On cohomological Hall algebras of quivers: Generators

A.1 Proof of Proposition 3.1

The proposition is proved in the literature under the assumption that Q does not carry oriented cycles (or edge loops). The following argument which works in full generality was explained to us by W. Crawley-Boevey. Let us begin by sketching the proof that Π is of homological dimension two. We may and will restrict ourselves to the case of a connected quiver. For any vertex i∈I we denote by ei the corresponding idempotent of Π. Consider the complex of (Π,Π)-bimodules

(A.1)

where all the tensor products are taken over ℂ, and where the maps are defined as follows: k is the multiplication map, g⁢(a⁢eh′′⊗eh′⁢b)=a⁢h⁢eh′⊗eh′⁢b-a⁢eh′′⊗eh′′⁢h⁢b and

where ϵh=1 if h∈Ω and ϵh=-1 if h∈Ω*. The fact that (A.1) is a complex is a direct consequence of the defining relations of Π. We claim that it is in addition exact. This exactness everywhere except for the leftmost term may be checked using standard arguments. Recall that we have assumed that Q is not of finite Dynkin type. Equipping Π with the ℕ-grading obtained by assigning degree 1 to any edge h∈Ω¯ we then have that Π is a Koszul algebra and that its Hilbert series

is equal to HQ⁢(t)=(Id+t⁢(𝐐+𝐐t)+t2⁢Id)-1, where 𝐐 is the adjacency matrix of Q, see [23, 14]. Observing that f,g,k are of respective degrees 1,1,0, we deduce that

as wanted. This proves that Π is of homological dimension two. Next, let M, N be any finite-dimensional Π-modules. Tensoring (A.1) by M yields the projective resolution of M

Applying the functor Hom⁡(N,⋅), we obtain the following complex computing Ext∙⁡(N,M):

It remains to observe that this complex is canonically isomorphic to the dual of the same complex in which the roles of M and N are exchanged.

A.2 Proof of Lemma 3.9

Being the fiber of the moment map 𝔤⁢𝔩⁢(v)2⁢q→𝔰⁢𝔩⁢(v), the set M⁡(v) has all its irreducible components of dimension at least (2⁢q-1)⁢v2+1. Note also that M⁡(v) is defined over ℤ and may be therefore reduced to any finite field. By the Lang–Weil theorem the irreducibility of M⁡(v) will thus be a consequence of the estimate as l→∞

By [26, Theorem 5.1] the number of points of M⁡(v)⁢(𝔽l) is given by the generating series

where Exp is the plethystic exponential and Av⁢(l) is the Kac polynomial, see, e.g., [37, Section 2]. It is well known that Av⁢(l) is a monic polynomial of degree 1-〈v,v〉=1+(g-1)⁢v2. The coefficient of zv in (A.2) reads

where the sum ranges over all r-tuples ((n1,v1),…,(nr,vr)) satisfying ∑ini⁢vi=v. We claim that only the first term on the right-hand side of (A.3) contributes to the leading term. Indeed, ll-1⁢Av⁢(l)∼l1+(q-1)⁢v2 while for any r⩾2 and tuple (ni,vi)i we have

and it is elementary to check that ∑ini⁢(1+(q-1)⁢vi2)<1+(q-1)⁢v2 whenever r⩾2. It now follows by taking the leading term of (A.3) that #⁢M⁡(v)⁢(𝔽l)∼l(2⁢q-1)⁢v2+1 as wanted.

A.3 Proof of Proposition 3.11

A simple representation S of Π~ will be called rigid if Ext1⁡(S,S)={0}. Since

a simple representation S is rigid if and only if (dim⁡S,dim⁡S)=2. Alternatively, a representation S is rigid if and only if every other simple representation of the same dimension is isomorphic to S. On the other hand, if S is non-rigid, then there are infinitely many non-isomorphic simple representations of the same dimension (recall that we work over the field k=ℂ), and we have (dim⁡S,dim⁡S)⩽0 (note that (u,u′)∈2⁢ℤ for any dimension vectors u,u′). For any dimension type d there exists a unique representation type τ=τ⁢(d) of dimension type d in which all non-rigid simple representations occur with multiplicity one. The stratum 𝔐0⁢(τ) is open and dense in 𝔐0⁢((d)). Since π is proper, the set Im⁡(π) is closed and the proposition will be proved once we show that for any d we have 𝔐0⁢(τ)⊂Im⁡(π) as soon as there exists τ′ of dimension type d such that 𝔐0⁢(τ′)⊂Im⁡(π). By definition, the existence of such a τ′ means that there exists a stable representation M∈Ms⁡(v,w) of semisimple type τ′. In other words, there exists a collection of simple Π~-modules S1,S2,…,Sr such that #⁢{i:dim⁡Si=u}=du for any u, and an iterated extension

which is stable. Note that we automatically have dim⁡S1∈ℕI×{w} and dim⁡Si∈ℕI×{0} for any i>1. Our aim is then to prove, under the above hypothesis, the existence of a similar iterated extension N1⊂N2⊂⋯⊂Nr which is stable, for which Ti:=Ni/Ni-1 is simple, satisfies dim⁡Ti=dim⁡Si and in which all non-rigid simple factors Ti are non-isomorphic. We will use the following simple observation, which might be of independent interest.

Let us fix some w and prove the proposition by induction on v. Let us also fix a dimension type d such that ∑udu=(v,w). Let us assume that Proposition 3.11 is proved for (w,v) for any dimension vector v′<v. Let τ′ be a representation type of dimension type d¯, and of dimension (v,w). Let M1⊂M2⊂⋯⊂Mr be a filtration as above of representation type τ′, and set Si=Mi/Mi-1. If r=1, there is nothing to prove, so we assume that r>1. There are two cases to consider.

A.4 Kirwan surjectivity for local quiver varieties

In this subsection, we provide an independent proof of the Kirwan surjectivity for the fixed point quiver varieties used in this paper.

A.4.3 The quivers Q⋄ and Q¯⋄

Consider a new quiver Q⋄=(I⋄,Ω⋄) defined as follows:

1. I⋄={i1:i∈I}⊔{i2:i∈I}≃I×{1,2},

2. Ω⋄={h:(h′)1→(h′′)2∣h∈Ω}⊔{i:i1→i2∣i∈I}.

Example

We have

The quiver Q⋄ is strongly bipartite: all arrows go from a vertex in I×{1} to one in I×{2}. In particular, it contains no oriented cycles. The moduli stacks of representations of Q and Q⋄ are related in the following fashion. For each v∈ℕI we define v⋄∈ℕI⋄ such that

vi1⋄=vi2⋄=vi.

Let

Rep(kQ⋄,v⋄)♡⊂Rep(kQ⋄,v⋄)

be the open subset consisting of the representations x⋄ such that xi⋄ is invertible for all i. Then the assignment x⋄↦(x,g), where

xh=(xh′′⋄)-1∘xh⋄,gi=xi⋄,

gives an isomorphism

Rep(kQ⋄,v⋄)♡≃Rep(kQ,v)×G(v),

hence an isomorphism of Artin stacks

Rep(kQ⋄,v⋄)♡/G(v⋄)≃Rep(kQ,v)/G(v).

This allows us to view Rep⁡(k⁡Q,v)/G⁢(v) as an open substack of Rep⁡(k⁡Q⋄,v⋄)/G⁢(v⋄).

We will now perform a similar construction for the double quiver. Let Q¯⋄ be the double quiver of Q⋄. For each v, w in ℕI we define v⋄, w⋄ in ℕI⋄ such that

(A.5)vi1⋄=vi2⋄=vi,

wi1⋄=wi,

wi2⋄=0.

The group G⁢(v⋄) acts in a Hamiltonian fashion on

R⁡(v⋄,w⋄)=Rep⁡(k⁡Q¯⋄,v⋄)⊕HomI⋄⁡(w⋄,v⋄)⊕HomI⋄⁡(v⋄,w⋄)

and we denote by μ⋄:R⁡(v⋄,w⋄)→𝔤⁢(v⋄) the associated moment map. Let R(v⋄,w⋄)♡ be the open subset of R⁡(v⋄,w⋄) consisting of pairs (x¯⋄,a¯⋄) such that xi⋄ is an isomorphism for all i. We also put

M⁡(v⋄,w⋄)=μ⋄-1⁢(0),

M(v⋄,w⋄)♡=R(v⋄,w⋄)♡∩M(v⋄,w⋄).

Now, let R⁡(v,w) and M⁡(v,w) be as in Section 3.7.1. Consider the map

ϕ:R(v,w)×G(v)→R(v⋄,w⋄)♡,((x¯,a¯),g)↦(x¯⋄,a¯⋄),

defined, for all h∈Ω and i∈I, by

(A.6)a¯i1⋄=a¯i,

xh⋄=gh′′∘xh,

xh*⋄=xh*∘(gh′′)-1,

xi⋄=gi,

xi*⋄=-∑h′′=i(xi⋄)-1∘xh⋄∘xh*⋄.

A direct computation gives:

Lemma A.5.

Given v,w let v⋄,w⋄ be as in (A.5). Then the map ϕ gives an isomorphism M(v,w)×G(v)→M(v⋄,w⋄)♡. ∎

We will next consider appropriate stability conditions on R⁡(v⋄,w⋄). Given a tuple of integers (θi), let θ be the character of G⁢(v⋄) defined by θ⁢(g)=∏i∈I⋄det⁡(gi)-θi. We set

𝔐θ(v⋄,w⋄)=Proj(⊕n∈ℕk[M(v⋄,w⋄)]θn),

which is a quasi-projective T×G⁢(w⋄)-variety equipped with a projective morphism

π:𝔐θ(v⋄,w⋄)→𝔐0(v⋄,w⋄)=M(v⋄,w⋄)//G(v⋄).

We can describe the set of closed points of 𝔐θ⁢(v⋄,w⋄) explicitly. Set

θ∞=-∑i∈I⋄θi⁢vi⋄,W⋄=⊕i∈I⋄Wi⋄,V⋄=⊕i∈I⋄Vi⋄.

We say that an element (x¯⋄,a¯⋄) is θ-semistable if the following conditions are satisfied:

1. For any (x¯⋄,a¯⋄)-stable I⋄-graded subspace U⊆V⋄ we have

   ∑i∈I⋄θi⁢dim⁡(Ui)⩽0.

2. For any I⋄-graded subspace U⊆V⋄ such that U⊕W⋄ is (x¯⋄,a¯⋄)-stable we have

   ∑i∈I⋄θi⁢dim⁡(Ui)+θ∞⩽0.

We will further say that (x¯⋄,a¯⋄) is stable if the above inequalities are strict for proper subspaces U. Let us denote by Rθ⁡(v⋄,w⋄) the open subset of R⁡(v⋄,w⋄) consisting of θ-semistable elements and by Mθ⁡(v⋄,w⋄) its intersection with M⁡(v⋄,w⋄). Then [32, Proposition 2.9] yields

𝔐θ(v⋄,w⋄)=Mθ(v⋄,w⋄)//G(v⋄).

We also set

Mθ(v⋄,w⋄)♡=Mθ(v⋄,w⋄)∩R(v⋄,w⋄)♡,

𝔐θ(v⋄,w⋄)♡=Mθ(v⋄,w⋄)♡//G(v⋄).

We say that θ is generic if neither equations

∑i∈I⋄θi⁢ui=0 and ∑i∈I⋄θi⁢ui+θ∞=0

have integer solutions (ui) satisfying 0⩽ui⩽vi⋄ other than the trivial solutions

(ui)=0⁢or⁢(ui)=(vi⋄).

If θ is generic, then any semistable pair (x¯⋄,a¯⋄) is stable and in that case the map

Mθ⁡(v⋄,w⋄)→𝔐θ⁢(v⋄,w⋄)

is a G⁢(v⋄)-torsor, hence the variety 𝔐θ⁢(v⋄,w⋄) is smooth.

Lemma A.6.

Given v,w let v⋄,w⋄ be as in (A.5). Let θ∈ZI⋄ be generic and such that

(A.7)θi1≫0,θi2≪0,θi1+θi2=1 for all ⁢i∈I.

Then the map ϕ restricts to an isomorphism

Mθ(v,w)×G(v)→∼Mθ(v⋄,w⋄)♡.

Proof.

Fix a tuple z=(x¯⋄,a¯⋄) in M(v⋄,w⋄)♡. We will prove that

z∈Mθ(v⋄,w⋄)♡⇔ϕ-1(z)∈Mθ(v,w)×G(v).

Let U be an I⋄-graded subspace of V⋄.

Let us first assume that U is z-stable and nonzero. Since xi⋄ is invertible, we have

dim⁡(Ui1)⩽dim⁡(Ui2),i∈I.

If there exists j∈I such that dim⁡(Uj1)<dim⁡(Uj2), then

∑i∈I⋄θi⁢dim⁡(Ui)=∑i∈I[θi2⁢(dim⁡(Ui2)-dim⁡(Ui1))+dim⁡(Ui1)]⩽θj2+∑i∈Ivi<0

because θj2≪0. Thus U is not destabilizing. On the other hand, if xi⋄ is an isomorphism Ui1→Ui2 for all i, then we have

∑i∈I⋄θi⁢dim⁡(Ui)=∑idim⁡(Ui1)>0

hence U is destabilizing.

Assume now that U⊕W⋄ is z-stable and proper. As above, we have

dim⁡(Ui1)⩽dim⁡(Ui2)

for all i. Thus

∑i∈I⋄θi⁢dim⁡(Ui)+θ∞=∑i∈I[θi2⁢(dim⁡(Ui2)-dim⁡(Ui1))+dim⁡(Ui1)]+θ∞⩽∑i∈Idim⁡(Ui1)-vi<0

for otherwise dim⁡(Ui1)=dim⁡(Ui2)=vi for all i and U is not a proper subspace of V⋄. Therefore U is not destabilizing.

By the above, we conclude that z is θ-semistable if and only if there does not exist a nonzero z-stable U⊂V⋄ stable under all the maps xi-1. This is easily seen to be equivalent to the condition that ϕ-1⁢(z) belongs to Ms⁡(v,w)×T×G⁢(v). ∎

Corollary A.7.

Given v,w let v⋄,w⋄ be as in (A.5). Let θ be a generic character as in (A.7). Then we have an isomorphism ψ:M⁢(v,w)→Mθ⁢(v⋄,w⋄)♡ and an open immersion i:M⁢(v,w)→Mθ⁢(v⋄,w⋄).∎