.
We denote the open dense subset Here, hom(T, d) is the dimension of the space of homomorphisms from the fixed representation T to a general representation of dimension vector d (see [3]).Moreover, γ T,M is the unique maximal rank of homomorphisms from T to M .
We consider the regular map . Obviously V T,d is irreducible and the fibers of π are irreducible of dimension hom(T, d).So there is a unique irreducible component V 0 of V of maximal dimension which dominates π, that is, we have π(V 0 ) = V T,d (Proposition 4.7)and Since the regular points in V form an open dense subset, there is regular point M in the irreducible component V 0 , and we have: Using the description of the tangent space by derivations, we thus find: for a general A-module homomorphism f : T → M .
Proposition 4.7.Let f : X → Y be a regular map of quasi-projective varieties.If Y is irreducible and all fibers of f are irreducible and of same dimension d (in particular f is surjective), then: In particular, we can conclude X is irreducible if either of the following holds: Proof.See [6].Proof.By Theorem 4.2, it remains to prove that Rep full (s,d) A[T ] is irreducible.We consider the dominant map induces a split mono for the differential π in M There is an open dense subset U ⊆ Rep full (s,d) A[T ] such that for each M ∈ U we have Hom A (T s , M ) ≃ ker(dπ M ) (Theorem 4.6).Since dπ M is surjective, by using Theorem 4.5 we obtain: ) is dense, too.The map π induces a surjective regular map dense (and thus irreducible).All fibers of π are irreducible of equal dimension and V is equidimensional, thus V has to be irreducible by Theorem 4.7.Using V = Rep full (s,d) (A[T ]), we can conclude the claim.

Semistability
For all notions concerning stability and moduli spaces of representations we refer to [9].For an Then M is (semi-)stable iff for all subspaces W ⊆ M∞ the following inequality is fulfilled: Mρ l,(i) (W ) .
Proof.For all subobjects Ũ = (U, W, g) M , we have: We can easily determine the total space of the smallest subobject W of M containing a given W ⊂ V .Namely, we have for each i ∈ Q 0 .

An observation on the Harder-Narasimhan filtration.
At first we recall the notion of Harder-Narasimhan filtration.
Since Ũ M is a subrepresentation, we have the following commutative diagram: Assume µ( M / Ũ ) = 0, i.e. dim V − dim W = 0.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, Ũ = M , contradicting our assumption.b) ⇒ a): Assume the structure morphism f : T ⊗ k V → M is not surjective.Then we can consider the proper subobject Ũ M induced by the commutative diagram M : Obviously then we have µ( M / Ũ ) = 0, and since f is not surjective, we neither have dim M / Ũ = 0.So M / Ũ is a semi-stable representation.Now look at the HN filtration of Ũ .Since the structure morphism of Ũ is surjective we can conclude from the first part of this proof that for the HN filtration of Since by definition the slope is always ≥ 0 we can conclude Using the uniqueness of the HN filtration we finally obtain that must be the HN filtration of M .But this contradicts our assumption about the HN filtration of M .So f has to be surjective.Consequences of Theorem 5.6 are: Corollary 5.7.For (s, d) ∈ N × N Q0 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 Proof.Let F * : 0 = F 0 ⊂ F 1 ⊂ . . .⊂ F r be a flag of type (s, d) * in the Q0 -graded vector space which are compatible with F * , i.e.Mγ (F l i ) ⊂ F l j for l = 1, . . ., r and for all arrows (γ : i → j) in Q.We have the regular map given by the projection p(s,d) * mapping M ∈ Z(s,d) * to the sequence of subquotients with respect to F * .The map p(s,d) * induces a regular map and it is a locally closed subset.By Theorem 4.8, Rep (s,d) , Rep (s 1 ,d 1 ) , . . ., Rep (s r ,d r ) are irreducible.
We set e * 2 := (s 1 , d 1 ), (s 2 , d 2 ) .We obtain the regular map Thus all fibers are irreducible and of equal dimension (Corollary 5.7 and 3.8).By Theorem 4.7 we have Therefore we obtain the regular map ), the map p e * 3 induces a regular map As above, we see that this regular map has irreducible fibers of equal dimension, that is, for ( M1 , M3 ) ∈ Z e * 2 × Rep (s 3 ,d 3 ) , we have: dim . Inductively, we obtain in this way a regular map with irreducible fibers of equal dimension, where . Together with the formula in Corollary 3.8, we find: To simplify the notation in the following, we write Z (resp.From this description, we can derive a recursive criterion for the existence of semi-stable representations: 1) 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 dim M st (2,4,1) (A[T ]) = 4. 2) Analogous statements as in 1) hold for the dimension vector (3,6,2).And we can deduce dim M st (3,6,2) A[T ] = 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 (s, d) ∈ N Q0 on Rep (s,d) (A[T ]) under the G (s,d) base change action.
We start with a general observation: Lemma 6.1.Let G be a linear reductive group and X an affine G-variety.Let A ⊆ X be a closed and G-stable subset.Then, for every semi-invariant function Proof.Let χ be a character of G.We consider the action of G on X × k given by g.(x, λ) := (gx, χ(g)λ), In the following, to simplify the notation we denote by Â the path algebra of the one-point extended quiver Q.
Let Ñ be a representation of Q with projective resolution For M ∈ Rep (s,d) ( Q) we apply the functor Hom Â(?, M ) to this resolution and get that is, in this case we end up with a linear map between vector spaces of equal dimension.Schofield and Van den Bergh proved ( [13]) that all semi-invariant functions arise as linear combination of functions 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.
Obviously Ñ0 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: From the homological properties we can further conclude: Theorem 6.4.For a character χ of G (s,d) , a representation M = (M, V, f : Proof.Take the standard resolution as described as in Theorem 3.3 and consider the cochain complex C := Hom A[T ] P • ( Ñ ), M : . In this way, we achieve the relation: Both conditions in the theorem are equivalent to φ 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: 1) The regular maps Rep full

Higher Gel'fand MacPherson correspondence
We first recall the definition of quiver Grassmannians (see for example [1]).For a quiver Q, a representation X of Q of dimension vector d and another dimension vector e ≤ d, we define Gr e Q (X) 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 G d of the set Hom Q (X, e) e of surjections (that is, rank e maps) from X to a representation of dimension vector e.
From now on, we assume End A (T ) = k, Ext A (T, T ) = 0 and γ T s ,d = d., and the claim follows from the previous theorem.

Theorem 4 . 2 .
If we assume γ T s ,d = d and Ext A (T, T ) = 0, then Rep full (s,d) A[T ] is smooth and every component has dimension

Corollary 4 . 4 . 4 . 4 .
We have hom(T, d) = γ T,d , d Q + dim Hom A (ker(f ), M ) for a general A-module homomorphism f : T → M .Proof.Use the the Euler form of Q and the identity dim Hom A (ker(f ), M ) − dim Ext A (ker(f ), M ) = dim ker(f ), d .Corollary 4.5.If we assume γ T,d = d and Ext A (T, T ) = 0, then hom(T, d) = t, d Q .An irreducible component in the rigid case.We need the following facts from algebraic geometry: Proposition 4.6.Let π : X → Y be a regular map of affine varieties.Then there is an open dense subset U ⊆ X such that for all x ∈ U T x π −1 π(x) = ker(dπ x ).

Theorem 4. 8 .
If we assume γ T s ,d = d and Ext A (T, T ) = 0, then Rep full (s,d) A[T ] is irreducible and smooth of dimension dim Rep full (s,d) A[T ] = dim Rep d (Q) + s • t, d Q .

Theorem 5 . 8 .Theorem 5 . 9 .
d) -action.By definition, the G (s,d) -orbits in Rep (s,d) (A[T ]) correspond bijectively to isomorphism classes [ M ] of representations of A[T ] of dimension vector (s, d).We consider the stability function Θ for Q given by Θ∞ = 1 and Θi = 0 for i ∈ Q 0 .The associated slope function on representations of Q coincides with the slope function µ on A[T ]-modules.This allows us to define moduli spaces M sst (s,d) A[T ] resp.M st (s,d) A[T ] as the algebraic quotient of Rep sst (s,d) (A[T ]), resp.the geometric quotient of Rep st (s,d) (A[T ]), by G (s,d) .Assume Ext A (T, T ) = 0 and γ T s ,d = d.If Rep st (s,d) A[T ] = ∅, then both M sst (s,d) A[T ] and M st (s,d) A[T ] are irreducible and smooth of dimension dim M sst (s,d) A[T ] = 1 − (s, d), (s, d) A[T ] .Proof.We calculate fibre dimensions for the geometric quotient π : Rep st (s,d) A[T ] → M st (s,d) A[T ] , 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 Rep (s,d) for Rep full (s,d) (A[T ]) to simplify notation.Let assume Ext A (T, T ) = 0 and γ T s ,d = d.The HN-strata for the HN-types (s, d) * = (s 1 , d 1 ), . . ., (s r , d r ) with weight (s, d), i.e. (s, d) = r i=1 (s i , d i ), and s r = 0 define a stratification of Rep (s,d) .The codimension of Rep HN (s,d) * in Rep (s,d) is given by: r, and denote by F l i the i-component of F l .Denote by Z(s,d) * the closed subvariety of Rep (s,d) (A[T ]) of representations M p) for Z (s,d) * (resp.p (s,d) * ).The preimage of Rep sst (s 1 ,d 1 ) × . . .× Rep sst (s r ,d r ) under p gives us an open subvariety Z 0 of Z and Z.Since the varieties Rep (s 1 ,d 1 ) , . . ., Rep (s r ,d r ) are irreducible, we see in a similar manner to Z that dim Z 0 = dim Z holds.The action of G (s,d) on Rep (s,d) (A[T ]) induces actions of the parabolic subgroup P (s,d) * of G (s,d) , consisting of elements fixing the flag F * , on Z 0 and Z.The image of the associated fiber bundle G (s,d) × P (s,d) * Z under the action morphism m equals Rep (s,d) * (s,d) (A[T ]), which is thus a closed subvariety of Rep (s,d) (A[T ]).The image of G (s,d) × P (s,d) * Z 0 under m equals Rep HN (s,d) * , and G (s,d) × P (s,d) * Z 0 is the full preimage.By the uniqueness of the HN filtration, the morphism m is bijective over Rep HN (s,d) * , which therefore is a locally closed subvariety of Rep (s,d) (A[T ]).The canonical mapG (s,d) × P (s,d) * Z 0 → G (s,d) P (s,d) * is (Zariski) locally trivial.Therefore dim G (s,d) × P (s,d) * Z 0 = (dim G (s,d) − P (s,d) * ) + dim Z 0 .The codimension of RepHN (s,d) * in Rep (s,d) is now easily computed as − n<l (s n , d n ), (s l , d l ) A[T ] , using the identity d, d Q = dim G d − dim Rep d (Q) and the above description of Z 0 .

3 :
Let Ñ be a representation of A[T ].Then we have the canonical

Theorem 6 . 3 .
dim Ñ , (s, d) Q = 0 and ω Ñ = 0 hold, we can conclude from the diagram that the determinants ω Ñ0 and ω Ñfull can be formed.This discussion shows: The ring of semi-invariant functions on Rep (s,d) (A[T ]) is generated by the functions ω = ω L • ω Ñ induced by representations L of Q such that dim L, d Q = 0 and full representations Ñ of A[T ] such that dim Ñ , (s, d) Q = 0.