Semi-Stable Chow-Hall Algebras of Quivers and Quantized Donaldson-Thomas Invariants
aa r X i v : . [ m a t h . R T ] A p r Semi-Stable Chow–Hall Algebras of Quivers and QuantizedDonaldson–Thomas Invariants
H. Franzen M. Reineke Abstract
The semi-stable ChowHa of a quiver with stability is defined as an analog of the CohomologicalHall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semi-stable loci in representation varieties of quivers. We prove several structural results on the semi-stable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and atautological presentation. For symmetric quivers, this leads to an identification of their quantizedDonaldson–Thomas invariants with the Chow–Betti numbers of moduli spaces.
The Cohomological Hall algebra, or CoHa for short, of a quiver is defined in [12] as an analog ofthe Hall algebra construction of [18] in equivariant cohomology of representation varieties. In [12]the CoHa serves as a tool for the study of quantized Donaldson–Thomas invariants of quivers, andin particular their integrality properties, since it admits a purely algebraic description as a shufflealgebra “with kernel” on spaces of symmetric polynomials. In [4], the CoHa of a symmetric quiver isshown to be a free super-commutative algebra, proving the positivity of quantized Donaldson–Thomasinvariants in this case.In another direction, the CoHa is used in [6, 8] to determine the ring structure on the cohomologyof non-commutative Hilbert schemes and more general framed moduli spaces of quiver representations,as defined in [5].Already in [8] it turns out that a “local” version of the CoHa (the semi-stable CoHa), constructedvia convolution on semistable loci of representation varieties with respect to a stability, is particularlyuseful, and that it is also convenient to replace equivariant cohomology by equivariant Chow groups.In the present paper, we study this local version, called the semi-stable ChowHa, more system-atically and demonstrate their utility both for understanding the structure of the CoHa and for thestudy of quantized Donaldson–Thomas invariants.We prove the following structural properties of the semi-stable ChowHa:The equivariant cycle map between the semi-stable ChowHa and the semi-stable CoHa is anisomorphism (Corollary 5.6), which can be viewed as a generalization of a result of [10] on the cycle Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 [email protected] Faculty of Mathematics, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 [email protected]
Acknowledgements
The authors would like to thank M. Brion, B. Davison, M. Ehrig, and M. Young for valuable discus-sions and remarks. While doing this research, H.F. was supported by the DFG SFB / Transregio 45“Perioden, Modulräume und Arithmetik algebraischer Varietäten”.2 A Reminder on Quiver Representations
Let Q be a quiver — i.e. a finite oriented graph — whose set of vertices resp. arrows we denote by Q resp. Q . We will often suppress the dependency on Q in the notation. The bilinear form χ = χ Q on Z Q defined by χ ( d, e ) = X i ∈ Q d i e e − X α : i → j d i e j = X i,j ( δ i,j − a i,j ) d i e j is called the Euler form of Q . Here, a i,j is the number of arrows from i to j in Q . We denote theanti-symmetrization χ ( d, e ) − χ ( e, d ) of the Euler form by h d, e i . Let Γ = Z Q ≥ be the monoid ofdimension vectors of Q .Let k be a field. A representation M of Q over k is a collection of finite-dimensional vector spaces M i with i ∈ Q together with linear maps M α : M i → M j for every arrow α : i → j . See [1] formore details. The tuple dim M = (dim M i | i ∈ Q ) ∈ Γ is called the dimension vector of M . For adimension vector d ∈ Γ , we define R d ( k ) to be the vector space R d ( k ) = M α : i → j Hom( k d i , k d j ) on which we have an action of the group G d ( k ) = Q i ∈ Q Gl d i ( k ) via base change. An element of R d ( k ) is a representation of Q on the vector spaces ( k d i ) i . Being an affine space, R d ( k ) admits a Z -model, i.e. there exists a scheme R d whose set of k -valued points is R d ( k ) . Likewise, there is agroup scheme G d which is a Z -model for G d ( k ) .We introduce a stability condition θ of Q , that is, a linear form Z Q → Z . For a non-zero dimensionvector d , the rational number θ ( d ) P i d i is called the θ -slope of d . For a rational number µ , let Γ θ,µ be the submonoid of all d ∈ Γ with d = 0 or whose θ -slope is µ . If M is a non-zero representation of M of Q over k , the θ -slope of M is definedas the slope of its dimension vector. A representation M of Q over k is called θ - semi-stable if nonon-zero subrepresentation of M has larger θ -slope than M . It is called θ - stable if the θ -slope of everynon-zero subrepresentation M ′ is strictly less than the slope of M , unless M ′ agrees with M . There isa Zariski-open subset R θ − sst d of the scheme R d whose set of k -valued points is the set of θ -semi-stablerepresentations of Q . There is also an open subset R θ − st d of R θ − sst d parametrizing absolutely θ -stablerepresentations, that means R θ − st d ( k ) consists of those M ∈ R d ( k ) such that M ⊗ k K is θ -stable forevery finite extension K | k . Fix a prime power q and let F = F q be the finite field with q elements. Then R d ( F ) and G d ( F ) arefinite sets, and we consider the set of orbits R d ( F ) /G d ( F ) . We define the completed Hall algebra of Q as the vector space H q = H q (( Q )) = { f | f : G d ∈ Γ R d ( F ) /G d ( F ) → Q } f and g , we define ( f ∗ g )( X ) = X U ⊆ X f ( U ) g ( X/U ) , the sum ranging over all subrepresentations of X . Note that this sum is finite. This multiplicationturns H q into an associative algebra. We define yet another algebra. Set T q := Q ( q / )[[ t i | i ∈ Q ]] and let the multiplication be given by t d ◦ t e = ( − q / ) h d,e i t d + e . It is shown in [14] that the so-called integration map R : H q → T q defined by Z f = X [ X ] ( − q / ) χ (dim X, dim X ) ♯ Aut( X ) · f ( X ) · t dim X is a homomorphism of algebras. Define ∈ H q to be the function with ( X ) = 1 for all [ X ] . An easycomputation shows that A ( q, t ) := R equals A ( q, t ) = X d ( − q / ) − χ ( d,d ) Y i d i Y ν =1 (1 − q − ν ) − t d . For a stability condition θ of Q and a rational number µ , we define θ,µ ∈ H q as the sum ofthe characteristic functions on R θ − sst d ( F ) /G d ( F ) over all d ∈ Γ θ,µ . Set A θ,µ ( q, t ) = R θ,µ . Using aHarder–Narasimhan type recursion, it is shown in [14] that Theorem 3.1 ([14]) . In H q , we have = ← Q µ ∈ Q θ,µ . This implies that the the series A and A θ,µ relate in the same way in the twisted power series ring T q . Let R be the power series ring Q ( q / )[[ t i | i ∈ Q ]] with the usual multiplication. Let R + be theset of power series without constant coefficient. There exists a unique bijection Exp : R + → R + such that Exp( f + g ) = Exp( f ) Exp( g ) and Exp( q k/ t d ) = 11 − q k/ t d for every k ∈ Z and d ∈ Γ . This function is called the plethystic exponential. We call the stabilitycondition θ generic for the slope µ ∈ Q (or µ -generic) if h d, e i = 0 for all d, e ∈ Γ θ,µ . Assuming that θ is µ -generic, the series A θ,µ can be displayed as a plethystic exponential: Theorem 3.2 ([11]) . For a µ -generic stability condition θ , there are polynomials ˜Ω θd ( q ) = P k ˜Ω θd, k q k in Z [ q ] for every non-zero dimension vector d of slope µ such that A θ,µ ( q − , t ) = Exp − q X d ( − q / ) χ ( d,d ) ˜Ω θd ( q ) t d ! . Definition. If θ is a µ -generic stability condition and d is a non-zero dimension vector of slope µ then the coefficients of Ω θd ( q ) = X k ∈ Z Ω θd,k q k/ := q χ ( d,d )2 ˜Ω θd ( q ) ∈ Z [ q ± / ] are called the quantum Donaldson–Thomas invariants of Q with respect to θ .When the quiver Q is symmetric, that means the Euler form of Q is a symmetric bilinear form,then every stability condition is µ -generic for every value µ ∈ Q . For example, the trivial stabilitycondition is -generic and we can define the Donaldson–Thomas invariants Ω d,k for every d = 0 . Notethat Theorem 3.1 implies Ω d,k = Ω θd,k for every θ . We may therefore write Ω d,k in this case. Fix an algebraically closed field k . Abusing the notation from the first section, we will use the symbols R d , R θ − sst d , R θ − st d , and G d for the base extensions of the respective Z -models to Spec k .Let d be a dimension vector for Q . We define A θ − sst d to be the G d -equivariant Chow ring withrational coefficients of the semi-stable locus A θ − sst d ( Q ) = A ∗ G d ( R θ − sst d ) Q . For the definition of equivariant Chow groups/rings, see Edidin–Graham’s article [3]. As we willalways work with rational coefficients, we will often omit it in the notation. We define A θ − sst ,µ asthe graded vector space A θ − sst ,µ ( Q ) = M d ∈ Γ θ,µ A θ − sst d We mimic Kontsevich–Soibelman’s construction of the Cohomological Hall algebra (CoHa) of a quiverwith stability and trivial potential from [12].For two dimension vectors d and e of the same slope µ , we set Z d,e as the subspace of R d + e ofrepresentations M which have a block upper triangular structure as indicated, i.e. for every arrow α : i → j , the linear map M α sends the first d i coordinate vectors of k d i + e i into the subspace of k d j + e j spanned by the first d j coordinate vectors. We consider R d × R e ← Z d,e → R d + e , the left hand map sending a representation M = (cid:0) M ′ ∗ M ′′ (cid:1) to the pair ( M ′ , M ′′ ) , and the right handmap being the inclusion. These maps are called Hecke correspondences. For a short exact sequence → M ′ → M → M ′′ → of representations of the same slope , M is θ -semi-stable if and only if both M ′ and M ′′ are. We thus obtain cartesian squares R d × R e Z d,e R d + e ⊆ ⊆ ⊆ R sst d × R sst e Z d,e ∩ R sst d + e R sst d + e . G = G d + e on R d + e restricts to an action of the parabolic P = (cid:0) G d ∗ G e (cid:1) on Z d,e , andthe map Z d,e → R d × R e is compatible with the action of its Levi L = G d × G e on R d × R e . Withrespect to these actions, the respective semi-stable loci are invariant. This gives rise to morphisms ( R sst d × R sst e ) × L G ← Z sst d,e × L G → Z sst d,e × P G → R sst d + e × P G → R d + e . We see that • ( R sst d × R sst e ) × L G ← Z sst d,e × L G is a G -equivariant (trivial) vector bundle, • Z sst d,e × L G → Z sst d,e × P G is a fibration whose fiber P/L is an affine space (as L is the Levi of P in G ) and thus induces an isomorphism in G -equivariant intersection theory, • Z sst d,e × P G → R sst d + e × P G is the zero section of a G -equivariant vector bundle whose rank is s = P α : i → j d i e j , and • R sst d + e × P G → R d + e is proper as G/P is complete, and the dimension of
G/P is s = P i d i e i .The above morphisms give rise to maps in equivariant intersection theory A nL ( R sst d × R sst e ) ∼ = −→ A nL (cid:0) Z sst d,e (cid:1) ∼ = ←− A nP (cid:0) Z sst d,e (cid:1) → A n + s P ( R sst d + e ) → A n + s − s G ( R sst d + e ) . Note that s − s precisely equals − χ ( d, e ) , the negative of the Euler form of d and e . Composingwith the equivariant exterior product map A ∗ G d ( R sst d ) ⊗ A ∗ G e ( R sst e ) → A ∗ L ( R sst d × R sst e ) , we obtain alinear map A sst d ⊗ A sst e → A sst d + e . The proof of [12, Thm. 1] also shows that we thus obtain an associative Γ θ,µ -graded algebra. Inanalogy to Kontsevich–Soibelman’s terminology, we define: Definition.
The algebra A θ − sst ,µ ( Q ) is called the θ -semi-stable Chow–Hall algebra (ChowHa) ofslope µ of Q .For the special case that θ is zero (i.e. R sst d = R d ) and µ = 0 , we write A instead of A − sst , andcall it the ChowHa of Q . We discuss the relation between the semi-stable ChowHa and Kontsevich–Soibelman’s semi-stableCoHa. Let k be the field of complex numbers. There is an equivariant analog (see [3, 2.8]) of thecycle map from [9, Chap. 19]. Concretely, there is a homomorphism of rings which doubles degrees A ∗ G d ( R θ − sst d ) → H ∗ G d ( R θ − sst d ) for every stability condition θ and every dimension vector d . Theorem 5.1.
The equivariant cycle map A ∗ G d ( R θ − sst d ) → H ∗ G d ( R θ − sst d ) is an isomorphism. In par-ticular, R θ − sst d has no odd-dimensional G d -equivariant cohomology. P G d -quotient R θ − (s)st d → M θd , and the G d -equivariant Chow/cohomology groups agree with the tensor product of the ordinaryChow/cohomology groups of the quotient with a polynomial ring Q [ z ] ∼ = A ∗ G m (pt) ∼ = H ∗ G m (pt) .We need a general lemma in order to prove Theorem 5.1. Let X be a complex algebraic schemeembedded into a non-singular variety X of complex dimension N ; for the rest of this section, a scheme will be a complex algebraic scheme (cf. [9, B.1.1]) which admits such an embedding. The Borel–Moorehomology H k ( X ) is isomorphic to the singular cohomology H N − k ( X, X − X ) . We consider the cyclemap cl : A k ( X ) → H k ( X ) . Lemma 5.2.
Let X be a scheme, Y a closed subscheme of X and U the open complement.1. If both Y and U have no odd-dimensional homology then X does not have odd-dimensionalhomology.2. If A k ( Y ) → H k ( Y ) and A k ( U ) → H k ( U ) are isomorphisms and H k +1 ( U ) = 0 then A k ( X ) → H k ( X ) is an isomorphism.3. Suppose that A k ( Y ) → H k ( Y ) and A k ( X ) → H k ( X ) are isomorphisms and H k − ( Y ) = 0 .Then A k ( U ) → H k ( U ) is also an isomorphism.Proof. The first assertion is clear by the long exact sequence in homology. To prove the secondstatement, we consider the diagram A k ( Y ) A k ( X ) A k ( U ) 0 . . . H k +1 ( U ) H k ( Y ) H k ( X ) H k ( U ) H k − ( Y ) . . . The left and right vertical maps being isomorphisms and the leftmost term in the lower row being zeroby assumption, the claim follows by applying the snake lemma. The third claim follows by diagramchase in the same diagram. We give the proof for completeness. Let u ∈ H k ( U ) . There exists x ∈ H k ( X ) with j ∗ x = u where j : U → X is the open embedding. We find a unique ξ ∈ A k ( X ) with cl X ξ = x , so u = j ∗ cl X ξ = cl U j ∗ ξ which proves the surjectivity of cl U . To show that cl U isinjective, let υ ∈ A k ( U ) with cl U υ = 0 . For an inverse image ξ ∈ A k ( X ) of υ under j ∗ , we obtain j ∗ cl X ξ = 0 , whence there exists y ∈ H k ( Y ) such that i ∗ y = cl X ξ . Here i : Y → X denotes theclosed immersion. Let η ∈ A k ( Y ) be the unique cycle with cl Y η = y . We get cl X i ∗ η = i ∗ y = cl X ξ and thus i ∗ η = ξ by injectivity of cl X . This implies υ = j ∗ i ∗ η = 0 .An immediate consequence of the above lemma is the following Lemma 5.3.
Suppose that a scheme X has a filtration X = X N ⊇ . . . ⊇ X ⊇ X = ∅ by closedsubschemes such that the cycle map for the successive complements S n = X n − X n − is an isomorphismfor all n . Then cl X is an isomorphism and, moreover, we have A ∗ ( X ) ∼ = N M i =0 A ∗ ( S i ) and A ∗ ( X ) ∼ = N M i =0 A ∗− codim X S i ( S i ) . G be a reductive linear algebraic group acting on ascheme X of complex dimension n . For an index i , we choose a representation V of G and an opensubset E ⊆ V such that a principal bundle quotient E/G exists and such that codim V ( V − E ) > n − i .Then, the group A Gk ( X ) = A i +dim V − dim G ( X × G E ) is independent of the choice of E and V . In the same vein, for an index j with V ( V − E ) > n − j , we can define equivariant Borel–Moore homology via ordinary Borel–Moore homology, namely H Gj ( X ) = H j +2 dim V − G ( X × G E ) (see [3]). If X is smooth then H Gj ( X ) is dual to H n − j ( X × G E ) which is isomorphic to H n − j ( X × G EG ) = H n − jG ( X ) (where EG is the classifying space for G ).We consider the equivariant cycle map cl : A Gk ( X ) → H G k ( X ) which is defined as the ordinary cyclemap cl : A i +dim V − dim G ( X × G E ) → H k +2 dim V − G ( X × G E ) (again independent of E ⊆ V ). Forcomplementary open/closed subschemes U and Y of X which are G -invariant, we choose E ⊆ V suchthat the principal bundle quotient E/G exists and codim V ( V − E ) > n − i (note that all the equivariantversions of the groups appearing in the diagram in the proof of Lemma 5.2 can be defined using E )and apply Lemma 5.2 to the complementary open/closed subschemes U × G E and Y × G E . We thusobtain Corollary 5.4.
In the above equivariant situation, Lemmas 5.2 and 5.3 hold for equivariant Chow/Borel–Moore homology groups.Proof of Theorem 5.1.
We prove Theorem 5.1 in two steps:1. Prove that the odd-dimensional equivariant cohomology of the θ -semi-stable locus vanishes byreducing the arbitrary case to a situation where stability and semi-stability agree. There, thestatement is known thanks to [14].2. Prove that the equivariant cycle map A G d k ( R θ − sst d ) → H G d k ( R θ − sst d ) is an isomorphism by induc-tion over the Harder–Narasimhan strata. Step 1:
First, assume that d is a θ -coprime dimension vector. This means that there is no sub-dimension vector = d ′ ≤ d with the same slope as d apart from d itself. In this case, θ -semi-stabilityand θ -stability on R d agree and there exists a smooth geometric P G d -quotient R θ − (s)st d → M θd . Here, P G d = G d / C × . In [14, Thm. 6.7] it is shown that the odd-dimensional cohomology of M θd vanishes.But as by the existence of a geometric quotient H ∗ G d ( R θ − sst d ) ∼ = H ∗ ( M θd ) ⊗ H ∗ C × (pt) , it follows that R θ − sst d has no odd-dimensional cohomology.Now, let d be arbitrary. We show that for a fixed — not necessarily positive — integer k , thereexist an acyclic quiver b Q , a stability condition b θ , a b θ -coprime dimension vector b d , and an integer s ≥ (all depending on k ) for which H G d k ( R θ − sst d ) ∼ = H P G b d k +2 s ( R b θ − (s)st b d ( b Q )) . (1)8 ChowHa vs. CoHaFor a dimension vector n of Q , we consider b R d,n = R d × F n where F n = L i Hom( C n i , C d i ) . Thespace b R d,n is the space of representations of the framed quiver b Q of dimension vector b d which ariseas follows (cf. [5, Def. 3.1]): we add an extra vertex ∞ to the vertexes of Q , i.e. b Q = Q ⊔ {∞} and,in addition to the arrows of Q , we have n i arrows from ∞ heading to i for all i ∈ Q . The dimensionvector b d is defined by b d i = d i for i ∈ Q and b d ∞ = 1 and is indivisible. The structure group G b d is C × × G d , whence we can identify P G b d with G d . We define b θ in the same way as in [5, Def. 3.1]. Thefollowing are equivalent for a framed representation ( M, f ) ∈ b R d,n (cf. [5, Prop. 3.3]): • ( M, f ) is b θ -semi-stable, • ( M, f ) is b θ -stable, • M is θ -semi-stable and the ( θ -)slope of every proper subrepresentation M ′ of M which containsthe image of f is strictly less than the slope of M .We denote the set of b θ -(semi-)stable points of b R d,n with b R θd,n . It is, by the above characterization, anopen subset of R θ − sst d × F n . Let b R xd,n denote the complement of b R θd,n inside R θ − sst d × F n . As R θ − sst d × F n is a G d -equivariant vector bundle over R θ − sst d , we obtain H G d k ( R θ − sst d ) ∼ = H G d k +2 d · n ( R θ − sst d × F n ) where d · n := P i d i n i = dim C F n . We thus obtain a long exact sequence . . . → H G d k +2 d · n ( b R xd,n ) → H G d k ( R θ − sst d ) → H G d k +2 d · n ( b R θd,n ) → H G d k − d · n ( b R xd,n ) → . . . in equivariant Borel–Moore homology. The equivariant BM homology groups H G d l ( b R xd,n ) vanish if l exceeds b R xd,n . So in order to show that (1) is an isomorphism, it suffices to find a framing datum n such that the (complex) dimension of b R xd,n is smaller than ( k − / d · n . As shown in the proofof [8, Thm. 3.2], b R xd,n is the union of Harder–Narasimhan strata b R xd,n = G b R HN( b p,q ) ,n over all proper sub-dimension vectors p of d which have the same slope (and q = d − p ). The set b R HN( b p,q ) ,n is defined as follows: let L ( M, f ) be minimal among those representations of the same slopeas M which contain im f . We set b R HN( b p,q ) ,n as the set of all ( M, f ) ∈ R θ − sst d × F n with dim L ( M, f ) = p .As b R HN( b p,q ) ,n ∼ = (cid:16)(cid:0) R sst p ∗ R sst q (cid:1) × (cid:0) F p (cid:1)(cid:17) × P p,q G d , the dimension of this stratum — if non-empty — equals X α : i → j ( d i d j − p i q j ) + X i p i n i − X i ( d i − p i q i ) + X i d i = dim( R d ) + d · n + χ ( p, q ) − q · n. Choosing n large enough such that q · n > dim( R d ) − k −
12 + χ ( d − q, q ) = q ≤ d of the same slope as d (which is possible as these are finitelymany non-zero dimension vectors q ), we find that the dimension of b R xd,n is smaller than ( k − / d · n ,as desired.Similar arguments were also used by Davison–Meinhardt in [2, Le. 4.1]. Step 2:
Let Q , θ and d be arbitrary. We consider the open/closed complementary subsets R sst d and R unst d . As R sst d is smooth (of dimension n = P α : i → j d i d j ), we have A G d i ( R sst d ) ∼ = A n − iG d ( R sst d ) and H G d j ( R sst d ) ∼ = H n − jG d ( R sst d ) . By Corollary 5.4 (concretely, the equivariant analog of part (3) of Lemma5.2), it suffices to show that cl : A G d ∗ ( R unst d ) → H G d ∗ ( R unst d ) is an isomorphism. If R unst d = ∅ , then the assertion is clear. So let us assume that R unst d is non-empty.The unstable locus admits a stratification into locally closed (irreducible) subsets R HN d ∗ , the Harder–Narasimhan strata, by [14, Prop. 3.4]. By [14, Prop. 3.7], they can be ordered in such a way that theunion of the first n strata is closed for all n , thus yielding a filtration by G d -invariant closed subsetslike in Lemma 5.3. Thus it suffices to prove that A G d ∗ ( R HN d ∗ ) → H G d ∗ ( R HN d ∗ ) is an isomorphism for all HN types d ∗ = ( d , . . . , d l ) of d ; this includes showing that all HN stratahave even cohomology. But by the proof of [14, Prop. 3.4], we have R HN d ∗ ∼ = Z d ∗ × P d ∗ G d , where Z d ∗ is a (trivial) vector bundle over R sst d × . . . × R sst d l , and P d ∗ is a parabolic subgroup of G d with Levi G d × . . . × G d l .In particular, R HN d ∗ is smooth, therefore we can again identify Borel–Moore homology with coho-mology. Moreover, A iG d ( R HN d ∗ ) ∼ = A iG d × ... × G dl ( R sst d × . . . × R sst d l ) , and similarly for equivariant coho-mology. This shows in particular that the equivariant odd-dimensional cohomology of R HN d ∗ vanishesby the first step of the proof. Now we argue by induction on the dimension vector d , where the setof dimension vectors is partially ordered by d ≤ e if d i ≤ e i for all i ; with respect to this order, all d ν ’s are strictly smaller than d . We thus assume that the equivariant cycle map for each R sst d ν is anisomorphism. Then, [19, Le. 6.2], which can be generalized to equivariant Chow groups, implies thatthe equivariant exterior product map A ∗ G d ( R sst d ) ⊗ . . . ⊗ A ∗ G dl ( R sst d l ) → A ∗ G d × ... × G dl ( R sst d × . . . × R sst d l ) is an isomorphism (even with integral coefficients). As the R sst d ν ’s have even cohomology, the Künnethmap is an isomorphism. We are thus reduced to proving the assertion for minimal dimension vectors d , i.e. d = 0 . But there the statement is obviously true. Remark . Theorem 5.1 is valid for integer coefficients.
Corollary 5.6.
The cycle map induces an isomorphism A sst ,µ → H sst ,µ of algebras.Proof. The multiplication both in the semi-stable ChowHa and in the semi-stable CoHa H sst ,µ areconstructed by means of the same Hecke correspondences. Moreover, the cycle map is compatiblewith push-forward and pull-back. 10 Tensor Product Decomposition We apply Corollary 5.4 to the Harder–Narasimhan filtration, like in the proof of Theorem 5.1. Weobtain that A G d ∗ ( R d ) ∼ = L d ∗ A G d ∗ ( R HN d ∗ ) , where the sum ranges over all Harder–Narasimhan typeswhich sum to d . Under the above isomorphism, the natural surjection A G d ∗ ( R d ) → A G d ∗ ( R sst d ) is theprojection to the summand A G d ∗ ( R HN( d ) ) = A G d ∗ ( R sst d ) and the push-forward of the closed embeddingof the unstable locus corresponds to the embedding of the direct summand L d ∗ =( d ) A G d ∗ ( R HN d ∗ ) . Inparticular, both maps are split epi-/monomorphisms. Using the cohomological grading of the Chowgroups, we get A ∗ G d ( R d ) ∼ = M d ∗ A ∗− codim Rd ( R HN d ∗ ) G d ( R HN d ∗ ) . The codimension of R HN d ∗ in R d can easily be computed as χ ( d ∗ ) := P r . . . > µ l such that d + . . . + d l = d ). The aboveconsiderations then prove Theorem 6.1.
The ChowHa-multiplication induces an isomorphism ← O µ ∈ Q A θ − sst ,µ ∼ = −→ A of Γ -graded vector spaces between the descending tensor product of the θ -semi-stable ChowHa’s overall possible slopes and the ChowHa.Remark . The theorem is valid with integral coefficients, for an arbitrary quiver, and does notrequire the stability condition to be generic. Theorem 6.1 has been proved with different methods byRimányi in [17] for the CoHa of a Dynkin quiver which is not an orientation of E .11 Structure of the CoHa of a Symmetric Quiver The CoHa/ChowHa of a quiver is described explicitly in [12]. Since we will make use of this descrip-tion, we recall it here: The equivariant Chow ring A ∗ G d ( R d ) ∼ = A ∗ G d (pt) is isomorphic to Q [ x i,r | i ∈ Q , ≤ r ≤ d i ] W d , where W d = Q i S d i is the Weyl group of a maximal torus of G d . We may regard the variables x i,r (located in degree ) as a basis for the character group of this torus or as the Chern roots of the G d -linear vector bundle R d × k d i → R d with G d acting on k d i by its i th factor. Theorem 7.1 ([12, Thm. 2]) . For f ∈ A d and g ∈ A e , the product f ∗ g equals the function X f ( x i,σ i ( r ) | i, ≤ r ≤ d i ) · g ( x i,σ i ( d i + s ) | i, ≤ s ≤ e i ) · Y i,j ∈ Q d i Y r =1 e j Y s =1 ( x j,σ j ( d j + s ) − x i,σ i ( r ) ) a i,j − δ i,j . The sum ranges over all ( d, e ) -shuffles σ = ( σ i | i ) ∈ W d + e , that means each σ i is a ( d i , e i ) -shufflepermutation. We assume that the stability condition θ is µ -generic. In this case, we can equip the semi-stableChowHa of slope µ with a refined grading: setting A sst( d,n ) = ( A ( n − χ ( d,d )) G d ( R sst d ) , n ≡ χ ( d, d ) (mod 2) , n χ ( d, d ) (mod 2),it is easy to see that the multiplication map becomes bigraded, thus A sst( d,n ) ⊗ A sst( e,m ) → A sst( d + e,n + m ) .Like in Section 3, we consider again the case of a symmetric quiver and the trivial stabilitycondition. In this situation, it is immediate from the formula in the above theorem that f ∗ g =( − χ ( d,e ) g ∗ f for f ∈ A d and g ∈ A e . One can show (cf. [12, Sect. 2.6]) that there exists a bilinearform ψ on the Z / Z -vector space ( Z / Z ) Q such that f ⋆ g = ( − ψ ( d,e ) f ∗ g is a super-commutativemultiplication, when defining the parity of an element of bidegree ( d, n ) to be the parity of n . We seethat the generating series P ( q, t ) = P d P k ( − k dim A ( d,k ) q k/ t d is X d ( − q / ) χ ( d,d ) Y i d i Y ν =1 (1 − q ν ) − t d . So, P ( q, t ) = A ( q − , t ) . By Theorem 3.2, the generating series has a product expansion P ( q, t ) = Y d Y k Y n ≥ (1 − q n + k/ t d ) ( − k − Ω d,k . As a free super-commutative algebra with a generator in bidegree ( d, k ) has the generating series (1 − q k/ t d ) ( − k − = Exp(( − k q k/ t d ) , Kontsevich–Soibelman made a conjecture in [12] which waseventually proved by Efimov. 12 Tautological Presentation of the Semi-Stable ChowHa Theorem 7.2 ([4, Thm. 1.1]) . For a symmetric quiver Q , the algebra A ( Q ) , equipped with the super-commutative multiplication ⋆ , is isomorphic to a free super-commutative algebra over a (Γ × Z ) -gradedvector space V = V prim ⊗ Q [ z ] , where z lives in bidegree (0 , , and L k V prim d,k is finite-dimensionalfor every d . This result implies that the DT invariants Ω d,k must agree with the dimension of V prim( d,k ) and musttherefore be non-negative. We will give another characterization of the primitive part of the CoHa inTheorem 9.2. We investigate the relation between the semi-stable ChowHa A θ − sst ,µ and the ChowHa A of a quiver Q . For a dimension vector d of slope µ , we consider the open embedding R sst d → R d which gives rise to a surjective map A ∗ G d ( R d ) → A ∗ G d ( R sst d ) . As the Hecke correspondences for the semi-stable ChowHa are given by restricting the Hecke correspondences of A to the semi-stable loci, theseopen pull-backs are compatible with the multiplication, i.e. they induce a surjective homomorphismof Γ -graded algebras A → A θ − sst ,µ . Here, we regard A θ − sst ,µ as a Γ -graded algebra by extending it trivially to every dimension vectorwhose slope is not s . We can describe the kernel explicitly: Theorem 8.1.
The kernel of the natural map A d → A θ − sst d equals the sum X A p ∗ A q over all pairs ( p, q ) of dimension vectors of Q which sum to d and such that µ ( p ) > µ ( q ) . The key ingredient of the proof of this result is a purely intersection-theoretic lemma. Following[9, B.1.1], we call a k -scheme algebraic if it is separated and of finite type over Spec k . Thus, a varietyis an algebraic scheme which is integral. Lemma 8.2.
Let f : X → Y be a surjective, proper morphism of algebraic k -schemes. Then thepush-forward f ∗ : A ∗ ( X ) Q → A ∗ ( Y ) Q is surjective.Proof. It is obviously sufficient to prove that, for every dominant morphism f : X → Y of an algebraicscheme X to a variety Y , there exists a subvariety W of X of dimension dim W = dim Y whichdominates Y . This is a local statement, so we may assume X and Y to be affine, say X = Spec B and Y = Spec A . The morphism f corresponds to an extension A ֒ → B of rings. We therefore needto show that there exists a prime ideal q of B with q ∩ A = (0) such that the induced extension Q ( B/ q ) | Q ( A )
13 Tautological Presentation of the Semi-Stable ChowHais finite. Let K = Q ( A ) and R = B ⊗ A K . By Noether Normalization, there exist b , . . . , b n ∈ R ,algebraically independent over K , such that K [ b , . . . , b n ] ⊆ R is a finite (and hence integral) ring-extension. Without loss of generality, we may assume b , . . . , b n ∈ B . Choose a set of generators c , . . . , c s of R as a K [ b , . . . , b n ] -algebra and polynomials p i ( T ) ∈ K [ b , . . . , b n ][ T ] such that p i ( c i ) = 0 .We find an element s ∈ A − { } such that the coefficients of all the p i ’s lie in A s [ b , . . . , b n ] and p i ( c i ) = 0 holds in B s . This implies that B s is an integral A s [ b , . . . , b n ] -algebra which yields thesurjectivity of the map Spec B s → Spec A s [ b , . . . , b n ] . We consider the prime ideal p ′ = ( b , . . . , b n ) of A s [ b , . . . , b n ] and find a prime ideal q ′ of B s which lies above it. Then B s / q ′ is an integral extensionof A s [ b , . . . , b n ] / p ′ = A s and therefore, setting q = q ′ ∩ B , the extension Q ( B/ q ) = Q ( B s / q ′ ) | Q ( A s ) = Q ( A ) is finite. Proof of Theorem 8.1 . Let R unst d be the complement of R sst d in R d . Then, we have an exact sequence A Gm ( R unst d ) → A Gm ( R d ) → A Gm ( R sst d ) → , where G = G d . For a decomposition d = p + q , let R p,q be the closed subset of R d of all representationswhich possess a subrepresentation of dimension vector p . It is the G -saturation of Z p,q . The G -actiongives a surjective, proper morphism Z p,q × P p,q G → R p,q , where P p,q is the parabolic (cid:0) G p ∗ G q (cid:1) . The unstable locus R unst d equals the union S R p,q over alldecompositions d = p + q where the slope of p is larger than the slope of q . Let us call thesedecompositions θ -forbidden. We obtain, using Lemma 8.2 and [9, Ex. 1.3.1 (c)], that the sequence M A Gm (cid:0) Z p,q × P p,q G (cid:1) → A Gm ( R d ) → A Gm ( R sst d ) → is exact when passing to rational coefficients — the direct sum being taken over all forbidden decom-positions d = p + q . Setting n = dim R d − m , we identify A Gm (cid:0) Z p,q × P p,q G (cid:1) ∼ = A n + χ ( p,q ) P p,q ( Z p,q ) ∼ = A n + χ ( p,q ) G p × G q ( R p × R q ) like in Section 4. As the equivariant product map A ∗ G p ( R p ) ⊗ A ∗ G q ( R q ) → A ∗ G p × G q ( R p × R q ) is anisomorphism (which is clear from the explicit description given above), we have shown that M p + q = d forbidden M k + l = n + χ ( p,q ) A kG p ( R p ) Q ⊗ A lG q ( R q ) Q → A nG d ( R d ) Q → A nG d ( R sst d ) Q → is an exact sequence. The first map in this sequence is precisely the ChowHa-multiplication. Thisproves the theorem. 14 The Primitive Part of the Semi-Stable ChowHa As a next step, we analyze the kernel of the pull-back A ∗ G ( R sst d ) → A ∗ G ( R st d ) induced by the openembedding of the stable locus into the semi-stable locus. A semi-stable representation M ∈ R d is notstable if and only if there exists a proper subrepresentation of the same slope. For a decomposition d = p + q into sub-dimension vectors of the same slope, we define R sst p,q as the subset of those M ∈ R sst d which admit a subrepresentation of dimension vector p . Therefore, the set of properly semi-stablerepresentations is the union R sst d − R st d = [ R sst p,q over all decompositions d = p + q such that p and q have the same slope and are both non-zero. Wehave, yet again, a surjective, proper morphism (cid:0) R sst p ∗ R sst q (cid:1) → R sst p,q . A result of Totaro (see [19, Le. 6.1]), which can easily be transferred to equivariant Chow rings, showsthat the exterior product A ∗ G p ( R sst p ) ⊗ A ∗ G q ( R sst q ) → A ∗ G p × G q ( R sst p × R sst q ) is an isomorphism. Followingthe arguments of the proof of Theorem 8.1, we obtain: Theorem 9.1.
The kernel of the surjection A θ − sst d → A θ − st d is the sum X A θ − sst p ∗ A θ − sst q over all decompositions d = p + q into non-zero sub-dimension vectors of the same θ -slope. In other words, the graded vector space A θ − st ,µ = L d ∈ Γ θ,µ A θ − st d equipped with the trivialmultiplication (by which we mean that the product of two homogeneous elements of positive degree isset to be zero) is isomorphic to the quotient A θ − sst ,µ / ( A θ − sst ,µ + ∗ A θ − sst ,µ + ) of the semi-stable ChowHamodulo the square of its augmentation ideal A θ − sst+ .Again, we consider the case of a symmetric quiver Q . We have deduced from Theorem 8.1 that A θ − sst ,µ is free super-commutative over V θ,µ = L d ∈ Γ θ,µ V d . The quotient of the augmentation idealof a free super-commutative algebra by its square is isomorphic to the primitive part of the algebra,i.e. in our case V d ∼ = A θ − st d = A ∗ G d ( R θ − st d ) ∼ = A ∗ P G d ( R θ − st d ) ⊗ A ∗ G m (pt) for every d = 0 . As V d = V prim d ⊗ Q [ z ] , we deduce that V prim d,k = ( A ( k − χ ( d,d )) P G d ( R st d ) , k ≡ χ ( d, d ) (mod 2) , k χ ( d, d ) (mod 2).Assuming that R θ − st d is non-empty and denoting by M θ − st d the geometric quotient R θ − st d /P G d (whichwe call the stable moduli space), we get A jP G d ( R θ − st d ) = A P G d dim R d − j ( R θ − st d ) = A dim R d − dim P G d − j ( M θ − st d ) and dim M θ − st d = dim R d − dim P G d = 1 − χ ( d, d ) . This yields that the Donaldson–Thomas invariantsof Q are given by the Chow–Betti numbers of the stable moduli spaces, more precisely:150 Examples Theorem 9.2.
For a symmetric quiver Q , a stability condition θ and a dimension vector d = 0 , theDonaldson–Thomas invariant Ω d,k equals Ω d,k = ( dim A − ( k + χ ( d,d )) ( M st d ) , if k ≡ χ ( d, d ) (mod 2) and M st d = ∅ and , otherwise.In particular, Ω d,k can only be non-zero if χ ( d, d ) ≤ k ≤ − χ ( d, d ) .Remark . The range for the non-vanishing of the DT invariants from the above theorem yieldsthat the number N d ( Q ) in [4, Cor. 4.1] can be chosen as − χ ( d, d ) , i.e. the dimension of M θ − st d .
10 Examples
We start by illustrating the tensor product decomposition Corollary 6.1. There are exactly threeconnected symmetric quivers which are not wild, that is, for which a classification of their finite-dimensional representations up to isomorphism is known. Namely, these are • the quiver L of Dynkin type A with a single vertex and no arrows, • the quiver L of extended Dynkin type e A consisting of a single vertex and a single loop, • and the quiver Q of extended Dynkin type e A with two vertices i and j and single arrows i → j resp. j → i .For the quivers L and L , the structure of the CoHa is determined in [12]. Namely, we have A ( L ) ∼ = S ∗ ( Q (1 , z ]) and A ( L ) ∼ = S ∗ ( Q (1 , z ]) , where S ∗ denotes the free super-commutative algebra, Q ( d, i ) denotes a one-dimensional Q -spaceplaced in bidegree ( d, i ) , and z denotes the element in bidegree (0 , whose existence is guaranteedby Theorem 7.2.The structure of the CoHa of Q is described in [8, Cor. 2.5]; here we give a simplified derivationof this result using the present methods. We consider the stability θ given by θ ( d i , d j ) = d i (notethat any non-trivial stability is equivalent to θ or − θ in the sense that the class of (semi-)stablerepresentation is the same). Let a representation M of Q of dimension vector d be given by vectorspaces V i and V j and linear maps f : V i → V j , g : V j → V i . We claim that this representation is θ -semi-stable if and only if V i = 0 , or V j = 0 , or dim V i = dim V j and f is an isomorphism; moreover,it is θ -stable if it is θ -semi-stable and dim V i , dim V j ≤ .The case (dim V i ) · (dim V j ) = 0 being trivial, we assume dim V i , dim V j ≥ . Suppose M is θ -semi-stable. If f is not injective, we choose a vector = v ∈ V i in the kernel of f , yielding a subrep-resentation U of dimension vector (1 , . Then we find µ ( U ) ≤ µ ( M ) = dim V i / (dim V i + dim V j ) ,thus dim V j = 0 , a contradiction. Thus f is injective, and ( V i , f ( V i )) defines a subrepresentation U ′ of dimension vector (dim V i , dim V i ) of M . Then we find / µ ( U ′ ) ≤ µ ( M ) , thus dim V j ≤ dim V i ,160.2 The Kronecker Quiverwhich already implies dim V i = dim V j and shows that f is an isomorphism. In this case, M is θ -semi-stable since the subrepresentations are of the form ( U, f ( U )) , and thus of the same slope as M ,for U ⊂ V i a gf -stable subspace. Since such subspaces always exist, we also see that stability forces dim V i = 1 = dim V j .This analysis provides identifications A ∗ G d ( R θ − sst d ) ∼ = A ∗ Gl n ( k ) (pt) for d = ( n, or d = (0 , n ) ,A ∗ G d ( R θ − sst d ) ∼ = A ∗ Gl n ( k ) ( M n × n ( k )) for d = ( n, n ) , which we recognize as the homogeneous parts of the CoHa of L and L , respectively. These identifi-cations obviously being compatible with the respective Hecke correspondences defining the multipli-cations, we see that A θ − sst , ( Q ) ∼ = A ( L ) ∼ = A θ − sst , ( Q ) and A θ − sst , / ( Q ) ∼ = A ( L ) . By Corollary 6.1, we thus arrive at A ( Q ) ∼ = S ∗ (cid:0) ( Q ((1 , , ⊕ Q ((0 , , ⊕ Q ((1 , , z ] (cid:1) . Now we consider the Kronecker quiver K with two vertices i and j and two arrows from i to j . Aswe will use results from Section 5, we work over the field of complex numbers. Again we considerthe stability θ ( d i , d j ) = d i . This is again a case where the representation theory of the quiver isknown: up to isomorphism, there exist unique ( θ -stable) indecomposable representations P n resp. I n for each of the dimension vectors ( n, n + 1) resp. ( n + 1 , n ) for n ≥ , and there exist one-parametricfamilies R n ( λ ) of ( θ -semi-stable) indecomposables for each of the dimension vectors ( n, n ) for n ≥ and λ ∈ P ( C ) . Arguing as in the first example, we can conclude that A θ − sst ,µ ( d ) ( K ) ∼ = S ∗ ( Q ( d, z ]) for d = ( n, n + 1) or d = ( n + 1 , n ) , and A θ − sst ,µ ( K ) = 0 if µ (cid:26) , , , , . . . , , . . . , , , (cid:27) . It remains to consider A θ − sst , / ( K ) .We construct a stratification of the θ -semi-stable locus in R ( n,n ) ( K ) ∼ = M n × n ( C ) × M n × n ( C ) , onwhich G = Gl n ( C ) × Gl n ( C ) acts via ( g, h ) · ( A, B ) = ( hAg − , hBg − ) . For ≤ r ≤ n , we define S r as the G -saturation of the set of pairs of matrices (cid:18)(cid:18) E r N (cid:19) , (cid:18) A E n − r (cid:19)(cid:19) , where E i denotes an i × i -identity matrix, A denotes an arbitrary r × r -matrix, and N denotes anilpotent ( n − r ) × ( n − r ) -matrix. We claim that every S r is locally closed, their union equals the θ -semi-stable locus, and the closure of S r equals the union of the S r ′ for r ′ ≤ r .170.2 The Kronecker QuiverThe representation R n ( λ ) is given explicitly by the matrices ( E n , λE n + J n ) for λ = ∞ , and by ( J n , E n ) for λ = ∞ , where J n is the nilpotent n × n -Jordan block. As noted above, a θ -semi-stablerepresentation of M of dimension vector ( n, n ) is of the form M = R n ( λ ) ⊕ . . . ⊕ R n k ( λ k ) for n = n + . . . + n k and λ , . . . , λ k ∈ P ( C ) , uniquely defined up to reordering. Now we reorder thedirect sum and assume that λ , . . . , λ j = ∞ and λ j +1 = . . . = λ k = ∞ . Using the above explicit formof the representations R n ( λ ) , we see that M is represented by a pair of block matrices of the form (cid:18)(cid:18) E r N (cid:19) , (cid:18) A E n − r (cid:19)(cid:19) with N nilpotent and A arbitrary. All claimed properties of the stratification follow.Now we claim that S r ∼ = (Gl n ( C ) × Gl n ( C )) × Gl r ( C ) × Gl n − r ( C ) ( M r ( C ) × N n − r ( C )) , where the group Gl r ( C ) × Gl n − r ( C ) is considered as a subgroup of Gl n ( C ) × Gl n ( C ) by mapping apair ( g , h ) to (cid:0)(cid:0) g h (cid:1) , (cid:0) g h (cid:1)(cid:1) . We consider the stabilizer of the set of matrices in the above blockform. So we take g, h ∈ Gl n ( C ) , written as block matrices g = (cid:18) g g g g (cid:19) , h = (cid:18) h h h h (cid:19) , and assume we are given matrices A, A ′ ∈ M r × r ( C ) and N, N ′ ∈ N n − r ( C ) , the nilpotent cone of ( n − r ) × ( n − r ) -matrices, such that (cid:18) h h h h (cid:19) (cid:18) E r N (cid:19) = (cid:18) E r N ′ (cid:19) (cid:18) g g g g (cid:19) , (cid:18) h h h h (cid:19) (cid:18) A E n − r (cid:19) = (cid:18) A ′ E n − r (cid:19) (cid:18) g g g g (cid:19) . From these equations we first conclude h = g and h = g , thus h = A ′ g and g = h N , whichyields h = A ′ h N . By induction, this implies h = ( A ′ ) k h N k for all k ≥ . But N is nilpotent,thus h = 0 , thus g = 0 . Similarly, we can conclude h = 0 and g = 0 . But then g and g areinvertible, and A ′ = g Ag − as well as N ′ = g N g − . This proves the claim.To obtain information on the Chow groups from this stratification using Lemma 5.3, we first haveto analyze the Chow groups of nilpotent cones.The nilpotent cone N d ( C ) is irreducible of dimension d − d , and the Gl d ( C ) -orbits O λ in N d areparametrized by partitions λ in P d , the set of partitions of d (we denote by P the union of all P d ’s).The stabilizer G λ of a point in O λ has dimension h λ, λ i = P i,j min( m i , m j ) m i m j , and its reductivepart is isomorphic to Q i Gl m i ( C ) , where m i = m i ( λ ) denotes the multiplicity of i as a part of λ , for i ≥ . We can thus apply Lemma 5.3 and reduce the structure group — note that in characteristiczero, an orbit is isomorphic to the quotient of the group by the stabilizer of a point — to get A ∗ Gl d ( C ) ( N d ( C )) ∼ = M λ ∈ P d A ∗ + d −h λ,λ i G λ (pt) , N d ( C ) is an isomorphism.This enables us to again apply Lemma 5.3, this time to the stratification ( S r ) r . We compute(using codim S r = n − r ): A ∗ Gl n ( C ) × Gl n ( C ) ( R θ − sst( n,n ) ( K )) ∼ = n M r =0 A ∗− n + r Gl n ( C ) × Gl n ( C ) ( S r ) ∼ = n M r =0 A ∗− n + r Gl r ( C ) × Gl n − r ( C ) ( M r ( C ) × N n − r ( C )) ∼ = n M r =0 A ∗ Gl r ( C ) ( M r ( C )) ⊗ A ∗− n + r Gl n − r ( C ) ( N n − r ( C )) ∼ = n M r =0 A ∗ Gl r ( C ) (pt) ⊗ M λ ∈ P n − r A ∗−h λ,λ i G λ (pt) Summing over all n , we obtain A θ − sst , / ∗ , ∗ ) ( K ) ∼ = M n ≥ A ∗ Gl n ( C ) × Gl n ( C ) ( R θ − sst( n,n ) ( K )) ∼ = (cid:18) M r ≥ A ∗ Gl r ( C ) (pt) (cid:19) ⊗ (cid:18) M λ ∈ P A ∗−h λ,λ i G λ (pt) (cid:19) . The generating function of the bigraded space A θ − sst , / ( K ) therefore equals (cid:18) X n ≥ t n (1 − q ) . . . (1 − q n ) (cid:19) · (cid:18) X λ q −h λ,λ i t | λ | Q i ≥ ((1 − q ) . . . (1 − q m i )) (cid:19) = Y i ≥ − q i t · Y i ≥ − q i t by standard identities. We thus arrive at an isomorphism of bigraded Q -spaces A θ − sst , / ( K ) ∼ = S ∗ (( Q ((1 , , ⊕ Q ((1 , , z ]) . However, this is not an isomorphism of algebras, since we will now exhibit an example showingthat the algebra A θ − sst , / ( K ) is not super-commutative.We use the algebraic description of the CoHa of Section 7 together with Theorem 8.1. We have A ( m,n ) ( K ) ∼ = Q [ x , . . . , x m , y , . . . , y n ] S m × S n with multiplication given as in Section 7. By Theorem 8.1, A θ − sst , / , ( K ) is the factor of A (1 , ( K ) ∼ = Q [ x , y ] by the image of the multiplication map A (1 , ( K ) ⊗ A (0 , ( K ) → A (1 , ( K ) , thus A θ − sst , / , ( K ) ∼ = Q [ x, y ] / ( x − y ) . Again by Theorem 8.1, A θ − sst , / , ( K ) is the factor of A (2 , ( K ) by the image of the multiplicationmap A (2 , ( K ) ⊗ A (0 , ( K ) ⊕ A (1 , ( K ) ⊗ A (1 , ( K ) → A (1 , ( K ) . − χ ((2 , , (0 , − χ ((1 , , (1 , . A directcalculation shows that for the elements , x, y ∈ A θ − sst , / , ( K ) , we have in A θ − sst , / , ( K ) : ∗ , ∗ x = y y ,x ∗ x x − ( y y , ∗ y = − ( x x
2) + 2( y y , and y ∗ x x . In particular, the (anti-)commutator of and x does not vanish.In light of the previous description of A θ − sst , / ( K ) , it can be expected that there exists a naturalfiltration on A θ − sst , / ( K ) such that the associated graded algebra is isomorphic, as a bigradedalgebra, to S ∗ (cid:0) ( Q ((1 , , ⊕ Q ((1 , , z ] (cid:1) . Next, we illustrate Theorem 9.2. We consider the symmetric quiver Q with two vertices i and j and n ≥ arrows from i to j and from j to i , and the dimension vector d = (1 , r ) for r ≤ n . To determinethe quantized Donaldson–Thomas invariant Ω d,k , we use the stability θ = ( r, − , for which d iscoprime. Therefore, Ω d,k = Ω θd,k equals the (suitably shifted) Poincaré polynomial of the cohomologyof the moduli space R θ − sst d ( Q ) /P G d , which is isomorphic to a vector bundle over the Grassmannian Gr r ( k n ) [16, Sect. 6.1]. By Theorem 9.2, we can also compute Ω d,k as the (suitably shifted) Poincarépolynomial of the Chow ring of the moduli space R − st d ( Q ) /P G d . Again by [16], this moduli space isisomorphic to the space X of n × n -matrices of rank r . Mapping such a matrix to its image definesa Gl n ( k ) -equivariant fibration X → Gr r ( k n ) , whose fibre is isomorphic to the space of r × n -matricesof highest rank. The latter being open in an affine space, its Chow ring reduces to Q , thus the Chowring of X is isomorphic to the Chow ring of Gr r ( k n ) as expected. Finally, we consider the quiver L m with a single vertex and m ≥ loops. The quantized Donaldson–Thomas invariants are computed explicitly in [15].All stability conditions are equivalent for this quiver. Let M simp d be the moduli space of simple(which is the same as stable) representations of L m of dimension d . It is obtained as the geometricquotient R simp d / PGl d . The Chow ring A ∗ ( M simp d ) Q = A ∗ PGl d ( R simp d ) Q (we will always work withrational coefficients in this subsection and therefore neglect it in the notation) is a quotient of theequivariant Chow ring A ∗ PGl d ( R d ) = A ∗ PGl d (pt) . The group of characters of a maximal torus of Gl d identifies with the free abelian group in letters x , . . . , x d , the natural action of the Weyl group W = S d being the permutation action. A maximal torus of PGl d is given by the quotient of thechosen maximal torus of Gl d by the diagonally embedded multiplicative group. The correspondingWeyl group is also S d and the character group is then the submodule X d = Sp ( d − , = { a x + . . . + a d x d | a + . . . + a d = 0 } . Sym( X d ) over X d is the subalgebra of Q [ x , . . . , x d ] generated by x j − x i (with i < j ) and the equivariant Chow ring A ∗ PGl d ( R d ) is therefore Sym( X d ) S d which identifies witha subalgebra of A ∗ G d (pt) = Q [ x , . . . , x d ] S d . As in the proofs of Theorems 8.1 and 9.1, the kernel of A ∗ PGl d ( R d ) → A ∗ PGl d ( R simp d ) is then given by the image of M p + q = dp,q> A ∗ PGl d ( Z p,q × P p,q PGl d ) → A ∗ PGl d ( R d ) , where P p,q is the obvious parabolic subgroup of PGl d — this, by the way, can be done for an arbitraryquiver and for the kernels A ∗ P G d ( R d ) → A ∗ P G d ( R θ − sst d ) and A ∗ P G d ( R θ − sst d ) → A ∗ P G d ( R θ − st d ) . The ring A ∗ PGl d ( Z p,q ) is isomorphic to Sym( X d ) S p × S q and the push-forward map m p,q : A ∗ PGl d ( Z p,q × P p,q PGl d ) → A ∗ PGl d ( R d ) can be described algebraically and looks just like the explicit formula from [12, Thm. 2], i.e. given bya shuffle product with kernel Q pi =1 Q qj =1 ( x p + j − x i ) m − . The relations in A ∗ PGl d ( R d ) which present A ∗ ( M simp d ) thus have at least degree ( m − d − . In other words, for every ≤ i < ( m − d − ,we get A i ( M simp d ) ∼ = A i PGl d ( R d ) = Sym i ( X d ) S d . The generating series of
Sym( X d ) S d is − q ) . . . (1 − q d ) = X i ≥ ♯ { ( k , . . . , k d ) | k + . . . + dk d } q i and using Theorem 9.2, we obtain a description of the first few Donaldson–Thomas invariants: Proposition 10.4.1.
For the m -loop quiver, the Donaldson–Thomas invariant Ω d,k for a non-negativeinteger d and an integer k of the same parity as (1 − m ) d satisfying (1 − m ) d ≤ k < (1 − m )( d − d + 2) computes as Ω d,k = ♯ (cid:8) ( k , . . . , k d ) | k + . . . + dk d = 12 (( m − d + k ) (cid:9) . We conclude the subsection with a computation of the numbers Ω ,k . The ring Sym( X ) S is thesubalgebra of Q [ x , x ] S which is generated by ( x − x ) . Abbreviate ∆ = x − x . As a Sym( X ) S -module, Sym( X ) is generated by and ∆ . The push-forward map m , : Sym( X ) → Sym( X ) S sends f ( x , x ) to (cid:0) f ( x , x ) + ( − m − f ( x , x ) (cid:1) ∆ m − and therefore, the image of m , is the ideal of Sym( X ) S = Q [∆ ] which is generated by ∆ ⌊ m/ ⌋ (i.e. ∆ m if m is even and ∆ m − if m is odd). We have shown that Ω ,k = ( , if k ≡ (mod ) and − m ) − ≤ k ≤ ⌊ m/ ⌋ − m ) , and , otherwise. 21eferences References [1] I. Assem, D. Simson, and A. Skowroński.
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