Multiplicative preprojective algebras are 2-Calabi-Yau
aa r X i v : . [ m a t h . R A ] M a y The 2-Calabi–Yau property for multiplicativepreprojective algebras
Daniel Kaplan and Travis SchedlerMay 2019
Abstract
We prove that multiplicative preprojective algebras, defined by Crawley-Boevey andShaw, are 2-Calabi–Yau algebras for quivers containing an unoriented cycle. We alsoprove that the dg versions of these algebras (arising in Fukaya categories of certain We-instein 4-manifolds) are formal with homology the multiplicative preprojective algebraconcentrated in degree zero. We prove that the center is trivial in the case that the cy-cle is properly contained, and hence the Calabi–Yau structure is unique. We conjecturethat the same properties hold for all non-Dynkin, non-extended Dynkin quivers, andexplain how to reduce the conjecture to extended-Dynkin quivers (note that the cycleis the type A case). As an application, corresponding multiplicative quiver varietiesare normal and are locally isomorphic to ordinary quiver varieties. This includes char-acter varieties of Riemann surfaces with punctures and monodromy conditions; quiverscontaining cycles cover all positive genus cases. Contents θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Introduction
Multiplicative preprojective algebras have recently gained attention in geometry and topol-ogy. These algebras appear in the study of certain wrapped Fukaya categories, see [EL17c],[EL17b], in the study of microlocal sheaves on rational curves, see [BK16], and in thestudy of generalized affine Hecke algebras, see Appendix 1 in [EOR07]. Their modulispaces of representations are called multiplicative quiver varieties, an analogue of Naka-jima’s quiver varieties, which includes character varieties of rank n local systems on closedRiemann surfaces, or on open Riemann surfaces with punctures and monodromy conditions[CBS06, Yam08, ST18]. Multiplicative quiver varieties have also been studied from variousviewpoints in [VdB07], [Boa13], [CB13], and [CF17]. A quantization was defined in [Jor14]and further studied in [GJS19].Historically, the multiplicative preprojective algebra was defined by Crawley-Boevey andShaw, in [CBS06], to view solutions of the Deligne–Simpson problem as irreducible represen-tations of multiplicative preprojective algebras of certain star-shaped quivers. Their paperestablishes the foundations for much of this work. For a fixed field k and a quiver Q withvertex set Q and arrow set Q and q P p k ˆ q Q , Crawley-Boevey and Shaw defineΛ q p Q q : “ L Q J Q : “ kQ rp ` aa ˚ q ´ s a P Q ´ r : “ ś a P Q p ` aa ˚ qp ` a ˚ a q ´ ´ q ¯ , a quotient of the localized path algebra of the double quiver, L Q , by the two-sided ideal J Q generated by a single relation, r .In section 7.5 of [ST18], Tirelli and the second author conjecture that multiplicativepreprojective algebras of connected, non-Dynkin quivers are 2-Calabi–Yau. They observe,following [BGV16], that the 2-Calabi–Yau property would determine the (formal) localstructure of the moduli space of representations. Namely, any formal neighborhood canbe identified with the formal neighborhood of the zero representation of the moduli spaceof representations of some (additive) preprojective algebra. Among other applications men-tioned in [ST18], it would follow that multiplicative quiver varieties are normal. In light ofthese applications, and others mentioned below, the goal of this paper is to work towards aproof of the following conjecture: Conjecture 1.1. Λ q p Q q is 2-Calabi–Yau for all q P p k ˆ q Q and all Q connected and notDynkin; moreover, it is a prime ring. If Q is furthermore not extended Dynkin, then Z p Λ q p Q qq “ k , and the Calabi–Yau structure is unique. We explain how one can reduce these conjectures to the extended Dynkin cases below,see Section 6.5 for more detail. We carry this procedure out for Q “ ˜ A n , thereby provingthe conjecture for all connected quivers containing it: Theorem 1.2. Λ q p Q q is 2-Calabi–Yau and prime for any q P p k ˆ q Q and any k a field,and Q connected and containing an unoriented cycle. If the containment is proper, then Z p Λ q p Q qq “ k , and the Calabi–Yau structure is unique. As observed, it follows the multiplicative quiver varieties for quivers containing cyclesare formally locally isomorphic to ordinary quiver varieties, and in particular are normal.This includes (as an open subset) character varieties of Riemann surfaces of positive genuswith punctures and prescribed monodromy conditions, as explained in [ST18, §
3] (following[CBS06, Yam08]). (In the case of closed Riemann surfaces, as observed in [BS16, Remark8.8], this statement does not require our result, since the group algebra C r π p Σ qs is well-known to be 2-Calabi–Yau.) Recall that a prime ring is a noncommutative analogue of an integral domain, being a ring R in which aRb “ a “ b “
2n addition to the above perspective of giving a (formal) local description of multiplicativequiver varieties, there are various other perspectives on this work. Namely, multiplicativepreprojective algebras arise from studying certain wrapped Fukaya categories. If X Γ is aplumbing of cotangent bundles according to the graph Γ, then Ekholm and Lekili [EL17a]and Etg¨u and Lekili [EL17c], [EL17b] produced quasi-isomorphisms, W p X Γ q r EL17a s– / / B Γ r EL17c s , r EL17b s– / / L Γ where W p X Γ q denotes the partially wrapped Fukaya category of X Γ , B Γ denotes the Chekanovand Eliashberg dg-algebra and L Γ is a derived multiplicative preprojective algebra. It fol-lows from Ganatra’s thesis [Gan12] that L Γ is 2-Calabi–Yau, as a dg-algebra. We establishthis result purely algebraically, in the case Γ contains a cycle. We show that H ˚ p L Γ q Section 4 “ / / H p L Γ q “ Λ p Γ q and hence L Γ is formal. Note that for the ordinary preprojective algebra, Π p Q q , the Ginzburgdg-algebra has homology concentrated in degree zero: Π p Q q has a length two bimoduleresolution (see [BBK02, MV96] for the characteristic zero case, and [EE07] in general) whichAnick shows is equivalent for graded connected algebras in [Ani82, Theorems 2.6 & 2.9], and[EG07] observes this extends to the quiver case.Consequently, deformations of the wrapped Fukaya category W p X Γ q as an A (respec-tively A Calabi–Yau) category over a degree zero base are given by deformations of Λ p Γ q as an associative (respectively Calabi–Yau) algebra. The infinitesimal deformations can beidentified with HH p Λ p Γ qq , which by Van den Bergh duality assuming Conjecture 1.1 isisomorphic to HH p Λ p Γ qq . The techniques in [Sch16] can likely be adapted to compute thezeroth Hochschild homology using an explicit basis for Λ q p Q q computed here. Furthermore,by the 2-Calabi-Yau property, HH p Λ q p Q qq “
0, so there are no obstructions to extend-ing to infinite order deformations. We conjecture that the same holds for every connected,non-Dynkin quiver. More precisely:
Conjecture 1.3. If Q is a connected, non-Dynkin quiver, then the dg multiplicative pre-projective algebra Λ dg,q p Q q is quasi-isomorphic to Λ q p Q q , in degree zero. We give the precise definitions and details in Section 4.Another perspective on this work involves specializing to Q E an extended Dynkin quiver and q “
1. In Shaw’s thesis, [Sha05], he shows that for the extended vertex v , in analogy withthe additive case, the subalgebra e v Λ p Q E q e v is commutative of dimension 2 and is smoothaway from the origin. In further analogy, it is reasonable to pose the following conjecture: Conjecture 1.4.
The algebra Λ p Q E q is a 2-dimensional NCCR (non-commutative crepantresolution) over its center, Z p Λ p Q E qq . The Satake map Z p Λ p Q E qq Ñ e v Λ p Q E q e v , z ÞÑ e v z , is an isomorphism. For Q E “ ˜ A n , our results imply the conjecture as follows: One can identifyΛ p ˜ A n q – ÝÑ End e Λ p ˜ A n q e p e Λ p ˜ A n qq , by identifying e Λ p ˜ A n q e – k r X, Y, Z s{p Z n ` ` XY ` XY Z q as an algebra (which is com-mutative) and e Λ p ˜ A n q e i – M i : “ p Z i , Y q , the ideal. The isomorphism is then explicitlygiven: We use the terminology “Satake” following the analogous one for symplectic reflection algebras at t “ on vertices by i ÞÑ id : M i Ñ M i ; • on arrows by a i ÞÑ Z : M i Ñ M i ` for i ă n and a n ÞÑ Z {p ` Y q Z : M n Ñ M ; • on reverse arrows by a ˚ i ÞÑ inc : M i ã Ñ M i ´ for i ą a ˚ ÞÑ Y : M Ñ M n .The surjectivity follows from the observation that every e Λ p ˜ A n q e -module map of idealsis given by left multiplication by some element of the field of fractions of e Λ p ˜ A n q e . Theinjectivity follows from the fact that Λ p ˜ A n q is prime (see Proposition 6.8) and injectivityon e Λ p ˜ A n q e .Now, to prove the second statement, generally, if A is an algebra and e an idempotent,then End A op p eA q – eAe . In the case End eAe p eA q – A op also, we obtain an identification Z p eAe q – End eAe b A op p eA q – Z p A q . Explicitly, the map Z p A q Ñ Z p eAe q is given by z ÞÑ ez .If in addition, as here, eAe is commutative, we obtain Z p A q – eAe , via the Satake map.Finally, to complete the proof of the first statement, the computation above shows thatHom e Λ p ˜ A n q e p M i , M j q – e i Λ p ˜ A n q e j – M j ´ i as a module over e Λ p ˜ A n q e – Z p Λ p ˜ A n qq – e i Λ p ˜ A n q e i , so in particular e Λ p ˜ A n q is self-dual and hence reflexive as a e Λ p ˜ A n q e -module. Since furthermore Λ p ˜ A n q is 2-Calabi–Yau (by Theorem 1.2) and e Λ p ˜ A n q e is Gorenstein (as it is a hypersurface in A ), it follows that Λ p ˜ A n q is an NCCR over e Λ p ˜ A n q e .Note that Λ p Q E q can only be an NCCR in the extended Dynkin case, as in the Dynkincase it has infinite global dimension, and in the non-Dynkin, non-extended Dynkin case itis expected to have trivial center (by Conjecture 1.1); see Section 5.A final perspective on this work involves the Kontsevich-Rosenberg Principle which says:a non-commutative geometric structure on an associative algebra A should induce a geo-metric structure on the representation spaces Rep n p A q , for all n ě
1. This principle needsadjusting for structures living in the derived category of A -modules, as the representationfunctor is not exact. For a d-Calabi–Yau structure on A , it is shown in [BD18] and [Yeu18]that the derived moduli stack of perfect complexes of A -modules, R Perf p A q , has a canonical p ´ d q - shifted symplectic structure. Since the dg Ginzburg algebra is 2-Calabi–Yau, thisimplies that its moduli stack of representations has a 0-shifted symplectic structure. ByConjecture 1.1, it is the same as the moduli stack of representations of Λ q p Q q itself. Notethat the multiplicative quiver variety can be viewed as a coarse moduli space of stemistablerepresentations; so the aforementioned result that this variety locally has the structure ofan ordinary quiver variety is a singular analogue of the statement on the moduli stack.We prove Theorem 1.2 using a complex: P ‚ : “ Λ q p Q q b kQ kQ b kQ Λ q p Q q α ÝÑ Λ q p Q q b kQ kQ b kQ Λ q p Q q β ÝÑ Λ q p Q q b Λ q p Q q defined originally in [CBS06] and shown to resolve Λ q p Q q , except for the injectivity of themap α . We show α is injective and then show the dual complex P _‚ is a resolution ofΛ q p Q qr´ s , which implies Λ q p Q q is 2-Calabi–Yau.First, we establish a chain of implications to reduce the proof to a presentation of thelocalization L Q that we call the strong free product property . (The strong free product prop-erty is a version of Anick’s weak summand property in the ungraded case, see [Ani82].) Weexplain a procedure to show that the strong free product property holds for any connectedquiver containing a subquiver where the strong free product property holds. Then we es-tablish the strong free product property explicitly using the Diamond Lemma in the case Q “ ˜ A n and carry out the procedure in this case.To prove the 2-Calabi–Yau property from the strong free product property we establish4he chain of implications: Strong Free Product Property for Q : σ : Λ q p Q q ˚ kQ kQ r t, p q ` t q ´ s Ñ L Q is a linear isomorphism Section . ó Section 6.1Weak Free Product Property for Q :gr p σ q : gr p Λ q p Q q ˚ kQ kQ r t sq Ñ gr p L Q q is an algebra isomorphism P rop . ó Prop 3.4gr p σ q : Λ q p Q q b kQ kQ r t s b kQ Λ q p Q q Ñ J Q { J Q is an isomorphism of Λ q p Q q -bimodules P rop . P rop . . . ó Prop 3.1 & Prop 3.4 P ‚ is a length two projective Λ q p Q q -bimodule resolution of Λ q p Q q P rop . . ó Thm 3.5Λ q p Q q is 2-Calabi–YauHere the isomorphism σ is determined by a choice of kQ -bimodule section Λ q p Q q Ñ L Q of the quotient map L Q ։ Λ q p Q q , but gr p σ q is independent of this choice. The element t maps to the relation, and the filtrations used are the t -adic one on the source and the J Q -adic filtration on L Q . Remark 1.5.
Note that for ordinary preprojective algebras, the free product propertywas observed in [Sch16]. In fact, as these algebras are nonnegatively graded augmentedwith positively graded augmentation ideal, the strong and weak free product properties areequivalent (and independent of the choice of graded section σ ), and equivalent to the con-dition that the algebra has a projective bimodule resolution of length two, as was alreadyobserved by Anick in [Ani82]. Such algebras were called noncommutative complete intersec-tions in [EG07] due to their close relationship to the condition that representation varietiesbe complete intersections.However, in the ungraded case, we only have the implication above, that the free productproperty implies the existence of a length-two projective bimodule resolution. Indeed, aswe see above, the latter property only depends on a piece of the associated graded algebrawith respect to the p r q -adic filtration, and in the ungraded case this filtration need not evenbe Hausdorff. In contrast, the strong free product property implies the Hausdorff conditionand gives information about the algebra itself.Motivated by this, we believe that the strong free product property can be viewed asan ungraded analogue of the noncommutative complete intersection property. Following[EG07], it makes sense to ask if the strong free product property can be used to show thatrepresentation varieties are (asymptotic) complete intersections under certain conditions.In Section 4, we show that the strong free product property implies that the dg multi-plicative preprojective algebra Λ dg,q is formal. Therefore, by the results of this paper, bothConjectures 1.1 and 1.3 would follow from the following more general statement (see Section6 for precise details): Conjecture 1.6. If Q is a connected, non-Dynkin quiver, then σ as above is a linearisomorphism ( p L Q , r, σ, kQ r t, p t ` q q ´ sq satisfies the strong free product property). Let us outline an inductive strategy to prove the strong free product property. Let Q “ Q Y Q be a connected quiver containing Q . Q is chosen so that Q X Q “ H anddefine B : “ Q X Q . Assume we have the strong free product property for Q . Filter Λ q p Q q by the descending filtration given by powers of the ideal generated by those arrows in Q incident to B . We expect that this filtration is Hausdorff and that the associated gradedalgebra with respect to this filtration is isomorphic to a free product Λ q p Q q ˚ kQ Λ q p Q , Bq .Here Λ q p Q z Q, Bq is the partial preprojective algebra with no relations at the vertices in B Ă Q . We prove the strong free product property for arbitrary partial multiplicative5reprojective algebras. Put together we obtain linear isomorphisms L Q – L Q ˚ kQ L Q – Λ q p Q q ˚ kQ kQ r t, p q ` t q ´ s ˚ kQ Λ q p Q , Bq ˚ kQ kQ r t, p q ` t q ´ s– gr Λ q p Q q ˚ kQ kQ r t, p q ` t q ´ s , and since the filtration is Hausdorff we can also pick a section, eliminating the gr.In Section 6, we prove these statements when Q “ ˜ A n is a cycle. To do so, we build asection Λ q p ˜ A n q ã Ñ L ˜ A n and hence obtain a map ϕ : Λ q p ˜ A n q ˚ k p ˜ A n q k p ˜ A n q r t, p q ` t q ´ s Ñ L ˜ A n . We show that ϕ is an isomorphism with a carefully chosen basis for L ˜ A n , whose linearindependence and spanning are proven with the Diamond Lemma. Putting this togetherwith the strong free product property for partial multiplicative preprojective algebras, weobtain all of the statements for connected quivers Q containing Q “ ˜ A n . This is thetechnical heart of the paper.An outline of the paper is as follows. In Section 2, we give elementary background infor-mation on multiplicative preprojective algebras and produce an alternative generating setcrucial for our approach to the 2-Calabi–Yau property. In Section 3, we prove the 2-Calabi–Yau property for Λ q p Q q assuming the strong free product property. In Section 4, using thestrong free product property, we show that the multiplicative Ginzburg dg-algebra has ho-mology Λ q p Q q , concentrated in degree zero. In Section 5, we compute that Z p Λ q p Q qq “ k for Q connected and properly containing a cycle. This shows that the Calabi–Yau structure inthese cases are unique, up to scaling. Finally, in Section 6, we precisely define the weak andstrong free product properties, and prove them first for multiplicative preprojective algebrasof cycles, then for partial multiplicative preprojective algebras. Putting the two together,we deduce these properties for connected quivers containing cycles. Acknowledgements
The first author was supported by the Roth Scholarship through the Department of Math-ematics at Imperial College London. We thank the Max Planck Institute for Mathematicsin Bonn for their support and ideal working conditions. We’d like to thank Yankı Lekilifor bringing the problem to our attention and discussing the Fukaya category perspective.We’re grateful to Michael Wemyss for explaining the NCCR perspective.
Throughout the paper we fix an arbitrary field k . For each quiver (i.e. directed graph) Q ,let Q be the vertex set, Q be the arrow set, and h, t : Q Ñ Q the head and tail maps,respectively. We will assume that Q and Q are finite for convenience, but really onlyneed finitely many arrows incident to each vertex. Q op denotes the quiver with the sameunderlying graph of vertices and arrows, but with every arrow in the opposite direction. Q denotes the quiver with the same vertex set as Q and Q op and with arrow set Q \ Q op .For each arrow a P Q , we write a ˚ for the corresponding arrow in Q op , and vice versa. In Q we distinguish between arrows in Q and Q op using a function ǫ : Q Ñ t˘ u defined by ǫ p a q “ a P Q and ǫ p a q “ ´ a P Q op1 .Next define g a : “ ` aa ˚ , viewed as elements in the algebra of paths, kQ , and considerthe localization L : “ kQ r g ´ a s a P Q . Notice, for all a P Q : g a a “ a ` aa ˚ a “ ag a ˚ . (2.1)6his implies: g a ˚ a ˚ “ a ˚ g a g ´ a a “ ag ´ a ˚ g ´ a ˚ a ˚ “ a ˚ g ´ a . Fixing a total ordering ď on the set of arrows Q , one can make sense of a product over(subsets of) the arrow set. Using ď and ǫ we define: ρ : “ ź a P Q g ǫ p a q a l a : “ ź b P Q,b ă a g ǫ p b q b r a : “ ź b P Q,b ą a g ǫ p b q b . When we need to make the role of the total ordering ď more explicit, we write ρ ď , l a, ď , and r a, ď for ρ, l a , and r a respectively. By definition, l a and r a are truncations of ρ to the rightand left of a , respectively. Therefore, ρ “ l a g ǫ p a q a r a (2.2)for all a P Q . Definition 2.1.
Let k be a field, Q a quiver, and q P p k ˆ q Q . Fixing an ordering ď onthe arrows of Q and ǫ as defined above, the multiplicative preprojective algebra , Λ q p Q q , isdefined to be Λ q p Q q : “ L { J where L “ kQ r g ´ a s a P Q is the localization and J is the two-sided ideal generated by theelement ρ ´ q .Note that q is viewed as an element of kQ via ř i P Q q i e i P kQ Ă kQ , and as ρ isinvertible we need q i ‰ e i Λ q p Q q ‰
0, for all i . Remark 2.2.
The isomorphism class of Λ q p Q q is independent of both the orientation ofthe quiver and the choice of an ordering on the arrows, by Section 2 in [CBS06].In the multiplicative preprojective algebra, (2.2) becomes the identity: l a g ǫ p a q a r a “ q. Hence: r a l a “ qg ´ ǫ p a q a . (2.3) θ In [CBS06], the important map θ : Q Ñ Λ q p Q q is defined by θ p a q “ q ´ l a ar a ˚ and extendedto kQ by the identity on Q and by requiring θ to be an algebra map. Then Lemma 3.3 in[CBS06] shows that θ p g a q “ l a g a l ´ a (2.4) “ r ´ a g a r a (2.5)is invertible, and hence θ factors through the localization L : “ kQ r g ´ a s a P Q . We will show θ descends to the quotient Λ q p Q q , with the ordering of the arrows reversed, using the followingresult. Lemma 2.3.
Let ď denote a total order on Q and let ě denote its opposite ordering, i.e. a ě b if b ď a . Such an order fixes a bijection Q – t a , a , . . . , a | Q | u . Then θ p r a j , ě q “ l a j , ď “ : l a j and θ p l a j , ě q “ r a j , ď “ : r a j for any a j P Q . roof. We prove θ p r a j , ě q “ l a j by induction on j , where j “ θ p q “ θ p r a j ` , ě q “ θ p g ǫ p a j q a j q θ p r a j , ě q (IH) “ θ p g ǫ p a j q a j q l a j p . q “ l a j g ǫ p a j q a j l ´ a j l a j “ l a j g ǫ p a j q a j “ l a j ` . The second identity is similar and one can formally obtain a proof from the above byexchanging the symbols r and l , the identity 2.4 for 2.5, and the order of the multiplication. Corollary 2.4. θ p ρ ě q “ ρ . This corollary implies θ descends to a map Λ q p Q, ěq Ñ Λ q p Q, ďq . Notice that we cansimilarly define θ ě : Λ q p Q, ďq Ñ Λ q p Q, ěq . Proposition 2.5. θ ě ˝ θ “ Id Λ q p Q, ďq .Proof. It suffices to check θ ě ˝ θ is the identity on arrows in Q and indeed: θ ě p θ p a qq “ θ p q ´ l a ar a ˚ q“ q ´ θ ě p l a q θ ě p a q θ ě p r a ˚ qp . q “ q ´ r a θ ě p a q l a ˚ “ q ´ r a p q ´ l a ar a ˚ q l a ˚ p . q “ g ´ ǫ p a q a ag ´ ǫ p a ˚ q a ˚ p . q “ ag ´ ǫ p a q a ˚ g ´ ǫ p a ˚ q a ˚ “ a The goal of this section is to prove that Λ q p Q q is 2-Calabi–Yau for Q containing an unorientedcycle. We do so by exhibiting a length two, projective, Λ q p Q q -bimodule resolution P ‚ ofΛ q p Q q , whose bimodule dual complex P _‚ is quasi-isomorphic to Λ q p Q q . This resolution isdue to Crawley-Boevey and Shaw, but they don’t state nor prove that it is exact. The mainnew ingredient we provide is the injectivity of α which relies on the free product property.We have kept most of the notation for ease of toggling between them. This includes thedecision to fix q and Q , connected and non-Dynkin, and write Λ : “ Λ q p Q q . Λ Crawley-Boevey and Shaw build a chain complex of Λ-bimodules P ‚ “ P α Ñ P β Ñ P where, P “ P : “ Λ b kQ kQ b kQ Λ “ x η v y v P Q P : “ Λ b kQ kQ b kQ Λ “ x η a y a P Q α p η v q : “ ÿ a P Q : t p a q“ v l a ∆ a r a where ∆ a “ η a a ˚ ` aη a ˚ if a P Q ´ g ´ a p η a a ˚ ` aη a ˚ q g ´ a if a P Q op .β p η a q : “ aη t p a q ´ η h p a q a. We claim that it is a resolution of Λ. To see this, following [CBS06], we first write downan explicit chain map of Λ-bimodule complexes ψ : P ‚ Ñ Q ‚ , where Q ‚ is quasi-isomorphicto Λ; we then prove it is an isomorphism. Q ‚ is the cotangent exact sequence in Corollary2.11 in [CQ95], but in this context it was defined earlier (and shown quasi-isomorphic to Λ)by Schofield in [Sch85]. So we have the maps: P ‚ ψ ÝÑ CBS Q ‚ q.i. ÝÑ Schofield Λ . Proposition 3.1 (Lemma 3.1 in [CBS06]) . The following diagram commutes: P α / / ψ (cid:15) (cid:15) P β / / ψ – (cid:15) (cid:15) P γ / / ψ – (cid:15) (cid:15) Λ id “ (cid:15) (cid:15) J { J κ / / Λ b L Ω kQ p Λ q b L Λ λ / / Λ b kQ Λ µ / / Λ where the vertical maps are Λ -bimodule maps defined on generators by, ψ p η v q : “ ρe v ´ qe v ψ p η a q : “ b L r a b kQ ´ b kQ a s b L ψ p η v q “ e v b e v and the horizontal maps are defined by, κ p x ` J q “ b L δ p x q b for x P J where δ p x q “ x b ´ b xλ p b L r ab b kQ c ´ a b kQ bc s b L q “ ab b kQ c ´ a b kQ bc for a, b, c P L and µ is the multiplication map in Λ . Since ψ and ψ are Λ-bimodule isomorphisms, it remains to show ψ is a Λ-bimoduleisomorphism. We show this using the strong free product property. Theorem 3.2. (Weak free product property)Let Q be a connected quiver containing an unoriented cycle. Then, there exists an isomor-phism of graded algebras: ÿ i ϕ i : gr p Λ q p Q q ˚ kQ kQ r t sq Ñ gr p L Q q where the associated graded algebras are taken with respect to the t -adic and J Q -adic filtra-tions on Λ q p Q q ˚ kQ kQ r t s and L Q respectively. Remark 3.3.
In Section 6 we prove a stronger theorem, giving a linear isomorphism on thefiltered algebras, which descends to this statement on the level of associated graded algebras.
Proposition 3.4.
Suppose Λ satisfies the weak free product property. Then P ‚ is a bimoduleresolution of Λ .Proof. Taking the i “ p ϕ q “ ÿ i ϕ i : gr p Λ ˚ kQ kQ r t, p t ` q q ´ sq ÝÑ gr p L Q q ϕ : Λ b kQ kQ ¨ t b kQ Λ Ñ J Q { J Q . Since ϕ sends t ÞÑ r , it sends te v ÞÑ re v “ p ρ ´ q q e v and hence ϕ “ ψ . We concludethat ψ is an isomorphism of Λ-bimodules and hence ψ ‚ : P ‚ Ñ Q ‚ is an isomorphism ofΛ-bimodule complexes. In particular, P ‚ is a resolution since Q ‚ is a resolution.Therefore, Λ has Hochschild dimension at most 2 and hence global dimension at most 2. Recall that a d -Calabi–Yau structure is an A -bimodule isomorphism η : A Ñ Ext dA b A op p A, A b A op q “ : HH d p A, A b A op q . For dg-algebras, one further equips this structure with a class in negative cyclic homologythat lifts the Hochschild homology class of the isomorphism. But, as shown in Proposition5.7 and explained in Definition 5.9 of [VdBdTdV12], for ordinary algebras this requirementis automatic.We have established P ‚ as a Λ-bimodule resolution of Λ. To show Λ is 2-Calabi–Yau, itsuffices to show that its dual complexRHom Λ ´ bimod p Λ , Λ b Λ op q : “ Hom Λ ´ bimod p P ‚ , Λ b Λ op q “ : P _‚ is quasi-isomorphic to Λ r´ s .Define η _ v P Hom Λ ´ bimod p P , Λ b Λ q and η _ a P Hom Λ ´ bimod p P , Λ b Λ q by, η _ v p η w q : “ e v b e v if v “ w η _ a p η b q : “ e t p a q b e h p a q if b “ a ˚ . These are generators of P _ and P _ respectively and give isomorphisms, P _ – Λ b kQ kQ b kQ Λ “ x η _ v y , P _ – Λ b kQ kQ b kQ Λ “ x η _ a y . Rather than directly study the dual complex P _‚ , we modify the formulas for α _ and β _ using the map θ , in a way that doesn’t effect the homology of the complex. Namely, afterchoosing generators t ξ v u for P _ and t ξ a u for P _ , defined below, one can expand: α _ p ξ a q “ ÿ v P Q a v ξ v a v β _ p ξ v q “ ÿ a P Q b a ξ a b a , for some a v , a v , b a , b a P Λ and then define α _ θ p ξ a q : “ ÿ v P Q θ p a v q ξ v θ p a v q β _ θ p ξ v q : “ ÿ a P Q θ p b a q ξ a θ p b a q . It suffices to show that p P _‚ q θ : “ P _ ´ β _ θ / / P _ α _ θ / / P _ is quasi-isomorphic to Λ r´ s .We prove this by establishing an isomorphism of Λ-bimodule complexes ϕ ‚ : P ‚ r s Ñ p P _‚ q θ following Crawley-Boevey and Shaw, so p P _‚ q θ ϕ ´ ‚ / / P ‚ r s ψ ‚ r s / / Q ‚ r s quasi-iso / / Λ r s . heorem 3.5. The following diagram commutes: P _ ´ β _ θ / / (II) P _ α _ θ / / (I) P _ γ ˝ ϕ ´ / / Λ P α / / ϕ – O O P β / / ϕ – O O P γ / / ϕ – O O Λ id “ O O where the vertical maps are Λ -bimodule isomorphisms defined on generators by, ϕ p η v q : “ ξ v : “ qη _ v ϕ p η a q : “ ξ a ˚ : “ l a η _ a ˚ l ´ a ˚ if a P Q op ´ r ´ a ˚ η _ a ˚ r a if a P Q . Note that ϕ is an invertible map since r a and l a are invertible elements of Λ for all a P Q . The commuting of the above diagram becomes clear once we compute the maps α _ and β _ explicitly, the content of the next two lemmas. Lemma 3.6 (Lemma 3.2 in [CBS06]) . α _ p η _ a q “ a ˚ r a η _ h p a q l a ´ g ´ a ˚ r a ˚ η _ t p a q l a ˚ g ´ a ˚ a ˚ if a P Q r a ˚ η _ t p a q l a ˚ a ˚ ´ a ˚ g ´ a r a η _ h p a q l a g ´ a if a P Q op α _ p ξ a q “ θ p a ˚ q ξ t p a ˚ q ´ ξ h p a ˚ q θ p a ˚ q α _ θ p ξ a ˚ q “ aξ t p a q ´ ξ h p a q aβ “ ϕ ´ ˝ α _ θ ˝ ϕ So square (I) in Theorem 3.5 commutes.
Proof.
The first two equalities are shown directly in [CBS06] and the last two are clear fromthe definitions together with Proposition 2.5.
Theorem 3.5.
By Lemma 3.6, it suffices to show that (II) commutes. While one can similarlycompute β _ directly, such a calculation is unnecessary as the commuting of (II) follows from thatof (I) .Indeed, dualizing and applying p´q θ to the maps in (I) , produce a still commuting diagram: P p ϕ q _ θ (cid:15) (cid:15) (I) _ θ P p α _ θ q _ θ o o p ϕ q _ θ (cid:15) (cid:15) = P ´ ϕ (cid:15) (cid:15) (I) _ θ P α o o ϕ (cid:15) (cid:15) P _ P _ β _ θ o o P _ P _ β _ θ o o which indeed shows ϕ ˝ α “ ´ β _ θ ˝ ϕ , i.e. (II) commutes.The equality of maps p α _ θ q _ θ “ α follows from Proposition 2.5, and p ϕ q _ θ “ ϕ _ “ ϕ followsfrom the definitions. For p ϕ q _ θ “ ´ ϕ , observe that it suffices to show p ϕ q θ “ p ϕ q _ and indeed, p ϕ q θ p η a ˚ q “ p ξ a q θ “ θ p l a ˚ q η _ a θ p l ´ a q if ǫ p a q “ ´ θ p r ´ a ˚ q η _ a θ p r a q if ǫ p a q “ “ r a ˚ η _ a r ´ a if ǫ p a q “ ´ l ´ a ˚ η _ a l a if ǫ p a q “ ´ “ ´p ϕ q _ p η a ˚ q . Corollary 3.7. If P ‚ Ñ Λ is exact then p P _‚ q θ Ñ Λ r´ s is exact and hence P _‚ Ñ Λ r´ s is exact. Corollary 3.8. If Q is connected and contains an unoriented cycle then Λ q p Q q is 2-Calabi–Yau. Formality of the multiplicative Ginzburg dg-algebras
In Conjecture 1.3, we claim that a multiplicative version of this result should hold. In thissection we show that if Q satisfies the strong free product property, then the multiplicativeGinzburg dg-algebra is formal. In particular this proves Conjecture 1.3 in the case Q is con-nected and contains a cycle. Moreover, it reduces the conjecture to the remaining extendedDynkin cases and Conjecture 1.6.If one views the dg multiplicative preprojective algebra as the central object of study,as in [EL17c] and [EL17b], then we are showing one can formally replace it by the non-dgversion.We begin with a standard lemma. Lemma 4.1.
Let k be a commutative ring. Let A be the dg-algebra defined as a gradedalgebra to be k r r s ˚ k r s s with | r | “ and | s | “ ´ , product given by concatenation of words,and differential extended as a derivation from the generators d p s q “ r and d p r q “ . Then A is quasi-isomorphic (in fact chain homotopy equivalent) to its cohomology H ˚ p A q “ k concentrated in degree zero.Proof. Let h : A Ñ A r´ s be the homotopy with the property h p rf q “ sf and h p sf q “ f P A , and h p k q “
0. Then h ˝ d ` d ˝ h ´ A is the projection with kernel k to theaugmentation ideal of A . Therefore, it defines a contracting homotopy from A to k .In other words, the lemma is merely observing that A , as the tensor algebra on an acycliccomplex kr ‘ ks , is itself quasi-isomorphic to k . Corollary 4.2.
The dg-algebra A given by Λ q p Q q ˚ kQ kQ r r, p r ` q q ´ s ˚ kQ kQ r s s with | r | “ and | s | “ and with differential determined by d p s q “ r is quasi-isomorphic to Λ q p Q q concentrated indegree zero.Proof. Define a homotopy by, for f, g P Λ q p Q q : h p f rg q “ f sg, h p f sg q “ , h p f p r ` q q ´ g q “ q ´ h p f g q ´ q ´ f s p r ` q q ´ g where the definition of h p f p r ` q q ´ g q is chosen to match the formula for h p f rg q in the r -adiccompletion. There is an augmentation A ։ Λ q p Q q with kernel p r, s, r : “ p r ` q q ´ ´ q ´ q .Notice that h ˝ d ` d ˝ h is a homotopy from the identity on A to the augmentation Λ q p Q q ,as it annihilates Λ q p Q q and is the identity on s , r , and r . Definition 4.3.
The
Ginzburg dg-algebra Λ dg,q p Q q is defined to beΛ dg,q p Q q : “ p L Q ˚ kQ kQ r s s , d p s q “ ρ ´ q q . Proposition 4.4. If Λ q p Q q satisfies the strong free product property then H ˚ p Λ dg,q p Q qq – H p Λ dg,q p Q q – Λ q p Q q so in particular Λ dg,q p Q q is formal.Proof. The strong free product property yields an isomorphism of complexes, L Q – Λ q p Q q ˚ kQ k r r, p r ` q q ´ s . Hence, as complexesΛ dg,q p Q q – L Q ˚ kQ kQ r s s – Λ q p Q q ˚ kQ k r r, p r ` q q ´ s ˚ kQ kQ r s s , q p Q q , concentrated in degree zero. It followsthat, Λ dg,q p Q q – H ˚ p Λ dg,q p Q qq – H p Λ dg,q p Q q – Λ q p Q q as dg-algebras. Remark 4.5.
In the presence of Conjecture 1.1, formality of Λ dg,q p Q q implies Λ dg,q p Q q is2-Calabi–Yau. Hence by Theorem 1.2, we have shown that Λ dg,q p Q q is 2-Calabi–Yau, when Q is connected and contains a cycle. One may be able to adapt the techniques in Section3 to prove that Λ dq,q p Q q is 2-Calabi–Yau, in general. In more detail, the role of the Λ q p Q q -bimodule resolution, P ‚ , should now be played by the Λ dg -dg-bimodule given by the totalcomplex of:Λ dg b kQ k ¨ s b kQ Λ dg α dg / / β dg Λ dg b kQ kQ b kQ Λ dg β dg / / Λ dg b kQ Λ dg , where β p a b s b b q “ as b b ´ a b sb and β p a b x b b q “ ax b b ´ a b xb . First we present some general theory of the center of free products. Let B “ k I for someset I and let R and S be two (unital) k -algebras which contain B as a nonunital subalgebra(we do not assume that 1 P B Ď S is the unit in S and similarly for R ). We consider thefree product A : “ R ˚ B S .Under mild conditions, such a free product has trivial center. We call an element s P S a weak nonzerodivisor if for all s P S , we have sSs ‰ s Ss ‰
0. Note that a ring isprime if and only if every nonzero element is a weak nonzerodivisor.
Proposition 5.1.
Suppose that R ‰ B is prime and R fl Mat p k q , and also R is not afield extension of k of degree two. Suppose that S ‰ B has the property that B is a weaknonzerodivisor in S . Then A is a prime ring with trivial center. The proof of the proposition involves more general, but somewhat technical, considera-tions. Let s R, s S be vector space complements to B in R and S . We have a decomposition A “ B ‘ A R ‘ A S ‘ A RS ‘ A SR , where (5.1) A RS : “ ÿ m ě p s R s S q m , A SR : “ ÿ m ě p s S s R q m , A R : “ s R ‘ s R ¨ A SR , A S : “ s S ‘ s S ¨ A RS . (5.2) Lemma 5.2.
Let z P Z p A q be central.(i) Assume that, for some i P I , at least one of e i s S, s Se i is nonzero. Then the part z R of z in A R satisfies e i z R “ z R e i “ .(ii) Assume that z R “ and furthermore that, for some i P I , e i s R has dimension greaterthan one. Then the part z RS of z in A SR satisfies z RS e i “ “ e i z RS .(iii) Again assume that z R “ . Assume that for some i P I , s Re i has dimension greaterthan one. Then the part z SR of z in A SR satisfies e i z SR “ “ z SR e i . roof. (i) Note first that e i z “ ze i , which implies that e i z R “ z R e i . Suppose that this isnonzero. Suppose that e i s ‰ s P s S . Then z R e i s P A RS is nonzero. This is alsothe component of ze i s in A RS . On the other hand, e i s z has no component in A RS . Thiscontradicts centrality of z . The same argument works if s e i ‰
0. So e i z R “ e i s R has dimension greater than one. Note that A RS “ s RA S and A SR “ A S s R . Take a basis f j of A S , such that each basis element is in some summand e i A S e i . Write z RS “ ř j r j f j and z SR “ ř j f j r j , for r j , r j P s R . Suppose that for some ℓ we have r ℓ f ℓ e i ‰
0. Let r ℓ P e i s R not be a scalar multiple of r ℓ . Then the part of zr ℓ in A R is of the form c ` ř j r j f j r ℓ , for some c P R . On the other hand, the part of r ℓ z in A R is ofthe form c ` ř j r ℓ f j r j , for some c P R . This does not have the same term in s Rf ℓ s R . Thus z RS e i “ Corollary 5.3.
Assume:(a) For every i P I , one of e i s R or s Re i is nonzero;(b) Conditions (i), (ii), and (iii) of Lemma 5.2 hold for some vertices i, i , i P I , notnecessarily equal, such that e i , e i , and e i are weak nonzerodivisors in R .Then B Z p A q “ B k .Proof. Under assumption (a), we have that e i z S “ “ z S e i for all i P I . Summing, we get1 B z S “ “ z S B .Assume condition (b). Under condition (i) for i P I , we get e i z R “
0. Up to subtractinga scalar from z , we can assume that e i z B “ e j z R ‰ j P I . Since e i is a weak nonzerodivisor, we have e j z R f e i ‰ f . By centrality of z , we have e i z R ‰ e j z R “
0. Summing over j we get 1 B z R “
0. By thesame proof, under the subsequent assumptions, we get 1 B z SR “ B z RS “ Proof of Proposition 5.1.
Without loss of generality, we may assume that 1 B “ R . Oth-erwise, we can simply enlarge I , adding in a new element i such that e i : “ ´ B , andreplace S by S ‘ ke i .We first verify conditions (a) and (b) of Corollary 5.3.Let us begin with (b). Since R is prime, every e i is a weak nonzerodivisor in R . So weonly have to verify (i)–(iii) of Lemma 5.2. For (i), we claim that, for some i P I , e i s S ‰
0. Ifnot, 1 B s S “
0. Since S ‰ B and 1 B is a weak nonzerodivisor in S , we can find some s P s S and some s P S with 1 B s s ‰
0. This implies that s P B , but that implies that 1 B s ‰
0, acontradiction.For the remaining parts, first assume that | I | ě
2. Then by the prime property for R ,we have for i ‰ j that e i Re j “ e i s Re j ‰
0. This takes care of condition (a). Condition (b)also follows unless | I | “ e i Re j is one-dimensional. As R is prime this can onlyhappen if R – Mat p k q , which we assumed was not the case.Next assume that | I | “
1. Then R ‰ B implies that e i s R “ s R “ s Re i is nonzero for i P I ,taking care of condition (a). Condition (b) follows unless R is two-dimensional. But since R is prime, this implies R is a field extension of k of degree two, which we assumed was notthe case.Therefore, Corollary 5.3 applies and 1 B Z p A q “ B k “ Z p A q B . Now let z P Z p A q . Upto subtracting a multiple of 1, we can assume that 1 B z “
0. We claim that z “
0. Since 1 B is a weak nonzerodivisor in both R and S , we can either find r P R with 0 ‰ B rz “ B zr or s P S with 0 ‰ B sz “ B zs , both contradictions. This proves that Z p A q “ k .That fact that A is prime follows from the definition of the free product together withthe fact that 1 B is a weak nonzerodivisor in both R and S .14 emark 5.4. The assumption that R not be a degree two field extension is necessary as xy ` yx P Z p R r x s{p x ` q ˚ R R r y s{p y ` qq . The assumption that R fl Mat p k q is alsonecessary as rs ` sr P Z p Mat p R q˚ R S q where S Ă Mat p R q is the algebra of upper-triangularmatrices, with s P s S “ e Se strictly upper-triangular and r P e Mat p R q e . Using the decomposition for Λ q p Q q produced in the final section, the above results implythe following: Corollary 5.5. If Q is connected and contains an unoriented cycle, then Λ q p Q q is prime.If the containment is proper, then Z p Λ q p Q qq “ k . Remark 5.6.
Conjecture 1.1 predicts Z p Λ q p Q q “ k for Q non-Dynkin and non-extendedDynkin, while Conjecture 1.4 predicts Z p Λ p Q qq – e v Λ p Q q e v for Q extended Dynkin and q “ Q “ ˜ A n ). In the Dynkin case, for q “ p Q q – Π p Q q first established in [CB13]. Proof.
Suppose ˜ A n Ĺ Q . By Corollary 6.11, one has a decomposition,grΛ q p Q q – Λ q n p ˜ A n q ˚ kJ Λ q p Q z ˜ A n , W q . using the descending filtration by powers of the ideal generated by those arrows in Q z ˜ A n which are incident to W . Therefore gr p Z p Λ q p Q qqq Ď Z p Λ q n p ˜ A n q ˚ k W Λ q p Q z ˜ A n , W qq .By Proposition 6.8, Λ q p ˜ A n q is prime. We now check the remaining hypotheses of Propo-sition 5.1: Λ q p ˜ A n q is not equal to k W , Mat p k q , nor is it a field extension of k . Moreover,Λ q p Q z ˜ A n , W q ‰ kJ has 1 k W a weak nonzerodivisor since every vertex in Q z ˜ A n is connectedto some vertex in W (as Q is connected). Thus the hypotheses of Proposition 5.1 are sat-isfied, and gr Λ q p Q q is prime and has trivial center. Since the filtration is Hausdorff (againby Corollary 6.11), we conclude also that Z p Λ q p Q qq “ k . Corollary 5.7. If Q is connected and properly contains a non-oriented cycle, then Λ q p Q q has a unique, up to scaling, Calabi–Yau structure.Proof. Any two Calabi–Yau structures differ by an invertible map Hom Λ ´ bimod p Λ , Λ q , whichis determined by the image of the unit, a central invertible element. So the set of Calabi–Yaustructures, when non-empty, forms a Z p Λ q ˆ -torsor, which in this case is k ˆ . So any twoCalabi–Yau structures differ by an invertible scalar. The goal of this section is to prove the strong free product property for connected quiverscontaining a cycle. We first establish the strong free product property for the quivers ˜ A n for n ě π : L Ñ Λ q p ˜ A n q . Then we establish the more general result using the corresponding result for partial multiplicative preprojective algebras. Generally, if A is an algebra over a semisimple ring S and J “ p r q an ideal generated by asingle relation r , we can form a canonical algebra map,Φ : A { J ˚ S S r t s Ñ gr J A, Φ | A { J “ Id , Φ p t q “ r, (6.1)where gr J means the associated graded algebra with respect to the J -adic filtration.15 efinition 6.1. The pair p A, r q satisfies the weak free product property if Φ is an isomor-phism.Next, given an S -bimodule section σ : A { J Ñ A of the quotient map that is the identityon S , we can form an associated linear map, r σ : A { J ˚ S S r t s Ñ A, p a t m a t m ¨ ¨ ¨ t m n a n q ÞÑ σ p a q r m σ p a q r m ¨ ¨ ¨ r m n σ p a n q , (6.2)for m i ą
0, for all i .By construction, this is S r t s -bilinear, where t acts on A by multiplication by r . It alsoreduces to the identity modulo p t q on the source and J on the target. If p A, r q satisfies theweak free product property, then moreover the completion pr σ : { A { J ˚ S S r t s Ñ p A, (6.3)with respect to the t -adic and J -adic filtrations, is an isomorphism.In nice cases we can identify a subalgebra on the left-hand side mapping isomorphicallyto A (which is a subalgebra of p A under the Hausdorff condition Ş m ě J m “ t u ). In sucha case we say that p A, r, σ q satisfies the strong free product property . Remark 6.2.
The choice of σ is important. Let A “ k x x, y y and J “ p y q so A { J – k r x s . Consider two different choices σ , σ : k r x s Ñ k x x, y y σ p x ` p y qq “ x, σ p x ` p y qq “ x ´ yx. Then σ is a linear isomorphism while σ is not surjective as x “ σ p x ` p y qqp ´ y q ´ “ σ p x ` p y qq ÿ i ě y i R σ p k r x s ˚ k k r t sq . Remark 6.3.
The strong free product property has already appeared, albeit in the gradedcontext working over a field, in [Ani82]: Anick defines a notion of weak summand B Ă A ,meaning there exists a linear isomorphism A – B ˚ A {p B q , where B : “ ‘ i ą B i is theaugmentation ideal. In the graded setting he observes that the strong free product propertydoes not depend on the choice of σ and it is easy to see it is implied by the weak free productproperty.Next, let B be a localization of S r t s obtained by inverting finitely many elements of S ˆ ` p t q , such that S r t s Ñ A extends (uniquely) to an algebra map τ : B Ñ A . Let B : “ tB , so that we have an S -bimodule decomposition B “ S ‘ B . Then r σ extends to amap σ : A { J ˚ S B Ñ A , which has the form a b a ¨ ¨ ¨ b n a n ÞÑ σ p a q τ p b q ¨ ¨ ¨ τ p b n q σ p a n q , a i P A { J, b i P B. (6.4) Definition 6.4.
The data p A, r, σ, B q satisfies the strong free product property if σ is alinear isomorphism.In this case, it follows by taking associated graded algebras that p A, r q satisfies the weakfree product property. Moreover, A is Hausdorff in the J -adic filtration (because the sourceof σ is Hausdorff in the t -adic filtration), and σ is indeed a restriction of pr σ . Remark 6.5.
The definition of σ in (6.4) depends on the choice B “ tB of complement. Infact the property of being an isomorphim, i.e. the strong free product property, depends onthis choice. To see this return to the example of p A “ k x x, y y , r “ y, σ p x ` p y qq “ x ´ yx q in Remark 6.2. Let B “ k r t, p ´ t q ´ s with B : “ tB and B : “ p ´ t q ´ B . Then σ as defined using B is not surjective, as x R σ p k r x s ˚ k r t, p ´ t q ´ sq , as before. But σ asdefined using B is an isomorphism. See Remark 6.7 below for another example.Now back to the setup of Q a connected, non-Dynkin quiver and q “ p k ˆ q Q . Let B : “ kQ r t, p q ` t q ´ s and B : “ tB “ Span p t m , p t q m | m ě u , for t : “ p q ` t q ´ ´ q ´ .We conjecture that, for every connected, non-Dynkin quiver Q, for some choice of σ , thequadruple p L Q , r, σ, B q satisfies the free product property. The rest of the paper will provethis conjecture in the case where Q contains a cycle.16 .2 The case of cycles Consider the quiver ˜ A n ´ : with vertex set p ˜ A n ´ q : “ t , , . . . n ´ u and arrow set p ˜ A n ´ q “ t a , a ˚ , a , a ˚ , . . . , a n ´ , a ˚ n ´ u with s p a i q “ i and t p a i q “ i ` n q . Themultiplicative preprojective algebra for this quiver is defined, with respect to the orderinggiven by a i ă a i ` ă a ˚ j ă a ˚ j ` for all i, j P t , , . . . , n ´ u , to beΛ q p ˜ A n ´ q : “ k ˜ A n ´ rp ` a i a ˚ i q ´ , p ` a ˚ i a i q ´ s i “ ,...,n ´ Aś ni “ p ` a i a ˚ i q ś n ´ i “ p ` a ˚ i a i q ´ ´ ř ni “ q i e i E “ : LJ Writing a : “ ř i a i , a ˚ : “ ř i a ˚ i , and q “ ř i q i e i since1 ` aa ˚ “ ` ÿ i a i a ˚ i “ n ´ ź i “ p ` a i a ˚ i q ` a ˚ a “ ` ÿ i a ˚ i a i “ n ´ ź i “ p ` a ˚ i a i q we have, Λ q p ˜ A n ´ q : “ k ˜ A n ´ rp ` aa ˚ q ´ , p ` a ˚ a q ´ sxp ` aa ˚ qp ` a ˚ a q ´ ´ q y . We write r : “ p ` aa ˚ qp ` a ˚ a q ´ ´ q for this relation, S for the degree zero piece k p ˜ A n ´ q of Λ q p ˜ A n ´ q . As before let B : “ kQ r t, p q ` t q ´ s and B “ tB , spanned over kQ by t m , p t q m , m ě
1, for t : “ p q ` t q ´ ´ q ´ . Let r : “ p q ` r q ´ ´ q ´ .We prove the free product property by producing an explicit basis: Proposition 6.6. L is a free left kQ -module with basis consisting of together with allalternating products of elements of the following two sets, for x : “ p ` aa ˚ q : B : “ x m a ℓ , x m p a ˚ q ℓ | m P Z , ℓ P N ( , R : “ t r m , p r q m | m P N u . In particular, B forms a basis for Λ q p ˜ A n ´ q “ L {p r q , and p L, r, σ, B q satisfies the strongfree product property, with σ induced from the inclusion of B into L .Proof. Note that, for every vertex i , we have e i a “ ae j for a unique j , and similarly forthe elements a ˚ , x, y : “ p ` a ˚ a q ´ , x ´ , y ´ , and by definition, e i r “ re i . Therefore L isspanned as a left S -module by words in a, a ˚ , x, y, x ´ , y ´ , and r, r (of course, this set isredundant, but we will need it this way to obtain the desired basis). The relations are: xx ´ “ “ x ´ x, yy ´ “ “ y ´ y, x “ ` aa ˚ , y “ ` a ˚ a, (6.5) r “ xy ´ ´ q, r “ yx ´ ´ q ´ . (6.6)The first step is to show that we can get into a normal form in finitely many steps. After-wards, we will show that this form is unique. We can form the following system of reductionswhich implement these relations: • Inverse Reductions: xx ´ , x ´ x, yy ´ , y ´ y ÞÑ • Short Cycle Reductions: aa ˚ ÞÑ x ´ , a ˚ a ÞÑ y ´ • Reordering Reductions: a ˚ x ˘ ÞÑ y ˘ a ˚ , ay ˘ ÞÑ x ˘ a . • Substitution Reductions: y ´ ÞÑ x ´ p r ` q q , y ÞÑ p r ` q ´ q x (if not preceded by a ); ax ÞÑ a p r ` q q y, ax ´ ÞÑ ay ´ p r ` q ´ q . • Reductions in B : rr , r r ÞÑ ´ qr ´ q ´ r .17bserve that we can apply a reduction to a monomial unless the monomial has the form w u w ¨ ¨ ¨ u n w ℓ ` for some ℓ P Z ě where u i P R and w i P B .We can form an ordering on monomials such that these reductions always replace mono-mials with a linear combination of smaller monomials. The ordering is as follows. For everygenerator or word in generators z , let n z p w q denote the number of occurrences of z in w .For every pair of generators (or words in generators) z, z , let n z,z p w q denote the numberof pairs of occurrences of z and z in w with z appearing to the left of z . For subsets ofgenerators (or words in generators) Z, Z let n Z : “ ř z P Z n z and n Z,Z : “ ř z P Z,z P Z n z,z .For a word w in the generators, define the function N p w q : “ p n a p w q , n t a,a ˚ u , t x,x ´ ,y,y ´ u p w q , n t ax,ax ´ u p w q , n t y,y ´ u p w q , n t r,r u p w qq P N . (6.7)Now we say that w ă w if N p w q ă N p w q in the lexicographical ordering on N . Observethat, if w is obtained from w by applying a reduction, then N p w q ă N p w q . This impliesthat any sequence of reductions terminates in finitely many steps.It remains to show that the procedure above produces a unique result. More precisely, wewish to show that any nontrivial linear combination of monomials in normal form is nonzero.First note that, because the generators y ˘ are nonreduced, they can be substituted outeverywhere (except in the substitution reductions for y ˘ themselves). As long as we checkthat the defining relations are still satisfied, this will produce another valid reduction system.The result is:Inverse Reductions:(1) xx ´ r ÞÝÑ x ´ x r ÞÝÑ aa ˚ r ÞÝÑ x ´ a ˚ a r ÞÝÑ p r ` q ´ q x ´ a ˚ x r ÞÝÑ p r ` q ´ q xa ˚ (6) ax r ÞÝÑ qxa ´ qar x (7) ax ´ r ÞÝÑ ax ´ x ´ a p r ` q ´ q (8) a ˚ x ´ r ÞÝÑ x ´ p r ` q q a ˚ Substitution Reductions:(9) y ´ ÞÑ x ´ p r ` q q (10) y ÞÑ p r ` q ´ q x Reductions in B :(11) rr ÞÑ ´ qr ´ q ´ r (12) r r ÞÑ ´ qr ´ q ´ r It is clear that this still implies all of the defining relations: for instance, p ` aa ˚ q stillreduces to x and the relations for r and r are implied by substitution.The advantage of doing it this way is that the final reductions (substitution and B )don’t overlap with any others, so the only overlaps between the nonreduced words aboveare amongst the (1)–(8), involving the generators a, a ˚ , x, x ´ only. (As we stated, such aprocedure works in general, whenever some generators are nonreduced.)To prove uniqueness, we apply the Diamond Lemma (see [Ber78]). The Diamond Lemmasays it suffices to check overlap ambiguities, i.e., where two non-reduced subwords overlap ina larger word. In this case, it suffices to show that the 12 cubic terms formed by overlappingthe leading terms of the quadratic reductions uniquely resolve. These terms are:(I) xx ´ x (II) x ´ xx ´ (III) aa ˚ a (IV) a ˚ aa ˚ (V) a ˚ xx ´ (VI) axx ´ (VII) a ˚ x ´ x (VIII) ax ´ x (IX) aa ˚ x (X) a ˚ ax (XI) aa ˚ x ´ (XII) a ˚ ax ´ .The resolution of (I) and (II) are immediate (and are completely general, having to do18ith a basis for k r x, x ´ s ). Here is a summary of the remaining resolutions of ambiguities:(III) p r ´ r ˝ r qp aa ˚ a q “ p r ˝ r ´ r qp ax ´ x q “ p r ´ r ˝ r qp a ˚ aa ˚ q “ p r ´ r ˝ r ˝ r qp aa ˚ x q “ p r ˝ r ´ r qp a ˚ xx ´ q “ p r ´ r ˝ r ˝ r ˝ r qp a ˚ ax q “ p r ˝ r ´ r qp axx ´ q “ p r ´ r ˝ r ˝ r qp aa ˚ x ´ q “ p r ˝ r ´ r qp a ˚ x ´ x q “ p r ´ r ˝ r ˝ r qp a ˚ ax ´ q “ . We explicitly demonstrate (X), one of the more involved resolutions. a ˚ ax “ p a ˚ a q x r ÞÝÑ rp r ` q ´ q x ´ s x “ p r ` q ´ q x ´ x and a ˚ ax “ a ˚ p ax q r ÞÝÑ a ˚ p qxa ´ qar x q r ˝ r ÞÝÝÝÑ q p r ` q ´ q xa ˚ a ´ q pp r ` q ´ q x ´ q r x r ÞÝÑ q p r ` q ´ q x pp r ` q ´ q x ´ q ´ q pp r ` q ´ q x ´ q r x “ q p r ` q ´ q x p q ´ x ´ q ` qr x “ p r ` q ´ q x ´ x. Remark 6.7.
The choice of B was important here. If we instead had defined it so that p q ` t q ´ P B , i.e., if we replace r “ p q ` r q ´ ´ q ´ P R by p q ` r q ´ , then our desiredbasis would no longer be linearly independent. Indeed, reducing aa ˚ a one way, we get p x ´ q a “ xa ´ a , which is reduced, whereas the other way we get a p y ´ q “ a p q ` r q ´ x ´ a ,also reduced. That is, xa “ a p q ` r q ´ x , an equality of two distinct reduced elements. Proposition 6.8. Λ q p ˜ A n q is prime for all n ě and all q P p k ˆ q n ` .Proof. We need to show, for every pair f, g P Λ q p ˜ A n q , both nonzero, there exists some h P Λ q p ˜ A n q such that f hg ‰
0. It suffices to take f and g to be linear combinations of basiselements that all begin at some vertex i and end at some vertex j . By right multiplicationby a n ´ j or p a ˚ q j , one can take f and g to be linear combinations of basis elements ending atvertex 0. By left multiplication by a i or p a ˚ q n ´ i and then applying Reordering reductions,one can take f or g to be linear combinations of basis elements starting and ending at vertex0. In fact by doing this we can assume that f is of the form e f p x, x ´ q f p a n ` q , where f ‰ f has nonzero constant term, and similarly for g . Then their product is of thesame form, and hence is nonzero. Generally, for any quiver Q , let W Ď Q be a nonempty subset of white vertices, and B : “ Q z W the black vertices. Then the partial multiplicative preprojective algebras of p Q, W q is Λ q p Q, W q : “ L Q {p r B q , where r B “ B r B , for 1 B : “ ř j P B e j . This algebrainterpolates between Λ q p Q, Q q “ L Q and Λ q p Q, Hq “ Λ q p Q q . Proposition 6.9.
Let Q be a connected quiver and Q “ B \ W a decomposition into blackand white vertices with W ‰ H . Then p L, r B , σ, B q satisfies the strong free product propertyfor some choice of σ .In more detail, let F Ď Q be a forest rooted at the vertices of W with F “ Q and arrowsdirected towards the roots (we call this a spanning forest ). For convenience we assume (upto reorienting the arrows of Q) that F Ď Q . basis for L is given by concatenable words in the arrows a P Q , the elements x ˘ a : “p ` aa ˚ q ˘ for a P Q z F , and the elements r B , r B : “ p q ` r B q ´ ´ q ´ , such that thefollowing subwords do not occur: x a x ´ a , x ´ a x a , aa ˚ , ax ˘ a ˚ , for a P Q ; x ´ a ˚ , x a ˚ a, x a ˚ , for a P F . (6.8) The words in which r B do not occur form a basis for Λ q p Q, W q “ L {p r B q , and the section σ is given by the inclusion of these elements.Proof of Proposition 6.9. The proof parallels that of Proposition 6.6.Note that L is spanned by concatenable words in a, x a : “ p ` aa ˚ q , x ´ a , r, r subject tothe relations, depending on the choice of ordering ď on the arrows a P Q : x a x ´ a “ “ x ´ a x a , x a “ ` aa ˚ , (6.9) r “ ź ď x ǫ p a q a ´ q, r “ ź a, ě x ´ ǫ p a q a ´ q ´ (6.10) rr “ r r “ ´ qr ´ q ´ r. (6.11)Define red ǫ p a q a “ ℓ ´ a p r ` q q r ´ a . We implement the above relations with the following reduc-tions: • Inverse Reductions: x a x ´ a , x ´ a x a ÞÑ a P Q . • Short Cycle Reductions: aa ˚ ÞÑ x a ´ a P Q . • Reordering Reductions: a ˚ x ˘ a ÞÑ x ˘ a ˚ a ˚ for a P Q . • Substitution Reductions: x ˘ a ÞÑ red ˘ a , x ´ a ˚ ÞÑ ´ a ˚ red ´ a a ,Substitution Reductions: x a ˚ ÞÑ x a ˚ ` a ˚ red a a, x a ˚ a ˚ ÞÑ a ˚ red a , for a P F • Reductions in B : rr , r r ÞÑ ´ qr ´ q ´ r .For each monomial in L , define a weighted size, ϕ a : “ n t a,a ˚ u ` n t x a ,x a ˚ u ` n t x ´ a ,x ´ a ˚ u for each a P Q . From this, one can define a notion of size among monomials in x x ˘ a , a | a P Q y , by first defining a total ordering on the arrows p Q , ă q such that, a ă a if a P F , a P Q z F , or if a, a P F with a disconnected from W in F zt a u . Intuitively, we are saying that arrows in the spanning forest come before the rest in theordering, with arrows closer to the white vertices coming first. Using ă , define N : x x ˘ a , a, r, r | a P Q y Ñ N p Q , ă q ˆ N N p w q : “p ϕ a p w q , n t a | a P Q u , t x a | a P Q u p w q , n t r,r u p w qq , from which we say w ă w if N p w q ă N p w q in the lexicographical ordering on N | Q |` .Notice, as in Proposition 6.6, that N p r i p w qq ă N p w q for any word w and reduction r i with r i p w q ‰ w . First notice that, by design, ϕ a decreases under the following reductions: • Inverse Reductions: ϕ a p x a x ´ a q “ ϕ a p x ´ a x a q “ ` { ą “ ϕ a p q • Short Cycle Reductions: ϕ a p aa ˚ q “ ą { “ ϕ a p x a q Substitution Reductions: ϕ a p x a q “ { ą “ ϕ a p red a q ,Substitution: ϕ a p x ´ a q “ ą “ ϕ a p red ´ a q , ϕ p x ´ a ˚ q “ ą “ ϕ a p a ˚ red ´ a a q ,Substitution: ϕ p x a ˚ q “ ą “ ϕ a p a ˚ red a a q and ϕ p x a ˚ q “ ą { “ ϕ a p x a ˚ q , Substitution: ϕ a p x a ˚ a ˚ q “ { ą “ ϕ a p a ˚ red a q .For the Substitution Reductions observe that red a for a P F has subwords x b , x b ˚ foronly a single b P F which is necessarily farther from the root than a and the remainingarrows are not in the spanning forest. Consequently, ϕ a decreasing, despite ϕ b increasing forlarger b , implies that N decreases. The Reordering Reductions preserve all ϕ a but decrease n t a | a P Q u , t x a | a P Q u p w q by definition, and hence decrease N . The reductions in B preserveall ϕ a and n t a | a P Q u , t x a | a P Q u p w q but decrease n t r,r u p w q .We conclude that every monomial in L reduces to a linear combination of monomialswithout subwords in t x a x ´ a , x ´ a x a , aa ˚ , ax a ˚ , ax ´ a ˚ | a P Q u Y t x ´ a ˚ , x a ˚ a, x a ˚ | a P F u after applying finitely many reductions.Note that some generators are nonreduced: x a , x ´ a , and x ´ a ˚ for a P F . Therefore, wecan put in reductions for each of these and throw out all other reductions involving thesegenerators, provided we check that all the defining relations still reduce to zero. We havethe reductions:(1) x a x ´ a r ÞÝÑ a P Q (2) x ´ a x a r ÞÝÑ a P Q (3) aa ˚ r ÞÝÑ red a ´ a P F (4) aa ˚ r ÞÝÑ x a ´ a R F (5) ax ˘ a ˚ r ÞÝÑ x ˘ a a for a R F (6) ax a ˚ r ÞÝÑ red a a for a P F (7) x a ˚ r ÞÝÑ x a ˚ ` a ˚ red a a for a P F (8) x a ˚ a ˚ r ÞÝÑ a ˚ red a for a P F which don’t overlap with the remaining reductions:Substitution reductions: x ˘ a ÞÑ red ˘ a , x ´ a ˚ ÞÑ ´ a ˚ x ´ a a, a P F ;Reductions in B: rr , r r ÞÑ ´ qr ´ q ´ r .As before, reductions (3) and (4) imply the relations x a “ ` aa ˚ , whereas the substi-tution reductions imply the defining relations for r, r . So this is a valid reduction system.This reduction system has thirteen ambiguities:(I) x a x ´ a x a for a R F (II) x ´ a x a x ´ a for a R F (III) ax a ˚ x ´ a ˚ for a R F (IV) ax ´ a ˚ x a ˚ for a R F (V) ax a ˚ for a P F (VI) x a ˚ a ˚ for a P F (VII) x a ˚ a ˚ a for a P F (VIII) ax a ˚ a ˚ for a P F (IX) a ˚ ax a ˚ for a P F (X) a ˚ ax a ˚ for a P Q z F (XI) aa ˚ a for a P Q z F (XII) aa ˚ a for a P F (XIII) aa ˚ a for a ˚ P F which all resolve by the resolutions(I) p r ´ r qp x a x ´ a x a q “ p r ´ r qp x ´ a x a x ´ a q “ p r ´ r ˝ r ˝ r qp ax a ˚ x ´ a ˚ q “ p r ´ r ˝ r ˝ r qp ax ´ a ˚ x a ˚ q “ p r ˝ r ˝ r ´ r ˝ r qp ax a ˚ q “ p r ˝ r ˝ r ´ r ˝ r qp x a ˚ a ˚ q “ p r ˝ r ´ r qp x a ˚ a ˚ a q “ p r ˝ r ´ r ˝ r qp ax a ˚ a ˚ q “ p r ˝ r ´ r qp a ˚ ax a ˚ q “ p r ˝ r ˝ r ´ r qp a ˚ ax a ˚ q “ p r ´ r ˝ r qp aa ˚ a q “ p r ˝ r ´ r qp aa ˚ a q “ p r ˝ r ´ r qp aa ˚ a q “ r and r , and similarlyfor (IX) and (VII), leaving three calculations: (V), (VIII), and (IX). These ambiguities ex-press the overlap of r with r , r , and r respectively and further reduce uniquely to red a a ,red a p red a ´ q , and a ˚ red a a . It will be convenient for us to make the substitutions: x ˘ a : “ x ˘ a ´ . (6.12)The motivation for this is as follows: let A be either a multiplicative preprojective algebraor a partial version for a connected quiver Q . Let I be the ideal generated by all pathsbeginning and ending at vertices having either q “ A { I is nonzero,and we can make use of the I -adic filtration. The modified generators x ˘ a , for a an arrow in I , have the advantage of lying in the ideal I . As we will see, at least in the case where Q either contains a cycle or there is a white vertex, the I -adic filtration is Hausdorff. Thus,we get an embedding of A into the completion p A I , realising x ˘ a as power series with zeroconstant term. In the special case where q “ λ “ q as a deformation parameter based at q “ Ă A n (although we donot strictly need it in that case). We formally set x ˘ : “ x ˘ ´ y ˘ : “ y ˘ ´
1; thenthe modified reductions from Section 6.2 are the following ones: • Inverse Reductions: x ` x ´ , x ´ x ` ÞÑ ´ x ` ´ x ´ and y ` y ´ , y ´ y ` ÞÑ ´ y ` ´ y ´ • Short Cycle Reductions: aa ˚ ÞÑ x ` , a ˚ a ÞÑ y ` . • Reordering Reductions: a ˚ x ˘ ÞÑ x ˘ a ˚ , ay ˘ ÞÑ y ˘ a . • Substitution Reductions: y ´ ÞÑ x ´ p r ` q q` r `p q ´ q , y ` ÞÑ p r ` q ´ q x ` ` r `p q ´ ´ q (if not preceded by a ); ax ` ÞÑ a p r ` q q y ` ` ar ` p q ´ q a , ax ´ ÞÑ ay ´ p r ` q ´ q ` ar ` p q ´ ´ q a .This produces the same ambiguities as before, which resolve in the same way after eliminatingthe nonreduced generators y ˘ (another way to say this is that the reductions are the sameup to the change of variables, so ambiguities resolve if and only if they did before). Themodified ordering function, N z p w q : “ p n a p w q , n t a,a ˚ u , t x ` ,x ´ ,y ` ,y ´ u p w q , n t ax ` ,ax ´ u p w q , n t y ` ,y ´ u p w qq ,
22s strictly decreasing under applications of reductions and hence every term reduces afterapplying finitely many reductions. The new left free kQ -module basis, as in Proposition6.6, is given by alternating words in R and B : “ tp x ˘ q m a ℓ , p x ˘ q m p a ˚ q ℓ | m P N , ℓ P N u . In the case of the partial multiplicative preprojective algebra, the modified reductions areas follows: • Inverse Reductions: x ` a x ´ a , x ´ a x a ÞÑ ´ x ` a ´ x ´ a for a P Q . • Short Cycle Reductions: aa ˚ ÞÑ x a for a P Q . • Reordering Reductions: a ˚ x ˘ a ÞÑ x ˘ a ˚ a ˚ for a P Q . • Substitution Reductions: x ˘ a ÞÑ red ˘ a ´ , x ´ a ˚ ÞÑ ´ a ˚ red ´ a a ,Substitution Reductions: x a ˚ ÞÑ ´ x a ˚ ` a ˚ red a a, x a ˚ a ˚ ÞÑ a ˚ p red a ´ q , for a P F .Again, the same ordering function applies here and strictly decreases under these reductions.The ambiguities must resolve since they did before. Note that, for the following subsection,we only require the substitutions x a in the case where the arrow a begins at a white vertex(which in particular implies that a R F , although it could be that a ˚ P F ). If we only makethese substitutions, it is similarly easy to write the above reductions in the case where forcertain arrows x a appears and for others x a appears; we leave this to the reader.The only thing that we require from the above in the next subsection is the followingobservation: reductions on W L Q W preserve the augmentation ideal , that is to say, the idealgenerated by the generators. In other words, any monomial of positive length beginning andending at white vertices reduces to a linear combination of other such monomials. This wasnot true with the original generators (e.g., looking at the inverse reductions). Finally, we prove the strong free product property for a connected quiver containing a cycle,along with providing a natural decomposition and basis. The technique involves showingthat, in these cases, the multiplicative preprojective algebra decomposes (as a vector space)into a free product of the multiplicative preprojective algebra for the cycle and a partial multiplicative preprojective algebra for the complement of the cycle, where “partial” meansthat we turn off the relations at the vertices of the cycle. This technique should extend tothe case of general extended Dynkin quivers, hence reducing Conjecture 1.1 to the extendedDynkin case.We will show furthermore that, if Q is a connected quiver containing a cycle Q E , thenthere is a linear isomorphism, for W : “ p Q E q :Λ q E p Q E q ˚ kQ Λ q p Q , W q Ñ Λ q p Q q . (6.13)Let us explain how such an isomorphism arises. Let Q Ď Q be the subset of arrows incidentto W . Consider the p Q q -adic filtration on both sides. Then the associated graded relationto r Q is the sum r Q E ` r Q , B . As a result, there is a canonical surjection Λ q E p Q E q ˚ kQ Λ q p Q , W q ։ gr p Q q Λ q p Q q . The isomorphism above is filtered and induces this canonicalmap on associated graded map algebras, which is hence also an isomorphism. We note that,in the additive case, the isomorphism (6.13) follows from the proof of [EE07, Theorem 3.4.2](see also [Sch16, Section 5], particularly Corollary 5.2.9.(ii)). Also, note that the fact thatthe associated graded of (6.13) is an isomorphism does not imply that the original map is anisomorphism (there would be no way to deduce surjectivity, since the descending filtrationis not finite; if we knew a priori that the filtration were Hausdorff then we could deduce23njectivity, although we only know this as a consequence of the proof of the isomorphism(6.13).)As before, let B : “ kQ r t, p q ` t q ´ s and B “ tB , which is spanned by elements t m , p t q m , m ě t : “ p q ` t q ´ ´ q ´ . Proposition 6.10.
Let Q be a connected quiver containing a cycle Q E Ď Q ( Q E – r A n ´ ).Then a k -linear basis for L is obtained by concatenable products of arrows a and elements x a , for a P p Q E q , and x a , for a P Q , and the elements r, r : “ p q ` r q ´ ´ q ´ , suchthat maximal subwords in a, x ˘ a , a P p Q E q are basis elements for Λ q E p Q E q , and maximalsubwords in a, x ˘ a , a P Q are basis elements of Λ q p Q , W q .In particular, the inclusion on basis elements not containing r, r defines a filtered iso-morphism (6.13) , as well as a section σ : Λ q p Q q Ñ L , such that p L, r, σ, B q satisfies thestrong free product property. Note in the proposition that we only need the elements x a instead of x a for a P Q beginning at a vertex of Q E (and making this change to the statement does not affect theproof). On the other hand, we could freely replace all generators by the modified ones x a ,even for a P p Q E q , again without changing the proof. Corollary 6.11.
The filtration on Λ q p Q q generated by arrows in Q z Q E incident to p Q E q is Hausdorff, and the associated graded map of (6.13) (using the same filtration on the LHS)is an algebra isomorphism, Λ q E p Q E q ˚ kQ Λ q p Q , p Q E q q Ñ gr Λ q p Q q . (6.14) Proof of Corollary 6.11.
Let X Ď Q z Q E be the set of arrows incident to p Q E q not in Q E itself. Note that gr Λ q p Q , p Q E q q – Λ q p Q , p Q E q q , since there are no relations at p Q E q ;also the filtration in Λ q E p Q E q ˚ kQ Λ q p Q , p Q E q q is entirely induced by the second factor,so the associated graded algebra of the free product is again the free product itself.Next, observe that the filtration on Λ q p Q , p Q E q q is Hausdorff, as again there are norelations at p Q E q . So the filtration is Hausdorff on the source of (6.13), hence also on thetarget, as the isomorphism is filtered. This yields the desired statement. Proof of Proposition 6.10.
Using the strong free product properties for L p Q E q and L p Q q ,we obtain the strong free product property for p L, r p Q E q ` r p Q q B , σ, B q , with σ definedby extending the sections σ p Q E q , σ p Q q . In more detail, combining the reduction systemsfor L Q E in Proposition 6.6 and for L Q in Proposition 6.9, one gets a reduction system for L Q – L Q E ˚ kQ L Q . However, Λ q p Q q “ L {p ρ p Q E q ρ p Q q ´ q q is not the quotient L {p ρ p Q E q ` ρ p Q q B ´ q q . So we need to perturb the relation to r “ ρ p Q q ρ p Q E q ´ q .First observe that this change does nothing to the reductions for L Q , since there therelation r is unchanged. For L Q E , this change perturbs only the substitution reductions,which become the following (applying a reordering relation as well for clarity). For conve-nience we use the original generators x, y, a, a ˚ for Q E but the modified ones x a for L Q (weonly need this for a beginning at a vertex of Q E ): • y ´ ÞÑ x ´ p ρ p Q q ´ ´ qp r ` q q ` x ´ p r ` q q , y ÞÑ p r ` q ´ qp ρ p Q q ´ q x ` p r ` q ´ q x (if not preceded by a ); • ax ÞÑ a p ρ p Q q ´ ´ qp r ` q q y ` ary ` qya , ax ´ ÞÑ ay ´ p r ` q ´ qp ρ p Q q ´ q ` ay ´ r ` q ´ y ´ a .The reason why we wrote these relations as above is because, at every vertex v P p Q E q , e v p ρ p Q q ˘ ´ q is a linear combination of positive-length monomials in the generators x ˘ a , a , a P Q (in fact we only need the x ˘ a ).Combining these systems of reductions one gets a system of reductions for L “ L Q E ˚ kQ L Q . Order monomials in L lexicographically in the orderings N and N of Propositions24.6 and 6.9. Then the above reductions strictly decrease the ordering (we use here the factthat the ideal of positive-length monomials beginning and ending at vertices of p Q E q ispreserved under reductions). All ambiguities lie either entirely in L Q E or entirely in L Q and hence they all resolve.We conclude that the strong free product property holds for p L, σ, ρ p Q E q ρ p Q q ´ q, B q and hence 6.13 is an isomorphism. References [Ani82] David J. Anick. Noncommutative graded algebras and their Hilbert series.
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D. Kaplan,
Department of Mathematics, Imperial College London, London, UK SW7 2AZ
E-mail address , D. Kaplan: [email protected]
T. Schedler,
Department of Mathematics, Imperial College London, London, UK SW7 2AZ
E-mail address , T. Schedler: [email protected]@imperial.ac.uk