Simple Z -graded domains of Gelfand-Kirillov dimension two
aa r X i v : . [ m a t h . R A ] M a y SIMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOVDIMENSION TWO LUIGI FERRARO, JASON GADDIS, AND ROBERT WON
Abstract.
Let k be an algebraically closed field and A a Z -graded finitelygenerated simple k -algebra which is a domain of Gelfand-Kirillov dimension2. We show that the category of Z -graded right A -modules is equivalent tothe category of quasicoherent sheaves on a certain quotient stack. The the-ory of these simple algebras is closely related to that of a class of generalizedWeyl algebras (GWAs). We prove a translation principle for the noncommu-tative schemes of these GWAs, shedding new light on the classical translationprinciple for the infinite-dimensional primitive quotients of U ( sl ). Introduction
Let k denote an algebraically closed field of characteristic zero. Throughout thispaper, all vector spaces are taken over k , all rings are k -algebras, and all categoriesand equivalences of categories are k -linear.Suppose A = L i ∈ N A i is a right noetherian N -graded k -algebra. There is anotion of a noncommutative projective scheme of A , defined by Artin and Zhangin [AZ94]. Let GrMod - A denote the category of N -graded right A -modules andlet grmod - A denote its full subcategory of finitely generated modules. Let Tors - A (tors - A , respectively) denote the full subcategory of GrMod - A (grmod - A , respec-tively) consisting of torsion modules (see Section 2.2).Let QGrMod - A denote the quotient category GrMod - A/ Tors - A , let A de-note the image of A A in QGrMod - A , and let S denote the shift operator onQGrMod - A . The general noncommutative projective scheme of A is the triple(QGrMod - A, A , S ). Similarly, if we let qgrmod - A = grmod - A/ tors - A , then the noetherian noncommutative projective scheme of A is the triple (qgrmod - A, A , S ).These two categories play the same role as the categories of coherent and quasico-herent sheaves on Proj R for a commutative k -algebra R . By abuse of language,we will refer to either of the categories QGrMod - A or qgrmod - A as the noncom-mutative projective scheme of A .If A = k , then A is called connected graded . In [AS95], Artin and Staffordclassified the noncommutative projective schemes of connected graded domains ofGelfand-Kirillov (GK) dimension 2. Since a connected graded domain of GK di-mension 2 is a generalization of a projective curve, Artin and Stafford’s theoremcan be viewed as a classification of noncommutative projective curves. Theorem (Artin and Stafford, [AS95, Corollary 0.3]) . Let A be a connected N -graded domain of GK dimension 2 which is finitely generated in degree . Thenthere exists a projective curve X such that qgrmod - A ≡ coh( X ) . Mathematics Subject Classification.
The focus of this paper is on Z -graded k -algebras. A fundamental example ofsuch a ring is the (first) Weyl algebra, A = k h x, y i / ( yx − xy − N -grading but does admit a Z -grading by letting deg x = 1 anddeg y = −
1. This Z -grading is natural in light of the fact that A is isomorphic tothe ring of differential operators on the polynomial ring k [ t ], where x correspondsto multiplication by t and y corresponds to differentiation by t . In [Smi11], Smithshowed that there exists an abelian group Γ and a commutative Γ-graded ring C such that the category of Γ-graded C -modules is equivalent to GrMod - A . As acorollary, there exists a quotient stack χ such that GrMod - A is equivalent to thecategory Qcoh( χ ) of quasicoherent sheaves on χ [Smi11, Corollary 5.15].As the Weyl algebra is a domain of GK dimension 2, Smith’s result can be seenas evidence for a Z -graded version of Artin and Stafford’s classification. Furtherevidence is found in [Won18a], in which the third-named author studied the gradedmodule categories over the infinite-dimensional primitive quotients { R λ | λ ∈ k } of U ( sl ). Each R λ is a Z -graded domain of GK dimension 2, and for each R λ ,there exists an abelian group Γ and a commutative Γ-graded ring B λ such thatQGrMod - R λ is equivalent to the category of Γ-graded B λ -modules [Won18a, The-orems 4.2 and 4.10, Corollary 4.20].In this paper, we prove a Z -graded analogue of Artin and Stafford’s theoremfor simple domains of GK dimension 2. A key ingredient in our result is Bell andRoglaski’s classification of simple Z -graded domains. In [BR16, Theorem 5.8], theyproved that for any simple finitely generated Z -graded domain A of GK dimension2, there is a generalized Weyl algebra (GWA) A ′ satisfying Hypothesis 2.3 suchthat GrMod - A ≡ GrMod - A ′ . We are therefore able to reduce to studying theseGWAs and prove the following theorem. Theorem (Theorem 5.3 and Corollary 5.5) . Let A = L i ∈ Z A i be a simple finitelygenerated Z -graded domain of GK dimension 2 with A i = 0 for all i ∈ Z . Then thereexists an abelian group Γ and a commutative Γ -graded ring B such that GrMod - A is equivalent to the category of Γ -graded B -modules. Hence, if χ is the quotientstack h Spec B Spec k Γ i then QGrMod - A ≡ Qcoh( χ ) . We also study properties of the commutative rings B that arise in the abovetheorem. For instance, we prove that B is non-noetherian of Krull dimension 1,and that B is coherent Gorenstein (see Section 5).In Section 3, we prove a translation principle for GWAs. This builds upon theclassical translation principle for U ( sl ), which we describe as follows. Over k , theassociative algebra U ( sl ) is generated by E, F, H subject to the relations [
E, F ] = H , [ H, E ] = 2 E , and [ H, F ] = − F . The Casimir element Ω = 4 F E + H + 2 H generates the center of U ( sl ). The infinite-dimensional primitive factors of U ( sl )are given by R λ = U/ (Ω − λ + 1) U for each λ ∈ k . The classical result states that R λ is Morita equivalent to R λ +1 unless λ = − ,
0. We refer the reader to [Sta82]for a proof.The rings R λ are isomorphic to GWAs with base ring k [ z ], defining automor-phism σ ( z ) = z + 1, and quadratic defining polynomial f ∈ k [ z ]. We study aclass of GWAs which includes all of the R λ (see Hypothesis 2.1). In particu-lar, we prove that for each GWA A satisfying these hypotheses, if a new GWA A ′ is obtained by shifting the factors of the defining polynomial of A by the IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 3 automorphism σ , then QGrMod - A ≡ QGrMod - A ′ (Theorem 3.6). One con-sequence of this result is that although Mod - R − Mod - R Mod - R andGrMod - R − GrMod - R GrMod - R , nevertheless there is an equivalence ofnoncommutative projective schemesQGrMod - R − ≡ QGrMod - R ≡ QGrMod - R . Hence, for any λ ∈ k , QGrMod - R λ ≡ QGrMod - R λ +1 . We remark that thisparticular consequence follows from a more specific result proved in [Won18a], andwas first observed by Sierra [Sie17]. Acknowledgments.
The authors would like to thank Daniel Chan, W. FrankMoore, Daniel Rogalski, S. Paul Smith, and James Zhang for helpful conversa-tions. We particularly thank Calum Spicer for his help on an earlier version of thismanuscript and Susan Sierra for directing us to references on Morita contexts andsuggesting the relationship with quotient categories that we prove in Section 3.2.
Preliminaries
In this section, we fix basic notation, definitions, and terminology which will bein use for the remainder of the paper.2.1.
Graded rings and modules.
Let Γ be an abelian semigroup. We say thata k -algebra R is Γ -graded if there is a k -vector space decomposition R = L γ ∈ Γ R γ such that R γ · R δ ⊆ R γ + δ for all γ, δ ∈ Γ. Each R γ is called the γ -graded component of R and each r ∈ R γ is called homogeneous of degree γ . Similarly, a (right) R -module M is Γ -graded if it has a k -vector space decomposition M = L γ ∈ Γ M γ suchthat M γ · R δ ⊆ M γ + δ for all γ, δ ∈ Γ. Each M γ is called the γ -graded component of M . When the group Γ is clear from context, we will call R and M simply graded .A homomorphism f : M → N of Γ-graded right R -modules is called a gradedhomomorphism of degree δ if f ( M γ ) ⊆ N γ + δ for all γ ∈ Γ. We denoteHom R ( M, N ) = M δ ∈ Γ Hom R ( M, N ) δ where Hom R ( M, N ) δ is the set of all graded homomorphisms M → N of degree δ .A graded homomorphism is a graded homomorphism of degree 0.The Γ-graded right R -modules together with the graded right R -module homo-morphisms (of degree 0) form a category which we denote GrMod -( R, Γ). There-fore, Hom
GrMod -( R, Γ) ( M, N ) = Hom R ( M, N ) . We use lower-case letters to denote full subcategories consisting of finitely-generatedobjects, so grmod -( R, Γ) denotes the category of finitely-generated Γ-graded right R -modules. When Γ = Z , we omit the group from our notation and refer to thesecategories as simply GrMod - R and grmod - R .We say two algebras R and S are Morita equivalent if there is an equivalenceof categories Mod - R ≡ Mod - S . If R is Γ-graded and S is a Λ-graded, thenwe say that R and S are graded Morita equivalent if there exists an equivalenceMod - R → Mod - S that restricts to an equivalence between their subcategories ofgraded modules.The group of autoequivalences of grmod - R modulo natural transformation iscalled the Picard group of grmod - R and is denoted Pic(grmod - R ). For a Z -graded k -algebra R , the shift functor is an autoequivalence of grmod - R which sends a FERRARO, GADDIS, AND WON graded right module M to the new module M h i = L j ∈ Z M h i j , defined by M h i j = M j +1 . We write this functor as S R . We use the notation M h i i forthe module S iR ( M ) and note that M h i i j = M j + i . This is the standard conventionfor shifted modules, although it is the opposite of the convention that is used in[Sie09, Won18a, Won18b].2.2. Noncommutative projective schemes of Z -graded algebras. Now sup-pose that A is a noetherian Z -graded k -algebra and let M ∈ GrMod - A . As in[Smi00], define the torsion submodule of M by τ ( M ) = { sum of the finite-dimensional submodules of M } . The module M is said to be torsion if τ ( M ) = M and torsion-free if τ ( M ) = 0. LetTors - A (tors - A , respectively) denote the full subcategory of GrMod - A (grmod - A ,respectively) consisting of torsion modules. This is a Serre subcategory and so wemay form the quotient category QGrMod - A = GrMod - A/ Tors - A (qgrmod - A =grmod - A/ tors - A , respectively).The shift functor S of GrMod - A descends to an autoequivalence of QGrMod - A and qgrmod - A , which we also denote by S . Let A denote the image of A inthe quotient categories. Then (QGrMod - A, A , S ) ((qgrmod - A, A , S ), respectively)is the noncommutative projective scheme ( noetherian noncommutative projectivescheme , respectively) of A . When A is actually N -graded, this definition coincideswith the noncommutative projective scheme Proj A defined by Artin and Zhang[AZ94].It is clear that tors - A = fdim - A , the subcategory of grmod - A consisting ofmodules of finite k -dimension. Since every A -module is a union of its finitelygenerated submodules, Tors - A can be described as the subcategory of GrMod - A consisting of modules which are unions of their finite-dimensional submodules.2.3. Generalized Weyl algebras and Z -graded simple rings. Let R be aring, let σ : R → R an automorphism of R , and fix a central element f ∈ R .The generalized Weyl algebra (GWA) of degree one A = R ( σ, f ) is the quotient of R h x, y i by the relations xy = f, yx = σ − ( f ) , xr = σ ( r ) x, yr = σ − ( r ) y for all r ∈ R . We call R the base ring and σ the defining automorphism of A . Gener-alized Weyl algebras were so-named by Bavula [Bav93], and many well-studied ringscan be realized as GWAs, including the classical Weyl algebras, ambiskew polyno-mial rings, and generalized down-up algebras. There is a Z -grading on R ( σ, f ) givenby deg x = 1, deg y = −
1, and deg r = 0 for all r ∈ R . We also remark that if R is commutative, then every Z -graded right A -module M has an ( R, A )-bimodulestructure as follows: if m ∈ M is homogeneous of degree i , then the left R -actionof r ∈ R is given by r · m = m · σ − i ( r ) . In this paper, every GWA satisfies the following hypothesis.
Hypothesis 2.1.
Let A = R ( σ, f ) be the GWA with(1) base ring R = k [ z ] and defining automorphism σ ( z ) = z + 1 , or(2) base ring R = k [ z, z − ] and defining automorphism σ ( z ) = ξz for somenonroot of unity ξ ∈ k × . IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 5 Assume that f ∈ k [ z ] is monic and in case 2 assume that is not a root of f . Let Zer( f ) denote the set of roots of f and for α ∈ Zer( f ) , let n α denote the multiplicityof α as a root of f . For any η ∈ R × , there is an isomorphism A ∼ = R ( σ, ηf ) mapping x to ηx , y to y and z to z . Hence, by adjusting by an appropriate unit in R , every GWA withbase ring and defining automorphism as above is isomorphic to one satisfying theadditional assumptions in Hypothesis 2.1.Since we are assuming that R = k [ z ] or k [ z, z − ], the GWA A is a noetheriandomain [Bav93, Proposition 1.3] of Krull dimension of 1 [Bav92, Theorem 2]. Hence, A is an Ore domain. We denote by Q ( A ) the quotient division ring of A , obtained bylocalizing A at all nonzero elements. The rank of an A -module M is the dimensionof M ⊗ A Q ( A ) over the division ring Q ( A ). Since A is Z -graded, we also consider Q gr ( A ), the graded quotient division ring of A , obtained by localizing A at allnonzero homogeneous elements. The field of fractions of A = R is k ( z ), so Q gr ( A )is a skew Laurent ring over k ( z ): Q gr ( A ) = k ( z )[ x, x − ; σ ] = M i ∈ Z k ( z ) x i . We use the notation σ k to denote the action of σ on the k -points of Spec R ; i.e.,for λ ∈ k , if σ ( z ) = z + 1 then σ k ( λ ) = λ − σ ( z ) = ξz then σ k ( λ ) = ξ − λ .We say two roots of f are congruent if they are on the same σ k -orbit. For a GWA A satisfying Hypothesis 2.1, by [Hod93, Bav96], the global dimension of A dependsonly on the roots of f :gldim A = f has no multiple roots and no congruent roots2 if f has a congruent root but no multiple roots ∞ if f has multiple roots.It follows from [Bav92, Theorem 3] that A is simple if and only no two distinctroots of f are congruent.In [BR16], Bell and Rogalski showed that simple Z -graded domains are closelyrelated to GWAs satisfying Hypothesis 2.1. Theorem 2.2 (Bell and Rogalski, [BR16, Theorem 5.8]) . Let S = L i ∈ Z S i be asimple finitely generated Z -graded domain of GK dimension 2 with S i = 0 for all i ∈ Z . Then S is graded Morita equivalent to a GWA R ( σ, f ) satisfying Hypothesis 2.1where additionally no two distinct roots of f ∈ R are congruent. Hence, if we are interested only in the category of graded modules over simple Z -graded rings of GK dimension 2, it suffices to restrict our attention to these GWAs.Starting in Section 4, we operate under the following additional hypothesis: Hypothesis 2.3.
Let A = R ( σ, f ) be a GWA satisfying Hypothesis 2.1. Furtherassume that f ∈ k [ z ] has no distinct two roots on the same σ k -orbit so that A issimple. Thus, when we restrict to Hypothesis 2.3, then it follows that gldim A = 2.3. A translation principle for GWAs
Throughout, suppose A is a GWA satisfying Hypothesis 2.1. The simple gradedright A -modules were described by Bavula in [Bav92] and we adopt his terminology FERRARO, GADDIS, AND WON here. The group h σ i acts on MaxSpec R , the set of maximal ideals of R . Specifically,the orbit of ( z − λ ) ∈ MaxSpec R is given by O λ = (cid:8) σ i ( z − λ ) (cid:12)(cid:12) i ∈ Z (cid:9) = (cid:8) ( z − σ i k ( λ )) (cid:12)(cid:12) i ∈ Z (cid:9) . If the σ -orbit of ( z − λ ) contains no factors of f , it is called nondegenerate , otherwiseit is called degenerate . If two distinct factors ( z − λ ) and σ i ( z − λ ) lie on the same σ -orbit, then λ and σ i k ( λ ) are congruent roots of f and we call O λ a congruentorbit . Otherwise, a degenerate orbit is called a non-congruent orbit . Lemma 3.1 ([Bav92, Theorem 1]) . Let A = R ( σ, f ) . The simple modules of grmod - A are given as follows.(1) For each nondegenerate orbit O λ of MaxSpec R , one has the module M λ = A ( z − λ ) A and its shifts M λ h n i for each n ∈ Z .(2) For each degenerate non-congruent orbit O α , one has(a) the module M − α = A ( z − α ) A + xA and its shifts M − α h n i for each n ∈ Z and(b) the module M + α = Aσ − ( z − α ) A + yA h− i and its shifts M + α h n i foreach n ∈ Z .(3) For each degenerate congruent orbit O α , label the roots on the orbit so thatthey are given by α, σ i k ( α ) , . . . , σ i r k ( α ) where > i > · · · > i r . Set i = 0 .Then one has(a) the module M − α = A ( z − α ) A + xA and its shifts M − α h n i for each n ∈ Z ,(b) for each k = 1 , . . . , r , the module M ( i k − ,i k ] α = Aσ i k (( z − α )) A + xA + y i k − i k − A h i k i and its shifts M ( i k − ,i k ] α h n i for each n ∈ Z , and(c) the module M + α = Aσ i r − ( z − α ) A + yA h i r − i and its shifts M + α h n i for each n ∈ Z . The notation M ± α is intended to reflect the fact that M − α is nonzero only insufficiently negative degree while M + α is nonzero only in sufficiently positive degree.The module M ( i k − ,i k ] α is nonzero in degrees ( i k − , i k ] ∩ Z . Example 3.2. If R = k [ z ], σ ( z ) = z + 1, and f = z ( z − ( z − O with roots 0 , σ − k (0) = 1 , σ − k (0) = 3. Inthis case, the simple modules are given by the shifts of M λ = A/ ( z − λ ) A for each λ ∈ k / Z as well as all shifts of the modules • M − = AzA + xA , • M (0 , = A ( z − A + xA + yA h− i , • M (1 , = A ( z − A + xA + y A h− i , and • M +0 = A ( z − A + xA + yA h− i . IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 7 These modules are exactly the simple modules appearing in the composition seriesof the module
A/zA .As described in the introduction, Stafford proved a translation principle for theinfinite dimensional primitive factors R λ of U ( sl ) [Sta82]. These rings are theGWAs k [ z ]( σ, f ) of type (1) in Hypothesis 2.1 with quadratic defining polynomial f = z ( z − λ ) ∈ k [ z ]. Stafford showed that for λ ∈ k , the rings k [ z ]( σ, z ( z − λ ))and k [ z ]( σ, z ( z − λ − λ = 0 or −
1. The cases λ = 0 , − K -theoretic techniques, along with Stafford’s result, to prove that twoprimitive factors R λ and R µ , are Morita equivalent if and only if λ ± µ ∈ Z [Hod92].Several authors have studied Morita equivalences between GWAs R ( σ, f ) satis-fying Hypothesis 2.1(1) with higher degree defining polynomials f . Jordan gavesufficient conditions on f and f ′ for R ( σ, f ) and R ( σ, f ′ ) to be Morita equivalent[Jor92, Lemma 7.3(iv)]. Richard and Solotar gave necessary conditions for theMorita equivalence of R ( σ, f ) and R ( σ, f ′ ) with the additional hypotheses that theGWAs are simple and have finite global dimension [RS10]. Shipman fully classifiedthe equivalence classes of GWAs satisfying Hypothesis 2.1(1) under the strongernotion of strongly graded Morita equivalences [Shi10].Our treatment builds upon [Sta82]. We extend Stafford’s translation principleto all GWAs satisfying Hypothesis 2.1. This was essentially done by Jordan [Jor92,Lemma 7.3(iv)]—our contribution is to pay close attention to those situations inwhich Stafford’s techniques do not give Morita equivalences. In these cases, weshow that Stafford’s methods still yield a graded Morita context. Using this Moritacontext, we prove that there is an equivalence between the quotient categories whichplay the role of the noncommutative projective schemes for these Z -graded rings.Hence, the results of this section can be viewed as a translation principle for the non-commutative projective schemes of GWAs, which is analogous to Van den Bergh’stranslation principle for central quotients of the four-dimensional Sklyanin algebra[VdB96].Before we proceed, we recall some details on graded Morita contexts and thegraded Kato-M¨uller Theorem in the Z -graded setting. Recall that a Morita context between two algebras T and S is a 6-tuple ( T, S, S M T , T N S , φ, ψ ) where S M T and T N S are bimodules and φ : M ⊗ T N → S and ψ : N ⊗ S M → T are bimodulemorphisms satisfying • ψ ( m ⊗ n ) m ′ = mφ ( n ⊗ m ′ ) • φ ( n ⊗ m ) n ′ = nψ ( m ⊗ n ′ )for all m, m ′ ∈ M , n, n ′ ∈ N . We refer to I = im φ and J = im ψ as the trace ideals of the Morita context. If both φ and ψ are surjective, then this Morita contextgives a Morita equivalence between T and S and so Mod - T ≡ Mod - S .Let T be an algebra and let M be a right T -module. One important constructionof a Morita context is given by ( T, M, M ∗ , S, φ, ψ ) where M ∗ = Hom T ( M, T ), S = Hom T ( M, M ), and where φ : M ∗ ⊗ S M → T and ψ : M ⊗ T M ∗ → S aredefined by φ ( f, m ) = f ( m ) and ψ ( m, f )( n ) = mf ( n ) for f ∈ M ∗ and m, n ∈ M .This Morita context is a Morita equivalence if and only if M is a progenerator ofMod - T .This theory carries over to the graded setting (see [Haz16, Chapter 2]). If T is a Z -graded algebra and M is a finitely generated graded right A -module, then FERRARO, GADDIS, AND WON both M ∗ = Hom T ( M, T ) and S = Hom T ( M, M ) are, in fact, graded with M ∗ =Hom T ( M, T ) and S = Hom T ( M, M ). Then (
T, M, M ∗ , S, φ, ψ ) is a graded Moritacontext . If M is a graded progenerator then this graded Morita context gives agraded Morita equivalence so Mod - T ≡ Mod - S and GrMod - T ≡ GrMod - T .In [IN05], Iglesias and Nˇastˇasescu establish a graded version of the Kato-M¨ullerTheorem as follows. Let T be Z -graded ring. For any two-sided graded ideal I of T , the class P I = { M ∈ GrMod - T | IM = 0 } is a rigid closed subcategory of GrMod - T . Let C I be the smallest localizing sub-category of GrMod - T containing P I . By [IN05, Remark 2.5], if I is idempotent( I = I ) then C I = P I . For a graded Morita context ( T, S, T M S , S N T , φ, ψ ) withtrace ideals I and J , the graded functorsHom T ( M, − ) : GrMod - T ⇄ GrMod - S : Hom S ( N, − )induce functors between the quotient categoriesHom T ( M, − ) : GrMod - T / C I ⇄ GrMod - S/ C J : Hom S ( N, − ) . (3.3)By [IN05, Theorem 4.2], these induced functors give graded equivalences betweenGrMod - T / C I and GrMod - S/ C J . Note that Hom T ( M, − ) commutes with the shiftfunctors of GrMod - T and GrMod - S . In particular, if X is a graded T -module, thenHom T ( M, S T X ) = S S Hom T ( M, X ) [GG82, Lemma 2.2]. This result also descendsto the quotient categories: the functor induced by Hom T ( M, − ) commutes with thefunctors induced by the shift functors of GrMod - T and GrMod - S .Let A = R ( σ, f ) be a GWA satisfying Hypothesis 2.1 with n = deg( f ) >
0. Let h ∈ k [ z ] divide f and define g = h − f . Define the graded module M = hA + xA and construct a graded Morita context( A, B, B M A , A M ∗ B , φ, ψ )( † )as discussed above with M ∗ = Hom A ( M, A ) and B = Hom A ( M, M ). We make theidentification M ∗ = { p ∈ Q gr ( A ) | pM ⊂ A } where Q gr ( A ) = k ( z )[ x, x − ; σ ] is the graded quotient division ring of A [MR87,Proposition 3.1.15]. Similarly, we make the identification B = { p ∈ Q gr ( A ) | pM ⊂ M } . Under this identification, the graded bimodule maps φ : M ⊗ A M ∗ → B and ψ : M ∗ ⊗ B M → A are given by φ ( m, m ′ ) = mm ′ and ψ ( m ′ , m ) = m ′ m . Since M is not a graded progenerator in general, this graded Morita context is not alwaysgraded Morita equivalence. However, we will show that it induces a graded equiv-alence between certain quotient categories of GrMod - A and GrMod - B . Part (3)of the lemma below recovers [Sta82, Corollary 3.3] when R = k [ z ] and f ∈ k [ z ] isquadratic. Lemma 3.4.
Retain notation as above. Let A ′ = R ( σ, f ′ ) be the GWA with f ′ = σ ( h ) g .(1) Then B ∼ = A ′ so ( † ) gives a graded Morita context between A and A ′ .(2) Suppose β is a root of f of multiplicity one. If h = z − β , then ∈ im φ and so M = hA + xA is a projective right A -module. IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 9 (3) Set d = gcd( σ − ( g ) , h ) in A . Then ∈ im ψ if and only if d = 1 if andonly if M is a generator for A .Proof. (1) It is clear from the identification above that A ⊂ M ∗ . Observe that x − gx = σ − ( g ) ∈ A and x − gh = x − f = x − xy = y ∈ A . Hence, x − g ∈ M ∗ and so A + Ax − g ⊂ M ∗ .Note that M M ∗ ⊂ B . Clearly 1 , x ∈ B . Since zx = xσ − ( z ) ∈ xA , then z ∈ B .As x − g ∈ M ∗ , then x − σ ( h ) g = hx − g ∈ M M ∗ ⊂ B . Let S be the subalgebra of B ⊂ Q gr ( A ) generated by X = x , Y = x − σ ( h ) g , and Z = z . We check that thesegenerators satisfy the relations for A ′ . It is clear that XZ = σ ( Z ) X and XY = f ′ .A computation shows that Y Z = x − σ ( h ) gz = σ − ( z ) x − σ ( h ) g = σ − ( Z ) Y,Y X = x − σ ( h ) gx = hσ − ( g ) = σ − ( f ′ )and so S ∼ = A ′ .Given any q ∈ M , Bq ⊂ M ⊂ S . By [Jor92, Corollary 4.4], S is a maximal orderin its quotient ring and by [VdBVO89, Lemma 2], it is also a graded maximal orderin its graded quotient ring, whence B = S ∼ = A ′ .(2) By the Dual Basis Lemma, M is projective if and only if 1 ∈ M M ∗ [MR87,Lemma 3.5.2]. By the proof of (1), A + Ax − g ⊂ M ∗ . Thus, it is clear that X, h ( Z ) ∈ im φ . We have x ( x − g ) = g ( Z ) ∈ im φ , so 1 = gcd( g ( Z ) , h ( Z )) ∈ im φ .(3) By (2), M is projective and so again by the Dual Basis Lemma, M ∗ = A + Ax − g . Thus, im ψ is generated by products of generators in M ∗ and M .Since A ⊂ M ∗ , then x, h ∈ M ∗ M . Furthermore, y = ( x − g ) h ∈ M ∗ M and σ − ( g ) = ( x − g ) x ∈ M ∗ M , so d ∈ im ψ . Since ( xA ) = f R and ( yA ) = σ − ( g ) R ,therefore ( M ∗ M ) is the ideal of R generated by h and σ − ( g ). Hence, 1 ∈ im ψ ifand only if d = 1. (cid:3) We are now prepared to prove the translation principle for GWAs which is themain result of this section. The result in (1) below should also be compared to aresult of Shipman [Shi10] in the strongly graded Morita equivalence setting.
Proposition 3.5.
Let A = R ( σ, f ) . Choose a root β of f , and consider its σ k -orbit, O . Write f = ( z − β ) · · · ( z − β n ) · ˜ f where β , . . . , β n ∈ O and ˜ f has no rootson O . For each ≤ j ≤ n , write β j = σ − i j k ( β ) for some i j ∈ Z . After reordering,assume that i j ≥ i j − for < j ≤ n .(1) Suppose that either k = 1 or ≤ k ≤ n and i k − i k − > . Suppose furtherthat n β k = 1 . Let A ′ = R ( σ, f ′ ) where f ′ is the same as f except that thefactor ( z − β k ) of f has been replaced by a factor of ( z − σ k ( β k )) in f ′ .Then there is an equivalence of categories GrMod - A ≡ GrMod - A ′ .(2) Suppose that either k = n or ≤ k ≤ n − and i k +1 − i k > . Supposefurther that n β k = 1 . Let A ′ = R ( σ, f ′ ) where f ′ is the same as f exceptthat the factor ( z − β k ) of f has been replaced by a factor of ( z − σ − k ( β k )) in f ′ . Then there is an equivalence of categories GrMod - A ≡ GrMod - A ′ .(3) Suppose for some ≤ k ≤ n that i k − i k − = 1 , and that n β k = 1 . Let A ′ = R ( σ, f ′ ) where f ′ is the same as f except that the factor ( z − β k ) of f hasbeen replaced by a factor of ( z − β k − ) in f ′ . Then there is an equivalence ofcategories GrMod - A/ C I ≡ GrMod - A ′ where I = ( x, y, z − β k ) . Therefore, QGrMod - A ≡ QGrMod - A ′ . Proof. (1) Set h ( z ) = z − β k . Then by hypothesis, gcd( σ − ( g ) , h ) = 1 and soLemma 3.4 (2) and (3) imply that M is a progenerator for A . Thus the Moritacontext in (1) is an equivalence.(2) By the hypotheses, σ − k ( β k ) has multiplicity 1 as a root of f ′ and so we mayapply (1) to obtain GrMod - A ′ ≡ GrMod - A .(3) Again set h ( z ) = z − β k so that gcd( σ − ( g ) , h ) = h and gcd( g, h ) = 1.Set I = im ψ = xA + yA + hA to be the trace ideal corresponding to ψ . ThenGrMod - A/ C I ≡ GrMod - A ′ by the graded Kato-M¨uller Theorem [IN05, Theorem4.2]. We claim that I is idempotent so that C I = P I . We will then shows that C I contains only torsion modules to arrive at the intended equivalence.Since x, z − β k ∈ I , then ( σ ( z ) − z ) x = [ x, z − β k ] ∈ I . In case R = k [ z ],we have σ ( z ) − z = 1 so x ∈ I . If R = k [ z, z − ], then σ ( z ) − z = ( ξ − z , so x = ( ξ − − z − ( σ ( z ) − z ) x ∈ I . Similarly, y ∈ I . Now since xy = f ∈ I and yx = σ − ( f ) ∈ I , then z − α = gcd( f, σ − ( f )) ∈ I . As z − β k is a consecutivenon-repeated root, we have gcd( f, σ − ( f ) , z − β k ) = z − β k ∈ I and so I ⊂ I ⊂ I as claimed.We apply [IN05, Theorem 4.2] and our description of the finite-dimensional sim-ple modules. It is clear that I is the annihilator of M ( i k − ,i k ] (and its shifts) butthat ann N I = { n ∈ N : nx = 0 for all x ∈ I } = 0 for any other simple A -module N , including those corresponding to nondegenerate orbits.As M is a finitely generated graded right A -module, there is a graded surjectionfrom the graded free module L ni =1 A h d i i → M which induces a graded injectionHom A ( M, N ) → Hom A n M i =1 A h d i i , N ! = n M i =1 N h− d i i . Thus, Hom A ( M, N ) is isomorphic to a graded submodule of L ni =1 N h− d i i , whichis finite-dimensional if and only if N is finite-dimensional. Therefore, Hom A ( M, − )maps finite-dimensional modules to finite-dimensional modules. Since it also pre-serves submodule inclusion, Hom A ( M, − ) also maps torsion modules to torsionmodules.Finally, let M ∈ C I and let m ∈ M . Since Im = 0 and A/I is finite-dimensional,then dim k ( Am ) < ∞ . It follows that M ∈ Tors - A , so C I is a subcategory ofTors - A , whence the equivalence GrMod - A/ C I ≡ GrMod - A ′ induces an equivalenceQGrMod - A ≡ QGrMod - A ′ . (cid:3) An immediate consequence of the above proposition is the following translationprinciple for GWAs.
Theorem 3.6.
Let A = R ( σ, f ) be a GWA satisfying Hypothesis 2.1. Let A ′ = R ( σ, f ′ ) where f ′ is obtained from f by replacing any irreducible factor ( z − β ) of f by σ ( z − β ) in f ′ . Then QGrMod - A ≡ QGrMod - A ′ . The following example illustrates our methods from Proposition 3.5.
Example 3.7.
Let A = k [ z, z − ]( σ, f ) where σ ( z ) = pz for some nonroot of unity p ∈ k × and f = ( z − z − p ) ( z − p ). Let A ′ = k [ z, z − ]( σ, f ′ ) with σ as before but f ′ = ( z − z − p )( z − p )( z − p ). Applying Proposition 3.5 (2), we can replace f ′ with f ′ = ( z − z − p ) ( z − p ) up to equivalence of the quotient categoryQGrMod - A . Then applying Proposition 3.5 (1) to the last factor we have thatQGrMod - A ≡ QGrMod - A ′ . On the other hand, let A ′′ = k [ z, z − ]( σ, f ′′ ) with σ IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 11 as before but f ′′ = ( z − . We can collapse f ′ to f ′′ in the following way, wherethe labels on the arrows indicate the applicable part of Proposition 3.5: f ′ (2) −−→ ( z − ( z − p )( z − p ) (1) −−→ ( z − ( z − p )( z − p ) → · · ·· · · (2) −−→ ( z − ( z − p ) (1) , (1) −−−−→ ( z − ( z − p ) (2) −−→ ( z − = f ′′ Hence, QGrMod - A ′ ≡ GrMod - A ′′ and, transitively, QGrMod - A ≡ GrMod - A ′′ . Corollary 3.8.
Suppose A = R ( σ, f ) is a GWA satisfying Hypothesis 2.1. Then QGrMod - A ≡ GrMod - A ′ for some simple GWA A ′ = R ( σ, f ′ ) satisfying Hypoth-esis 2.3. Furthermore, the noncommutative projective schemes of A and A ′ areequivalent.Proof. Repeatedly apply Theorem 3.6 to each distinct σ -orbit of the irreduciblefactors of f until the resulting polynomial f ′ has no distinct irreducible factors onthe same σ -orbit. Then f ′ has no congruent roots so A ′ is simple and QGrMod - A ≡ QGrMod - A ′ . Since A ′ is simple, GrMod - A ′ has no finite-dimensional modules, andhence QGrMod - A ≡ QGrMod - A ′ = GrMod - A ′ .It remains to prove that (QGrMod - A, A , S A ) ≡ (QGrMod - A ′ , A ′ , S A ′ ). Notethat at each step of the reduction from A to A ′ , the equivalence functor is givenby (3.3). The discussion following (3.3) shows that this equivalence commutes withthe shift functors. Finally, by construction, the equivalence functor maps A to A ′ ,and so maps A and A ′ and so the noncommutative projective schemes of A and A ′ are equivalent. (cid:3) Graded modules over simple GWAs
Henceforth we assume Hypothesis 2.3 unless otherwise stated. In [Won18b,Section 3], the third-named author studied the finite-length modules and projectivemodules in grmod - R ( σ, f ) when R = k [ z ] and f ∈ k [ z ] is quadratic. Here, we provesimilar results for the simple GWAs of interest, thereby generalizing these resultsin two directions: first, by considering both base rings R = k [ z ] and R = k [ z, z − ]and second, by considering polynomials of arbitrary degree with no two distinctroots on the same σ k -orbit. In contrast with the treatment in [Won18b], ratherthan trying to understand all extensions of simple graded modules, we will focusonly on certain indecomposable modules, which we will need in Section 4.4.4.1. Finite-length indecomposable modules.
Recall the description of the sim-ple graded A -modules from Lemma 3.1. Since we are assuming Hypothesis 2.3, thereare no degenerate congruent orbits, so the third case in Lemma 3.1 does not occur.Hence, GrMod - A has no modules of finite k -dimension, so throughout this section,GrMod - A = QGrMod - A .By [Bav96, Lemma 3.6], both M − α and M + α have projective dimension 1 if n α = 1and infinite projective dimension if n α >
1. Then following lemma shows thatthere are indecomposable modules of projective dimension 1 and length n α whosecomposition factors are all M − α or all M + α . Lemma 4.1.
For each α ∈ Zer( f ) and each j = 1 , . . . , n α , there is an indecompos-able graded right A -module of length jM − jα = A ( z − α ) j A + xA whose composition factors are j copies of M − α . Similarly, there is an indecomposablegraded right A -module of length jM + jα = Aσ − (( z − α ) j ) A + yA h− i whose composition factors are j copies of M + α . Further, both M − n α α and M + n α α have projective dimension .Proof. We prove the statements for M − α . The proofs of the statements for M + α aresimilar.We proceed by induction on j . For the base case, M − α = M − α . For 2 ≤ j ≤ n α ,consider the natural projection M − jα → M − ( j − α whose kernel is given by K = ( z − α ) j − A + xA ( z − α ) j A + xA ∼ = ( z − α ) j − A ( z − α ) j − A ∩ [( z − α ) j A + xA ] . There is an isomorphism M − α ∼ = K given by left multiplication by ( z − α ) j − . Hence, M − jα is a length j module whose composition factors are all isomorphic to M − α .It suffices to show that M − jα has a unique submodule isomorphic to M − α andis thus indecomposable. Any homomorphism ϕ : M − α → M − jα is determined by ϕ (1) ∈ ( M − jα ) . Since, in M − jα , ( z − α ) j = 0, every element of ( M − jα ) can bewritten as P j − ℓ =0 β ℓ ( z − α ) ℓ for some β ℓ ∈ k . For ϕ to be well-defined, we must have ϕ (1) = β ( z − α ) j − for some β ∈ k , and so Hom grmod - A ( M − α , M − jα ) = k .Finally, to see that M − n α α has projective dimension 1, we use a technique similarto [Bav93, Theorem 5]. Observe that there is an exact sequence0 −→ ( z − α ) n α A + xA φ −→ A ⊕ A h− i ψ −→ (cid:2) σ − (( z − α ) n α ) A + yA (cid:3) h− i −→ φ ( w ) = ( w, [ σ − (( z − α ) n α )] − yw ) and ψ ( u, v ) = yu − σ − (( z − α ) n α ) v .Then M − n α α and M + n α α will have projective dimension 1 if this exact sequencesplits; i.e., if there is a splitting map ν : A ⊕ A → ( z − α ) n α A + xA such that ν ◦ φ = Id. This is equivalent to the existence of a, b ∈ ( z − α ) n α A + xA so that 1 = a + b [ σ − (( z − α ) n α )] − y (if so, we can define the splitting map ν ( u, v ) = au + bv ).But this is true since gcd (( z − α ) n α , f / ( z − α ) n α ) = 1 in R . (cid:3) The indecomposable modules M − n α α and M + n α α will play an important role inconstructing autoequivalences of grmod - A in section 4.4. The proof of the followinglemma is the same as that of [Won18b, Lemma 3.4]. Lemma 4.2.
Let n ∈ Z . Then for each α ∈ Zer( f ) and each j = 1 , . . . , n α , ann R ( M − jα h n i ) i = ann R ( M + jα h n i ) i = σ − n (( z − α ) j ) R, for i ≤ − n, and M − jα h n i = M + jα h n i = 0 for i > − n . As graded left R -modules, we have A ( z − α ) A h n i ∼ = Aσ − ( z − α ) A h n − i ∼ = M i ∈ Z Rσ − n ( z − α ) R .
It is easy to see that as a left R -module, M λ ∼ = L j ∈ Z R/ ( z − λ ) R . This fact, whencombined with the previous lemma and Lemma 3.1, means that any finite-lengthgraded A -module is supported at finitely many k -points of Spec R when consideredas a left R -module. IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 13 Definition 4.3. If M is a graded A -module of finite length, define the support of M ,Supp M , to be the support of M as a left R -module. If Supp M ⊂ { σ i k ( α ) | i ∈ Z } for some α ∈ k , we say that M is supported along the σ k -orbit of α .By Lemmas 3.1 and 4.2, for any λ ∈ k \ { σ i k ( α ) | α ∈ Zer( f ) , i ∈ Z } , the simplemodule M λ is the unique simple supported at λ and for each α ∈ Zer( f ) and n ∈ Z , M − α h n i and M + α h n i are the unique simple modules supported at σ − n k ( α ).4.2. Rank one projective modules.
In this section, we seek to understand therank one graded projective right A -modules. We begin by studying the gradedsubmodules of Q gr ( A ).Let I be a finitely generated graded right A -submodule of Q gr ( A ). Since R is aPID, for each i ∈ Z , there exists a i ∈ k ( z ) so that as a left R -module, I = M i ∈ Z Ra i x i . For every i ∈ Z , multiplying I i on the right by x shows that Ra i ⊆ Ra i +1 , whilemultiplying I i on the right by y shows that Ra i +1 σ i ( f ) ⊆ Ra i .Define c i = a i a − i +1 . Since Ra i ⊆ Ra i +1 , therefore c i ∈ R , and since Ra i +1 σ i ( f ) ⊆ Ra i , we conclude that c i divides σ i ( f ) in R . By multiplying by an appropriate unitin R , we may assume that c i ∈ k [ z ] and that c i is monic. Hence, c i actually divides σ i ( f ) in k [ z ]. Definition 4.4.
For a finitely generated graded submodule I = L i ∈ Z Ra i x i of Q gr ( A ), we call the sequence { c i = a i a − i +1 } i ∈ Z described above the structure con-stants of I .Many properties of I can be deduced by examining its structure constants.The proofs of the next two lemmas are immediate generalizations of the proofsof [Won18b, Lemmas 3.8 and 3.9]. Lemma 4.5.
Let I and J be finitely generated graded submodules of Q gr ( A ) withstructure constants { c i } and { d i } , respectively. Then I ∼ = J as graded right A -modules if and only if c i = d i for all i ∈ Z . Lemma 4.6.
Suppose I = L i ∈ Z Ra i x i is a finitely generated graded right A -submodule of Q gr ( A ) with structure constants { c i } . Then for n ≫ , c n = 1 and c − n = σ − n ( f ) . Further, for any choice { c i } i ∈ Z satisfying(1) for each n ∈ Z , c n ∈ k [ z ] , c n | σ n ( f ) and(2) for n ≫ , c n = 1 and c − n = σ − n ( f ) ,there is a finitely generated module I with structure constants { c i } . The structure constants of I also determine which of the indecomposable modulesdescribed in Lemma 4.1 are homomorphic images of I . Lemma 4.7.
Let I = L i ∈ Z Ra i x i be a finitely generated graded right A -submoduleof Q gr ( A ) with structure constants { c i } . For α ∈ Zer( f ) and j = 1 , . . . , n α (1) there is a surjective graded right R -module homomorphism I → M − jα h n i ifand only if σ − n (cid:0) ( z − α ) n α − j +1 (cid:1) ∤ c − n ,(2) there is a surjective graded right R -module homomorphism I → M + jα h n i ifand only if σ − n (cid:0) ( z − α ) j (cid:1) | c − n .Moreover, these surjections are unique up to a scalar. Proof.
We prove statement (1); the proof of statement (2) is similar. Suppose thereis a graded surjective homomorphism I → M − jα h n i . The kernel of this morphism isagain a graded submodule of Q gr ( A ), so we can write the kernel as J = L i ∈ Z Rb i x i with each b i ∈ k ( z ). Let { d i } denote the structure constants of J . Since, M − jα h n i is nonzero in precisely degrees i ≤ − n , it follows that b i = a i for all i > − n .Further, by Lemma 4.2, for all i ≤ − n , ( M − jα h n i ) i ∼ = R/σ − n (( z − α ) j ) R as left R -modules. Therefore, for all i ≤ − n , Rb i ⊇ Rσ − n (( z − α ) j ) a i . We also noticethat dim k Ra i /Rb i = dim k R/Rσ − n (( z − α ) j ) = j , since Rb i ⊆ Ra i and both a i and b i are monic, we deduce b i = σ − n (( z − α ) j ) a i . Computing structure constants,we see that when i = − n , c i = d i , while d − n = σ − n (( z − α ) j ) c − n . Since by thediscussion before Definition 4.4, d − n | σ − n ( f ), we conclude that σ − n (cid:0) ( z − α ) n α − j +1 (cid:1) ∤ c − n , otherwise ( z − α ) n α +1 would divide f .Conversely, suppose that σ − n (cid:0) ( z − α ) n α − j +1 (cid:1) ∤ c − n . Then we can construct afinitely generated graded right A -submodule of Q gr ( A ) by setting J = L i ∈ Z Rb i x i ⊆ I and b i = a i for all i > − n and b i = σ − n (( z − α ) j ) a i for all i ≤ − n . By multi-plying by a scalar, we may assume that b i is monic for all i . For all i ∈ Z , define d i = b i b − i +1 . Observe that for i = − n , d i = c i , while d − n = σ − n (( z − α ) j ) c − n . The { d i } are the structure constants of J , which is finitely generated by Lemma 4.6.Now by [Bav92, Theorem 2], I/J has finite length. Further,
I/J is a gradedmodule such that (
I/J ) i = 0 for each i > − n and dim k ( I/J ) i = j for each i ≤ − n ,since in this case Ra i /Rb i ∼ = R/σ − n (( z − α ) j ). Hence, by Lemma 3.1 and bylooking at the degrees in which the graded simple modules are nonzero, we deducethat the composition factors of I/J are j copies of M − α h n i . Finally, I/J has aunique submodule which is isomorphic to M − α h n i , corresponding to the module J ⊆ L i ∈ Z Rb ′ i x i ⊆ I where b ′ i = a i for all i > − n and b ′ i = σ − n (( z − α ) j − ) a i for all i ≤ − n . Therefore, I/J is indecomposable and
I/J ∼ = M − jα h n i . We note that wehave constructed the unique kernel of a morphism I → M − jα and so the surjectionis unique up to a scalar. (cid:3) We now focus on the rank one projective modules. Let P be a finitely generatedgraded projective right A -module of rank one. Since P embeds in Q ( A ) and isgraded it follows that P embeds in Q gr ( A ), therefore P has a sequence of struc-ture constants. We will be able to detect the projectivity of P from its structureconstants. Lemma 4.8.
Let P be a graded projective right A -module. Let α ∈ Zer( f ) and j = 1 , . . . , n α . Any graded surjection P → M ± α h n i lifts to a graded surjection P → M ± jα h n i which is unique up to a scalar.Proof. It suffices to prove that for any j = 2 , . . . , n α any graded surjection P → M − ( j − α lifts to a graded surjection P → M − jα .Let π be the projection π : M − jα → M − ( j − α . Since P is projective, any gradedsurjection P → M − ( j − α lifts to a graded morphism g : P → M − jα . Now sincethe kernel of π is ( z − α ) j − A + xA , any preimage of 1 under π has the form p = 1 + ( z − α ) j − a for some a ∈ A . Observe that p (1 − ( z − α ) j − a ) = 1 in M − jα ,so any preimage of 1 generates M − jα and hence g is an surjection. By Lemma 4.7, g is unique up to a scalar. In the other case, the proof is analogous. (cid:3) IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 15 The following lemma generalizes [Won18b, Lemma 3.32].
Lemma 4.9.
Let P be a graded submodule of Q gr ( A ) with structure constants { c i } .Then P is projective if and only if for each n ∈ Z , c n = Q α ∈ I σ n (( z − α ) n α ) forsome (possibly empty) subset I ⊆ Zer( f ) .Proof. We first assume that for each n ∈ Z , c n = Q α ∈ I σ n (( z − α ) n α ) for some I ⊆ Zer( f ) and prove that P must be projective. As in [Won18b, Lemma 3.32], weprove the projectivity of P by constructing a finitely generated projective modulewith the same structure constants as P , the claim would then follow by Lemma 4.5.Note that by Lemma 4.6, there exists N ∈ Z such that for all n ≥ N , c n = 1and for all n ≤ − N , c n = σ n ( f ). An elementary computation shows that if P = A h N i and the structure constants of P are denoted by { d i } , then d n = 1for n ≥ N and d n = σ n ( f ) for n < N . If c n = d n for all n then we are done.Otherwise let i be the largest integer such that c i = d i . By hypothesis, thereis some I ⊆ Zer( f ) such that c i = Q α ∈ I σ i (( z − α ) n α ) and by construction, d i = σ i ( f ). Let J = Zer( f ) \ I .Since σ i (( z − α ) n α ) | σ i ( f ) = d i then, by Lemma 4.7, there is a gradedsurjection π α : P → M + n α α h− i i for each α ∈ Zer( f ). Let P be the kernel of themap L α ∈ J π α : P → L α ∈ J M + n α α h− i i . By Lemma 4.1, the modules M + n α α h− i i have projective dimension 1, therefore the projectivity of P implies the projectivityof P . Following the same strategy used in the first paragraph of the proof ofLemma 4.7 we see that P is a submodule of Q gr ( A ) and the structure constants of P and P are equal except in degree i , where P has structure constant c i . Wecontinue this process for the finitely many indices where c i differs from d i until wereach a projective module which has the same structure constants as P .Conversely suppose that P is projective. Using Lemma 4.7 and the above lemma,it follows that if σ n ( z − α ) | c n , then σ n (( z − α ) n α ) | c n . Since by the paragraphbefore Definition 4.4 we know that c n | σ n ( f ), the assertion follows. (cid:3) Since projective graded submodules of Q gr ( A ) have rank one, as a corollary, wesee that a graded rank one projective module has a unique simple factor supportedat σ n k ( α ) for each α ∈ Z and each n ∈ Z . Corollary 4.10.
Let P be a rank one graded projective A -module, let α ∈ Zer( f ) ,and let n ∈ Z . Then P surjects onto exactly one of M − α h− n i and M + α h− n i .Proof. By Lemma 4.9 each structure constant c n of P is either relatively prime to σ n (( z − α ) n α ) or else has a factor of σ n (( z − α ) n α ). By Lemma 4.7, in the firstcase, P surjects onto M − α h− n i and not M + α h− n i , and in the second case P surjectsonto M + α h− n i and not M − α h− n i . (cid:3) Corollary 4.11.
A rank one graded projective A -module is determined up to iso-morphism by its simple factors which are supported at the σ k -orbits of the roots α of f .Proof. Let P be a rank one graded projective A -module with structure constants { c i } . By Lemma 4.7 and Corollary 4.10, the simple factors of P supported atthe σ k -orbits of the roots α of f determine the { c i } and by Lemma 4.5 thereforedetermine P . (cid:3) We have shown that for a rank one projective P for each α ∈ Zer( f ) and each n ∈ Z , P has a unique simple factor supported at σ n k ( α )—either M − α h− n i or M + α h− n i . The next result show that if for each n ∈ Z and each α ∈ Zer( f ), we choose one of M − α h− n i and M + α h− n i , as long as our choice is consistent with Lemma 4.6, thereexists a projective module with the prescribed simple factors. Lemma 4.12.
For each α ∈ Zer( f ) and each n ∈ Z choose S α,n ∈ { M − α h− n i , M + α h− n i} such that for n ≫ , S α,n = M − α h− n i and S α, − n = M + α h n i . Then there exists afinitely generated rank one graded projective P such that for all n ∈ Z , the uniquesimple factor of P supported at σ n k ( α ) is S α,n .Proof. We construct P via its structure constants { c i } as follows. For each α ∈ Zer( f ) and i ∈ Z , let d α,i = ( S α,i = M − α h− i i σ − i (( z − α ) n α ) if S α,i = M + α h− i i and let c i = Q α ∈ Zer( f ) d α,i . By Lemmas 4.6 and 4.9, P is a finitely generated rankone graded projective module. (cid:3) Morphisms between rank one projectives.
In [Won18b, § A were described in the casethat A = k [ z ]( σ, f ) for σ ( z ) = z + 1 and f ∈ k [ z ] quadratic. The hypotheses usedtherein were that k [ z ] is a PID and A is 1-critical, i.e., A has Krull dimension 1 andfor any proper submodule B ⊆ A , A/B has Krull dimension 0. Since both k [ z ] and k [ z, z − ] are PIDs and all of the simple GWAs in this paper are 1-critical, the re-sults generalize immediately. We briefly review the pertinent results and definitionsfrom [Won18b]. Definition 4.13.
Given a graded rank one projective module P with structureconstants { c i } , the canonical representation of P is the module P ′ = M i ∈ Z Rp i x i ⊆ Q gr ( A )where p i = Q j ≥ i c j . We note that P ′ ∼ = P and call P ′ a canonical rank one gradedprojective module . Every rank one graded projective right A -module is isomorphicto a unique canonical one in this way.Let P and Q be finitely generated right A -modules. An A -module homomor-phism f : P → Q is called a maximal embedding if there does not exist an A -modulehomomorphism g : P → Q such that f ( P ) ( g ( P ). In [Won18b], it was proved thatif P and Q are finitely generated graded rank one projectives, then there exists amaximal embedding P → Q which is unique up to a scalar. Further, if P and Q are canonical rank one graded projectives, this maximal embedding can be computedexplicitly. Proposition 4.14 ([Won18b, Proposition 3.38]) . Let P and Q be finitely generatedgraded rank one projective A -modules embedded in Q gr ( A ) . Then every homomor-phism P → Q is given by left multiplication by some element of k ( z ) and as a left R -module, Hom grmod - A ( P, Q ) is free of rank one. Lemma 4.15 ([Won18b, Lemma 3.40 and Corollary 3.41]) . Let P and Q be rankone graded projective A -modules with structure constants { c i } and { d i } , respectively. IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 17 As above, write P = M i ∈ Z Rp i x i = M i ∈ Z R Y j ≥ i c j x i and Q = M i ∈ Z Rq i x i = M i ∈ Z R Y j ≥ i d j x i . Then the maximal embedding P → Q is given by multiplication by θ P,Q = lcm i ∈ Z (cid:18) q i gcd( p i , q i ) (cid:19) = lcm i ∈ Z Q j ≥ i d j gcd (cid:16)Q j ≥ i c j , Q j ≥ i d j (cid:17) where lcm is the unique monic least common multiple in R . There exists some N ∈ Z such that θ P,Q = q N gcd( p N , q N ) . Finally, we give necessary and sufficient conditions for a set of projective objectsin grmod - A to generate the category. This is the natural generalization of theconditions in [Won18b] and follows from the same proof. Lemma 4.16 ([Won18b, Proposition 3.43]) . A set of rank one graded projective A -modules P = { P i } i ∈ I generates grmod - A if and only if for every M ± α h n i with α ∈ Zer( f ) and n ∈ Z , there exists a graded surjection to M ± α h n i from a direct sumof modules in P . Involutions of grmod - A . We now construct autoequivalences of grmod - A which are analogous to those constructed in [Sie09, Proposition 5.7] and [Won18b,Propositions 5.13 and 5.14]. Recall that S denotes the shift functor on grmod - A . Proposition 4.17.
Let A = R ( σ, f ) and let Zer( f ) be the set of roots of f . Forany α ∈ Zer( f ) and any j ∈ Z , there is an autoequivalence ι αj of grmod - A suchthat ι αj ( M ± α h j i ) ∼ = M ∓ α h j i and ι αj ( S ) ∼ = S for all other graded simple A -modules S .For any j, k ∈ Z , S jA ι αk ∼ = ι αj + k S jA , and ( ι αj ) ∼ = Id grmod - A .Proof. For each α ∈ Zer( f ), we will construct ι α and then define ι αj as S jA ι α S − jA .Let R denote the full subcategory of grmod - A consisting of the canonical rankone graded projective modules. We first define ι α on R . Consider the set D α = (cid:8) M − iα , M + iα (cid:12)(cid:12) i = 1 , . . . , n α (cid:9) and let D α denote the full subcategory of grmod - A whose objects are the elementsof D α .Let P be an object of R . By Lemma 4.8 and Corollary 4.10, there is an unique(up to a scalar) surjection from P onto M − n α α or M + n α α , let N be the kernel ofthis surjection (which does not depend on the particular choice of the surjection),so P/N ∈ D α . As D α is closed under subobjects, the functor R → grmod - A thatmaps an object P to this unique kernel and acts on morphisms by restriction is awell-defined additive functor by [Won18b, Lemma 4.4]. By [Won18b, Lemma 4.3],this then extends to an additive functor defined over the full subcategory of directsums of rank one projective modules. By [Won18b, Lemma 4.2] this functor furtherextends to an additive functor ι α : grmod - A → grmod - A .We now show that ι α has the claimed properties. We begin by describing theaction on structure constants. Suppose P ∈ R has structure constants { c i } anddenote the structure constants of ι α P by { d i } . By Lemma 4.7, P surjects onto exactly one of M − n α α and M + n α α . If P surjects onto M − n α α , then a similar strategyto the one used in the first paragraph of the proof of Lemma 4.7 can be used toshow that ( ι α P ) n = ( ( z − α ) n α P n if n ≤ P n if n > . Similarly, if P surjects onto M + n α α then( ι α P ) n = ( P n if n ≤ z − α ) n α P n if n > . In the first case, d = c ( z − α ) n α while in the second case, d = c / ( z − α ) n α .Therefore by Lemma 4.7 if P surjects onto M ± n α α then ι α P surjects onto M ∓ n α α .Hence ( ι α ) P = ( z − α ) n α P , which is isomorphic to P .Let P ′ be another object in R . Consider map induced by ( ι α ) Hom grmod - A ( P, P ′ ) → Hom grmod - A (( z − α ) n α P, ( z − α ) n α P ′ ) , given by g g | ( z − α ) nα P . By Proposition 4.14, every element of Hom grmod - A ( P, P ′ )and of Hom grmod - A (( z − α ) n α P, ( z − α ) n α P ′ ) is given by left multiplication by someelement of k ( z ). Therefore the map in the above display must be an isomorphism. If Q is a finite direct sum of rank one graded projective modules then, by the additivityof ι α , ( ι α ) is given by multiplication by ( z − α ) n α in each component of Q . Hence,( ι α ) is naturally isomorphic to the identity functor on the full subcategory of finitedirect sums of rank one projectives. Therefore, by [Won18b, Lemma 4.2], ι α is anautoequivalence of grmod - A which is its own quasi-inverse.If ι αj is defined as ι αj = S jA ι α S − jA then S jA ι αk = S jA S kA ι α S − lA = S j + kA ι α S − j − kA S jA = ι αj + k S jA . Because for any P ∈ R the structure constants for ι α P differ from those of P onlyin degree 0, where they differ only by a factor of ( z − α ) n α , by Lemma 4.7, ι α P and P have the same simple factors supported along { σ i k ( α ) | i ∈ Z , α ∈ Zer( f ) } exceptif P has a factor of M − α h j i then ι α P has a factor of M + α h j i and vice versa. Let S bea simple module not of the form M ± α h j i , i.e. S is M ± β h j i for some β ∈ Zer( f ) \{ α } ,or S is M λ h j i for some λ . By looking at the way ι α is defined on grmod - A , in[Won18b, Lemmas 4.2 and 4.3], it follows that ι α ( S ) = S . (cid:3) Remark . Since we defined ι αj = S jA ι α S − jA , we can construct ι αj by adjusting theconstruction in the previous proof by shifting all of the modules in D by j . It followsthat for a rank one graded projective module P , ( ι αj ) P = σ − j (( z − α ) n α ) P . Lemma 4.19.
For any α, β ∈ Zer( f ) and any j, k ∈ Z , ι αj ι βk = ι βk ι αj and so theautoequivalences { ι αj | α ∈ Zer( f ) , j ∈ Z } generate a subgroup of Pic(grmod - A ) isomorphic to M α ∈ Zer( f ) M j ∈ Z Z / Z . Proof.
Since, for each α ∈ Zer( f ) and j ∈ Z , ( ι αj ) ∼ = Id grmod - A , each ι αj generatesa subgroup of Pic(grmod - A ) isomorphic to Z / Z . It remains to show that for any IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 19 β ∈ Zer( f ) and k ∈ Z that ι αj and ι βk commute. Tracing through the constructionin the proof of Proposition 4.17 shows that ( ι βk ) − ( ι αj ) − ι βk ι αj ∼ = Id grmod - A . (cid:3) Quotient stacks as noncommutative schemes of GWAs
Let A = R ( σ, f ) and fix a labeling of the distinct roots of f Zer( f ) = { α , . . . , α r } where α i has multiplicity n i in f . The main aim of this section is to use theautoequivalences constructed in the previous section to construct a Γ-graded com-mutative ring B whose category of Γ-graded modules is equivalent to grmod - A .We will then study properties of the ring B .We identify L i ∈ Z Z / Z with the group Z fin of finite subsets of the integers. Theoperation on Z fin is given by exclusive or, denoted ⊕ . For convenience, we usesimply j to denote the singleton set { j } ∈ Z fin . Throughout, we write the group ofautoequivalences described in Lemma 4.19 asΓ = r M i =1 Z fin . We note that Z fin ⊕ Z fin ∼ = Z fin and therefore Γ ∼ = Z fin , but it will be convenientto index this group by the roots of f . We use the notation [1 , r ] to refer to the set { , , . . . , r } .For i ∈ [1 , r ] and j ∈ Z , let e i,j = {∅ , . . . , j, . . . , ∅} ∈ Γ, where j is in the i th component. We denote the identity ( ∅ , . . . , ∅ ) of Γ by ∅ .For each J = ( J , J , . . . , J r ) ∈ Γ, let ι J = r Y i =1 Y j ∈ J i ι α i j . Since, for each j ∈ Z , the autoequivalence ι α i j is its own quasi-inverse, therefore ι J is also its own quasi-inverse. Lemma 5.1.
The set { ι J A | J ∈ Γ } generates GrMod - A .Proof. We remark that by the construction in Proposition 4.17, if P is a rank oneprojective, then ι J P is a graded submodule of Q gr ( A ) with the same structureconstants as P except in certain degrees, where the structure consants of ι J P differfrom those of P by a factor of Q i ∈ I ( z − α i ) n i for some I ⊆ { , , . . . r } . Hence, byLemma 4.9, ι J P is also a rank one graded projective module. By Lemma 4.16, itis enough to show that for every i = 1 , . . . , r and every n ∈ Z , there is a surjectionfrom some ι J A to M − α i h n i and similarly for M + α i h n i . By Corollary 4.11, A surjectsonto exactly one of M − α i h n i and M + α i h n i . Then ι e i,n A surjects onto the other, so { ι J A | J ∈ Γ } generates grmod - A and hence generates GrMod - A . (cid:3) For J = ( J , . . . , J r ) , K = ( K , . . . , K r ) ∈ Γ, we define J ∩ K = ( J ∩ K , . . . , J r ∩ K r ) and J ∪ K = ( J ∪ K , . . . , J r ∪ K r ) , in the natural way, and similarly define J ⊕ K = ( J ⊕ K , . . . , J r ⊕ K r ) . For each i = 1 , . . . , r and each j ∈ Z , we define the polynomial h α i j = σ − j (( z − α i ) n i ).If J ∈ Γ, let h J = r Y i =1 Y j ∈ J i h α i j . For completeness, empty products are defined to be 1.Let I , J ∈ Γ. Since, in Pic(grmod - A ), ι I ι J = ι I ⊕ J , therefore ι I ι J is naturallyisomorphic to ι I ⊕ J and in particular, ι I ι J A ∼ = ι I ⊕ J A . We now explicitly describethis isomorphism. By Remark 4.18, ι J A = h J A . Denote by τ J the isomorphism ι J A → A given by left multiplication by h − J . Define Θ I , J : ι I ι J A = ι I ⊕ J ι I ∩ J A → ι I ⊕ J A by Θ I , J = ι I ⊕ J ( τ I ∩ J ) = Θ J , I . By a proof that is identical to that of [Won18a, Lemma 4.1], we have that forany I , J , K ∈ Γ and any ϕ ∈ Hom grmod - A ( A, ι I A ),Θ K , I ⊕ J ◦ ι K (Θ J , I ) ◦ ι K ι J ( ϕ ) = Θ J ⊕ K , I ◦ ι J ⊕ K ( ϕ ) ◦ Θ K , J . Therefore, by [Won18a, Proposition 3.4] we can define the Γ-graded ring(5.2) B ( R, σ, f ) = B = M J ∈ Γ B J = M J ∈ Γ Hom grmod - A ( A, ι J A ) . For a ∈ B I and b ∈ B J multiplication in B is given by a · b = ι I ⊕ J ( τ I ∩ J ) ◦ ι J ( a ) ◦ b .In the remainder of this section, we assume that B = B ( R, σ, f ) is defined as in(5.2).
Theorem 5.3.
There is an equivalence of categories
GrMod - A ≡ GrMod -( B, Γ) . Proof.
This is an immediate corollary of Lemma 5.1 and [Won18a, Proposition3.6]. (cid:3)
Theorem 5.4.
The ring B is a commutative ring with presentation B ∼ = R [ b i,j | i ∈ [1 , r ] , j ∈ Z ] (cid:0) b i,j = σ − j (( z − α i ) n i ) (cid:12)(cid:12) i ∈ [1 , r ] , j ∈ Z (cid:1) . There is a Γ -grading on B , given by R = B ∅ and deg b i,j = e i,j .Proof. By the construction Proposition 4.17, for each α ∈ Zer( f ) and each j ∈ Z , ι αj A is the kernel of the unique graded surjection from A to M − n α α h j i ⊕ M + n α α h j i .Both M − n α α h j i and M + n α α h j i have k -dimension n α in all graded components inwhich they are nonzero. Therefore, ( ι αj A ) has codimension n α in A . Further,(( ι αj ) A ) = ( h αj A ) and since (( ι αj ) A ) ⊆ ( ι αj A ) ⊆ A , by comparing codimensionswe conclude that ( ι αj A ) = ( h αj A ) = h αj R. Thus, for J ∈ Γ, ( ι J A ) = h J R . We use the isomorphism ϕ J : ( ι J A ) → Hom grmod - A ( A, ι J A ), to identify ( ι J A ) = h J R with B J . Let b J = ϕ J ( h J ). Sincethe autoequivalences { ι J | J ∈ Γ } act on morphisms by restriction, using thedefinition of multiplication in B , one can check that for I , J ∈ Γ, b I b J = b I ∩ J b I ⊕ J . For each i ∈ [1 , r ] and j ∈ Z , let b i,j = b e i,j . IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 21 By the above computation, the b i,j commute and b J = Q ri =1 Q j ∈ J i b i,j . Since h J freely generates ( ι J A ) as an R -module, b J freely generates B J as a B ∅ = R -module. Hence, it is clear that the b i,j generate B as an R -module. Again usingthe definition of multiplication in B , b i,j = h α i j .Finally, the proof of [Won18a, Proposition 4.4] shows that the ideal (cid:0) b i,j = σ − j (( z − α i ) n i ) (cid:12)(cid:12) i ∈ [1 , r ] , j ∈ Z (cid:1) contains all of the relations among the b i,j and hence B has the claimed presenta-tion. (cid:3) Corollary 5.5.
Let A = L i ∈ Z A i be a simple Z -graded domain of GK dimension2 with A i = 0 for all i ∈ Z . Then there is a commutative Z fin -graded ring B so that GrMod - A ≡ Qcoh (cid:20)
Spec B Spec k Z fin (cid:21) . Proof.
This follows from standard results [Sta19, Tag 06WS], along with Theo-rems 5.3 and 5.4. The Z fin -grading on B gives an action of Spec k Z fin on Spec B .We can therefore take the quotient stack χ = h Spec B Spec k Z fin i . The category of Z fin -equivariant quasicoherent sheaves on Spec B is equivalent to the category Qcoh( χ )of quasicoherent sheaves on χ . Hence, GrMod - A ≡ GrMod -( B, Z fin ) ≡ Qcoh( χ ). (cid:3) Unfortunately, since the ring k Z fin is not locally of finite type, the quotient stack χ is not easy to study. We devote the remainder of the paper to the study of thering B . Corollary 5.6.
Suppose that A is a GWA satisfying Hypothesis 2.1. Then thereexists a quotient stack χ such that QGrMod - A ≡ Qcoh χ .Proof. This follows immediately from Corollary 3.8 and Corollary 5.5. (cid:3)
Theorem 5.7.
The ring B is non-noetherian.Proof. It is enough to prove that the quotient of B by the ideal generated by z − B ( z −
1) = k [ b i,j | i ∈ [1 , r ] , j ∈ Z ] (cid:0) b i,j − γ i,j (cid:12)(cid:12) i ∈ [1 , r ] , j ∈ Z (cid:1) , where the γ i,j are elements of k depending on α i , j , n i , and on the defining automor-phism σ of A . In particular, if R = k [ z ] and σ ( z ) = z − γ i,j = (1 − j − α i ) n i while if R = k [ z, z − ] and σ ( z ) = ξz , then γ i,j = ( ξ − j − α i ) n i . In either case, weclaim that the ideals I n = ( b ,j + √ γ ,j | ≤ j < n )form a non-stabilizing ascending chain of ideals in B/ ( z − b ,n + √ γ ,n is not in I n . Indeed if we further quotient B/ ( z −
1) by I n we geta ring of the form B ( z −
1) + I n = k [ b i,j | i ∈ [1 , r ] , j ∈ Z ] b ,j + √ γ ,j | ≤ j < nb ,j − γ ,j | j ≥ nb i,j − γ i,j | i ∈ [2 , r ] . We prove that b ,j + √ γ ,j is not zero in this quotient by proving that it does notbelong to the ideal J n = b ,j + √ γ ,j | ≤ j < nb ,j − γ ,j | j ≥ nb i,j − γ i,j | i ∈ [2 , r ] of the polynomial ring k [ b i,j | i ∈ [1 , r ] , j ∈ Z ]. Fix a monomial order on thepolynomial ring by setting b i,j < b p,q if i < p or if i = p and j < q , and using lexi-cographic order. With this ordering the generators of the ideal J n form a Gr¨obnerbasis because their leading terms are relatively prime. The reduction of b ,j + √ γ ,j by this Gr¨obner basis does not yield zero, showing that I n ( I n +1 . (cid:3) Theorem 5.8.
The Krull dimension of the ring B is 1.Proof. We first claim that R ⊆ B is an integral extension. Indeed, it is clear thatfor each i ∈ [1 , r ] and each j ∈ Z , that b i,j is integral over R . Since B is generatedby the b i,j over R then B is integral over R by [AM69, Corollary 5.3]. Finally, by[Mat89, Exercise 9.2] the Krull dimension of B is the same as the Krull dimensionof R , which is 1. (cid:3) Remark . Let X be the set { ( i, j ) | i ∈ [1 , r ] and j ∈ Z } . Since X is countable,we can fix an enumeration X = { x m | m ∈ N } . If x m = ( i, j ) then we write b i,j as b x m and σ − j (( z − α i ) n i ) as g m . Denote by B m the following ring B m = R [ b x , . . . , b x m ] (cid:0) b x t − g t | t ∈ [1 , m ] (cid:1) . Then B m +1 = B m [ b x m +1 ] (cid:16) b x m +1 − g m +1 (cid:17) with obvious maps B m → B m +1 , and we have that(5.10) B = lim −→ m B m . Definition 5.11.
A commutative ring is said to be coherent if every finitely gen-erated ideal is finitely presented.
Proposition 5.12. (1) If n i is odd for all i ∈ [1 , r ] , then B is a domain.(2) The ring B is coherent.Proof. (1) Since the direct limit of domains is a domain, by (5.10) it is enoughto show that the rings B m are all domains. We set B − to be R (which isa domain) and proceed by induction on m . Since B m +1 = B m [ b x m +1 ] (cid:16) b x m +1 − g m +1 (cid:17) , and by assumption √ g m +1 B m , it is straightforward to see that b x m − g m +1 is a prime element in the domain B m [ b x m +1 ] and therefore B m +1 is adomain.(2) By [Gla89, Theorem 2.3.3] it suffices to show that B m is a flat extension of B m − . Indeed B m = B m − ⊕ b x m B m − , and hence the extension is free. (cid:3) IMPLE Z -GRADED DOMAINS OF GELFAND-KIRILLOV DIMENSION TWO 23 Let I ⊆ [1 , r ] consist of those i such that n i is even. If I is empty, then theproposition above says that B is a domain. In the following lemma, we classify theminimal prime ideals of B when I is not empty. Lemma 5.13.
Suppose I = ∅ . An ideal of B is a minimal prime if and only if itis of the form p = (cid:16) b i,j − ( − ε i,j σ − j (cid:16) ( z − α i ) ni (cid:17) (cid:12)(cid:12)(cid:12) i ∈ I, j ∈ Z (cid:17) for some choice of ε i,j ∈ { , } for each i ∈ I and j ∈ Z .Proof. Let S be the ring R [ b i,j | i ∈ [1 , r ] , j ∈ Z ] and let S ′ be the ring R [ b i,j | i ∈ [1 , r ] \ I, j ∈ Z ]. The prime ideals of B correspond to prime ideals of S containingthe ideal(5.14) (cid:0) b i,j − σ − j (( z − α i ) n i ) | i ∈ [1 , r ] , j ∈ Z (cid:1) . For each i ∈ I and j ∈ Z , choose ε i,j ∈ { , } . Now define the map ρ : S → S ′ → S ′ ( b i,j − σ − j (( z − α i ) n i ) | n i odd) , where the first map is the evaluation map that sends b i,j to ( − ε i,j σ − j (cid:16) ( z − α i ) ni (cid:17) when n i is even and the second map is the canonical projection. The kernel of ρ isthe ideal in (5.13). Since S ′ (cid:0) b i,j − σ − j (( z − α i ) n i ) (cid:12)(cid:12) i ∈ [1 , r ] \ I, j ∈ Z (cid:1) is a domain by Proposition 5.12, we deduce that the ideal p is prime. To see that p is a minimal prime we observe that if a prime ideal contains b i,j − σ − j (( z − α i ) n i )with n i even then it contains b i,j − σ − j (cid:16) ( z − α i ) ni (cid:17) or b i,j + σ − j (cid:16) ( z − α i ) ni (cid:17) . (cid:3) The proof of the following result is a straightforward generalization of the proofof [Won18a, Proposition 4.6]. We reproduce the argument here for completeness.
Corollary 5.15.
The ring B is reduced.Proof. Let I ⊆ [1 , r ] consist of those i ∈ [1 , r ] such that n i is even. If I = ∅ then B is a domain so we may assume that I = ∅ . We show that the intersectionof all the minimal prime ideals of B is (0). We first set the following notation:let J = ( J , . . . , J r ) ∈ Γ and set B ( J ) to be the k -subalgebra of B generated by { , z } ∪ { b i,j | i ∈ [1 , r ] , j ∈ J i } if R = k [ z ] or by { , z, z − } ∪ { b i,j | i ∈ [1 , r ] , j ∈ J i } if R = k [ z, z − ]. Hence B ( J ) ∼ = R [ b i,j | i ∈ [1 , r ] , j ∈ J i ] (cid:0) b i,j − σ − j (( z − α i ) n i ) (cid:12)(cid:12) i ∈ [1 , r ] , j ∈ J i (cid:1) . Let a be an element in the intersection of all minimal prime ideals of B . We canwrite a as a sum of finitely many Γ-homogeneous terms, so a is in an element of B ( J ) for some J ∈ Γ. Fix i ∈ I and suppose j ∈ J i . Since a ∈ (cid:16) b i,j + σ − j (( z − α i ) ni ) , b i ′ ,j ′ − σ − j ′ (( z − α i ′ ) ni ′ ) (cid:12)(cid:12)(cid:12) j ′ ∈ J i \ { j } or j ′ ∈ J i ′ for i ′ ∈ I \ { i } (cid:17) and a ∈ (cid:16) b i,j − σ − j (cid:16) ( z − α i ) ni (cid:17) , b i ′ ,j ′ − σ − j ′ (cid:16) ( z − α i ′ ) ni ′ (cid:17) (cid:12)(cid:12)(cid:12) j ′ ∈ J i \ { j } or j ′ ∈ J i ′ for i ′ ∈ I \ { i } (cid:17) we can write(5.16) a = (cid:16) b i,j + σ − j (cid:16) ( z − α i ) ni (cid:17)(cid:17) r + s = (cid:16) b i,j − σ − j (cid:16) ( z − α i ) ni (cid:17)(cid:17) r ′ + s ′ with r, r ′ ∈ B ( J ) and s, s ′ ∈ (cid:16) b i ′ ,j ′ − σ − j ′ (cid:16) ( z − α i ′ ) ni ′ (cid:17) (cid:12)(cid:12)(cid:12) j ′ ∈ J i \ { j } or j ′ ∈ J i ′ for i ′ ∈ I \ { i } (cid:17) . Setting b k,ℓ = σ − ℓ (cid:16) ( z − α k ) nk (cid:17) for all k ∈ I and ℓ ∈ J k in (5.16), the right handside becomes identically 0, and hence r ∈ (cid:16) b k,ℓ − σ − ℓ (cid:16) ( z − α k ) nk (cid:17) (cid:12)(cid:12)(cid:12) k ∈ I, ℓ ∈ J k (cid:17) . So a ∈ (cid:16) b i,j − σ − j (( z − α i ) n i ) , b i ′ ,j ′ − σ − j ′ (cid:16) ( z − α i ′ ) ni ′ (cid:17) (cid:12)(cid:12)(cid:12) j ′ ∈ J i \ { j } or j ′ ∈ J i ′ for i ′ ∈ I \ { i } (cid:17) . But b i,j − σ − j (( z − α i ) n i ) = 0 in B , hence a ∈ (cid:16) b i ′ ,j ′ − σ − j ′ (cid:16) ( z − α i ′ ) ni ′ (cid:17) (cid:12)(cid:12)(cid:12) j ′ ∈ J i \ { j } or j ′ ∈ J i ′ for i ′ ∈ I \ { i } (cid:17) . Inducting on the size of J i and on the size of I , we conclude that a = 0. As a resultthe intersection of the minimal primes of B is (0). (cid:3) Corollary 5.17.
The minimal spectrum of B , Min B , is compact in Spec B withrespect to the Zariski topology.Proof. This follows from Proposition 5.12, Corollary 5.15, and [Mat85, Proposition1.1]. (cid:3)
We refer the reader to the paper [HM09] for the definition of coherent Gorensteinrings and their basic properties.
Proposition 5.18.
The ring B is a coherent Gorenstein ring.Proof. By [HM09, Proposition 5.11], it suffices to prove that the rings B m in (5.10)are complete intersections. If R = k [ z ] then B m is a polynomial ring in m + 1 vari-ables modulo m relations, if R = k [ z, z − ] then B m is isomorphic to a polynomialring in m + 2 variables modulo m + 1 relations. As in the proof of Theorem 5.8 therings B m have Krull dimension 1, hence they are complete intersections. (cid:3) References [AM69] M. F. Atiyah and I. G. Macdonald,
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