Derived Representation Schemes and Nakajima Quiver Varieties
DDERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES
STEFANO D’ALESIOA bstract . We introduce a derived representation scheme associated with a quiver, which may bethought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derivedrepresentation scheme as a Koszul complex and by doing so we show that it has vanishing higherhomology if and only if the moment map defining the corresponding Nakajima variety is flat. In thiscase we prove a comparison theorem relating isotypical components of the representation scheme toequivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of thisresult we obtain some integral formulas present in the mathematical and physical literature since a fewyears, such as the formula for Nekrasov partition function for the moduli space of framed instantonson S . On the technical side we extend the theory of relative derived representation schemes byintroducing derived partial character schemes associated with reductive subgroups of the generallinear group and constructing an equivariant version of the derived representation functor for algebraswith a rational action of an algebraic torus. C ontents
1. Introduction 12. Derived representation schemes of an algebra 63. The case of Nakajima quiver varieties 194. Main results 265. Examples 33Appendix A. Projective model structure on T-equivariant dg-algebras 38Appendix B. Representation theory of G = G v ntroduction Nakajima quiver varieties are certain Poisson varieties constructed from linear representationsof a quiver. They were firstly introduced by Nakajima ([28],[29]) as a geometric tool to studyrepresentations of Kac-Moody algebras. They are also interesting from a purely geometric pointof view, being a large class of examples of algebraic symplectic manifolds, many of which havebeen objects of study on their own (for example flag manifolds, framed moduli spaces of torsionfree sheaves on P , or a Lie algebra version of the character variety of a Riemann surface —see [3]). More recent studies have also supported the idea that symplectic resolutions, and inparticular hyperkÃd’hler reductions such as Nakajima quiver varieties, provide a bridge betweenenumerative geometry, representation theory and integrable systems ([1],[33],[35],[36],[37]).Quiver varieties are varieties of representations of a quiver: one fixes a vector space on eachvertex of the quiver and then consider the linear space of representations obtained by associatingto each arrow of the quiver a linear map. Kronheimer and Nakajima ([22]) have first introduced Mathematics Subject Classification.
Primary 14D21, 16G20; Secondary 16E05, 16E45, 19L47. a r X i v : . [ m a t h . K T ] J un STEFANO D’ALESIO a framed version, which amounts to doubling the set of vertices and drawing a new arrowfrom each new vertex to its corresponding old one. One of the reasons for considering framedrepresentations is that they appear naturally in the ADHM construction ([2]) of solutions ofself-dual or antiself-dual Yang-Mills equations on S . They are also interesting from the point ofview of representation theory of Lie algebras because dimension vectors of the framed verticesappear as highest weights of the representations ([30]). The framing is equivalent to a simpleroperation of adding just one vertex with dimension vector 1, together with as many arrows toeach vertex as the framing dimension (as pointed out in [9]), however in this paper we considerthe framed version of Nakajima quiver varieties.The framed quiver is then doubled, which means that each arrow gets doubled by an arrowthat goes in the opposite direction: the linear space of representations becomes now a linearcotangent bundle M ( Q , v , w ) := T ∗ L ( Q f r , v , w ) (where v , w are dimension vectors for, respectively,the original and framing vertices). The gauge group is a general linear group on the originalvertices G = G v and there is a moment map µ : M ( Q , v , w ) → g ∗ in the form of a generalised ADHM equation. Nakajima quiver varieties are defined as Hamiltonianreductions of this action G (cid:121) M ( Q , v , w ) : either affine Hamiltonian reductions, M ( Q , v , w ) = µ − ( ) (cid:12) G , or quasi-projective M χ ( Q , v , w ) = µ − ( ) (cid:12) χ G with the usual tools of geometricinvariant theory ([26]). For each choice of a (nontrivial) character χ : G → C × there is a properPoisson morphism(1.1) p : M χ ( Q , v , w ) → M ( Q , v , w ) ,which is often, but not always, a symplectic resolution of the singularities of M .1.1. Outline and results.
In this paper we link these varieties with some (derived) representationschemes. The idea of considering representation schemes is certainly not new, in fact it is motivatedby the very first algebraic origin of these varieties (see, for example, representation schemes ofpreprojective algebras in [10] and [13]). However the derived version of representation schemesintroduces some new invariants in a natural way.The theory of representation schemes is recalled in detail in § 2.1. To a (unital, associative)algebra A ∈ Alg k one associates Rep V ( A ) , the scheme of finite dimensional representations intoa fixed vector space V . There is a relative version in which the algebra A comes with a fixedstructure ι : S → A of algebra over another algebra S with a fixed representation ρ : S → End ( V ) and it is natural to define Rep V ( A ) as the scheme of only those finite dimensional representationswhich are compatible with ρ .General definitions and results on representation schemes work well over any field k ofcharacteristic zero, but it is necessary to specialise to k = C in order to relate them to (Nakajima)quiver varieties, which are algebraic varieties over the complex numbers. The (complex) linearspace of representations of a quiver Q is a representation scheme of the form Rep V ( A ) , where A = C Q is the path algebra of the quiver. This fact is a consequence of one of the basic results inthe theory of representations of quivers: There is an equivalence of categories between the category of C -linear representations of a quiver Q and thecategory of left C Q -modules. The construction can be easily adapted to include the framing and the doubling of the quiver,and also the operation of taking the fiber of zero through the moment map. In other words it ispossible to write the scheme µ − ( ) as a representation scheme for the path algebra of the framed, ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 3 doubled quiver, modulo the ideal I µ defined by the moment map: µ − ( ) = Rep C v ⊕ C w ( A ) , A = C Q f r / I µ ,where C v = ⊕ a C v a is the direct sum of the vector spaces placed on the original vertices of thequiver and C w = ⊕ a C w a is the one on the framing. We denote this representation scheme alsosimply by Rep v , w ( A ) . The gauge group by which we take the quotient is G = G v := (cid:81) a GL v a ( C ) ⊂ G v × G w . This group also arises naturally in the context of representation functors. It is possible toconstruct an invariant subfunctor by the group G and by doing so we obtain the affine Nakajimavariety as the partial character variety M ( Q , v , w ) = µ − ( ) (cid:12) G = Rep G v , w ( A ) .Now that we have such a model for this singular scheme we can try to resolve it using themachinery of model categories and in particular the theory of derived representation schemes ([4],[5]): we consider the derived schemeDRep v , w ( A ) ∼ = Rep v , w ( A c of ) ,where A c of ∼ (cid:16) A is a cofibrant replacement in the category of differential graded algebras. It is (thehomotopy class of) a differential graded scheme of the form X = ( X , O X , • ) , where X ∼ = M ( Q , v , w ) is the vector space of linear representations of the framed, doubled quiver, and O X , • is a sheaf ofdg-algebras whose zero homology gives: π ( X ) = Spec (cid:0) H ( O X , • ) (cid:1) = µ − ( ) .We exhibit an explicit (minimal) resolution A c of ∼ (cid:16) A for which this derived representation schemeis a well-known object when it comes to studying resolutions of a singular locus: Theorem (3.5.2 in § 3.5) . There is a cofibrant resolution A c of ∼ (cid:16) A ∈ DGA S which gives a model for thederived representation scheme as the (spectrum of the) Koszul complex on the moment map: (1.2) DRep v , w (cid:0) A (cid:1) ∼ = Rep v , w (cid:0) A c of ) = Spec (cid:0) O ( M ( Q , v , w )) ⊗ Λ • g (cid:1) .A somewhat natural question is whether or not there is any relationship between Nakajimaresolutions (1.1) and these derived schemes, and if it is possible to obtain informations about oneof the two from the other:(1.3) Quiver Q Affine Nakajimavariety M = µ − ( ) (cid:12) G Derived represen-tation schemesGIT Nakajimavarieties M χ geometricresolutionalgebraicresolution Relationship?A first answer is a close relationship (an equivalence) between the condition of flatness for themoment map (which assures that M χ → M is indeed a resolution, for well-behaved characters χ ), and the derived representation schemes to have vanishing higher homologies: Theorem (4.1.4 in § 4.1) . The derived representation scheme
DRep v , w (cid:0) A (cid:1) has vanishing higher homologiesif and only if µ − ( ) ⊂ M ( Q , v , w ) is a complete intersection, which happens if and only if the momentmap is flat. STEFANO D’ALESIO
We remark that in general it might not be easy to compute homologies of derived representationschemes, and even just to predict until which degree the homology is nontrivial. Nevertheless, inthis special situation it is possible to give a sufficient and necessary condition for the vanishing ofhigher homologies based on a geometric property (flatness) of the moment map. The importanceof Theorem 1.1 is that there is a combinatorial criterium on the dimension vectors v , w (provedby Crawley-Boevey, [9], based on the canonical decomposition of Kac, [20]) for the flatness of themoment map for representations of quivers.A second answer to the question in (1.3) comes when we compare some invariants associatedwith the derived representation schemes with others associated with the varieties M χ . A naturalchoice is to consider tautological sheaves on the GIT quotient M χ constructed with the usualmachinery developed by Kirwan (§ 4.2). Because of reductiveness of the gauge group G we restrictto consider only tautological sheaves of the form V λ induced from irreducible representations V λ of G . The push-forward of these sheaves in the K-theory of the affine Nakajima variety throughthe map (1.1) computes their ( T -)equivariant Euler characteristics:(1.4) χ T (M χ , V λ ) ∈ K T (M ) ,where T = T w × T (cid:32) h is the product of the standard maximal torus in the other general linear groupon the framing vertices T w ⊂ G w and a 2-dimensional torus T (cid:32) h rescaling the symplectic form andthe cotangent direction.On the other hand also the representation homology H • ( A , v , w ) (the homology of the derivedrepresentation scheme) is naturally a G -module and therefore decomposes into the direct sum ofits isotypical components:(1.5) H • ( A , v , w ) = (cid:77) λ Hom G (cid:0) V λ , H • ( A , v , w ) (cid:1) ⊗ V λ .The isotypical components Hom G ( V λ , H • ( A , v , w )) are modules over the G -invariant zerothhomology H ( A , v , w ) G = O ( µ − ( )) G and therefore their Euler characteristics define invariantsin(1.6) χ λT ( A , v , w ) = (cid:88) i (cid:62) (− ) i (cid:2) Hom G ( V λ , H i ( A , v , w )) (cid:3) ∈ K T (M ) .It is tempting to compare the invariants defined in (1.4) and (1.6), and the main results of thispaper go in this direction. First of all, when we consider the trivial representation V λ = C , weprove that if the moment map is flat, then the two invariants are indeed equal: Theorem (4.3.1 in § 4.3) . Let v , w be dimension vectors for which the moment map is flat and let χ suchthat M χ ( Q , v , w ) is a smooth variety (and therefore a resolution of M ( Q , v , w ) ). Then we have (1.7) p ∗ (cid:0) [ O M χ ( Q , v , w ) ] (cid:1) = [ O M ( Q , v , w ) ] = χ GT ( A , v , w ) ∈ K T (M ( Q , v , w )) .When we consider the Hilbert-PoincarÃl’ series of (1.7) we obtain an integral formula for the T -character of the ring of functions on the GIT quotient M χ , that has the following form(1.8) ch T ( O (M χ )) = ch T ( O (M )) = | W | (cid:90) T v (cid:81) i ( − (cid:32) h (cid:32) h r i ) (cid:81) j ( − s j ) ∆ ( x ) dx ,where r i = r i ( x ) and s j = s j ( x , t ) are characters for T v and T v × T , respectively, ∆ ( x ) is the Weylfactor for G v and the integration is over the compact real form of T v (see § 4.3 for a more detailedexplanation).Integral formulas of similar flavours have already appeared under different names, both in themathematical literature (Jeffrey-Kirwan integral/residue formula for GIT quotients — [19]) and in ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 5 the physical literature (integral formula for Nekrasov partition function — [31],[32] — proven, forexample, in Appendix A in [14]). We could say that this is not a coincidence, in fact recognising theright-hand side of (1.8) in the known example of the Jordan quiver (Nekrasov partition function)as the Euler characteristic of the representation homology was one of the motivations of thisproject.For what concerns other tautological sheaves V λ an equality of the same flavour of (1.7) is trueonly for large enough λ , where the definition of largeness depends on the quiver, the dimensionvectors v , w and, perhaps more importantly, also on the GIT parameter χ (see § 4.4): Theorem (4.4.1 in § 4.4) . Let v , w be dimension vectors for which the moment map is flat, and χ acharacter for which M χ ( Q , v , w ) is smooth. For λ large enough (Definition 4.4.1) we have (1.9) p ∗ ([ V λ ]) = [ H (M χ ( Q , v , w ) , V λ )] = χ λ ∗ T ( A , v , w ) ∈ K T (M ( Q , v , w )) .Once again by taking the Hilbert-PoincarÃl’ series of (1.9) we obtain a second integral formulafor tautological sheaves on the GIT quotient:(1.10) ch T ( χ T (M χ , V λ )) = ch T ( H (M χ , V λ )) = | W | (cid:90) T v (cid:81) i ( − (cid:32) h (cid:32) h r i ) (cid:81) j ( − s j ) f λ ( x ) ∆ ( x ) dx ,where f λ ( x ) = ch T v ( V λ ) is a product of Schur polynomials.1.2. Layout of the paper.
In § 2 we introduce the general theory of (derived) representationschemes of an algebra. First we recall the theory of representation schemes with some examples,in particular the linear space of representations of a quiver as a representation scheme for itspath algebra. Then we recall the derived version introduced by [5] and [4]. We introduce a moregeneral way to take invariant subfunctors and an equivariant version of derived representationschemes for an action of an algebraic torus which is useful for our purposes. We decompose therepresentation homology in isotypical components and define new invariants in the K-theory ofthe classical character scheme.In § 3 we recall the construction of Nakajima quiver varieties and we show how to view theaffine Nakajima variety M as a partial character scheme (a quotient of a representation scheme)for the algebra A := C Q f r / I µ . We construct the derived scheme associated to it and we use theinvariants defined in § 2 to decompose the representation homology into classes in the K-theory of M . In § 3.4 we construct an explicit cofibrant resolution A c of ∼ (cid:16) A that gives us a concrete modelfor the derived representation scheme as the (spectrum of the) Koszul complex on the momentmap. Therefore we recall some classical properties of the Koszul complex and commutativecomplete intersections.In § 4 we explain the main results of this paper. First we observe that, using the model foundin § 3.4, the derived representation scheme has vanishing higher homologies if and only if themoment map is flat, which is a combinatorical condition on the dimension vectors of the quiver([9]). We recall the definition of tautological sheaves on GIT quotients by the Kirwan map andprove results that compare them with the isotypical components of the representation homology((1.7) and (1.9)). In particular we obtain some interesting integral formulas ((1.8) and (1.10)).In § 5 we show some concrete examples, such as the quiver A for which Nakajima varietiesare cotangent spaces of Grassmannians, the Jordan quiver for which we obtain framed modulispace of torsion free sheaves on P , and the quiver A n − with some special dimension vectors forwhich we obtain the symplectic dual ( T ∗ P n − ) ˇ, and compute some of the integral formulas thatwe have proved before. STEFANO D’ALESIO
In Appendix A we construct a model structure on equivariant dg-algebras that we need in § 2.5,and in Appendix B we recall the theory of irreducible representations for a product of generallinear groups as multipartitions, and set some notation that we need in § 4.4.
Notation.
Throughout the paper we denote categories by the standard monospace font:
Sets , Grp , Vect k , Alg k , . . . The notation used is often both standard and self-explanatory, and when this isnot the case we usually recall it in the main body of the paper. Acknowledgements.
I want to express my gratitude to my advisor Giovanni Felder, who intro-duced me to this subject a couple of years ago and proved numerous times to be a patient, wiseand resourceful guide. I also want to mention other people with whom we shared our ideasand contributed with useful comments, in particular Yuri Berest during his brief stay in Zurich,Gabriele Rembado, Matteo Felder and Xiaomeng Xu.2. D erived representation schemes of an algebra
The family of schemes of finite dimensional representations { Rep n ( A ) } n (cid:62) of an algebra A hasbeen object of study for many years (see for example the early work of Procesi, [38]). With thedevelopment of noncommutative geometry, they have been seen in a new light when Kontsevichand Rosenberg ([21]) proposed the following principle:“ Any noncommutative structure of some kind on A should give an analogous commutative structure on allthe representation schemes
Rep n ( A ) , n (cid:62) Classical representation schemes.
Let k be an algebraically closed field of characteristiczero (later we fix k = C ). Let A ∈ Alg k be a unital, associative algebra and V ∈ Vect k a finitedimensional vector space. We consider the functor on unital commutative algebras:(2.1) Rep V ( A ) : CommAlg k (= Aff op k ) → Sets B (cid:55)−→ Hom
Alg k (cid:0) A , End ( V ) ⊗ k B (cid:1) .This functor is (co)-representable, by the commutative algebra A V := (cid:0) V √ A (cid:1) (cid:92)(cid:92) . The two functors V √ − and (−) (cid:92)(cid:92) are, respectively, the matrix reduction functor and the abelianisation functor, whichare left adjoints to the followings:(2.2) Alg k V √ − (cid:47) (cid:47) ⊥ Alg k End ( V ) ⊗ k (−) (cid:111) (cid:111) , Alg k (−) (cid:92)(cid:92) (cid:47) (cid:47) ⊥ CommAlg kU (cid:111) (cid:111) .Explicit formulas for them are V √ A = (cid:0) End ( V ) ∗ k A (cid:1) End ( V ) and ( C ) (cid:92)(cid:92) = C/ (cid:104) [ C , C ] (cid:105) , where (cid:104) [ C , C ] (cid:105) is the 2-sided ideal generated by the commutators. By combining the two adjunctions in (2.2) weget an adjunction for the representation functor:(2.3) Alg k (−) V (cid:47) (cid:47) ⊥ CommAlg k End ( V ) ⊗ k (−) (cid:111) (cid:111) , ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 7 so that the commutative algebra A V is uniquely defined by the natural isomorphisms:(2.4) Hom CommAlg k ( A V , B ) ∼ = Hom
Alg k ( A , End ( V ) ⊗ k B ) , ∀ B ∈ CommAlg k . Definition 2.1.1.
The affine scheme associated to A V ∈ CommAlg k is the representation scheme Rep V ( A ) = Spec ( A V ) ∈ Aff k (strictly speaking we identify it with its functor of points as weoriginally defined it Rep V ( A ) ∈ O ( Aff op k , Sets ) in (2.1)). We recover A V = O ( Rep V ( A )) as thealgebra of functions on the representation scheme.We can assume that V = k n and write simply Rep n ( A ) = Spec ( A n ) instead of Rep V ( A ) = Spec ( A V ) . Let us show some examples: Examples 1. (0) If A ∈ CommAlg k ⊂ Alg k is a commutative algebra then clearly from (2.4): A = A ↔ Rep ( A ) = Spec ( A ) .(1) The free algebra in m generators A = F m = k (cid:104) x , . . . , x m (cid:105) has no relations and thereforeRep n ( F m ) is the scheme of m -tuples of n × n matrices:Rep n ( F m ) = M n × n ( k ) m .(2) The polynomial algebra A = k [ x , . . . , x m ] can be expressed as the free algebra in m genera-tors modulo the ideal generated by all commutators [ x i , x j ] and therefore its representationscheme is the closed subscheme of m -tuples of n × n matrices that pairwise commute:Rep n ( A ) = C ( m , n ) := (cid:8) ( X , . . . , X m ) ∈ M n × n ( k ) m | [ X i , X j ] = ∀ i , j (cid:9) ⊂ Rep n ( F m ) .(3) The algebra of dual numbers A = k [ x ] / ( x ) gives the scheme of square-zero matrices:Rep n ( A ) = (cid:8) X ∈ M n × n ( k ) | X = (cid:9) .(4) The algebra of differential operators on the affine line A = Diff ( A k ) = k (cid:104) x , d (cid:105) / ([ d , x ] = ) has no representation because if X , D ∈ M n × n ( k ) are matrices satisfying [ D , X ] = n , thentaking traces we would get 0 = n , which is absurd:Rep n (cid:0) Diff ( A k ) (cid:1) = ∅ .(5) The algebra of Laurent polynomials in m variables A = k [ t ± , . . . , t ± m ] is similar to theexample of commuting matrices, except that now the matrices are required to be invertible:Rep n ( A ) = (cid:8) ( X , . . . , X m ) ∈ GL n ( k ) m | [ X i , X j ] = ∀ i , j (cid:9) .(6) More generally writing any finitely generated algebra as a free algebra modulo somerelations A = F m / (cid:104) r , . . . , r s (cid:105) , r , . . . , r s ∈ F m = k (cid:104) x , . . . , x m (cid:105) ,then its representation scheme is identified with the closed subschemeRep n ( A ) = (cid:8) ( X , . . . , X m ) ∈ M n × n ( k ) m | r i ( X , . . . , X m ) = ∀ i (cid:9) ⊂ Rep n ( F m ) of m -tuples of n × n matrices defined by the equations r , . . . , r s .Another fundamental example is that of path algebras of (finite) quivers. These algebras comewith an additional structure of algebras over the finite dimensional algebras of their empty paths,which is crucial when considering their representations, therefore we need to consider a relative version of representation schemes. Formally we fix an algebra S ∈ Alg k and we consider theunder category S ↓ Alg k (also denoted by Alg S following the notation of [4] and [5]) which is thecategory of algebras A ∈ Alg k together with a fixed morphism S → A . We also fix a representation ρ : S → End ( V ) . STEFANO D’ALESIO
Notation.
Sometimes when we want to remark that A comes with a map from S we denote thisobject as S \ A ∈ Alg S . However, when there is no risk of confusion, we just use A .With these ingredients it is natural to consider only those representations A → End ( V ) thatagree with ρ on S . In terms of functor of points this corresponds to(2.5) Rep V ( A ) : CommAlg k → Sets B (cid:55)−→ Hom
Alg S (cid:0) A , End ( V ) ⊗ k B (cid:1) .This functor is also (co)representable, by the commutative algebra A V defined as before except for ∗ k substituted by ∗ S , the coproduct in Alg S . Letting A vary we obtain a relative version of therepresentation functor (−) V , and a similar adjunction(2.6) Alg S (−) V (cid:47) (cid:47) ⊥ CommAlg k End ( V ) ⊗ k (−) (cid:111) (cid:111) . Example 2 (Path algebra of a quiver) . Let Q be a finite quiver and A = C Q ∈ Alg C its path algebraover the complex numbers. What follows works well for any field k of characteristic zero butlater we are interested only in k = C . We recall that the path algebra is the free vector space onthe admissible paths in the quiver, with product given by concatenation of paths. It has a set oforthogonal idempotents { e i } i ∈ Q ⊂ A : e i e j = δ ij e j ,which are the empty paths on the vertices, and their sum is the unit of the algebra: (cid:80) i ∈ Q e i = ∈ A .We can then consider the subalgebra generated by these idempotents S = (cid:104) e i (cid:105) i ∈ Q = Span C { e i } i ∈ Q ,with the natural inclusion ι : S → A . We now fix a dimension vector v ∈ N Q and we consider thelinear space of representations of the quiver Q with the complex vector space C v i placed at thevertex i ∈ Q :(2.7) L ( Q , v ) := (cid:77) γ ∈ Q Hom C (cid:0) C v s ( γ ) , C v t ( γ ) (cid:1) .where s , t : Q → Q are the source and target maps of the quiver. From the algebraic point ofview we fix the following representation of S in the vector space C v := ⊕ i C v i :(2.8) ρ = ρ v : S → ⊕ i End C ( C v i ) ⊂ End C ( C v ) e i (cid:55)−→ E i := ⊕ · · · ⊕ C vi (cid:124)(cid:123)(cid:122)(cid:125) i-th factor ⊕ · · · ⊕ Proposition 2.1.1.
The linear space of representations of the quiver Q with fixed dimension vector v isisomorphic to the (relative) representation scheme of its path algebra: (2.9) L ( Q , v ) ∼ = Rep C v ( C Q ) . Proof.
Let us consider the complex vector space with basis given by the set of arrows of the quiver M := Span C { x γ } γ ∈ Q . It has the structure of an S -bimodule, and its tensor algebra is the pathalgebra of the quiver: A = C Q = T S M := S ⊕ M ⊕ ( M ⊗ S M ) ⊕ . . . .For a dimension vector v ∈ N Q we consider the graded vector space C v = ⊕ i C v i , whoseendomorphism algebra End C ( C v ) is an S -bimodule via the map (2.8). By the universal property of ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 9 the tensor algebra, giving a representation T S M → End C ( C v ) that agrees with ρ on S , is equivalentto give a S -bimodule map M → End C ( C v ) :Hom Alg S ( A , End C ( C v )) ∼ = Hom S − Bimod ( M , End C ( C v ) ∼ = (cid:77) γ ∈ Q Hom C (cid:0) C v s ( γ ) , C v t ( γ ) (cid:1) = L ( Q , v ) . (cid:3) Derived representation schemes.
As already anticipated in the introduction of this Section,the noncommutative geometry principle of transferring a geometric property on an algebra A (e.g.noncommutative complete intersection, Cohen-Macaulay, etc.) on the corresponding commutativeone on Rep V ( A ) might fail when A is not a (formally) smooth algebra. This seems to be related tothe fact that the functor Rep V (−) is not exact.We discuss the following derived version of representation schemes firstly introduced in [5].The idea is to “resolve” the singularities of the representation schemes by using the tools ofhomological algebra, in the sense of Quillen’s derived functors on model categories.We enlarge the category of algebras to the one of differential graded algebras DGA k (in ourconventions differentials have always degree − DGA S := S ↓ DGA k of dg-algebras A with a fixed morphism S → A .We also fix a differential graded vector space V ∈ DGVect k of finite total dimension, and denoteby End ( V ) ∈ DGA k the differential graded algebra of endomorphisms, with differential(2.10) df = d V ◦ f − (− ) i f ◦ d V , f ∈ End ( V ) i .Moreover we need to fix a representation of S in V , that is a dga morphism ρ : S → End ( V ) ,which makes End ( V ) an object of DGA S . With these ingredients we can define a differential gradedversion of the representation functor for A ∈ DGA S as the functor from commutative dg-algebras:(2.11) Rep V ( A ) : CDGA k → Sets B (cid:55)−→ Hom
DGA S ( A , End ( V ) ⊗ k B ) . Remark 2.2.1.
We use the same notation as in the non-graded case because in the particular caseof S , A , V being concentrated in degree zero we recover the same functor as before (when restrictedto Alg k ⊂ DGA k ).This functor is also (co)-representable, by the object A V := (cid:0) V √ A (cid:1) (cid:92)(cid:92) constructed in the same wayas before, with(2.12) V √ A = ( End ( V ) ∗ S A ) End ( V ) ,where ∗ S is the free product over S , the categorical coproduct in DGA S . As before we obtain a pairof adjoint functors(2.13) DGA S (−) V (cid:47) (cid:47) ⊥ CDGA k End ( V ) ⊗ k (−) (cid:111) (cid:111) .These categories have model structures for which this adjunction is a Quillen adjunction, andtherefore produces a total right-derived functor R (cid:0) End ( V ) ⊗ k (−) (cid:1) , but more importantly a left-derived functor L (−) V that we use to define the derived representation scheme.We consider on DGA k and CDGA k the so-called projective model structures for which weakequivalences are quasi-isomorphisms of complexes and fibrations are degree-wise surjective maps(Theorem 4 in [4]). It is useful for later purposes to consider also the categories DGA + k and CDGA + k ,which are the categories of non-negatively graded differential graded and commutative differentialgraded algebras, respectively, and with their projective model structures with the only difference that now fibrations are degree-wise surjective maps in all (strictly) positive degrees. All thesecategories are fibrant (every object is fibrant), with initial object k and final object 0.The category DGA S is an example of an under category (category in which objects are objectsof the original category coming with a fixed morphism from the object S in this case). As suchit comes with a forgetful functor DGA S → DGA k and the model structure on DGA S is the one inwhich weak-equivalences, fibrations and cofibrations are exactly the maps which are sent toweak-equivalences, fibrations and cofibrations via the forgetful functor. Clearly also the undercategory DGA S is fibrant, with final object still 0 (viewed as an object of DGA S via the unique map S → S (viewed as an object of DGA S via the identity map Id S : S → S ).For a model category C , we denote by Ho ( C ) its homotopy category and by γ : C → Ho ( C ) thecanonical functor. Theorem 2.2.1 (Theorem 7 in [4]) . (i) The pair of functors in (2.13) form a Quillen pair.(ii) The representation functor (−) V has a total left derived functor given by (2.14) L (−) V : Ho ( DGA S ) → Ho ( CDGA k ) (cid:14) A (cid:55)−→ (cid:0) A c of (cid:1) V γf (cid:55)−→ γ ( ˜ f ) V where A c of ∼ (cid:16) A is a cofibrant replacement in DGA S , and for a morphism f : A → B , the morphism ˜ f : A c of → B c of is a lifting of f between the cofibrant replacements.(iii) For any A ∈ DGA S and any B ∈ CDGA k there is a canonical isomorphism: (2.15) Hom Ho ( CDGA k ) ( L ( A ) V , B ) ∼ = Hom Ho ( DGA S ) ( A , End ( V ) ⊗ k B ) . Definition 2.2.1.
For S ∈ Alg k concentrated in degree 0, the following composite functor(2.16) Alg S → Ho ( DGA S ) L (−) V −−−−→ Ho ( CDGA k ) A L ( A ) V is called derived representation functor . The homology of the (homotopy class of the) commutativedifferential graded algebra L ( A ) V ∈ Ho ( CDGA k ) depends only on A ∈ Alg S and V . It is called the representation homology of A with coefficients in V :(2.17) H • ( A , V ) := H • ( L ( A ) V ) . Remark 2.2.2.
By its definition, the zero-th homology recovers the classical representation scheme(see Theorem 9 in [4]):(2.18) H (cid:0) A , V ) ∼ = A V = O (cid:0) Rep V ( A ) (cid:1) .As we anticipated before, we are interested in a slightly different version of this story: if westart from a vector space V concentrated in degree 0 and S ∈ Alg k then the previous pair (2.13)restricts to a pair of functors(2.19) DGA + S (−) V (cid:47) (cid:47) ⊥ CDGA + k End ( V ) ⊗ k (−) (cid:111) (cid:111) ,which is still a Quillen pair, and the analogous result of Theorem 2.2.1 holds. We give a seconddefinition of: Definition 2.2.2.
The derived representation functor is the following functor:(2.20)
Alg S → Ho ( DGA + S ) L (−) V −−−−→ Ho ( CDGA + k ) .The representation homology of the relative algebra A ∈ Alg S is the homology of L ( A ) V ∈ Ho ( CDGA + k ) . ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 11
Remark 2.2.3.
Definition 2.2.1 and (2.2.2) are not really different. In fact, there is an adjunctionbetween the categories
DGA + S and DGA S (2.21) DGA + S ι (cid:47) (cid:47) ⊥ DGA Sτ (cid:111) (cid:111) ,where the functor ι is the obvious inclusion and the functor τ is the one that sends an unboundeddifferential graded algebra A ∈ DGA S to its truncation: τ ( A ) = [ · · · → A d −→ A d −→ ker ( d )] ∈ DGA + S .It is straightforward to see that τ preserves fibrations and weak equivalences, and dually themap ι preserves cofibrations and weak equivalences, in particular it sends cofibrant objects tocofibrant objects. Now let A ∈ Alg S and choose a cofibrant replacement Q ∼ (cid:16) A ∈ DGA + S . A priorithis map is only surjective in positive degrees, but because A is concentrated in degree 0, we have A = H ( A ) , and the isomorphism in homology H ( Q ) ∼ = H ( A ) proves that it is surjective also indegree 0, so still a fibration in DGA S . In other words the cofibrant replacement Q ∼ (cid:16) A is still acofibrant replacement in DGA S and therefore it can be used to compute the derived representationfunctor (2.16), showing that Definition 2.2.1 is equivalent to Definition 2.2.2. Remark 2.2.4 ( The dual language of dg-schemes ) . Another reason for considering the category
CDGA + k instead of CDGA k is that it is anti-equivalent to the category of differential graded schemes, asintroduced by I. Ciocan-Fontanine and M. Kapranov in [8]. We recall their definition of dg-schemes(over k ) as a pair X = ( X , O X , • ) , where X is an ordinary scheme over k and O X , • is a sheaf ofnon-negatively graded commutative dg-algebras on X such that the degree zero is O X ,0 = O X thestructure sheaf of the classical scheme X and each O X , i is quasicoherent over O X ,0 . A morphismof dg-schemes over k is just a morphism of dg-ringed spaces f : X = ( X , O X , • ) → Y = ( Y , O Y , • ) ,and this makes DGSch k into a category. A dg-scheme X is called affine if the underlying classicalscheme X is affine. The full subcategory of dg-affine schemes DGAff k ⊂ DGSch k is antiequivalentto the category CDGA + k , via the the equivalence of categories:(2.22) DGAff op k Γ (−) (cid:47) (cid:47) ⊥ CDGA + k Spec (cid:111) (cid:111) ,where Γ (−) is the functor taking a dg-affine X into the global sections of the sheaf O X , • (degreewise),and Spec is the dg-spectrum sending a commutative dg-algebra A to the classical scheme X = Spec ( A ) together with the quasicoherent sheaves O X , i associated to the modules A i via thecorrespondence QCoh X ∼ = Mod A . These names are motivated by the fact that the previousequivalence restricts to the classical equivalence of categories(2.23) Aff op k Γ (−) (cid:47) (cid:47) ⊥ CommAlg k Spec (cid:111) (cid:111) .This definition of dg-affine schemes coincides with Toà ´nn-Vezzosi’s definition of derived schemes d Aff op k = sCommAlg k as simplicial commutative algebras ([41]) because over a field k of character-istic zero they are equivalent to commutative dg-algebras.The equivalence of categories (2.22) can be trivially used to transfer the projective modelstructure on commutative dg-algebras to the category of dg-affine schemes. Obviously the pair ( Γ (−) , Spec ) becomes a Quillen equivalence, i.e. an equivalence on the homotopy categories:(2.24) Ho ( DGAff op k ) L Γ (−) (cid:47) (cid:47) ⊥ Ho ( CDGA + k ) R Spec (cid:111) (cid:111) .Moreover because every object in
CDGA + k is fibrant, the derived spectrum R Spec actually coincideswith the underived Spec on the objects.
Definition 2.2.3.
The derived representation scheme of the relative algebra A ∈ Alg S in a vector space V is the object DRep V ( A ) ∈ Ho ( DGAff k ) obtained applying to A the following composition offunctors:(2.25) DRep V (−) : Alg S → Ho ( DGA + S ) L (−) V −−−−→ Ho ( CDGA + k ) R Spec −−−−→ Ho ( DGAff k ) .This definition differs from the one given in [5] and [4] only from the last composition with thederived spectrum functor. The reason we have to do so is to be consistent with the notation forthe classical representation scheme Rep V ( A ) ∈ Aff k . Remark 2.2.5.
Because every object in
CDGA + k is fibrant, the derived representation schemeDRep V ( A ) is simply(2.26) DRep V ( A ) = R Spec ( L ( A ) V ) = Spec ( L ( A ) V ) = Spec (( A c of ) V ) = Rep V ( A c of ) ,where A c of ∼ (cid:16) A ∈ DGA + S is a cofibrant replacement. Different choices of cofibrant replacementsgive different models to DRep V ( A ) , which are weakly equivalent to each other. In what followswe choose one specific model for DRep V ( A ) obtained through a choice of a preferred cofibrantreplacement. Strictly speaking in (2.26) we should write DRep V ( A ) = γ Rep V ( A c of ) ∈ Ho ( DGAff k ) to remember that we are considering the homotopy class, but we make an abuse of notation bydropping γ . Examples 3.
In the following examples we describe explicit cofibrant resolutions for some ofthe algebras in the Examples 1 and give a model for their derived representation schemeswith value in a vector space V concentrated in degree 0 (therefore we still use the notationDRep n (−) = DRep V (−) for V = k n ).(1) The free algebra in m generators A = F m is already a cofibrant object in DGA + k because it isfree, therefore DRep n ( F m ) ∼ = Rep n ( F m ) ∼ = M n × n ( k ) m .(2) The commutative algebra in two variables A = k [ x , y ] is not cofibrant because of therelation [ x , y ] =
0. It turns out that it suffices to add one variable ϑ in homological degree1 that kills this relation ( dϑ = [ x , y ] ) to obtain a cofibrant replacement: A c of := k (cid:104) x , y , ϑ (cid:105) ∼ (cid:16) A = k [ x , y ] ,and therefore the derived representation scheme is the nothing else but the (spectrum ofthe) Koszul complex for the scheme of n × n commuting matrices:DRep n ( A ) ∼ = Rep n ( A c of ) = Spec (cid:0) k [ x ij , y ij , ϑ ij ] ni , j = (cid:1) , dϑ ij = (cid:88) k x ik y kj − y ik x kj .(3) Calabi–Yau algebras of dimension 3 (see [16, § 1.3]). Consider the free algebra F m andthe commutator quotient space of cyclic words: ( F m ) c yc = F m / [ F m , F m ] . M. Kontsevichintroduced linear maps ∂ i : ( F m ) c yc → F m for each i =
1, . . . , m which we can use, togetherwith a potential Φ ∈ ( F m ) c yc , to define the algebra(2.27) A = U( F m , Φ ) := F m / ( ∂ i Φ ) i = m , ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 13 which is the quotient of the free algebra F m by the two-sided ideal generated by the partialderivatives of the potential Φ . For example when m = F = k (cid:104) x , y , z (cid:105) and observe thatthe partial derivatives for the potential Φ = xyz − yxz give the commutators, therefore A = k [ x , y , z ] is the polynomial ring in 3 variables. For an algebra defined by a potential asabove in (2.27) we define the following dg-algebra:(2.28) D( F m , Φ ) := k (cid:104) x , . . . , x m , ϑ , . . . , ϑ m , t (cid:105) , ( deg ( x i , ϑ i , t ) = (
0, 1, 2 )) dϑ i = ∂ i Φ , dt = m (cid:88) i = [ x i , ϑ i ] .Ginzburg explains in [16] how Calabi–Yau algebras of dimension 3 are all of the form(2.27) and they are exactly those for which a suitable completion of D( F , Φ ) is a cofibrantresolution. This is in particular true for the example of polynomials in 3 variables (seeexample 6.3.2. in [4]), for which no completion is needed and:DRep n ( k [ x , y , z ]) ∼ = Rep n ( k [ x , y , z , ξ , ϑ , λ , t ]) = Spec (cid:0) k [ x ij , y ij , z ij , ξ ij , ϑ ij , λ ij , t ij ] ni , j = (cid:1) ,where the variables ξ , ϑ , λ are the ones we called ϑ , ϑ , ϑ in (2.28).2.3. G-invariants and isotypical components.
On the (derived) representation scheme there isa natural action of the general linear group GL ( V ) by which one can consider the associatedcharacter scheme of invariants. Later we consider only invariants by a subgroup G ⊂ GL ( V ) ,therefore we propose the following theory of partial invariant subfunctors by G that generalisesthe theory introduced in [5, § 2.3.5] and in [4, § 3.4] in the absolute case S = k . However we pointout that the results of this section are strongly inspired by [5] and [4], which already contain mostof the material needed.Suppose that both V and S are concentrated in degree 0, ρ : S → End ( V ) is a fixed representationand consider G S := { g ∈ GL ( V ) | g − ρ ( s ) g = ρ ( s ) ∀ s ∈ S } ,the subgroup of ρ -preserving transformations. Observe that in the absolute case S = k then G S = GL ( V ) . Now consider any reductive subgroup G ⊂ G S , whose right action on End ( V ) extends to the functor: End ( V ) ⊗ k (−) : CDGA k → DGA S (for this we need that G consists of transformations which all preserve ρ ). And consequently weobtain a left action on (−) V : DGA S → CDGA k and we can consider the invariant subfunctor(2.29) (−) GV : DGA S → CDGA k A (cid:55)−→ A GV .As explained in [5], unlike (−) V , the functor (−) GV does not seem to have a right adjoint, so wecannot prove that it has a left derived functor from Quillen’s adjunction theorem. Nevertheless wecan prove that such a left derived functor exists: Theorem 2.3.1. (a) (−) GV : DGA S → CDGA k has a total left derived functor L (−) GV .(b) For every A ∈ DGA S there is a natural isomorphism: (2.30) H • [ L ( A ) GV ] ∼ = H • ( A , V ) G .To prove this theorem it is convenient to recall a few notions/results. Let Ω = k [ t ] ⊕ k [ t ] dt be the algebraic de Rham complex of the affine line A k (in our conventions differentials havedegree − dt has the wrong degree − polynomial homotopy between f , g : A → B ∈ DGA S as a morphism h : A → B ⊗ Ω ∈ DGA S , such that h ( ) = f and h ( ) = g , wherefor each a ∈ k , h ( a ) is the following composite map: h ( a ) : A h −→ B ⊗ Ω π −→ B ⊗ Ω/ ( t − a ) ∼ = B ⊗ k = B .The reason why polynomial homotopy is equivalent to the homotopy equivalence relation in DGA S is explained in Proposition B.2. in [5]. Lemma 2.3.1.
Let h : A → B ⊗ Ω ∈ DGA S be a polynomial homotopy between f , g : A → B . Then:(1) There is a homotopy h V : A V → B V ⊗ Ω ∈ CDGA k between h V ( ) = f V and h V ( ) = g V .(2) h V restricts to a morphism h GV : A GV → B GV ⊗ Ω ∈ CDGA k . Remark 2.3.1.
It is important to observe that, despite the misleading notation, the map h V in part(1) is not the map obtained applying the functor (−) V to the map h . The latter would in fact be amap A V → ( B ⊗ Ω ) V (cid:54) = B V ⊗ Ω . The same thing applies for the map h GV in part (2), which is not the map obtained applying the functor (−) GV to the map h . Proof.
We omit the proof because it is analogous to the proof of Lemma 2.5 in [5]. (cid:3)
Proof of Theorem 2.3.1. (a) By Brown’s lemma (Lemma A.2 in [5]) it is sufficient to prove thatthe functor (−) GV maps acyclic cofibrations between cofibrant objects to weak equivalences.Let i : A ∼ (cid:44) → B such an acyclic cofibration between cofibrant objects A , B ∈ DGA S . Thenthere is a map p : B → A such that p ◦ i = Id A and i ◦ p is homotopic to Id B . The firstcomposition yields by functoriality p GV ◦ i GV = Id A GV and this proves that i GV is injective inhomology. The second fact that i ◦ p ∼ Id B yields, by Proposition B.2 in [5], an explicithomotopy h : B → B ⊗ Ω between i ◦ p = h ( ) and Id B = h ( ) . By Lemma 2.3.1 thereis a homotopy h GV between ( i ◦ p ) GV = i GV ◦ p GV and Id B GV and by Remark B.4.4 in [5] theyinduce the same maps at the level of homologies. This proves that i GV is also surjective inhomology.(b) The total left derived functor L (−) GV obtained in the previous point sends the class of γA ∈ Ho ( DGA S ) to the class of γ ( QA ) GV ∈ Ho ( CDGA k ) where QA is a cofibrant resolution in DGA S . Moreover H • (− , V ) G maps γA to H • ([ γ ( QA ) V ]) G . But because G is reductive (and k is a field of characteristic 0) there is an isomorphismH • ( γ [( QA ) GV ]) ∼ = [ H • ( γ ( QA V ))] G .and this concludes the proof. (cid:3) An analogous result holds also for the functor restricted on non-negatively graded objects, andit can be actually obtained as a corollary of Theorem 2.3.1:
Corollary 2.3.1. (a) (−) GV : DGA + S → CDGA + k has a total left derived functor L (−) GV .(b) For every A ∈ DGA + S there is a natural isomorphism: (2.31) H • [ L ( A ) GV ] ∼ = H • ( A , V ) G . Proof.
Using Brown’s lemma we just need to prove that (−) GV sends a trivial cofibrations betweencofibrant objects A ∼ (cid:44) → B to weak equivalences. We consider the following commutative diagram:(2.32) DGA + S CDGA + k DGA S CDGA k (−) GV ι ι (−) GV ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 15 and observe that ι ( A ∼ (cid:44) → B ) is still a trivial cofibration between cofibrant objects in DGA S , accordingto Remark 2.2.3. Now we can use the proof of Theorem 2.3.1 to conclude that the functor (−) GV : DGA S → CDGA k sends this map to a weak equivalence: ( ιA ) GV = ι ( A GV ) ∼ → ( ιB ) GV = ι ( B GV ) ∈ CDGA k .Finally from the very construction of ι we have that this map is a weak equivalence if and onlyif the map A GV ∼ → B GV ∈ CDGA + k is a weak equivalence. This concludes the proof of (a), while (b)follows from (a) as in Theorem 2.3.1. (cid:3) Now we derive also the other isotypical components of the representation functor. Let us fixany irreducible, finite-dimensional representation U λ of the reductive group G . We consider thefollowing functor:(2.33) (−) Gλ , V : DGA S → DGVect k A (cid:55)−→ (cid:0) U ∗ λ ⊗ k A V (cid:1) G .which is the invariant subfunctor of the functor (−) λ , V := U ∗ λ ⊗ k (−) V . Then we can prove thefollowing analogue to Theorem 2.3.1: Theorem 2.3.2. (a) The functor (2.33) has a total left derived functor L (−) Gλ , V .(b) For every A ∈ DGA S there is a natural isomorphism: (2.34) H • [ L ( A ) Gλ , V ] ∼ = (cid:0) U ∗ λ ⊗ H • ( A , V ) (cid:1) G .To prove it we need the following analogue of Lemma 2.3.1: Lemma 2.3.2.
Let h : A → B ⊗ Ω ∈ DGA S be a polynomial homotopy between f , g : A → B . Then:(1) There is a homotopy h λ , V : A λ , V → B λ , V ⊗ Ω ∈ DGVect k between h λ , V ( ) = f λ , V and h λ , V ( ) = g λ , V .(2) h λ , V restricts to a morphism h Gλ , V : A Gλ , V → B Gλ , V ⊗ Ω ∈ DGVect k .Proof. It is essentially a corollary of Lemma 2.3.1. In fact, we can define h λ , V to be h λ , V : A λ , V = U ∗ λ ⊗ A V Id U ∗ λ ⊗ h V −−−−−−→ U ∗ λ ⊗ B V ⊗ Ω = B λ , V ⊗ Ω ,where h V is the map from part (1) of Lemma 2.3.1. The map h V was G -equivariant, and thereforealso h λ , V = Id U ∗ λ ⊗ h V , from which part (2) follows. (cid:3) Proof of Theorem 2.3.2.
The proof works exactly as the proof of Theorem 2.3.1, using Lemma 2.3.2instead of Lemma 2.3.1. (cid:3)
The analogous results in the non-negative case also hold:
Corollary 2.3.2. (a) The functor (−) Gλ , V : DGA + S → DGVect + k has a total left derived functor L (−) Gλ , V .(b) For every A ∈ DGA + S there is a natural isomorphism: (2.35) H • [ L ( A ) Gλ , V ] ∼ = (cid:0) U ∗ λ ⊗ H • ( A , V ) (cid:1) G . Proof.
The proof follows from Theorem 2.3.2 in the same way as the proof of Corollary 2.3.1followed from Theorem 2.3.1. (cid:3)
K-theoretic classes.
We use the classical G -invariant subfunctor (−) GV : Alg S → CommAlg k todefine Definition 2.4.1.
The partial character scheme of an algebra A ∈ Alg S in a vector space V , relativeto a subgroup G ⊂ G S , is the affine quotient of the representation scheme:(2.36) Rep GV ( A ) := Rep V ( A ) (cid:12) G = Spec ( A GV ) ∈ Aff k .The name is motivated by the fact that in the absolute case S = k and G = GL ( V ) the full group,we would obtain the classical scheme of characters Rep GL ( V ) V ( A ) . The derived version is: Definition 2.4.2.
The derived partial character scheme of A ∈ Alg S in a vector space V , relative to asubgroup G ⊂ G S , is the affine quotient of the derived representation scheme:(2.37) DRep GV ( A ) := DRep V ( A ) (cid:12) G = R Spec (cid:0) L ( A ) GV (cid:1) ∈ Ho ( DGAff k ) .Let us recall that the obvious inclusion Sch k → DGSch k has for right adjoint the truncationfunctor π : DGSch k → Sch k that associates to a dg-scheme X = ( X , O X , • ) the closed subscheme π ( X ) := Spec ( H ( O X , • )) ⊂ X :(2.38) Sch k (cid:47) (cid:47) ⊥ DGSch kπ (cid:111) (cid:111) .Because the differential d : O X , i → O X , i − is O X -linear, the homologies H i ( O X , • ) are quasicoherentsheaves on X , and also on the closed subscheme π ( X ) ⊂ X . We can put these data together in adg-affine scheme: X h := (cid:0) π ( X ) , H • ( O X , • ) (cid:1) ∈ DGAff k ,which in the affine case X = Spec ( A ) is nothing but Spec ( H • ( A )) . Definition 2.4.3 (Definition 2.2.6. in [8]) . A dg-scheme X is of finite type if X is a scheme of finitetype and each O X , i is a coherent sheaf on X .Let now come to the case of our interest, a dg-affine scheme of finite type X = Spec ( B ) , forwhich the sheaves H i ( O X , • ) are coherent both over X and over π ( X ) = Spec ( H ( B )) , thereforethey define a class in the algebraic K-theory (2.39) [ H i ( O X , • )] ∈ K ( π ( X )) .We first consider the derived scheme X = DRep V ( A ) . Let us assume that A is an algebra suchthat, for each vector space V , the following two conditions are satisfied:(1) The derived representation scheme X = DRep V ( A ) is of finite type.(2) The structure sheaf O X , • of the derived representation scheme is bounded, in the sensethat O X , i = i (cid:29) π ( DRep V ( A )) = Rep V ( A ) .By condition (1) each homology defines a coherent sheaf on π ( X ) = Rep V ( A ) and therefore aclass (cid:2) H i ( A , V ) (cid:3) ∈ K ( Rep V ( A )) .By condition (2) there is only a finite number of them nonzero, therefore in particular the followingdefinition makes sense, because the sum in (2.40) is bounded: By algebraic K-theory of a scheme we mean the Grothendieck ring of the abelian category of coherent sheaves on it.
ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 17
Definition 2.4.4.
The virtual fundamental class - or Euler characteristic of the derived representationscheme X = DRep V ( A ) is the following invariant in the K -theory of the classical representationscheme:(2.40) [ X ] v ir = χ ( A , V ) := ∞ (cid:88) i = (− ) i (cid:2) H i ( A , V ) (cid:3) ∈ K ( Rep V ( A )) = K ( π ( X )) .This virtual fundamental class carries an action of the group G , which is reductive, and thereforeit decomposes into a direct sum of its irreducible components. To formalise this we first considerthe quotient by derived partial character scheme X G = DRep GV ( A ) , whose truncation is π ( X G ) = Rep GV ( A ) . For each finite-dimensional irreducible representation U λ of G we proved the existenceof the derived functor of the corresponding component L (−) Gλ , V : Ho ( DGA + S ) → Ho ( DGVect + k ) andobserved that H i (cid:0) L ( A ) Gλ , V (cid:1) ∼ = (cid:0) U ∗ λ ⊗ H i ( A , V ) (cid:1) G ∈ Mod H ( A , V ) G ,and therefore they define coherent sheaves on Rep GV ( A ) . Definition 2.4.5.
The Euler characteristic of the U λ -irreducible component of the derived partialcharacter scheme is(2.41) χ λ ( A , V ) := ∞ (cid:88) i = (− ) i [ H i ( L ( A ) Gλ , V )] ∈ K ( Rep GV ( A )) .We observe that the irreducible component corresponding to the trivial representation U = k is the virtual fundamental class of the derived partial character scheme, which we denote by χ G ( A , V ) = ∞ (cid:88) i = (− ) i [ H i ( A , V ) G )] = [ X G ] v ir ∈ K ( Rep GV ( A )) .2.5. T-equivariant enrichment.
So far we have worked only with a group G ⊂ G S ⊂ GL ( V ) thatacts on the representation scheme Rep V ( A ) because of the standard action on the vector space V .However, often the algebra A itself comes with an action of some algebraic torus T which helpswhen calculating its invariants (for example the corresponding decomposition of A might consistof finite dimensional weight spaces, allowing a graded dimensions count). In this section weexplain how such an action T (cid:121) A induces a well-defined group scheme action T (cid:121) DRep V ( A ) , inthe sense that different models for the derived representation scheme are linked by T -equivariantquasi-isomorphism, and therefore their homologies (and all the other invariants, as the Eulercharacteristics introduced in § 2.4) carry a well-defined induced T -action.First we give a notion of a rational T -action, for an algebraic group T ∈ Grp k on any (dg,commutative)algebra. Definition 2.5.1.
Let C be any of the following categories: DGVect k , DGA S , CDGA k or their non-negatively graded versions. A rational action of an algebraic group T over k on an object A ∈ C is amorphism of groups ρ : T → Aut C ( A ) with the additional property that every element a ∈ A iscontained in a finite dimensional T -stable vector subspace a ∈ V ⊂ A on which the induced action T → GL k ( V ) is a morphism of algebraic groups over k . We denote by C T the category with objectsthe objects in C with a rational T -action and morphisms the equivariant morphisms.This definition is motivated by the fact that the equivalence of categories (2.22) enriches to anequivalence of categories between ( CDGA + k ) T and the (opposite) category of dg-affine schemes witha group scheme action of T . Remark 2.5.1.
If we denote by T the one-object groupoid associated to the group T , then a rationalaction on an object in C is just a functor T → C with some additional properties, and a T -equivariantmorphism is a natural transformation of functors. Another way to say this is that we can view thecategory C T ⊂ [ T , C ] as a full subcategory of the category of functors. If C , D are two among thecategories mentioned in 2.5.1, and F : C → D is any functor between them, then we can consider theinduced functor on the functor categories F ∗ = F ◦ (−) : [ T , C ] → [ T , D ] . If this induced functor sendsobjects of C T ⊂ [ T , C ] into objects of D T , then it restricts to a functor that we denote by F T : C T → D T .This is true whenever F is defined purely in “algebraic terms” , which is the case of all the functorswe considered so far. The induced functor F T is an enrichment of the functor F in the sense that wecan recover F under the natural forgetful functors:(2.42) C T D T C D F T U UF
It is easy to see from the definition of the representation functor that a rational action T (cid:121) A induces (as explained in Remark 2.5.1), an action T (cid:121) A V which is still rational, and therefore agroup scheme action T (cid:121) Rep V ( A ) . To summarise the adjunction (2.19) enriches to an adjunction:(2.43) (cid:0) DGA + S (cid:1) T (−) V (cid:47) (cid:47) ⊥ (cid:0) CDGA + k (cid:1) T End ( V ) ⊗ k (−) (cid:111) (cid:111) .We did not add a superscript (−) T to the enriched functors in this diagram because we want toavoid confusion with the same symbols used with a different meaning in § 2.3.From now on we restrict ourselves to the case of our interest in this paper of an algebraic torus T = ( k × ) r . To do what we promised to do in the beginning of this section we need to provethat, roughly speaking, any T -equivariant algebra admits an equivariant cofibrant replacementin the model category DGA + S , and that any two such equivariant cofibrant replacements producequasi-isomorphic representation schemes. To do it we introduce a model structure on the category ( DGA + S ) T compatible with the model structures on DGA + S under the forgetful functor (in thefollowing Theorem we explain in which sense these model structures are compatible). We recallthat DGA + S is equipped with the projective model structure in which weak equivalences are quasi-isomorphisms and fibrations are surjections in positive degrees. We also observe that actually thecategory of T -equivariant dg-algebras over S is ( DGA + S ) T = S ↓ ( DGA + k ) T nothing else but the undercategory of T -equivariant dg-algebras over k receiving a map from S if we give S the trivial action,and therefore we only need to give a model structure in the absolute case S = k . Theorem 2.5.1.
There exists a model structure on ( DGA + k ) T with the following properties:(1) Weak equivalences / fibrations are exactly the maps that are weak equivalences / fibrations under theforgetful functor U : ( DGA + k ) T → DGA + k (and cofibrations are the maps with the left-lifting propertywith respect to acyclic fibrations defined in this way).(2) The forgetful functor preserves cofibrations.Proof. We refer the reader to Appendix A for the proof of this Theorem. (cid:3)
As a corollary of this result we can naturally equip the derived representation scheme of a T -equivariant algebra with a group scheme action of T . In fact, let S ∈ Alg k and ( A ∈ Alg S ) T = S ↓ ( Alg k ) T be a T -equivariant algebra. We leave intentionally this as an intuitive, not well-defined, notion.
ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 19
Corollary 2.5.1.
There is a well-defined action T (cid:121) DRep V ( A ) which is compatible with the one on Rep V ( A ) ∼ = π ( DRep V ( A )) induced by T (cid:121) A .Proof. First of all, we can pick up a T -equivariant cofibrant replacement Q ∼ (cid:16) A ∈ ( DGA + S ) T usingthe model structure we just defined. Because of Theorem 2.5.1 (1) and (2), when we forget the T -action we still have a cofibrant replacement for A , therefore we can use this Q as a model forDRep V ( A ) = Rep V ( Q ) . There is a natural T -action on this dg-scheme induced by T (cid:121) Q , which iscompatible with the one on its truncation π ( Rep V ( Q )) ∼ = Rep V ( A ) .To prove that the previous definition is well posed, we show that if Q (cid:48) ∼ (cid:16) A is any another T -equivariant cofibrant replacement, then there is a T -equivariant quasi-isomorphisms of dg-schemesRep V ( Q ) ∼ → Rep V ( Q (cid:48) ) . In fact by the general machinery of model categories we can lift the identitymap 1 A : A → A to a T -equivariant (weak equivalence) between the two cofibrant replacements f : Q ∼ → Q (cid:48) . When we forget the T -action, this is still a weak equivalence, therefore giving anisomorphism γf in the homotopy category Ho ( DGA + S ) and therefore L ( γf ) V is an isomorphism in Ho ( CDGA + k ) . But because both domain and codomain are cofibrant, L ( γf ) V = γf V , and therefore f V : Q V → ( Q (cid:48) ) V is a T -equivariant isomorphism of commutative dg-algebras, which dually givesthe desired T -equivariant map Rep V ( Q ) ∼ → Rep V ( Q (cid:48) ) . (cid:3) As a final consequence, the representation homology of a T -equivariant algebra, and all theother invariants defined in § 2.4, enrich to T -equivariant invariants. For example we can define the T -equivariant virtual fundamental class of the derived representation scheme X = DRep V ( A ) asthe following object in the equivariant K -theory of the classical representation scheme:(2.44) [ X ] v ir = χ T ( A , V ) := ∞ (cid:88) i = (− ) i [ H i ( A , V )] ∈ K T ( Rep V ( A )) ,and also all the other U λ -irreducible components for a reductive group G by which we take thequotient (see § 2.4) as(2.45) χ λT ( A , V ) := ∞ (cid:88) i = [ H i ( L ( A ) Gλ , V )] ∈ K T ( Rep GV ( A )) = K T ( Rep GV ( A ) .In particular for U = k the trivial representation, we obtain an equivariant version of the virtualfundamental class of the derived partial character scheme X G = DRep GV ( A ) , which we denote by:(2.46) χ GT ( A , V ) = ∞ (cid:88) i = [ H i ( A , V ) G ] = [ X G ] v ir ∈ K T ( Rep GV ( A )) .3. T he case of N akajima quiver varieties In this section we first recall the construction of Nakajima quiver varieties and secondly weconstruct some derived representation schemes related to them.3.1.
Nakajima quiver varieties.
We already recalled in Example 2 that a finite quiver is a finitedirected graph defined by its sets of vertices and edges Q = ( Q , Q ) with two maps (source andtarget of an arrow) s , t : Q → Q .We first frame the quiver, this means that we add a new vertex for each old one with a newarrow from the new to the old. Then we double the framed quiver, in order to obtain a cotangent(symplectic) space when we consider its representations. We denote this quiver by Q f r . To considerrepresentations of a framed (doubled) quiver, we need to fix two dimension vectors v , w ∈ N Q ,and usually one assumes that (at least one of the components of) the framing vector is nonzero: w (cid:54) = • vx Jordan quiver • vxy (cid:3) wij Figure 1.
Example: framing and doubling the Jordan quiver. The framed vertices are usuallydenoted by a square symbol.
Notation.
We denote the linear representations of the doubled, framed quiver by(3.1) M ( Q , v , w ) := L (cid:16) Q f r , v , w (cid:17) ∼ = T ∗ L (cid:0) Q f r , v , w (cid:1) .Explicitely it is the following cotangent linear space:(3.2) M ( Q , v , w ) = T ∗ (cid:77) γ ∈ Q Hom C ( C v s ( γ ) , C v t ( γ ) ) ⊕ (cid:77) a ∈ Q Hom C ( C w a , C v a ) .We denote elements of this space by quadruples ( X , Y , I , J ) = ( X γ , Y γ , I a , J a ) γ , a , where X γ , I a ∈ L ( Q f r , v , w ) are elements of the representation space of the framed quiver, and ( Y γ , J a ) arecotangent vectors to them. The gauge group is the general linear group on the set of vertices ofthe original quiver Q :(3.3) G = G v := (cid:89) a ∈ Q GL v a ( C ) ⊂ GL ( C v ⊕ C w ) ,which acts by conjugation in a Hamiltonian fashion on M ( Q , v , w ) . The moment map for thisaction is(3.4) µ : M ( Q , v , w ) → g ∗ v ∼ = g v ( via trace )( X , Y , I , J ) (cid:55)−→ [ X , Y ] + IJ ,where in the above equation [ X , Y ] + IJ is a shortened symbol for(3.5) [ X , Y ] + IJ = (cid:88) γ : t ( γ )= a X γ Y γ − (cid:88) γ : s ( γ )= a Y γ X γ + I a J a a ∈ Q ∈ (cid:77) a ∈ Q gl v a ( C ) = g v .Nakajima varieties are defined as symplectic reductions of M ( Q , v , w ) by this action. The affineNakajima quiver variety is the geometric quotient:(3.6) M ( Q , v , w ) := µ − ( ) (cid:12) G = Spec (cid:0) O ( µ − ( )) G (cid:1) .The GIT Nakajima variety is instead given by the choice of a character χ ∈ Hom
Grp C ( G , C × ) as theproj of the graded ring of χ -quasiinvariant functions on µ − ( ) :(3.7) M χ ( Q , v , w ) = µ − ( ) (cid:12) χ G = Proj (cid:0) O ( µ − ( )) G , χ (cid:1) (elements of degree n (cid:62) O ( µ − ( ) G , χ ) are functions f ∈ O ( µ − ( )) with the property f ( g · p ) = χ n ( g ) f ( p ) for all g ∈ G and p ∈ µ − ( ) ). The inclusion of G -invariant functions asdegree zero elements of the graded ring of χ -quasiinvariant functions O ( µ − ( )) G ⊂ O ( µ − ( )) G , χ induces a projective morphism:(3.8) p : M χ ( Q , v , w ) → M ( Q , v , w ) , ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 21 which is often a symplectic resolution of singularities. Sometimes we denote these varieties simplyby M χ , M implicitly fixing the quiver Q , and the dimension vectors v , w .3.2. Derived representation schemes models.
In Proposition 2.1.1 we showed how the linearspace of representations of a quiver is isomorphic to the representation scheme for its path algebra.The same thing holds for the doubled, framed quiver so that(3.9) M ( Q , v , w ) = L (cid:16) Q f r , v , w (cid:17) ∼ = Rep v , w (cid:16) C Q f r (cid:17) .To obtain the zero locus of the moment map, we consider the 2-sided ideal I µ ⊂ C Q f r generatedby the | Q | -elements of the path algebra described in (3.4), and consider the quotient algebra(3.10) A := C Q f r / I µ ∈ Alg S ,relative to the subalgebra S ⊂ A of idempotents, with fixed representation ρ = ρ v , w : S → End C ( C v ⊕ C w ) (as in (2.8)). The following result is an immediate consequence of the fact that taking the quotientby some ideal amounts simply to impose these new relations in the representation scheme (seeExamples 1.(6)): Proposition 3.2.1.
The zero locus of the moment map µ is the (relative) representation scheme for the pathalgebra of the framed, doubled quiver, modulo the Hamiltonian relation: (3.11) µ − ( ) ∼ = Rep v , w (cid:0) C Q f r / I µ (cid:1) . Notation.
We denote the corresponding derived representation scheme and representation homol-ogy by:(3.12) DRep v , w ( A ) = Spec (cid:0) L (cid:0) A (cid:1) v , w (cid:1) ∈ Ho ( DGAff C ) ,H • ( A , v , w ) = H • (cid:0) L (cid:0) A (cid:1) v , w (cid:1) ∈ CDGA + C .The representation homology H • ( A , v , w ) is a graded commutative algebra, so when we view it in CDGA + C we mean that the differential is zero.Remark 2.2.2, together with Proposition 3.2.1 tells us that the π of this derived scheme X = DRep v , w ( A ) is the zero locus of the moment map:(3.13) π ( X ) = Spec (cid:0) H (cid:0) A , v , w (cid:1)(cid:1) ∼ = µ − ( ) .In particular when we consider the invariant subfunctor only by the gauge group on the originalvertices G (3.3): Corollary 3.2.1.
The π of the partial character scheme X G = DRep G v , w ( A ) is the affine Nakajima variety M : (3.14) π ( X G ) = π ( DRep G v , w ( A )) ∼ = M . Proof.
It follows directly from the previous observation (3.13) and the Theorem 2.3.1. Moreprecisely: π ( X G ) ∼ = Spec (cid:0) H ( A , v , w ) G (cid:1) ∼ = Spec (cid:0) O ( µ − ( )) G (cid:1) = µ − ( ) (cid:12) G = M . (cid:3) K-theoretic classes in the affine Nakajima variety.
In § 3.4 we describe an explicit cofibrantresolution for our algebra A = C Q f r , A c of ∼ (cid:16) A and therefore a model for the derived representa-tion scheme DRep v , w (cid:0) A (cid:1) = Rep v , w (cid:0) A c of (cid:1) , but we can already use Corollary 3.2.1 to define someinteresting invariants in the K-theory of M = Rep G v , w ( A ) . Throughout this section we denote by X = DRep v , w ( A ) the derived representation scheme and by X G = DRep G v , w ( A ) the correspondingpartial character scheme, whose π ( X G ) = M is the affine Nakajima variety.There is a torus, the (standard) maximal torus of the gauge group on the framing vertices T w ⊂ G w acting on the linear space of representations Rep v , w ( A ) , and therefore as explainedin § 2.4 it induces an action T w (cid:121) DRep v , w ( A ) and on its quotient by the gauge group G v : T w (cid:121) DRep G v v , w ( A ) . There is an additional (2-dimensional) torus T (cid:32) h = ( C × ) (cid:121) A acting rationallyon the path algebra of the doubled framed quiver. This action can be described by assigning,respectively, the following Z -weights to the arrows ( x γ , y γ , i a , j a ) (see § 3.1 to recall the name ofthe arrows): (
1, 0 ) , (
0, 1 ) , (
1, 1 ) , (
0, 0 ) , or explicitly as x γ (cid:55)→ (cid:32) h x γ , y γ (cid:55)→ (cid:32) h y γ , i a (cid:55)→ (cid:32) h (cid:32) h i , j a (cid:55)→ j a .As explained in § 2.5, also this torus induces actions T (cid:32) h (cid:121) DRep v , w ( A ) , DRep G v v , w ( A ) . In otherwords, the whole torus T := T w × T (cid:32) h acts on the derived representation scheme X = DRep v , w ( A ) and its partial character scheme X G v = DRep G v v , w ( A ) .Using the definitions we gave in § 2.4 and § 2.5 we obtain the following invariants in the(equivariant) K theory of the affine Nakajima variety M = Rep G v v , w ( A ) , for example the virtualfundamental class(3.15) (cid:2) X G v (cid:3) v ir = ∞ (cid:88) i = (− ) i (cid:2) H i ( A , v , w ) G v (cid:3) ∈ K T (cid:0) M (cid:1) .More generally for each irreducible representation U λ of G v , the Euler characterstic of thecorresponding isotypical component as(3.16) χ λT ( A , v , w ) = ∞ (cid:88) i = (− ) i (cid:2) H i ( L ( A )) G v λ , v , w (cid:3) ∈ K T (cid:0) M (cid:1) .3.4. Explicit cofibrant resolution.
In this section we describe an explicit cofibrant resolution forthe S -algebra A constructed in the previous Section. Let us recall that(3.17) A = C Q f r / I µ ∈ Alg S (cid:44) → DGA + S ,where S is the subalgebra generated by the idempotents of the path algebra of the framed quiver.The main obstruction for this object to be cofibrant is the Hamiltonian relation described bythe ideal I µ . The simplest idea is then to add one more variable for each of the generatingrelations in I µ which kills the relation itself. This technique in general might not work due tohigher homologies, but we prove that this case is one of the well-behaved cases. We construct thefollowing quiver Q ϑ , which is obtained by adding to the framed, doubled quiver Q f r , one loopcalled ϑ a on each original vertex a ∈ Q .In the path algebra C Q ϑ we assign homological degree 0 to the original arrows, and homologicaldegree 1 to the new arrows ϑ a . The differential is induced by the moment map (equations as in(3.5)) dϑ a = µ a ( x , y , i , j ) .We denote the resulting differential graded algebra by(3.18) A c of := ( C Q ϑ , d ) ∈ DGA + C . ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 23
It sits in the following diagram(3.19) SA c of A ι ϕπ where π is the composition of the following two obvious projections: π : A c of → ( A c of ) = C Q f r → C Q f r / I µ = A . Theorem 3.4.1. A c of is a cofibrant replacement for A in DGA + S . This amounts to prove that, in the diagram (3.19), the map π is an acyclic fibration, and ι is acofibration. Lemma 3.4.1.
The map π : A c of → A is an acyclic fibration in DGA + C .Proof. We need to prove that:(i) π is degreewise surjective in degrees (cid:62) A is concentrated indegree 0).(ii) H i ( π ) : H i ( A c of ) → H i ( A ) is an isomorphism for each i (cid:62)
0, which becomes proving that (cid:14) H ( π ) : H ( A c of ) ∼ −→ A ,H i ( A c of ) = i (cid:62) ( π ) is an isomorphism, this is evident from the construction of A c of . We are left to prove that A c of has no higher homologies.We first decompose A c of as a direct sum of the subalgebras of paths starting and finishing at afixed couple of vertices:(3.20) A c of = (cid:77) a , b ∈ Q ϑ P a , b , P a , b = (cid:8) paths in Q ϑ starting at a and ending at b (cid:9) .This decomposition preserves the differential, so we only need to prove that each P a , b has nohigher homologies.Claim: If we substitute each j a , i a with the cycle c a = i a j a and prove that the resultingdg-algebras have no higher homologies, then neither P a , b have. Proof of the claim:
Let us call (cid:101) P a , b the dg-algebra of paths from a to b in the quiver Q ϑ , where wesubstitute each pair of arrows j a , i a with the cycle c a = i a j a . Then, for each fixed a , b ∈ Q , wehave four cases :(3.21) P a , b = (cid:101) P a , b P a , b = j b · (cid:101) P a , b ∼ = (cid:101) P a , b P a , b = (cid:101) P a , b · i a ∼ = (cid:101) P a , b P a , b = j b · (cid:101) P a , b · i a ∼ = (cid:101) P a , b (cid:3) We use the following notation, for a vertex a ∈ Q in the original quiver, we denote by a the associated framedvertex. Now we consider the following filtration on the algebras (cid:101) P a , b : F p := Span C { paths with x + y (cid:62) p } .Remember that the differential has the form “ dϑ = [ x , y ] + c ”, so that the associated graded hasdifferential of the form d gr ϑ a = c a , which involves only loops on the vertices a ∈ Q . But thenwe can decompose the dg-algebras (cid:101) P a , b into their word structure , and discover that the onlynon-trivial building blocks of which they are made of are dg-algebras of the form L = (cid:0) k (cid:104) ϑ , c (cid:105) , dϑ = c (cid:1) .which have no higher homologies . (cid:3) Lemma 3.4.2. ι : S → A c of is a cofibration in DGA + C , or equivalently A c of is a cofibrant object in DGA + S .Proof. We need to prove that ι has the left lifting property with respect to acyclic fibrations.(3.22) S BA c of C σ ∼ ∃ Let us observe that because A c of = C Q ϑ is the (dg) path algebra of a quiver with idempotents S ,we can view A c of = T S M := S ⊕ M ⊕ ( M ⊗ S M ) . . . as the tensor algebra of the S -(dg)bimodule M := Span C (cid:8) arrows in Q ϑ (cid:9) .But then find a lifting in the diagram (3.22) amounts to simply give a (linear) lifting of the (dg)vector space M , which is possible for the surjectivity of the map B ∼ (cid:16) C (acyclic fibrations aresurjective in every homological degree). (cid:3) Koszul complex and complete intersections.
Theorem 3.4.1 tells us that a model for thederived representation scheme for the algebra A is the representation scheme of the cofibrantreplacement A c of . In this section we recognise it as the Koszul complex for the moment map, andin order to do so, we first recall a few notions and important classical results about the latter.The Koszul complex can be thought of as one of the main examples of derived intersections ofsubschemes of a scheme. Classically, affine varieties are the simplest examples of intersections,being zero loci of some simultaneous polynomial equations f , . . . , f m ∈ O ( A n C ) = C [ x , . . . , x n ] :(3.23) ( X , O ) = Spec (cid:0) R/ ( f ) ⊗ R · · · ⊗ R R/ ( f m ) (cid:1) , R = C [ x , . . . , x n ] .Then the associated derived intersection can be defined as the derived scheme(3.24) ( X , O • ) = Spec (cid:0) R/ ( f ) ⊗ L R · · · ⊗ L R R/ ( f m ) (cid:1) ,where ⊗ L R is the derived tensor product of R -modules. The algebra of functions on this derivedscheme is the Koszul complex : K = R/ ( f ) ⊗ L R · · · ⊗ L R R/ ( f m ) ∈ CDGA + C . More precisely, we can decompose (cid:101) P a , b into the direct sum of those paths that, except for the arrows ϑ a and c a —meaning that we set these to 1 — are the same. An elementary argument is to observe that the derivation defined by the formula h ( c ) = ϑ and h ( c ) =
0, is ahomotopy between the 0 map and the map length (−) · Id, which is an isomorphism in (homological) degrees (cid:62)
1. Thisimplies that H i ( L ) = i (cid:62) ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 25
A more concrete way to describe it is the following: we can view the collection of functions f = ( f , . . . , f m ) as a map of affine schemes f : A n C → V := A m C ,and consider its dual map O ( f ) : O ( V ) = Sym ( V ∗ ) → O ( A n C ) = R .Then the Koszul complex is the commutative dg-algebra K = ( R ⊗ C Λ • ( V ∗ ) , d ) , where R is inhomological degree 0, the vector space V ∗ is in homological degree 1, and the differential d := O ( f ) | V ∗ : V ∗ (cid:44) → Sym ( V ∗ ) → R .An useful classical result on the Koszul complex is Theorem 3.5.1 ([23]) . The following are equivalent:(1) dim C ( Spec ( H ( K ))) = n − m .(2) The sequence f , . . . , f m ∈ R is a regular sequence.(3) H ( K ) = .(4) H i ( K ) = for all i (cid:62) . Let us turn back to the case of our interest, in which we want to recognise ( A c of ) v , w = ( C Q ϑ ) v , w as the Koszul complex on the moment map. We recall that the quiver Q ϑ is constructed fromthe quiver Q f r by adding a new loop in homological degree 1 on each of the original verticesof the quiver Q . Therefore, a representation of the path algebra C Q ϑ is just a representationof the subalgebra C Q f r (an element of the vector space M ( Q , v , w ) ), together with a family ofendomorphisms Θ = ( Θ a ) a ∈ Q ∈ (cid:77) a ∈ Q gl v a ( C ) = g v in homological degree 1. Putting everything together we obtain ( C Q ϑ ) v , w = O ( M ( Q , v , w )) ⊗ C Λ • g v ∈ CDGA + C ,which is nothing else but the Koszul complex for the zero locus defined by the moment map µ : M ( Q , v , w ) → g ∗ v .Its spectrum is a model for our derived representation scheme, as the derived intersection of themoment map equations: Theorem 3.5.2.
The cofibrant resolution A c of ∼ (cid:16) A in DGA + S gives a model for the derived representationscheme as the (spectrum of the) Koszul complex on the moment map: (3.25) DRep v , w (cid:0) A (cid:1) ∼ = Rep v , w (cid:0) A c of ) = Spec (cid:0) O ( M ( Q , v , w )) ⊗ Λ • g (cid:1) .In particular we can observe that this is a derived scheme of finite type (Definition 2.4.3) and thatthe Koszul complex is bounded, Therefore all the invariants defined in § 3.3 ((3.15),(3.16)) makesense, because the sums are bounded (by the dimension of the Lie algebra dim C g v = v = v · v ). Remark 3.5.1.
In § 3.4 we gave a self-contained proof of why the resolution provided by the pathalgebra of the quiver Q ϑ obtained by adding one loop on each vertex in which the correspondingcomponent of the moment map is considered (i.e. the original vertices) works. In § 3.5 weexplained why the resulting representation scheme is the Koszul complex on the moment map.We remark that the same results can be explained in a slightly different flavour through the theoryof noncommutative complete intersections (NCCI) and partial preprojective algebras ([10], [13]).
4. M ain results
Flat moment map and vanishing representation homology.
In this section we recall someclassical results on the flatness for the moment map of Nakajima quiver varieties which are usefulfor our purposes. We show how flatness is equivalent to the condition of vanishing of higherrepresentation homologies for the corresponding algebra.Remember that for each quiver Q and for each fixed dimensions v , w ∈ N Q we have the corre-sponding Nakajima varieties M (affine) and M χ (quasiprojective), where χ ∈ Hom
Grp C ( G v , C × ) is a given (nontrivial) character. We also recall that the group of all characters of the gauge group G = G v = (cid:81) a ∈ Q GL v a ( C ) is isomorphic to the lattice Z Q ∼ = Hom
Grp C ( G , C × ) ,via the assignment θ (cid:55)→ χ θ ( g ) = (cid:89) a ∈ Q det ( g a ) θ a .In this section we use the parameter θ for the characters and denote M χ θ simply by M θ .We recall that the Cartan matrix of the quiver Q is the matrix C Q = · Id − A Q , where A Q isthe adjacency matrix of the doubled quiver Q . For a fixed dimension vector v ∈ N Q , a vector θ ∈ Z Q is called v -regular , if for each α ∈ Z Q \{ } such that C Q α · α (cid:54) (cid:54) α (cid:54) v (component-wise) then θ α + · · · + θ | Q | α | Q | (cid:54) = R Q of v -regular vectors is the complement of some hyperplanes. Its connectedcomponents are called chambers , and the variety M θ depends only on the chamber of θ . Theorem 4.1.1 (Theorem 5.2.2. in [17]) . Let v ∈ N Q be a dimension vector and θ ∈ Z Q be v -regular,then any θ -semistable point in µ − ( ) is θ -stable and M θ is a smooth, connected, variety of (complex)dimension dim M θ = v · w − C Q v · v , (with the convention that M θ = ∅ when this dimension is negative). Remark 4.1.1.
Observe that the dimension counting is what we would expect. In factdim ( M ( Q , v , w )) = v · w + A Q v · v = v · w − C Q v · v + v · v .When we take the zero locus by µ we expect to decrease the dimension by the number of equationsof µ , which is v · v and then again by v · v when taking the G v -quotient.Let us consider for some v -regular θ the natural affinisation morphism(4.1) ϕ : M θ → Spec (cid:0) O (M θ ) (cid:1) .This morphism is a Poisson morphism (obviously, because ϕ ∗ is the identity) and it is a resolutionof singularities (i.e. projective and birational) ([7]). The variety M θ depends, a priori on thechamber of θ , but actually its affinisation Spec ( O (M θ )) is independent of the choice of v -regular θ .We can call this variety simply M and we obtain a diagram of the following form(4.2) M θ M M ϕ pψ The Poisson structure on Nakajima varieties comes from the general formalism of Hamiltonian reduction, andcoincides with the one induced by the symplectic form on the regular locus.
ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 27 which is the so-called Stein factorisation ([40]) of the proper morphism p . The pre-image of thepoint 0 ∈ M through ψ is always 0 ∈ M . In particular the fiber p − ( ) is equal to the central fiber ϕ − ( ) of the affinisation morphism p − ( ) = ( ψ ◦ ϕ ) − ( ) = ϕ − ( ψ − ( )) = ϕ − ( ) ,and therefore is a homotopy retract of the variety M θ . Theorem 4.1.2 ([7]) . If the moment map µ : M ( Q , v , w ) → g ∗ v is flat, then ψ is an isomorphism, and inparticular O (M θ ) ∼ = O (M ) . The combinatorial criterium for the flatness of the moment map proved in [9] is given in thesetting of a non-framed quiver Γ . For any dimension vector α ∈ N Γ we consider the linear spaceof representations of the doubled quiver L ( Γ , α ) . The gauge group acting a priori in a non-trivialway is now G α / C × because, without the framing, the diagonal torus C × ⊂ G α acts triviallyon the linear space of representations. The Lie algebra of this group can be identified with thesubalgebra g (cid:92) α ⊂ g α = ⊕ i gl α i ( C ) of matrices with sum of their traces equal to zero (the notation g (cid:92) α is borrowed from [13]). The moment map is now µ α : L ( Γ , α ) → g (cid:92) α x (cid:55)−→ [ x , x ∗ ] .Let us denote by p the following function p : N Γ → Z , p ( α ) := + (cid:88) γ ∈ Γ α s ( γ ) α t ( γ ) − α · α . Theorem 4.1.3 (Theorem 1.1 in [9]) . The following are equivalent:(1) µ α is a flat morphism.(2) µ − α ( ) has dimension α · α − + p ( α ) ( = dim L ( Γ , α ) − dim g (cid:92) α ).(3) p ( α ) (cid:62) (cid:80) rt = p ( β ( t ) ) for each decomposition α = β ( ) + · · · + β ( r ) with each β ( t ) positive root.(4) p ( α ) (cid:62) (cid:80) rt = p ( β ( t ) ) for each decomposition α = β ( ) + · · · + β ( r ) with each β ( t ) ∈ N Γ \{ } . In a remark in § 1 in [9], Crawley-Boevey explains how to adapt this setting to the situation of aframed quiver. From a quiver Q and a framing vector w we can construct a new quiver Γ := Q ∞ ,which is obtained by adding only one new vertex, denoted by ∞ , together with a number of w a arrows towards each vertex a ∈ Q . If we fix now a dimension vector v ∈ N Q and define thenew vector α := ( v , 1 ) ∈ N Γ , then(4.3) L ( Γ , α ) ∼ = L (cid:0) Q f r , v , w (cid:1) = M ( Q , v , w ) ,by splitting the v a × w a matrices in M ( Q , v , w ) in columns and the w a × v a matrices in rows. Thetwo gauge groups are also isomorphic: G α / C × ∼ = G v , and under this isomorphism their actionson L ( Γ , α ) ∼ = M ( Q , v , w ) are the same. Therefore also the moment maps are identified:(4.4) L ( Γ , α ) g (cid:92) α M ( Q , v , w ) g v µ α ∼ ∼ µ and we have the following criterium: Corollary 4.1.1.
Consider the quiver Q f r with dimension vectors v , w ∈ N Q , and the quiver Γ = Q ∞ with α = ( v , 1 ) . Then the following are equivalent:(1) µ is flat. (2) µ α is flat. For the condition (2) now we can use the combinatorical test given by Theorem 4.1.3, and usingthis result, we can prove that the derived representation scheme has vanishing higher homologiesif and only if the moment map µ is flat: Theorem 4.1.4.
The representation homology H • ( A , v , w ) for the algebra 3.17 vanishes if and only if themoment map µ is flat.Proof. Because of the diagram (4.4) the moment map µ is flat if and only if µ α is flat and byTheorem 4.1.3, condition (2), this happens if and only if(4.5) dim µ − ( ) = dim µ − α ( ) = dim L ( Γ , α ) − dim g (cid:92) α = dim M ( Q , v , w ) − dim g v .The representation homology is the homology of the Koszul complexH • ( A , v , w ) = H • (cid:0) O (cid:0) M ( Q , v , w ) (cid:1) ⊗ Λ • g v (cid:1) ,and therefore, by Theorem 3.5.1, it vanishes in degrees i (cid:62) (cid:3) In the following examples we use Theorem 4.1.3 for some quivers and we find the combinatoricalcondition on the dimension vectors for the moment map to be flat. It is convenient to observe thatfor the quiver Γ = Q ∞ the map p is, for vectors of the form ( β , 1 ) or ( β , 0 ) (that is the only type ofvectors that we need to decompose the dimension vector α = ( v , 1 ) ): p ( β , 1 ) = (cid:88) γ ∈ Q β s ( γ ) β t ( γ ) + β · w − β · β , p ( β , 0 ) = p ( β , 1 ) + Examples 4. (1) The first example is that of a single-vertex quiver Q = A with no arrows,whose Γ = Q ∞ becomes a quiver with 2 vertices and w arrows going from one to the other.We need to test for which v it holds that for each decomposition ( v , 1 ) = ( β , 1 ) + ( β , 0 ) + · · · + ( β r , 0 ) , β t (cid:62) v ( w − v ) (cid:62) β ( w − β ) + r − β − · · · − β r .We can observe that actually all β , . . . , β r (cid:62) r − β − · · · − β r reaches its maximum for β = · · · = β r = v ( w − v ) (cid:62) β ( w − β ) , ∀ β =
0, . . . , v − (cid:24)(cid:24)(cid:24)(cid:24) ( v − β ) w (cid:62) (cid:24)(cid:24)(cid:24)(cid:24) ( v − β )( v + β ) , ∀ β =
0, . . . , v − ⇔ w (cid:62) v − m loops ( m (cid:62) m = v (cid:62) w (cid:62) Γ = Q ∞ still has 2 vertices, the first one with m loops and w arrows connecting the 2 nd to the 1 st , so that: p ( α , α ) = + mα + wα α − α − α .We need to test that for each decomposition ( v , 1 ) = ( β , 1 ) + ( β , 0 ) + · · · + ( β r , 1 ) , β , . . . , β r (cid:62) ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 29 the following inequality holds ( m − ) v + vw (cid:62) ( m − ) β + β w + r + ( m − )( β + · · · + β r ) ,which is actually true component-wise because (cid:14) ( m − ) v (cid:62) ( m − )( β + · · · + β r ) vw = ( β + · · · + β r ) w (cid:62) β w + rw (cid:62) β w + r .Therefore the moment map is always flat.(3) The third example is the quiver Q = A n − with the following particular choice of vectors v = (
1, . . . , 1 ) and w a = δ a ,1 + δ a , n − (for which the Nakajima variety is the symplecticdual of T ∗ P n − , as explained in the next Section). The resulting quiver Γ = Q ∞ is thecyclic quiver with n vertices and dimension vector α = (
1, . . . , 1 ) constant to 1, for which itis easy to check that the moment map is flat. In fact p ( α ) = β ∈ N n ,0 (cid:54) = β (cid:54) = α we have p ( β ) (cid:54) Kirwan map and tautological sheaves.
Let M χ = M χ ( Q , v , w ) be a smooth Nakajima quivervariety (so χ = χ θ with θ being v -regular, see Theorem 4.1.1), then the locus of χ -semistable pointscoincides with the locus of χ -stable points, on which the action is free, and M χ = µ − ( ) (cid:12) χ G = µ − ( ) χ -s t /G .The equivariant Kirwan map (in cohomology) is the map(4.6) κ T : H • G × T (cid:16) µ − ( ) (cid:17) → H • T (M χ ) ,obtained by composing the natural pullback for the inclusion µ − ( ) χ -s t ι ⊂ µ − ( ) with the iso-morphism H • G × T (cid:0) µ − ( ) χ -s t (cid:1) ∼ = H • T (M χ ) due to the fact that the G -action on the χ -stable locus isfree: H • G × T (cid:16) µ − ( ) (cid:17) ι • −→ H • G × T (cid:16) µ − ( ) χ -s t (cid:17) ∼ = H • T (cid:16) µ − ( ) χ -s t /G (cid:17) = H • T (M χ ) .McGerty and Nevins have recently shown that the Kirwan map (4.6) is surjective ([24, Corollary1.5]), and that the same holds for other generalised cohomology theories such as K -theory andelliptic cohomology. We are particularly interested in the K-theory, so the Kirwan map is(4.7) κ T : K G × T (cid:16) µ − ( ) (cid:17) → K T (M χ ) .Moreover the zero locus of the moment map µ − ( ) is equivariantly contractible:K G × T (cid:16) µ − ( ) (cid:17) ∼ = K G × T ( p t ) = R ( G × T ) ∼ = R ( G ) ⊗ R ( T ) ,where R (−) is the representation ring (over C ), so the Kirwan map has the form:(4.8) κ T : R ( G ) ⊗ R ( T ) → K T (cid:0) M χ (cid:1) ,and it is a surjective map of R ( T ) -modules. K T (M χ ) is therefore generated by tautological classes ,because they come from classes of topologically trivial vector bundles: if U is a G × T -module,and [ U ] ∈ R ( G × T ) is its class, then(4.9) κ T ([ U ]) = (cid:2)(cid:0) µ − ( ) χ -s t × U (cid:1) /G (cid:3) ∈ K T (cid:0) µ − ( ) χ -st /G (cid:1) = K T (M χ ) .Moreover the map (4.8) is a map of R ( T ) -modules, so the only non-trivial part consist in its imageon vector spaces U that are only representations of G . For U = V λ irreducible representation of G , we denote by a calligraphic V λ the sheaf whose K-theoretic class is [ V λ ] = κ T ([ V λ ]) ∈ K T (M χ ) . We can use these tautological classes to define invariants in the K-theory of the affine Nakajimavariety by using the pushforward under the map p :(4.10) R ( G ) ⊗ R ( T ) κ T −−→ K T (M χ ) p ∗ −→ K T (M ) .It is important to recall that in general the push-forward of a proper map p in K -theory is given bythe alternate sums of right-derived functors of p ∗ . In this particular case the target variety M isaffine, therefore this alternate sum calculates the Euler characteristic of a sheaf F on M χ , underthe natural identifications:(4.11) p ∗ ([ F ]) = χ T (M χ , F ) ∈ K T ( O (M ) − Mod ) ∼ = K T (M ) .The structure of O (M ) -module comes from the fact that the cohomologies H i (M χ , F ) havea structure of O (M χ ) -modules and the map p : M χ → M gives to the latter a structure of O (M ) -module.For an irreducible representation U = V λ of G the composition (4.10) gives the Euler character-istic of the corresponding tautological sheaf V λ :(4.12) p ∗ (cid:0) κ T ([ V λ ]) (cid:1) = p ∗ (cid:0) [ V λ ] (cid:1) = χ T (M χ , V λ ) ∈ K T (M ) .The notable special case of U = V the trivial 1-dimensional representation of G , has image underthe Kirwan map the ( K -theoretic class of the) sheaf of functions on the GIT quotient V = O M χ ,and its Euler characteristic:(4.13) p ∗ ( κ T ([ V ])) = p ∗ ([ O M χ ]) = χ T (M χ , O M χ ) ∈ K T (M ) .4.3. Comparison theorem and first integral formula.
In § 3.3 we defined the virtual fundamentalclasses of the isotypical components of the derived character scheme(4.14) χ λT ( A , v , w ) = ∞ (cid:88) i = (− ) i (cid:104) ( V ∗ λ ⊗ H i ( A , v , w )]) G (cid:105) ∈ K T (M ) ,and in particular for V λ = V = C :(4.15) χ T ( A , v , w ) = χ GT ( A , v , w ) = ∞ (cid:88) i = (− ) i (cid:104) H i ( A , v , w ) G (cid:105) ∈ K T (M ) . Theorem 4.3.1.
Let v , w be dimension vectors for which the moment map is flat, and let χ = χ θ with θ v -regular, so that M χ ( Q , v , w ) is smooth. Then we have the following equality in the equivariant K -theoryof the affine Nakajima variety : (4.16) p ∗ ([ O M χ ( Q , v , w ) ]) = [ O M ( Q , v , w ) ] = χ GT ( A , v , w ) ∈ K T (cid:16) M ( Q , v , w ) (cid:17) . Proof.
The first equality is a somewhat classical result. Firstly the (derived) pushforward inK-theory coincides with the underived pushforward p ∗ ([ O M χ ]) = χ T (M χ , O M χ ) = (cid:88) i (cid:62) (− ) i [ H i (M χ , O M χ )] = (cid:2) O M χ (cid:3) ∈ K T (M ) ,because of the vanishing of higher cohomologies (Grauert-Riemenschneider theorem, ([18])).Moreover when the moment map is flat and M χ is smooth we can use Theorem (4.1.2): (cid:2) O M χ (cid:3) = [ O M ] ∈ K T (M ) Finally by Theorem 1.1 the representation homology H • ( A , v , w ) vanishes in positive degrees, sothat the Euler characteristic of its G -invariant part (4.15) is: χ GT ( A , v , w ) = [ H ( A , v , w ) G ] Cor 3.2.1 = [ O M ] . ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 31 (cid:3)
Remark 4.3.1.
In light of the previous explanations that we gave during the course of the paper,the result stated in Theorem 4.3.1 is not entirely surprising:(1) On one hand we have a symplectic resolution of singularities p : M χ → M therefore it isexpected that functions on the smooth variety M χ are equal to functions on the singular M .(2) On the other hand µ − ( ) is a complete intersection in the linear space of representations M ( Q , v , w ) , therefore the Koszul complex O ( DRep v , w ( A )) ∼ = O ( M ( Q , v , w )) ⊗ Λ • g v is aresolution of O ( µ − ( )) :(4.17) H i ( A , v , w ) = (cid:14) O ( µ − ( )) , i =
00 , i (cid:62) (cid:16) = ⇒ χ T ( A , v , w ) = O ( µ − ( )) (cid:17) and the subcomplex of G -invariants is a resolution of the functions on M :(4.18) H i ( A , v , w ) G = (cid:14) O ( µ − ( )) G , i =
00 , i (cid:62) (cid:16) = ⇒ χ GT ( A , v , w ) = O (M ) . (cid:17) As a corollary of Theorem 4.3.1, we can take Hilbert-PoincarÃl’ series (character for the torus) ofthe equality in (4.16) and obtain a equality between numerical (power) series counting the gradeddimensions. Formally, if M were compact, the Hilbert-PoincarÃl’ would be the pushforward tothe point: ch T : K T (M ) → K T ( p t ) = R ( T ) , instead in general we land in the field of fractions (see,for example, §4 in [31]) ch T : K T (M ) → Frac ( R ( T )) =: Q ( T ) . Remark 4.3.2.
If we consider the only fixed point for the torus action 0 ∈ M , and denote itsinclusion by ι : { } → M , then by functoriality we have ch T = ( ι ∗ ) − , and this tells us that isnot really necessary to invert all non-zero elements in R ( T ) , but only the ones of the form 1 − t β for non-zero weights β , so that we actually land in the following smaller localisation (see §2.1 and§2.3 in [34]): R ( T ) ,l oc := C (cid:20) t α , 11 − t β (cid:21) ,where α , β run over all weights of T and β (cid:54) = x ∈ T v ⊂ G the variables in the maximal torus of the gauge group ( KÃd’hlervariables ) and by t = ( a , (cid:32) h ) ∈ T = T w × T (cid:32) h the equivariant variables . Then we have, by Weyl’sintegral formula:(4.19)ch T (cid:16) χ GT ( A , v , w ) (cid:17) = | G | (cid:90) G ch G × T ( χ T ( A , v , w )) ( g , t ) dg = | W | (cid:90) T v ch T v × T ( χ T ( A , v , w )))( x , t ) ∆ ( x ) dx ,( W is the Weyl group of G , ∆ ( x ) is the Weyl factor,and integrations are over the compact real forms of G , T v )Moreover, because the Euler characteristic of the homology of a complex is equal to the Eulercharacteristic of the complex itself, we have(4.20) ch T v × T ( χ T ( A , v , w )) = ch T v × T (cid:0) O ( M ( Q , v , w )) ⊗ Λ • g (cid:1) = (cid:81) i ( − (cid:32) h (cid:32) h r i ) (cid:81) j ( − s j ) . where s j are the weights of M ( Q , v , w ) ∗ and r i are the weights of g :ch T v × T ( M ( Q , v , w )) = (cid:88) j s − j , ch T v (g) = (cid:88) i r i .To summarise: Corollary 4.3.1.
Under the same conditions of Theorem 4.3.1, and with the notation used in the previousequations (in particular (4.20) ), we have the following equality of PoincarÃl’-Hilbert series in the field offractions Q ( T ) : (4.21) ch T O (M χ ( Q , v , w )) = ch T O (M ( Q , v , w )) = | W | (cid:90) T v (cid:81) i ( − (cid:32) h (cid:32) h r i ) (cid:81) j ( − s j ) ∆ ( x ) dx .We calculate the above expression (4.21) in some concrete examples in § 5. Remark 4.3.3.
The right-hand side of (4.21) does not depend on the GIT parameter χ , while theleft-hand side a priori does. By picking different v -regular χ , χ (cid:48) we obtain a combinatorical identitych T O (M χ ( Q , v , w )) = ch T O (M χ (cid:48) ( Q , v , w )) ,which we will show to be non-trivial, also in simplest quiver cases (see § 5, specifically Remark 5.1.1in § 5.1).4.4. Other isotypical components and second integral formula.
In this section we prove a resultsimilar to Theorem 4.3.1 to relate other tautological sheaves with the corresponding isotypicalcomponents.Let us recall that to define M χ we fixed a character χ ∈ Hom
Grp C ( G , C × ) . This character definesa 1-dimensional representation C χ of G , whose image under the Kirwan map is the Serre twistingsheaf(4.22) κ T ([ C χ ]) = (cid:2) O M χ ( ) (cid:3) ∈ K T (M χ ) .For each V λ irreducible representation of G , we have a tautological sheaf V λ in the K -theory of M χ . By Serre vanishing theorem when we twist(4.23) V λ ( m ) := V λ ⊗ O M χ ( m ) ,by a sufficiently large power m (cid:29) χ T (M χ , V λ ( m )) = H (M χ , V λ ( m )) .Moreover, more or less by definition of the GIT quotient M χ , this is equal to the G -invariant globalsections of the trivial vector bundle V λ ⊗ C χ m over the stable locus:(4.25) H (M χ , V λ ( m )) = Γ (cid:16) µ − ( ) χ -s t , V λ ⊗ C χ m (cid:17) G .Finally for m (cid:29) Γ (cid:16) µ − ( ) , V λ ⊗ C χ m (cid:17) G ∼ −→ Γ (cid:16) µ − ( ) χ -s t , V λ ⊗ C χ m (cid:17) G ,but the left-hand side is nothing else but(4.27) Γ (cid:16) µ − ( ) , V λ ⊗ C χ m (cid:17) G = (cid:16) O ( µ − ( )) ⊗ V λ ⊗ C χ m (cid:17) G . ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 33
It is worth noticing at this point that irreducible representations V λ of G are labelled by collectionsof partitions λ = ( λ ( ) , . . . , λ ( n ) ) and that the representation V λ ⊗ C χ m is still a irreduciblerepresentation of G , corresponding to the shifted collection of partitions:(4.28) V λ ⊗ C χ m = V (cid:101) λ , (cid:101) λ := λ + mθ = ( λ ( ) + mθ , . . . , λ ( n ) + mθ n ) ( χ = χ θ ) ,(see Appendix B for the notation). We give the following definition: Definition 4.4.1.
We say that an irreducible representation V (cid:101) λ is large enough if (cid:101) λ = λ + mθ (see (4.28)) with m (cid:29) Q , on the dimension vectors v , w and on the v -regular χ = χ θ .Denoting by (cid:101) λ ∗ the partition corresponding to the dual representation, we can continueequation (4.27) to recognise:(4.29) (cid:16) O ( µ − ( )) ⊗ V (cid:101) λ (cid:17) G = (cid:16) O ( µ − ( )) ⊗ V ∗ (cid:101) λ ∗ (cid:17) G = H ( A , v , w ) G (cid:101) λ ∗ ,the isotypical component of (cid:101) λ ∗ of the (zeroth) representation homology. Finally if we observe thatwith flat moment map, higher homologies vanish, we obtain the following result: Theorem 4.4.1.
Let v , w be dimension vectors for which the moment map is flat, and fix χ = χ θ with θ v -regular. For λ large enough (in the sense of Definition 4.4.1) we have (4.30) p ∗ ([ V λ ]) = [ H (M χ , V λ )] = χ λ ∗ T ( A , v , w ) ∈ K T (M ( Q , v , w )) .The analogous integral formula to obtained by taking characters is Corollary 4.4.1.
Under the same conditions of Theorem 4.4.1, and with the notation used in (4.20) , wehave the following equality of PoincarÃl’-Hilbert series in the field of fractions Q ( T ) : (4.31) ch T ( χ T (M χ , V λ )) = ch T ( H (M χ , V λ )) = | W | (cid:90) T v (cid:81) i ( − (cid:32) h (cid:32) h r i ) (cid:81) j ( − s j ) f λ ( x ) ∆ ( x ) dx . where f λ ( x ) = ch T v ( V λ ) (it is the product of Schur polynomials associated to the partitions in λ ).
5. E xamples
In this section we explain some concrete examples, mainly from the easiest quivers alreadyconsidered in the previous sections. We see how such elementary quivers still produce varieties ofgreat interest in various fields of mathematics.5.1.
Cotangent bundle of Grassmannian.
The quiver Q = A with only one vertex and noarrows. The framed, doubled quiver has two vertices and two arrows connecting them in oppositedirections. • v Single-vertex quiver • v (cid:3) wij Figure 2.
Framing and doubling the single-vertex quiver.
Therefore: µ − ( ) = { ( I , J ) ∈ Hom C ( C w , C v ) ⊕ Hom C ( C v , C w ) | I ◦ J = } . Because we have only one vertex we have to choose the GIT parameter θ ∈ Z , and it is easy tocheck that the v -regularity condition means simply θ (cid:54) = v ). For θ (cid:54) = θ -semistable points = (cid:14) J injective, θ < I surjective, θ > T ∗ G r ( v , w ) of v -planes in C w in thecase θ < T ∗ G r ( w − v , w ) in the case θ >
0. The two varieties are isomorphic to each other,but we have the following different identifications of the points in the Grassmannian: θ <
0: Im ( J ) ∈ G r ( v , w ) , θ >
0: ker ( I ) ∈ G r ( w − v , w ) .The affine quotient can be identified (using some version of the fundamental theorem of invarianttheory): M = Spec (cid:0) O ( µ − ( )) GL v (cid:1) ∼ = { A ∈ M w × w ( C ) | A =
0, rk ( A ) (cid:54) v } ,where A represents the composition J ◦ I : C w → C w . The condition on the rank is due to the factthat A : W → V → W factorises through V , but sometimes it is superfluous. In fact in general A = ( A ) (cid:54) (cid:98) w/ (cid:99) . The moment map is flat if and only if 2 v − (cid:54) w (seeExamples 4), and only in this cases the projective morphism p : T ∗ G r ( v , w ) → M is a resolution of singularities. p M θ ∼ = T ∗ P M ∼ = Spec (cid:16) C [ a , b , c ]( a + bc ) (cid:17) Figure 3.
The (real) picture of the case ( v , w ) = (
1, 2 ) : this is also known as Springer resolution ofthe nilpotent cone of sl ( C ) . In this case in the torus T = T w × T (cid:32) h only the product (cid:32) h (cid:32) h appears and we denote it by (cid:32) h . Wecan use (4.21) for χ = χ − for which M χ = T ∗ G r ( v , w ) and obtain a formula for the character ofthe ring of functions on the cotangent bundle of Grassmannian:(5.1)ch T ( O ( T ∗ G r ( v , w ))) = v ! · (cid:73) | x α | = (cid:81) α , β ( − (cid:32) hx − α x β ) (cid:81) α , γ ( − (cid:32) hx − α a γ )( − x α a − γ ) · ∆ ( x ) (cid:122) (cid:125)(cid:124) (cid:123)(cid:89) α (cid:54) = β ( − x − α x β ) dx (cid:122) (cid:125)(cid:124) (cid:123)(cid:89) α dx α πix α ,where in the above x = ( x α ) = ( x , . . . , x v ) and a = ( a γ ) = ( a , . . . , a w ) . ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 35
The integral in the right-hand side can be computed by iterated residues, and by doing so wecan recognise the localisation formula in equivariant K-theory as a sum over the fixed points p ∈ ( T ∗ G r ( v , w )) T of the inverse of the K-theoretic Euler class of the tangent space at that point:(5.2)ch T ( O ( T ∗ G r ( v , w ))) = (cid:88) B ⊂ { w } B = v (cid:81) β ∈ Bγ/ ∈ B (cid:16) − a β a γ (cid:17) (cid:16) − (cid:32) h a γ a β (cid:17) = (cid:88) p ∈ ( T ∗ G r ( v , w )) T T (cid:0) Λ − T ∗ p ( T ∗ G r ( v , w )) (cid:1) .For what concerns other sheaves, let us consider the standard representation V = C v of G = GL v ( C ) .The associated tautological sheaf V on M χ − = T ∗ G r ( v , w ) is indeed the usual tautologicalsheaf of rank v . Irreducible representations are labelled by Schur functors V λ = S λ ( V ) where λ = ( λ (cid:62) · · · (cid:62) λ v ) is a integer partition of v parts, and we consider the corresponding tautologicalsheaves V λ . For example the (standard) tautological sheaf itself is V = V ( ) , or powers of theSerre twisting sheaf are:(5.3) O T ∗ G r ( v , w ) ( m ) = det − m ( V ) = V (− m ,..., − m ) .A partition λ becomes large (Definition 4.4.1) in the sense that we can apply Theorem 4.3.1 whenall its components are negative enough (because the character χ = χ − is negative), and it turnsout that it suffices to have λ (cid:54)
0, that is equivalent to say that the partition is made of non-positiveterms (an example is (5.3), in which for m > T H (cid:0) T ∗ G r ( v , w ) , V λ (cid:1) = v ! · (cid:73) | x | = (cid:0) (cid:81) α , β ( − (cid:32) hx − α x β ) (cid:1) s λ ( x ) (cid:81) α , γ ( − (cid:32) hx − α a γ )( − x α a − γ ) · (cid:89) α (cid:54) = β ( − x − α x β ) (cid:89) α dx α πix α ,where s λ ( x ) = ch T v ( V λ ) is the Schur polynomial associated to the partition λ . Again, the integralin the right-hand side can be computed by means of iterated residues, giving the localisationformula for the corresponding tautological sheaf:(5.5)ch T H (cid:0) T ∗ G r ( v , w ) , V λ (cid:1) = (cid:88) B ⊂ { w } B = v s λ ( a B ) (cid:81) β ∈ Bγ/ ∈ B (cid:16) − a β a γ (cid:17) (cid:16) − (cid:32) h a γ a β (cid:17) = (cid:88) p ∈ ( T ∗ G r ( v , w )) T ch T ( V λ ) | p ch T (cid:0) Λ − T ∗ p ( T ∗ G r ( v , w )) (cid:1) ,where the expression s λ ( a B ) means that we are evaluating the Schur polynomial s λ ( x , . . . , x v ) inthe point x = ( a β ) β ∈ B . Remark 5.1.1.
As already observed in Remark 4.3.3 the right-hand side of the integral for-mula (4.21) does not depend on the character χ , while a priori the left-hand side does. In (5.1)we used the character χ = χ − for which M χ = T ∗ G r ( v , w ) . If we use χ (cid:48) = χ we have M χ (cid:48) = T ∗ G r ( w − v , w ) . The fixed point formula for the first variety (5.2) can be comparedwith the one for the second variety, and it gives a non-trivial combinatorical identity:(5.6) (cid:88) B ⊂ { w } B = v (cid:81) β ∈ Bγ/ ∈ B (cid:16) − a β a γ (cid:17) (cid:16) − (cid:32) h a γ a β (cid:17) = (cid:88) B ⊂ { w } B = v (cid:81) β ∈ Bγ/ ∈ B (cid:16) − a γ a β (cid:17) (cid:16) − (cid:32) h a β a γ (cid:17) .5.2. Framed moduli space of torsion free sheaves on P . This is the case of the Jordan quiver,the quiver with one vertex and one loop, Figure 1. Therefore: µ − ( ) = { ( X , Y , I , J ) ∈ Hom C ( C v , C v ) ⊕ ⊕ Hom C ( C w , C v ) ⊕ Hom C ( C v , C w ) | [ X , Y ] + I ◦ J = } . For GIT paramater θ ∈ Z : θ -semistable points = (cid:14) (cid:64) (cid:54) = S ⊂ V s.t. C (cid:104) X , Y (cid:105) ( S ) ⊂ S and S ⊂ ker ( J ) , θ < (cid:64) S (cid:40) V s.t. C (cid:104) X , Y (cid:105) ( S ) ⊂ S and Im ( I ) ⊂ S , θ > M θ and M ( w , v ) , the(framed) moduli space of torsion free sheaves on CP of rank w , second Chern class c = v , andfixed trivialisation at the line at ∞ . The affine Nakajima variety is M ∼ = M ( w , v ) the framedmoduli space of ideal instantons on S = C ∪ { ∞ } . The map p : M ( w , v ) → M ( w , v ) is always aresolution of singularities because the moment map is always flat.When the framing is w = v points on C to the symmetric v -power: p : Hilb v ( C ) → Sym v ( C ) .For general v and w the integral formula looks like:(5.7) ch T O ( M ( w , v )) = v ! · (cid:73) | x | = I ( x , a , (cid:32) h ) · (cid:89) α (cid:54) = β ( − x − α x β ) (cid:89) α dx α πix α ,where I ( x , t , (cid:32) h ) = (cid:81) α , β ( − (cid:32) h (cid:32) h x − α x β ) (cid:81) α , β ( − (cid:32) h x − α x β )( − (cid:32) h x − α x β ) · (cid:81) α , γ ( − (cid:32) h (cid:32) h x − α t γ )( − x α t − γ ) ,and it is also known as the integral formula for Nekrasov partition function (proved for examplein Appendix A of [14]).For other isotypical components, let us say that we fixed χ = χ . Again we have a tautologicalsheaf of rank v , V , and other sheaves associated to irreducible representations are labelled by Schurfunctors V λ where λ is an integer partition of v parts. In this case the largeness condition indeedmeans that the partition is big enough, and it turns out that it suffices for it to be non-negative λ (cid:62) · · · (cid:62) λ v (cid:62)
0. In this range we have:(5.8) ch T H ( M ( w , v ) , V λ ) = v ! (cid:73) | x | = I ( x , a , (cid:32) h ) · s λ ( x ) · (cid:89) α (cid:54) = β ( − x − α x β ) (cid:89) α dx α πix α .For λ (cid:62) s λ ( x ) is indeed an actual polynomial (and not a Laurentpolynomial), and therefore with (5.8) we recover the integral formula for Nekrasov partitionfunction with matter fields (the matter field is represented by the sheaf V λ in this case) which wasproved for example in [25].5.3. Symplectic dual of T ∗ P n − . X = T ∗ G r ( k , n ) has a symplectic dual, X ˇ, which for the choiceof parameters 2 k (cid:54) n can be shown to be also a Nakajima quiver variety ([39]). Specifically it isthe Nakajima variety associated to the following A n − quiver, with dimension vectors: v = (
1, 2, . . . , k − k , . . . , k (cid:124) (cid:123)(cid:122) (cid:125) ( n − k + ) -times , k −
1, . . . , 2, 1 ) , w = ( w , . . . , w n − ) w i = δ i , k + δ i , n − k .We restrict to the case k =
1, for which dimension vectors are(5.9) (cid:14) v = (
1, . . . , 1 ) , w = (
1, 0, . . . , 0, 1 ) , ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 37 and the corresponding Nakajima quiver variety is the symplectic dual of T ∗ P n − . For n = A case with dimensions v = w =
2, so we find that T ∗ P is symplectic dualto itself. Let us study the other cases n (cid:62) x , . . . , x n − , their dual by y , . . . , y n − andthen we have i , j and i n − , j n − because of the non-trivial framing at the vertices 1 and n − n -dimensional affinespace µ − ( ) ∼ = Spec (cid:18) C [ x , y , . . . , x n − , y n − , i , j , i n − , j n − ] i j = x y = x y = · · · = x n − y n − = − i n − j n − (cid:19) .The gauge group is a n − G v = GL ( C ) n − = ( C × ) n − , and the affineNakajima variety is identified with the ADE singularity of type A n − :(5.10) M ∼ = Spec (cid:18) C [ x , y , z ] xy = z n (cid:19) ∼ = C / Z n ,where x = x · · · x n − i j n − , y = y · · · y n − i n − j , z = x y . We recall that the action Z n (cid:121) C that gives the corresponding ADE singularity of type A n − is given by the embedding Z n ⊂ SL ( C ) in which a n -th root of unity ξ ∈ Z n becomes the matrix diag ( ξ , ξ − ) ∈ SL ( C ) .We fix GIT parameter χ = χ θ + with θ + = (
1, 1, . . . , 1 ) . The corresponding smooth Nakajimaquiver variety is a consecutive ( n − x = y = z = p : M χ + = (cid:94) C / Z n −−−→ M = C / Z n ,with exceptional fiber p − ( ) given by n − P intersecting in such away that their underlying intersection graph is A n − (see [11]), as shown in Figure 4. Figure 4.
Every sphere is replaced by a vertex and two vertices are linked by as many arrows asintersection points of the corresponding spheres.
The associated derived representation scheme isDRep v , w = Spec ( C [ x , y , x , y , . . . , x n − , y n − , i , j , i n − , j n − , ϑ , . . . , ϑ n − ]) ,where ϑ i have homological degree 1 and differential(5.12) dϑ = − y x + i j , dϑ k = x k − y k − − y k x k , ( k =
2, . . . , n − ) , dϑ n − = x n − y n − + i n − j n − ,and they are invariants under the gauge group G v = GL n − , so that the associated characterscheme is simplyDRep G v v , w ∼ = Spec (cid:18)(cid:18) C [ x , y , z , . . . , z n − , z n − , z n ] xy = z · · · z n (cid:19) [ ϑ , . . . , ϑ n − ] (cid:19) , where x , y are the same classes as before in (5.10), z k = x k y k for k =
1, . . . , n − z n − = i j , z n = i n − j n − . We denote the variables in the equivariant torus T = T w × T (cid:32) h by ( a , ˜ a , (cid:32) h , (cid:32) h ) (where a is on the vertex 1 and ˜ a on the vertex n −
1) and we have:(5.13)ch T O (cid:16) (cid:94) C / Z n (cid:17) = ch T O (cid:16) C / Z n (cid:17) = ch T (cid:16) χ T ( DRep G v v , w ) (cid:17) = + (cid:32) h (cid:32) h · · · + (cid:32) h n − (cid:32) h n − (cid:16) − (cid:32) h n − (cid:32) h a ˜ a (cid:17) (cid:16) − (cid:32) h (cid:32) h n −
12 ˜ aa (cid:17) .A ppendix A. P rojective model structure on T- equivariant dg - algebras In this Appendix we give a proof of Theorem 2.5.1 that gives a projective-like model structureon the category of T -equivariant dg-algebras ( DGA + k ) T , for an algebraic torus T = ( k × ) r . Weuse the same strategy used in [6], in which the authors prove that the category of bigraded dg-algebras BiDGA k has a projective-like model structure . The key observation is to recognise that BiDGA k being the category of dg-algebras with an additional non-negative (polynomial) compatiblegrading, is equivalent to the category of T -equivariant dg-algebras with a polynomial torus action(i.e. weight spaces are only for non-negative weights), and that the polynomial condition can bedropped, and substituted by the rational condition, in which weights can be arbitrary integers.More precisely, weight spaces for a torus T = ( k × ) r are r -tuples of integers n ∈ Z r , and weobserve that the category of dg-algebras with a rational T -action ( DGA + k ) T is equivalent to thecategory of dg-algebras A ∈ DGA + k with:(1) An additional grading of the underlying chain complex A = ⊕ n ∈ Z r A ( n ) . This means thateach A ( n ) is a complex of vector spaces preserved by the differential in A : dA ( n ) ⊂ A ( n ) .(2) The grading is compatible with the multiplication in A : A ( n ) · A ( m ) ⊂ A ( n + m ) .In fact, on one hand if A ∈ ( DGA + k ) T then for n ∈ Z r we define A ( n ) = { a ∈ A | t · a = t n a , ∀ t ∈ T } as the corresponding weight space and the above 2 conditions are satisfied thanks to the rationalityof the action (recall, Definition 2.5.1). On the other hand, obviously if we have such a decompositionwe define the T -action on A accordingly by t · a := (cid:80) n t n a ( n ) , where a = (cid:80) n a ( n ) , and theresulting T -action is rational.The observation that ( DGA + k ) T is equivalent to the category of dg-algebras with an additionalgrading as described above will be also useful later, and we will use indifferently one or the otherproperty, according to what is more convenient from time to time.Let us also denote by k [ T ] = O ( T ) , a Laurent polynomial ring in r variables and observe that Lemma A.0.1.
The forgetful functor U : ( DGA + k ) T → DGA + k is left-adjoint to the “free T -equivariantextension” functor: (A.1) ( DGA + k ) T U (cid:47) (cid:47) ⊥ DGA + kk [ T ] ⊗ (−) (cid:111) (cid:111) . Proof.
The adjunction is given by the natural isomorphisms:Hom
DGA + k ( UA , B ) ∼ = Hom ( DGA + k ) T ( A , k [ T ] ⊗ B ) ,where to a T -equivariant morphism ϕ : A → k [ T ] ⊗ B we assign the composition with theevaluation map at 1 ∈ T : A ϕ −→ k [ T ] ⊗ B ev ⊗ B −−−−−→ k ⊗ B ∼ = B . Which ultimately follows the explicit proofs of the existence of the projective model structure on
DGA + k by [27] or [15]. ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 39
Conversely if we start from a map f : UA → B which is not necessarily T -equivariant, we canconstruct a T -equivariant map ϕ : A → k [ T ] ⊗ B by decomposing: ϕ : (cid:77) n ∈ Z r A ( n ) → (cid:77) n ∈ Z r B · t n ,and defining ϕ | A ( n ) : A ( n ) → B · t n as f | A ( n ) (−) · t n . (cid:3) In order to prove Theorem 2.5.1 we need a few definitions and lemmas. Throughout this sectionof the Appendix we denote by C = DGA + k and by C T = ( DGA + k ) T . Notation.
We denote by C of , WE , F ib the collection of cofibrations, weak equivalences, and fibra-tions in the projective model structure on C . So F ib are surjective maps in positive homologicaldegrees, WE are the quasi-isomorphisms, and C of = (cid:27) ( WE ∩ F ib ) , where (cid:27) (−) denotes the collec-tion of morphisms with the left lifting property with respect to another collection of morphisms.Finally recall that a fibration which is also a quasi-isomorphism is actually surjective in all homological degrees, so that WE ∩ F ib consists of surjective quasi-isomorphisms. Definition A.0.1.
A morphism i : S → R ∈ C T is a T -equivariant noncommutative Tate extension (alsosimply a Tate extension) if there is a (possibly infinite) sequence V ( ) ⊂ V ( ) ⊂ V ( ) ⊂ . . . of(homologically) graded, T -equivariant vector spaces such that(1) Each S ∗ k T ( V ( i ) ) has a differential and a compatible embedding S ∗ k T ( V ( i ) ) ⊂ R suchthat at the limit V = ∪ i V ( i ) : S ∗ k T V = lim −→ i S ∗ k T ( V ( i ) ) = R .(2) Each differential has the property that d ( V ( i ) ) ⊂ S ∗ k T ( V ( i − ) ) (and for i = d ( V ( ) ) ⊂ S ).We denote the collecion of such morphism by TE ⊂ M or ( C T ) . Lemma A.0.2. (i) Every morphism S → A in C T has a factorisation of the form S i −→ R p −→ A where i ∈ TE and p ∈ U − ( WE ∩ F ib ) (is a surjective quasi-isomorphism).(ii) Every Tate extension has the left lifting property with respect to morphisms that are surjectivequasi-isomorphisms: TE ⊂ (cid:27) ( U − ( WE ∩ F ib )) . For the proof one can check that the proof of Proposition 3.1 (which relies on Proposition 2.1(ii))of [15] can be used also in this case of T -equivariant (i.e. additionally graded) objects.Now let x be a variable of positive homological degree as well as of some weight n ∈ Z r forthe torus T , and set V x := [ → k · x → k · dx → ] , and its tensor algebra T ( V x ) ∈ C T . Extensionsby objects of this form play another important role: Definition A.0.2.
A morphism in C T of the form S → S ∗ k (cid:96) i ∈ I T ( V x i ) , where I is any, possiblyuncountably infinite, indexing set, is called a special extension . We denote the collection of specialextensions by SE ⊂ M or ( C T ) . Lemma A.0.3. (i) Every morphism S → A in C T has a factorisation of the form S i −→ R p −→ A where i ∈ SE and p ∈ U − ( F ib ) .(ii) SE ⊂ (cid:27) ( U − ( F ib )) .(iii) SE ⊂ U − ( WE ) .Proof. (i) It suffices to consider the set of elements of A of positive homological degrees as well asof some weight for the torus action: I := { a ∈ A ( n ) i | n ∈ Z r , i > } . For each a ∈ I we consider the obvious T V x a p a −−→ A given by p a ( x a ) = a (and consequently p a ( dx a ) = da ). Then S i −→ S ∗ k (cid:97) a ∈ I T ( V x a ) f ∗ k (cid:96) a ∈ I p a −−−−−−→ A is the desired factorisation. (ii) and (iii) are quite obvious. (cid:3) Now we have everything we need to prove that the following definition yields the desiredmodel structure on C T : Definition A.0.3.
We define weak equivalences, fibrations and cofibrations in C T as:(A.2) WE T := U − ( WE ) , F ib T := U − ( F ib ) , C of T := (cid:27) ( WE T ∩ F ib T ) = (cid:27) ( U − ( WE ∩ F ib )) .We observe that, by Lemma A.0.2(ii), Tate extensions are cofibrations: TE ⊂ C of T , and byLemma A.0.3, special extensions are acyclic cofibrations: SE ⊂ WE T ∩ C of T . In fact, it is useful toobserve that Proposition A.0.1.
Every acyclic cofibration in C T is a retract of a special extension.Proof. Let i : A → B be an acyclic cofibration and let us factor it as A (cid:101) i −→ R q −→ B where (cid:101) i is a specialextension and q is a fibration, according to Lemma A.0.3(i). q is also a weak equivalence, becauseof the 2-out-of-3 property (see (MC2) in the proof of the next Theorem), therefore it is an acyclicfibration, and we can find a lift of the diagram:(A.3) A RB B (cid:101) ii q B ∃ l which proves that i is a retract of the special extension (cid:101) i :(A.4) A A AB R B A i A (cid:101) i il B q (cid:3) Theorem (2.5.1) . (1) Definition A.0.3 defines a model structure on C T .(2) The forgetful functor U : C T → C preserves cofibrations.Proof. (1) (MC1) (notation of Definition 3.3 of [12]): finite limits and colimits exist in C T becauseequalizers, coequalizers, finite product and finite coproducts exist (the same constructions as in C work in the equivariant setting). (MC2) WE T has the 2-out-of-3 property because it is U − ( WE ) with WE having the 2-out-of-3 property. (MC3) WE T and F ib T are closed under retracts because,again, defined as U − of classes closed under retracts. C of T are closed under retracts because theyare defined as the morphisms with the left lifting property with respect to some class (cid:27) A , andthis is always closed under retracts (it does not matter what A is). (MC4) We need to prove thatfor a diagram in C T of the following form:(A.5) A CB D fi pg
ERIVED REPRESENTATION SCHEMES AND NAKAJIMA QUIVER VARIETIES 41 a lift exists in the following situations: (i) i is a cofibration and p is an acyclic fibration, (ii) i isan acyclic cofibration and p is a fibration. (i) is obviously true by the definition of cofibrations.To prove that a lift exists in the case (ii), thanks to Proposition A.0.1 we only need to find a liftwhen i is a special extension, but this is true by Lemma A.0.3(ii). (MC5) We need to prove thateach morphism S → A in C T has factorisations of the form: (i) cofibration followed by an acyclicfibration, (ii) acyclic cofibration followed by a fibration. (i) follows from Lemma A.0.2(i), and (ii)follows from Lemma A.0.3(i).(2) This follows from the fact that U is left adjoint to k [ T ] ⊗ (−) (Lemma A.0.1), and the latterpreserves weak equivalences and fibrations, therefore U preserves cofibrations. (cid:3) A ppendix B. R epresentation theory of G = G v In this Appendix we recall the theory of irreducible representations of (a product of) generallinear groups and we fix the notation. Polynomial irreducible representations of GL v ( C ) arelabelled by ordinary (non-negative) partitions λ = ( λ , . . . , λ v ) . More precisely, they are obtainedby applying the Schur functors S λ (−) : Vect C → Vect C to the standard representation V = C v :(B.1) S λ ( V ) .Their characters, the Schur polynomials, form a linear basis of the ring of symmetric polynomialsin v variables:(B.2) s λ ( x ) := ch ( S λ ( V )) ∈ Z [ x , . . . , x v ] Σ v .Examples are(1) V = C v itself is S ( ) ( V ) and s ( ) ( x ) = x + · · · + x v .(2) More generally S ( d ,0,...,0 ) ( V ) = Sym d ( V ) and s ( d ,0,...,0 ) ( x ) = h d ( x ) the complete symmetricpolynomial.(3) For λ = (
1, 1, . . . , 1, 0, . . . 0 ) with 1 repeated d -times, S λ ( V ) = Λ d ( V ) and s λ ( x ) = e d ( x ) theelementary symmetric polynomial.(4) 1-dimensional representations are given by m := ( m , m , . . . , m ) , for which S m ( V ) = det ( V ) m ,and s λ ( x ) = e v ( x ) m = x m · · · x mv .If we shift a partition λ to λ + m := ( λ + m , . . . , λ v + m ) we have(B.3) S λ + m ( V ) = S λ ( V ) ⊗ det ( V ) m ,which allows to extend the definition of Schur functors to partitions made possibly of somenegative parts λ ∈ P v := { λ ∈ Z v | λ (cid:62) · · · (cid:62) λ v } as(B.4) S λ ( V ) := S λ − λ v ( V ) ⊗ det ( V ) λ v .All irreducible rational representations of GL v ( C ) are of the form (B.4) for some integer-valuedpartition λ ∈ P v . Their characters are generalised Schur polynomials and they form a linear basisof the ring of symmetric Laurent polynomials in v variables:(B.5) s λ ( x ) = ch ( S λ ( V )) ∈ Z [ x , x − , . . . , x v , x − v ] Σ v .If now v = ( v , . . . , v n ) is a dimension vector and G v = (cid:81) i GL v i ( C ) is a product of generallinear groups, then its irreducible rational representations are labelled by n -tuples of partitions λ = ( λ ( ) , . . . , λ ( n ) ) ∈ (cid:81) i P v i , as the external tensor product of Schur modules:(B.6) V λ := S λ ( ) ( C v ) (cid:2) · · · (cid:2) S λ ( n ) ( C v n ) .Their characters are products of (generalised) Schur polynomials and we denote them by (thesame notation as in (4.31)):(B.7) f λ ( x ) := ch ( V λ ) = s λ ( ) ( x ( ) ) · · · s λ ( n ) ( x ( n ) ) , where x = ( x ( ) , . . . , x ( n ) ) and each x ( i ) is a set of v i variables: x ( i ) = ( x ( i ) , . . . , x ( i ) v i ) .R eferences
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