Examples of geodesic ghor algebras on hyperbolic surfaces
aa r X i v : . [ m a t h . R A ] J a n EXAMPLES OF GEODESIC GHOR ALGEBRASON HYPERBOLIC SURFACES
KARIN BAUR AND CHARLIE BEIL
Abstract.
Cancellative dimer algebras on a torus have many nice algebraic andhomological properties. However, these nice properties disappear for dimer algebrason higher genus surfaces. We consider a new class of quiver algebras on surfaces,called ‘geodesic ghor algebras’, that reduce to cancellative dimer algebras on atorus, yet continue to have nice properties on higher genus surfaces. These algebrasexhibit a rich interplay between their central geometry and the topology of thesurface. We show that (nontrivial) geodesic ghor algebras do in fact exist, and giveexplicit descriptions of their central geometry. This article serves a companion tothe article ‘A generalization of cancellative dimer aglebras to hyperbolic surfaces’,where the main statement is proven. Introduction
Cancellative dimer algebras on a torus form a prominent class of noncommutativecrepant resolutions and Calabi-Yau algebras, e.g., [Br, D, B2]. In particular, they arehomologically homogeneous endomorphism rings of modules over their centers, andtheir centers are 3-dimensional toric Gorenstein coordinate rings. These interestingproperties vanish, however, once dimer algebras are placed on higher genus surfaces.For example, the center of a dimer algebra on a surface of genus g ≥ In Section 2 we introduce a special property thatcertain ghor algebras possess, called ‘geodesic’. On a torus, a ghor algebra is geodesic
Mathematics Subject Classification.
Key words and phrases.
Dimer algebra, hyperbolic surface, non-noetherian ring, noncommutativealgebraic geometry. Ghor algebras were originally called ‘homotopy algebras’ (e.g., in [B4]) because their relations arehomotopy relations on the paths in the quiver when the surface is a torus. However, in the highergenus case their relations also identify homologous cycles, and therefore the name ‘homotopy’ is lesssuitable. The word ‘ghor’ is Klingon for surface. if and only if it is a cancellative dimer algebra (if and only if it is a noncommutativecrepant resolution, if and only if it is noetherian [D, B2]). On higher genus surfaces,geodesic ghor algebras remain endomorphism rings of modules over their centers, butnew features arise. The purpose of this article is to present explicit examples of ghoralgebras that exhibit these new features.In each example, we will describe the ‘cycle algebra’, which is a commutative alge-bra formed from the cycles in the quiver. The cycle algebra and center coincide forgeodesic ghor algebras on a torus, but may differ on higher genus surfaces. However,the two commutative algebras remain closely related: if the center is nonnoetherian,then the cycle algebra is a depiction of the center. The cycle algebra thus plays afundamental role in describing the geometry of the center when the center is non-noetherian.We will also show that polynomial rings in at least three variables are geodesicghor algebras. The results stated in Section 2 are proven in the companion article[BB]. 2.
Ghor algebras: background and main results
Notation 2.1.
Throughout, k is an uncountable algebraically closed field. We de-note by Spec S and Max S the prime ideal spectrum (or scheme) and maximal idealspectrum (or affine variety) of S , respectively. Given a quiver Q , we denote by kQ the path algebra of Q ; by Q ℓ the paths of length ℓ ; by t , h : Q → Q the tail andhead maps; and by e i the idempotent at vertex i ∈ Q . By cyclic subpath of a path p , we mean a subpath of p that is a nontrivial cycle.In this article will consider surfaces Σ that are obtained from a regular 2 N -gon P , N ≥
2, by identifying the opposite sides of P . This class of surfaces includes allsmooth orientable compact closed connected genus g ≥ • if P is a 4 g -gon, then Σ is a smooth genus g surface; and • if P is a 2(2 g + 1)-gon, then Σ is a genus g surface with a pinched point.The polygon P is then a fundamental polygon for Σ.If N = 2, then Σ is a torus, and the covering space of Σ is the plane R . For N ≥
3, the covering space of Σ is the hyperbolic plane H . The hyperbolic planemay be represented by the interior of the unit disc in R , where straight lines in H are segments of circles that meet the boundary of the disc orthogonally. In thecovering, the hyperbolic plane is tiled with regular 2 N -gons, with 2 N such polygonsmeeting at each vertex. In this case, Σ is said to be a hyperbolic surface. Definition 2.2. • A dimer quiver on Σ is a quiver Q whose underlying graph Q embeds in Σ, suchthat each connected component of Σ \ Q is simply connected and bounded by anoriented cycle, called a unit cycle . XAMPLES OF GEODESIC GHOR ALGEBRAS ON HYPERBOLIC SURFACES 3 - A perfect matching of a dimer quiver Q is a set of arrows x ⊂ Q such thateach unit cycle contains precisely one arrow in x . Throughout, we assumethat each arrow is contained in at least one perfect matching.- A perfect matching x is called simple if Q \ x contains a cycle that passesthrough each vertex of Q (equivalently, Q \ x supports a simple kQ -moduleof dimension vector (1 , , . . . , P and S the set of perfect and simple matchings of Q , respectively.We will consider the polynomial rings generated by these matchings, k [ P ] and k [ S ]. • Denote by e ij ∈ M n ( k ) the n × n matrix with a 1 in the ij -th slot and zeroselsewhere. Consider the two algebra homomorphisms η : kQ → M | Q | ( k [ P ]) and τ : kQ → M | Q | ( k [ S ])defined on the vertices i ∈ Q and arrows a ∈ Q by η ( e i ) = e ii , η ( a ) = e h( a ) , t( a ) Q x ∈P : x ∋ a x,τ ( e i ) = e ii , τ ( a ) = e h( a ) , t( a ) Q x ∈S : x ∋ a x, and extended multiplicatively and k -linearly to kQ . We call the quotient A := kQ/ ker η the ghor algebra of Q . • The dimer algebra of Q is the quotient of kQ by the ideal I = h p − q | ∃ a ∈ Q such that pa, qa are unit cycles i ⊂ kQ, where p, q are paths.A ghor algebra A = kQ/ ker η is the quotient of the dimer algebra kQ/I since I ⊆ ker η : if pa, qa are unit cycles with a ∈ Q , then η ( p ) = e h( p ) , t( p ) Y x ∈P : x a x = η ( q ) . Dimer algebras on non-torus surfaces have been considered in the context of, forexample, cluster categories [BKM, K], Belyi maps [BGH], and gauge theories [FGU,FH].
Notation 2.3.
Let π : Σ + → Σ be the projection from the covering space Σ + (here, R or H ) to the surface Σ. Denote by Q + := π − ( Q ) ⊂ Σ + the (infinite) coveringquiver of Q .We introduce the following special class of ghor algebras that generalizes cancella-tive dimer algebras on a torus. KARIN BAUR AND CHARLIE BEIL
Definition 2.4.
Given a cycle c , we define the class of c to be[ c ] := X k ∈ [ N ] ( n k − n k + N )( δ kℓ ) ℓ ∈ Z N , where for k ∈ [2 N ], c transversely intersects side k of P n k times. If Σ is smooth(that is, N is even), then [ c ] is the homology class of c in H (Σ) := H (Σ , Z ). • A cycle p ∈ A is geodesic if the lift to Q + of each cyclic permutation of eachrepresentative of p does not have a cyclic subpath. • Two cycles are parallel if they do not transversely intersect. • A ghor algebra is geodesic if for each k ∈ [2 N ] there is a geodesic cycle γ k with class [ γ k ] = ( δ kℓ − δ k + N,ℓ ) ℓ ∈ [ N ] ∈ Z N , with indices modulo 2 N , and a set of pairwise parallel geodesic cycles c i ∈ e i kQe i , i ∈ Q , such that c t( γ k ) = γ k . Proposition 2.5. [BB, Corollary 3.15]
If a ghor algebra A is geodesic, then A := kQ/ ker η ∼ = kQ/ ker τ. In particular, it suffices to only consider the simple matchings of Q to determine therelations of A . The algebra homomorphisms η and τ on kQ induce algebra homomorphisms on A , η : A → M | Q | ( k [ P ]) and τ : A → M | Q | ( k [ S ]) . For i, j ∈ Q , consider the k -linear maps¯ η : e j Ae i → k [ P ] and ¯ τ : e j Ae i → k [ S ]defined by sending p ∈ e j Ae i to the single nonzero matrix entry of η ( p ) and τ ( p )respectively; that is, η ( p ) = ¯ η ( p ) e ji and τ ( p ) = ¯ τ ( p ) e ji . Then ¯ η ( p ) | x =1: x = ¯ τ ( p ) . These maps are multiplicative on the paths of Q . Moreover, for a ∈ Q , we have a ∈ x if and only if x | ¯ η ( a ).By the definition of A , if two paths p, q ∈ e j Ae i satisfy ¯ η ( p ) = ¯ η ( q ), then p = q .Furthermore, if A is geodesic, then ¯ τ ( p ) = ¯ τ ( q ) implies p = q , by Proposition 2.5. Animportant monomial is the ¯ τ -image of each unit cycle in Q , namely σ := Y x ∈S x. The following lemma will be useful in the next section.
Lemma 2.6. If p ∈ A is a cycle satisfying σ ∤ ¯ τ ( p ) , then p is a geodesic cycle. XAMPLES OF GEODESIC GHOR ALGEBRAS ON HYPERBOLIC SURFACES 5
Proof. If p is not a geodesic cycle, then there is a cyclic permutation of a lift of arepresentative of p with a cyclic subpath q in Q + . Since q is a cycle in Q + , we have¯ τ ( q ) = σ ℓ for some ℓ ≥ (cid:3) Remark 2.7.
Let A be a geodesic ghor algebra on a surface Σ, and fix i ∈ Q . If Σis a torus, then the geodesic assumption implies that for each j, k ∈ π − ( i ), there isa path p + from j to k in Q + such that σ ∤ τ ( p ) [B1, Proposition 4.20.iii]. However, ifΣ is hyperbolic, then this implication no longer holds.Indeed, suppose Σ is hyperbolic, that is, N ≥
3. Fix unit vectors u , u ∈ Z N for which u = ± u . Let p , p , q , q be paths in kQ + e j that are lifts of cycles withclasses [ π ( p )] = − [ π ( p )] = u and [ π ( q )] = − [ π ( q )] = u . Consider the paths s := q p q p and t := q q p p in Q + . Then ¯ τ ( s ) = ¯ τ ( t ), and so π ( s ) = π ( t ) in A . Observe that t is a cycle in Q + ,whereas s is not. Furthermore, the ¯ τ -image of any cycle in Q + is a power of σ [BB,Lemma 3.1.i]. Thus, although s is not a cycle, we have ¯ τ ( s ) = ¯ τ ( t ) = σ ℓ for some ℓ ≥
1. It therefore follows that the monomial of each path in Q + from t( s ) to h( s ) isa power of σ [BB, Lemma 3.1.ii], a feature that never occurs if Σ is flat.If Σ is a torus, then a ghor algebra A is geodesic if and only if it is noetherian, if andonly if its center R is noetherian, if and only if A is a finitely generated R -module [B2,Theorem 1.1]. If Σ is hyperbolic, then only one direction of the implication survives: Proposition 2.8. [BB, Lemma 4.4]
If the center R of a ghor algebra A is noetherian,then(1) A is geodesic;(2) A is noetherian; and(3) A is a finitely generated R -module. In contrast to the torus case, the centers of geodesic ghor algebras on hyperbolicsurfaces are almost always nonnoetherian. We can nevertheless view such a centeras the coordinate ring on a geometric space, using the framework of nonnoetheriangeometry introduced in [B3] (see also [B5]). In short, the geometry of a nonnoetheriancoordinate ring of finite Krull dimension looks just like a finite type algebraic variety,except that it has some positive dimensional closed points.
Definition 2.9. A depiction of an integral domain k -algebra R is a finitely generatedoverring S such that the morphismSpec S → Spec R, q q ∩ R, It is possible that there is only a finite number of geodesic ghor algebras that are noetherian foreach genus g ≥ KARIN BAUR AND CHARLIE BEIL is surjective, and U S/R := { n ∈ Max S | R n ∩ R = S n } = { n ∈ Max S | R n ∩ R is noetherian } 6 = ∅ . For example, the algebra S = k [ x, y ] is a depiction of its nonnoetherian subalgebra R = k + xS . We thus view Max R as the variety Max S = A k , except that the line { x = 0 } is identified as a 1-dimensional (closed) point of Max R . In particular, thecomplement { x = 0 } ⊂ A k is the ‘noetherian locus’ U S/ ( k + xS ) [B3, Proposition 2.8].The following is the main theorem of the companion article [BB]. Theorem 2.10.
Suppose A = kQ/ ker η is a geodesic ghor algebra on a surface Σ obtained from a regular N -gon P by identifying the opposite sides, and vertices, of P . Set R = k [ ∩ i ∈ Q ¯ τ ( e i Ae i )] and S = k [ ∪ i ∈ Q ¯ τ ( e i Ae i )] ; then R is isomorphic to the center of A . Furthermore, the following holds.(1) If there is a cycle p such that p n R for each n ≥ , then A and R arenonnoetherian. In this case, R is depicted by the cycle algebra S .(2) The center R and cycle algebra S have Krull dimension dim R = dim S = N + 1 . In particular, if Σ is a smooth genus g ≥ surface, then dim R = rank H (Σ) + 1 = 2 g + 1 . (3) At each point m ∈ Max R for which the localization R m is noetherian, thelocalization A m := A ⊗ R R m is an endomorphism ring over its center: for each i ∈ Q , we have A m ∼ = End R m ( A m e i ) . The locus of such points lifts to the open dense subset U S/R ⊂ Max S of thecycle algebra. Examples
In the following, we consider explicit examples of geodesic ghor algebras. Recallthat Σ is a closed surface obtained from a regular 2 N -gon P by identifying theopposite sides, and vertices, of P . Set [ m ] := { , . . . , m } . For a path p , set p := ¯ τ ( p ).3.1. Polynomial rings are geodesic ghor algebras.
The simplest possible geodesic ghor algebra on a 2 N -gon P has one arrow alongeach side, and a single diagonal arrow in the interior of P . The cases N = 2 , , N + 1 variables. It follows that every polynomial ring in at least3 variables arises as a ghor algebra. XAMPLES OF GEODESIC GHOR ALGEBRAS ON HYPERBOLIC SURFACES 7 y y yx x x x x x x x x x x x x x x x x x x x x x yyx x x x yx Figure 1.
The polynomial ghor algebras k [ x , x , y ], k [ x , x , x , y ],and k [ x , x , x , x , y ].3.2. A geodesic ghor algebra on a genus surface. Consider the ghor algebra A = kQ/ ker η on a smooth genus 2 surface with quiver Q given in Figure 2.i (the identifications of the sides of the polygon P are indicatedby color). We will determine the center R and cycle algebra S of A explicitly, andshow that R is nonnoetherian of Krull dimension 5. Q has twelve simple matchings, shown in Figure 3. The nontrivial simple matchings y j and z j may be determined using special partitions of Q called subdivisions, whichare described in [BB, Section 3]. The polynomial ring k [ S ] is thus k [ S ] = k [ x , . . . , x , y , . . . , y , z , . . . , z ] . Note that x j +1 is the rotation of x j by π in the counter-clockwise direction, andsimilarly for the simple matchings y j and z j . Using the four x j simple matchings andLemma 2.6, it is straightforward to verify that A is geodesic.Consider the eight cycles α , . . . , α that run along the sides of the fundamentalpolygon P , as shown in Figure 2.ii. These cycles have ¯ τ -images α j = ( x j +1 x j +2 x j +3 y j y j +2 y j +3 z j z j +1 z j +2 if j is odd( x j +2 x j +3 y j +2 y j +3 z j +2 z j +3 ) if j is evenObserve that A has cycle algebra S = k [ σ, α j | j ∈ [8]] ⊂ k [ S ] , and center R = k [ σ ] + ( α j α j +1 α j +2 , σ | j ∈ [8]) S, with indices taken modulo 8.The center R is nonnoetherian since it contains the infinite ascending chain ofideals α σ R ⊂ ( α , α ) σ R ⊂ ( α , α , α ) σ R ⊂ · · · . KARIN BAUR AND CHARLIE BEIL α α α α α α α α ( i ) ( ii ) ( iii ) Figure 2. (i) The dimer quiver Q of Section 3.2. (ii) The cycles α j runalong the sides of the fundamental polygon P . (iii) The homologouscycles p, q are equal in the ghor algebra A , but not in the dimer algebraof Q .Nevertheless, we claim that R satisfies the ascending chain condition on prime ideals,and in particular has Krull dimension 5 = 2 g + 1.If n ∈ Max S is a point for which none of the monomial generators of S vanish(for example, if each monomial is set equal to 1), then R n ∩ R = S n . Consequently,the noetherian locus U S/R is nonempty. Therefore dim R = dim S , by [B3, Theorem2.5.4]. It thus suffices to show that S has Krull dimension 5.For j ∈ [5] set p j := ( σ, α , . . . , α j ) S, and consider the chain of ideals of S (1) 0 ⊆ p ⊆ p ⊆ · · · ⊆ p . By [BB, Theorem 3.11], two cycles p and q are in the same class if and only if p = qσ ℓ for some ℓ ∈ Z . In particular, if Σ is smooth, then two cycles are homologousif and only if p = q . Furthermore, suppose q is a prime ideal of S . Then α i ∈ q implies σ ∈ q since α i α i +4 = σ . But this then implies α k or α k +4 is in q for each k ∈ [4]. Therefore each p j is prime, and p is a height one prime. Consequently,dim S ≤
5. But the inclusions in (1) are strict, again since any two cycles p and q are homologous if and only if p = qσ ℓ for some ℓ ∈ Z . Whence dim S ≥
5. It followsthat dim R = dim S = 5.3.3. Flower of life: a geodesic ghor algebra on a pinched torus.
Finally, consider the ghor algebra A = kQ/ ker η on a pinched torus with quiver Q given in Figure 4.i (the identifications of the sides of the polygon P are indicated XAMPLES OF GEODESIC GHOR ALGEBRAS ON HYPERBOLIC SURFACES 9 x x x x y y y y z z z z Figure 3.
The simple matchings for the geodesic ghor algebra of Sec-tion 3.2.by color). We will determine the center R and cycle algebra S of A explicitly, andshow that R is nonnoetherian of Krull dimension 4. Q has ten simple matchings, three of which are shown in Figure 5. Specifically, Q has one simple matching consisting of concentric circles, three simple matchings forthe three straight directions of Q ( Q is symmetric upon rotations by π and π ), andsix simple matchings that are ‘wiggly’ (and identical up to rotation by a multiple of π ). Using these simple matchings and Lemma 2.6, it is straightforward to verify that A is geodesic.As shown in Figure 4.ii, denote by α j , j ∈ [6], the cycles that run along the sidesof the fundamental polygon P ; by γ k , k ∈ [3], the ‘straight’ cycles that pass throughthe center of P ; and by δ k , k ∈ [3], the cycles that run opposite to γ k and also passthrough the center of P . We claim that A has cycle algebra(2) S = k [ σ, α j , γ k | j ∈ [6] , k ∈ [3]] ⊂ k [ S ] . The quiver in this example fits into a pattern of overlapping circles called the ‘flower of life’ in theNew Age literature. α α α α α α γ γ γ ( i ) ( ii ) ( iii ) Figure 4. (i) The dimer quiver Q of Section 3.3. (ii) The cycles α j run along the sides of the fundamental polygon P . (iii) The cycles q j p j each have ¯ τ -image γ α (note that p = α and q = γ ). Figure 5.
The three types of the ten simple matchings for the geodesicghor algebra of Section 3.3.Indeed, consider the set of all vertical paths p j , q j , j ∈ [3], shown in Figure 4.iii. Theconcatenations q j p j are cycles in Q . These cycles satisfy q j p j = γ α , j ∈ [3] . Furthermore, the cycle δ satisfies δ = α α α . Similar equalities hold for the southeasterly and southwesterly directions. Thus theten monomials given in (2) generate S . XAMPLES OF GEODESIC GHOR ALGEBRAS ON HYPERBOLIC SURFACES 11
Observe that A has center R = k [ σ ] + ( α j ( α j +1 , α j +2 , α j +3 , σ ) | j ∈ [3]) S, with indices taken modulo 6. R is nonnoetherian since it contains the infinite ascend-ing chain of ideals α σ R ⊂ ( α , α ) σ R ⊂ ( α , α , α ) σ R ⊂ · · · . To show that R has Krull dimension 4, it suffices to show that S has Krull dimension4, as in Section 3.2. For j ∈ [6] and k ∈ [3], we have the relations α j α j +3 = σ and γ k δ k = σ . We also have the three relations γ α = α α , γ α = α α , γ α = α α , and the six homotopy relations γ α = α α , γ α = α α , . . . , γ α = α α , γ α = α α . Thus, following the arguments of Section 3.2, the chain of ideals of S ⊂ ( σ, α , α , α , α , α , γ ) S =: p ⊂ ( p , α ) S ⊂ ( p , α , γ ) S ⊂ ( p , α , γ , γ ) S is a maximal chain of primes. Therefore dim R = dim S = 4. Acknowledgments.
The authors were supported by the Austrian Science Fund(FWF) grant P 30549-N26. The first author was also supported by the AustrianScience Fund (FWF) grant W 1230 and by a Royal Society Wolfson Fellowship.
References [BB] K. Baur, C. Beil, A generalization of cancellative dimer algebras to hyperbolic surfaces, sub-mitted to arXiv.[BKM] K. Baur, A. King, B. Marsh, Dimer models and cluster categories of Grassmannians, Proc.London Math. Soc. (2016) no. 2, 213-260.[B1] C. Beil, Ghor algebras, dimer algebras, and cyclic contractions, arXiv:1711.09771.[B2] , Noetherian criteria for dimer algebras, arXiv:1805.08047.[B3] , Nonnoetherian geometry, J. Algebra Appl. (2016).[B4] , Nonnoetherian homotopy dimer algebras and noncommutative crepant resolutions,Glasgow Math. J. (2) (2018) 447-479.[B5] , On the central geometry of nonnoetherian dimer algebras, J. Pure Appl. Algebra (2020).[B6] , On the noncommutative geometry of square superpotential algebras, J. Algebra (2012) 207-249. [BGH] S. Bose, J. Gundry, Y. He, Gauge Theories and Dessins d’Enfants: Beyond the Torus, J.High Energy Phys. (2015) no. 1, 135.[Br] N. Broomhead, Dimer models and Calabi-Yau algebras, Memoirs AMS (2012) 1011.[D] B. Davison, Consistency conditions for brane tilings, J. Algebra (2011) 1-23.[FGU] S. Franco, E. Garc´ıa-Valdecasas, A. Uranga, Bipartite field theories and D-brane instantons,J. High Energy Phys. (2018) 098.[FH] S. Franco and A. Hasan, Graded Quivers, Generalized Dimer Models and Toric Geometry,arXiv:1904.07954.[K] M. Kulkarni, Dimer models on cylinders over Dynkin diagrams and cluster algebras, Proc.Amer. Math. Soc. (2019), 921-932. School of Mathematics, University of Leeds, Leeds, LS2 9JT, United KingdomOn leave from the University of Graz
Email address : [email protected] Institut f¨ur Mathematik und Wissenschaftliches Rechnen, Universit¨at Graz, Hein-richstrasse 36, 8010 Graz, Austria.
Email address ::