Nonnoetherian geometry

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of $A$ are all isomorphic if and only if $A$ is noetherian, if and only if the center $Z$ of $A$ is noetherian, if and only if $A$ is a finitely generated $Z$-module. Furthermore, we show that $Z$ is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide.


Introduction
The purpose of this paper is two-fold: first, to introduce a new framework for understanding the geometry of infinitely-generated subalgebras of affine coordinate rings in some non-abstract sense; and second, to apply this framework to geometries that arise from certain superpotential quiver algebras.
A superpotential algebra of a cancellative brane tiling (also called a dimer model) is a type of quiver algebra with potential that has the cancellation property (Definition 3.6). It is now well known that these algebras are noncommutative crepant resolutions and 3-Calabi-Yau algebras with 3-dimensional normal toric Gorenstein centers [MR, D, Br, M, Bo, B]. Much less is understood, however, about superpotential algebras of non-cancellative brane tilings. In contrast to cancellative brane tilings, they are not finitely-generated modules over their centers, and their centers are nonnoetherian. In this paper we study the central geometry of these non-cancellative algebras, as it is of interest from both a mathematical and a string theory perspective. 1 We briefly outline our main results. The results in section 1 are about nonnoetherian subalgebras of affine coordinate rings, and may be of independent interest from superpotential algebras.
Let R be a subalgebra of an affine coordinate ring S. We introduce the subsets U := {n ∈ Max S | R n∩R = S n } and W := n ∈ Max S | (n ∩ R)S = n , and show the following.
Theorem A. Suppose R is a subalgebra of an affine k-algebra S. Then the map is injective on U, and W is the unique largest subset of Max S that φ is injective on.
In particular, U ⊆ W . If U = ∅ then Max S and Max R are isomorphic on open dense subsets, and thus birationally equivalent. We then introduce the notions of depiction and geometric dimension: we say R is depicted by S if U = ∅ and the map φ : Max S → Max R, n → n ∩ R, is surjective. Furthermore, for p ∈ Spec R let codim S p denote the length of a longest chain of distinct prime ideals of R contained in p that lifts to a chain of prime ideals of S, and set dim S p := dim S R − codim S p (Definition 2.11). The geometric dimension of p is then defined to be the supremum dim • p = sup {dim S p | S a depiction of R} .
The following theorem relates dim S p to the set U, and implies that the geometric dimension of a point will always be finite and bounded by the transcendence degree of Frac R over k.
Theorem C. Suppose R is depicted by S. Then dim S R = trdeg k Frac R = dim S. In 4-dimensional theories these algebras often correspond to non-superconformal quiver gauge theories, which are perfectly reasonable physical theories. In 3-dimensional Chern-Simons quiver gauge theories, cancellative and non-cancellative brane tilings are usually on an equal footing; see for example [MS, BR].
If p ∈ Spec R and q ∈ φ −1 (p), then with equality on the left if Z(q) ∩ U = ∅.
In section 2 we turn our attention to quiver algebras: Theorem D. Suppose A = kQ/I is a finitely-generated quiver algebra that admits an impression τ : A ֒→ M |Q 0 | (B) (Definition 3.2), with B an affine integral domain and τ (e i ) = E ii for each i ∈ Q 0 . For p ∈ e j Ae i , defineτ (p) by τ (p) =τ (p)E ji . Then the center of A is isomorphic to and is depicted by We then characterize the central geometry of a class of superpotential algebras of non-cancellative brane tilings. To do this, we define a k-homomorphism that turns a set of arrows into vertices, called a contraction (Definitions 3.9 and 3.11). This formalizes an operation known as 'Higgsing' in quiver gauge theories (see Remark 3.10).
Theorem E. Let ψ : A → A ′ be an adequate contraction between superpotential algebras of brane tilings, where A ′ is cancellative and A is not. Further suppose A ′ admits an impression (τ ′ , B), with B a polynomial ring andτ ′ (a) ∈ B a monomial for each a ∈ Q ′ 1 . Define the k-homomorphism τ : A → M |Q 0 | (B) bȳ τ (a) :=τ ′ (ψ(a)) for each a ∈ e j Ae i , i, j ∈ Q 0 , and let R, S and R ′ , S ′ be as in (1), (2) with τ and τ ′ respectively. Then Furthermore, R is depicted by S. In particular, the 'mesonic chiral ring' of A, namely S, is a depiction of its center Z.
As a corollary, we conclude that the nonnoetherian center Z of A is birational to the normal toric Gorenstein singularity S = S ′ and contains a positive dimensional subvariety that is identified as a single (closed) point.
Finally, in Proposition 3.20 we give an infinite family of non-cancellative brane tilings for which Theorem E applies.
Notation: Throughout R is a subalgebra of an affine integral domain S, both of which contain an algebraically closed field k. We will denote by dim R the Krull dimension of R; by Frac R the ring of fractions of R; by Max R the set of maximal ideals of R; and by Spec R either the set of prime ideals of R or the affine k-scheme with global sections R. For a ⊂ R we denote by Z(a) := {m ∈ Max R | m ⊇ a}, V (a) := {p ∈ Spec R | p ⊇ a}, the respective Zariski-closed sets of Max R and Spec R. Finally, Q ℓ denotes the paths of length ℓ in a quiver Q.
2. Nonnoetherian geometry: positive dimensional and nonlocal points 2.1. The largest subset. We begin with the following basic fact.
Proof. Since S is finitely-generated over an algebraically closed field, S/q ∼ = k, and thus since 1 S ∈ R, the composition ψ : R ֒→ S → S/q is an epimorphism. Therefore R/ ker ψ ∼ = k, and so q ∩ R = ker ψ ∈ Max R.
The embedding ι : R ֒→ S induces the morphism of schemes where φ : Spec S → Spec R is given by q → q ∩ R. 2 We introduce the following subsets of the variety Max S.
Definition 2.2. For n ∈ Max S, set m := n ∩ R ∈ Max R. Define the subsets We will omit the subscript S when S is fixed. Recall that S is an overring of a domain R if R ⊆ S ⊆ Frac R.
Lemma 2.3. If U is nonempty then S is an overring of R. In particular, the function field of Spec R equals the function field of Spec S. Furthermore, U contains a nonempty open subset of Max S.
Proof. Suppose n ∈ U. Then since S is an integral domain, S ⊆ Frac S = Frac (S n We now show that U contains a nonempty open subset of Max S. We first claim that if A is a subalgebra of B, n ∈ Max A, and nB ∩ A = A, then A n ⊆ B nB . Consider a b ∈ A n with a, b ∈ A ⊆ B and b ∈ n. Then b ∈ nB: suppose b ∈ nB. Then b ∈ nB ∩ A. But nB ∩ A ⊇ n ∈ Max A, and nB ∩ A = A by assumption, so nB ∩ A = n, whence b ∈ n, contrary to our assumption. Therefore b ∈ nB, so a b ∈ B nB , proving our claim. Now suppose {a i } i∈I is a generating set for S and set J := {j ∈ I | a j ∈ R}. Since S ⊂ Frac R by Lemma 2.3, for each j ∈ J there is a c j ∈ R such that a j c j ∈ R. The subset of Max S is nonempty and open since |J| ≤ |I| < ∞. Suppose n ∈ U ′ and m = n ∩ R. Then c j ∈ n, hence c j ∈ m, for each j ∈ J. Therefore S ⊆ R m . But then S ⊆ R m ⊆ S n , so by our claim above we have In the following theorem we show that W is similar in spirit to the Azumaya locus of A when A is a noncommutative algebra, module-finite over its center Z. Recall that if n, n ′ ∈ Max A and n ∩ Z = n ′ ∩ Z is in the Azumaya locus of A, is injective on U, and W is the unique largest subset of Max S that φ is injective on. In particular, U ⊆ W . If U = ∅ then Max S and Max R are isomorphic on open dense subsets, and thus birationally equivalent.
Proof. We first show that φ is injective on U: if n, n ′ ∈ U and n ∩ R = n ′ ∩ R, then S n = R n∩R = R n ′ ∩R = S n ′ , so S n has unique maximal ideal n = n ′ .
We now claim that n ∈ W c if and only if there is a point n ′ ∈ Max S, not equal to n, such that φ(n) = φ(n ′ ). First note that for m ∈ Max R, m ⊆ mS ⊆ √ mS ⊆ n, so m ⊆ √ mS ∩ R ⊆ n ∩ R = m, which yields Set m := n ∩ R and suppose n = √ mS. Since S is Jacobson, √ mS = √ mS⊆q∈Max S q, so there exists a maximal ideal n ′ = n of S such that √ mS ⊆ n ′ . But then by Lemma 2.1, Conversely suppose there are distinct points n, n ′ ∈ Max S such that φ(n) = φ(n ′ ). Then n ∩ R = n ′ ∩ R =: m, and so √ mS ⊆ n ∩ n ′ n. Finally, Max S is irreducible so U contains an open dense subset by Lemma 2.3. Therefore Max S and Max R are birationally equivalent since φ is injective on U.
3 If S is finitely-generated over k but R is not, then S will not be a finitely-generated R-module; this follows, for example, from the Artin-Tate lemma.
The following proposition generalizes the fact that for n ∈ Max S, S = k + n.
Proposition 2.6. Let R ′ be a subalgebra of S, I an ideal of S, and form the algebra Proof. We claim that if q ∈ Spec S does not contain I then R q∩R = S q ; in particular, if q ∈ Max S then q ∈ U. Set p := q ∩ R. Then R p ⊆ S q , so suppose a ∈ S q , i.e., there is some f, g ∈ S, g ∈ q, such that a = f g . Since q does not contain I there is some c ∈ I \ q. Since c, g ∈ S \ q and q is prime, we have cg ∈ S \ q. Since c ∈ I, cg ∈ I ⊂ R, so cg ∈ R \ p. But also b := agc ∈ I ⊂ R, and thus a = b cg ∈ R p . For the second statement, clearly I ∈ Max R. Suppose I is not a maximal ideal of S. Let n ∈ Z(I). Then n ⊇ I, so n ∩ R ⊇ I ∩ R = I ∈ Max R, so n ∩ R = I. But then √ IS = IS = I n, and so Z(I) ⊆ W c . The converse follows from the previous paragraph, and so W c = Z(I). Since Z(I) ⊇ U c ⊇ W c = Z(I), we also have U c = Z(I).
Note that U may properly contain Z(I) c ; for example, take R ′ = S. The following definition formalizes these 'geometric pictures'.
Question 2.9. If R admits a depiction, then does R admit a unique maximal depiction with respect to inclusion?
A partial answer to this question is given in the next proposition. We say two elements a, b ∈ S are coprime if their only common divisors are in k.
Proposition 2.10. Suppose S is a depiction of R with the property that a, b ∈ S are not coprime whenever a|b n for some n ≥ 1. Then S is the unique maximal depiction of R.
Proof. Suppose R admits depictions S and S ′ , S ′ has the 'coprime property', and assume to the contrary that a ∈ S \ S ′ . Since U S ⊂ Max S is nonempty, Frac S = Frac R by Lemma 2.3. Therefore a = b c for some b, c ∈ R, which we can assume to be coprime in where (i) holds since S ′ is Jacobson. Therefore c|b n for some n ≥ 1. But b and c were chosen to be coprime, contradicting our assumption that S ′ has the coprime property. Therefore S ⊆ S ′ .
2.2. Geometric dimensions of points. Throughout this section we assume that R is depicted by S. We introduce the following modifications of height and Krull dimension.
Definition 2.11. Let p ∈ Spec R. Denote by codim S p the length d of a longest chain of prime ideals of R, p 0 · · · p d = p, that lifts to a chain of prime ideals of S, q 0 ⊆ · · · ⊆ q d , in the sense that We then define the geometric dimension of p to be Our main result of this section is Theorem 2.19. We first give a couple examples.
Fortunately this chain does not lift to a chain of prime ideals of S: we have (y)S ∩ R = (xy)S ∩ R, but (y)S ⊂ (x)S, and (xy)S is not prime in S. Therefore the geometric dimension of the closed point ( be the coordinate ring for an algebraic variety X, and let n 1 , . . . , n r be maximal ideals of S. Then by Proposition 2.6, is the coordinate ring for a space which is identical to X, with the exception that the r points n 1 , . . . , n r are identified as one single point. S is a depiction of R, and U c is zero dimensional.
But φ is injective on U by Theorem 2.4, and so n = n ′ . Therefore n ∈ Z(q) ∩ U.
For the following, set p m := pR m .
Lemma 2.16. Let p ∈ Spec R. Consider a maximal chain of distinct prime ideals in R containing p, Suppose m ∈ Z(p) ∩ φ(U) and let n ∈ φ −1 (m). Then pS n ∈ Spec S n , and (4) p 0 S n · · · p d S n is a maximal chain of prime ideals in S n containing pS n with the property that Proof. (i) Let p p ′ be prime ideals in R. We claim that p m p ′ m , and so it suffices to show that p m = p ′ m . Assume to the contrary that these ideals are equal and let a ∈ p ′ \ p. Then there exists a d ∈ p and b c ∈ R m , with b, c ∈ R, c ∈ m ⊃ p, such that a = db c ∈ p m . But then ac = db ∈ p while a and c are in R \ p, contradicting the fact that p is a prime ideal in R.
(ii) To show that pS n ∩ R = p, consider where (i) follows since m ∈ φ(U).
(iii) Since n ∈ U, p m = pS n , and so p m is an ideal in S n . To show that p m is prime in S n , consider ab ∈ p m and a, b Thus a 1 b 1 ∈ R ∩ p m = p, so a 1 or b 1 is in p, and so a or b is in p m .
Proposition 2.17. Let p ∈ Spec R be such that Z(p)∩φ(U) = ∅. Given any maximal chain of distinct prime ideals of R containing p, p 0 · · · p d = p, there exists a maximal chain of distinct prime ideals of S, We have thus shown that there is a chain of prime ideals q 0 ⊆ · · · ⊆ q d with the property that q i ∩ R = p i for each 0 ≤ i ≤ d, and these inclusions are strict by Lemma 2.14.
(i) First suppose Z(q) ∩ U = ∅, i.e., q is contained in some n ∈ U. Set m = n ∩ R. We claim that codim S p = codim q, so by Proposition 2.17 and Lemma 2.14 it suffices to show that codim S p ≥ codim q. Consider a maximal chain of distinct prime ideals q 0 · · · q d = q. Since q ⊆ n, there is a chain of distinct prime ideals q 0 S n · · · q d S n in S n . Since n ∈ U, this is a chain of distinct prime ideals q where (a) holds since n ∈ U and (b) holds since q ⊆ n. This proves our claim.
(ii) Now suppose Z(q) ∩ U = ∅. U contains a nonempty open set U ′ by Lemma 2.3, which is dense since Max S is irreducible. Thus there exists a chain of distinct contrary to assumption by Lemma 2.15. Otherwise q d−1 ∩ R p, and so codim S p ≥ codim q.
Theorem 2.19. Suppose R is depicted by S. Then If p ∈ Spec R and q ∈ φ −1 (p), then with equality on the left if Z(q) ∩ U = ∅.
Proof. We first prove (6). By Lemma 2.3 and our assumption that Max S is an algebraic variety, dim S = trdeg k Frac S = trdeg k Frac R. Since U is nonempty, there is some n ∈ U such that where (i) follows since Max S ∋ n is an (irreducible) algebraic variety. We now prove (7). dim S p ≤ dim S holds by (6). Moreover, If Z(q) ∩ U = ∅ then dim S p = dim q by (6) and Proposition 2.18.
Corollary 2.20. The geometric dimension of a point p ∈ Spec R is always finite, and bounded by the transcendence degree of Frac R over k.

Central geometry of non-cancellative toric superpotential algebras
Throughout let A = kQ/I be a finitely-generated quiver algebra and let B be an affine integral domain containing k.
3.1. Depictions from quiver algebras. Denote by E ji ∈ M |Q 0 | (B) the matrix whose (ji)-th entry is 1 and all other entries zero. Given an algebra homomorphism τ : be a k-homomorphism that is an algebra homomorphism on each e i Ae i , i ∈ Q 0 , with τ (e i ) = E ii . Suppose there is a cycle b ∈ A which contains each vertex as a subpath and whose τ -image is nonzero. Further suppose the map Max B → Max R, q → q ∩ R, is surjective. Then the subalgebra Proof. A is finitely-generated, so |Q 1 | < ∞, so S is finitely-generated. Furthermore, S is a domain since it is a subalgebra of the domain B. U S is nonempty: Fix i ∈ Q 0 . Let b i ∈ e i Ae i be a cycle that contains each vertex e j as a subpath, and let c i ∈ e i Ae i be an arbitrary cycle. For each j ∈ Q 0 , denote by b j and d j the respective cycles obtained by cyclically permuting b i and d i := b i c i so that their heads and tails are at j.
since τ is an algebra homomorphism on e i Ae i . Therefore β andτ (c i )β are in R. Let q be a point in the nonempty open subset of Max B defined by β = 0. By Lemma 2.1, n := q ∩ S and m := n ∩ R are maximal ideals of S and R respectively, and β ∈ R is invertible in the localization R m . Consequentlȳ We are interested in cases where the center of A is isomorphic to R. For the remainder of this section, let R and S be as in (8) and (9). The following definition was introduced in [B] to study a class of superpotential algebras of cancellative brane tilings (Definition 3.6). An impression is useful in part because it determines the center Z of A, and if A is a finitely-generated Z-module then it determines all simple A-module isoclasses of maximal k-dimension [B, Proposition 2.5].
Theorem 3.3. If τ : A ֒→ M |Q 0 | (B) is an impression of A with B an integral domain and τ (e i ) = E ii for each i ∈ Q 0 , then the center of A is isomorphic to R and is depicted by S.
Proof. By [B,Lemma 2.4], the maximal k-dimension of the simple A-modules is |Q 0 |. Thus there exists a path p ji ∈ I between any two vertices i, j of Q. Since τ is injective, τ (p ji ) = 0. We can therefore form a cycle p 1|Q 0 | · · · p 32 p 21 which contains each vertex a subpath and whose τ -image is nonzero: indeed, since B is an integral domain we haveτ p 1|Q 0 | · · · p 32 p 21 =τ p 1|Q 0 | · · ·τ (p 32 )τ (p 21 ) = 0. By the definition of impression, the morphism Max B → Max R, q → q ∩ R, is surjective. We may therefore apply Proposition 3.1. Finally, by [B,Lemma 2 Remark 3.4. The role of S is new: S is a commutative ring obtained from A that in most cases is not a central subring of A, but is closely related to the geometry of the center Z of A. When Z is noetherian then S is isomorphic to Z, while if Z is nonnoetherian then S properly contains Z.
Example 3.5. Consider the quiver algebra A = kQ/ yba − bay , with quiver given in figure 1. The labeling of arrows defines an impression (τ, B = k[a, b, z]), and so we may apply Theorem 3.3. Letting x := ab, we find that the center of A is isomorphic to R = k [τ (e 1 Ae 1 ) ∩τ (e 2 Ae 2 )] = k + (x)S and is depicted by S = k [τ (e 1 Ae 1 ) ∪τ (e 2 Ae 2 )] = k[x, y], with R and S as in Examples 2.5, 2.7, and 2.12. 4 We make two remarks: • Even though A does not posses certain nice properties such as being a finitelygenerated module over its center, the simple A-modules of maximal k-dimension (i.e., the simples modules with dimension vector (1, 1) by [B,Lemma 5.1]) are nevertheless still parameterized by the smooth locus of Max Z, namely (bc, a) ∈ k * × k, which we naturally identify with U ⊂ Max S. • The moduli space of θ-stable A-modules of dimension vector (1, 1), for generic stability parameter θ, is precisely the 'resolution' Max S.

3.2.
Depictions of non-cancellative toric superpotential algebras and the mesonic chiral ring. In this section we will consider superpotential algebras of non-cancellative brane tilings. Such algebras cannot admit impressions (τ, B) with B prime, but fortunately many of them (if not all) still have the property that their centers are isomorphic to R = k i∈Q 0τ (e i Ae i ) , whereτ is defined in (11) below.
Definition 3.6. A brane tiling is a quiver Q whose underlying graphQ embeds into a two-dimensional real torus T 2 , such that each connected component of T 2 \Q is simply connected and each cycle on the boundary of a connected component, called a unit cycle, is oriented and has length at least 2. A superpotential algebra A of a brane tiling Q is the quiver algebra kQ/I, where I is the (two-sided) ideal 5 (10) I := d − d ′ | ∃ a ∈ Q 1 such that da and d ′ a are unit cycles ⊂ kQ.
Denote p − q ∈ I by p ≡ q. A and Q are called cancellative if for all paths a, p, q with h(a) = t(p) = t(q) (resp. t(a) = h(p) = h(q)), we have p ≡ q whenever pa ≡ qa (resp. ap ≡ aq).
Lemma 3.8. Let A = kQ/I be a superpotential algebra of any brane tiling (cancellative or not). Then • The element u := i∈Q 0 u i + I is in the center of A.
• If u i , u ′ i are two unit cycles at i ∈ Q 0 then u i − u ′ i ∈ I. • If p ∈ e i kQe i is a cycle whose lift p + is a cycle in Q, then p ≡ u n i for some n ≥ 0.
The following definition formalizes a notion of 'Higgsing' in quiver gauge theories, and we will use this notion to determine depictions of the centers of superpotential algebras of non-cancellative brane tilings.
Definition 3.9. Given a brane tiling Q and a subset of arrows Q * 1 ⊆ Q 1 , form the contracted quiver Q ′ by identifying the three paths a, h(a), and t(a), for each a ∈ Q * 1 . Denote by ψ : kQ → kQ ′ the k-homomorphism defined by sending a path in kQ to the corresponding path in kQ ′ .
If d − d ′ is a minimal generator of I, that is, there exists an a ∈ Q 1 such that ad and ad ′ are unit cycles, then ψ 1 then this follows from Lemma 3.8, and is trivial otherwise. Therefore ψ(I) ⊆ I ′ , and so ψ descends to a k-homomorphism A = kQ/I → A ′ = kQ ′ /I ′ , which we also denote by ψ. It is clear that ψ is an algebra homomorphism on the corner rings e i Ae i , i ∈ Q 0 . We say ψ is a contraction on Q * 1 . For the remainder of this section, let A = kQ/I and A ′ = kQ ′ /I ′ be superpotential algebras of brane tilings that are respectively non-cancellative and cancellative. Denote by Z and Z ′ the respective centers of A and A ′ . Suppose A ′ admits an impression τ : Since τ ′ is an algebra homomorphism and ψ is an algebra homomorphism on each e i Ae i , i ∈ Q 0 , τ ′ is also an algebra homomorphism on each e i Ae i .
Let R, S and R ′ , S ′ be defined by (8), (9) with τ and τ ′ respectively. Since A ′ is cancellative, it is well known that Z ′ ∼ = R ′ = S ′ since there is an isomorphism of corner rings e i Ae i ∼ = e j Ae j for each i, j ∈ Q 0 . In Theorem 3.18 we will show that (i) Z ∼ = R as algebras; (ii) S = S ′ ; and (iii) R is depicted by S.
Physics Remark 3.10. In a 4-dimensional N = 1 quiver gauge theory with quiver Q, the algebra generated by the cycles in Q modulo the F-flatness constraints, that is, the defining generators of I, is known as the 'mesonic chiral ring'. In the context of these theories, the algebra S is similar to, if not a formalized definition of, the mesonic chiral ring.
Furthermore, the Higgsing considered here is presumably related to RG flow: we start with a non-superconformal (strongly coupled) quiver theory Q, give an arrow δ ∈ Q * 1 a nonzero vev, and end with a theory Q ′ that lies at a conformal fixed point. It is also possible to Higgs between two cancellative brane tilings, where quite often a P n in a partial resolution of Max S is blown-down. In this case we expect that S = S ′ .
Let π : R 2 → T 2 be the canonical projection, and Q := π −1 (Q) ⊂ R 2 the covering quiver (or 'periodic quiver') of Q. Fix a fundamental domain D of Q. For each path p in Q, denote by p + the unique path in Q with tail in D satisfying π(p + ) = p. For a vertex j in the covering quiver Q, denote by j the corresponding vector in R 2 . We introduce the following definition.
Definition 3.11. We say ψ or Q * 1 is adequate if the following conditions hold: (1) If the lifts of two paths p and q contain no cyclic proper subpaths modulo I and bound a compact region whose interior does not contain the lift of any arrow in Q * 1 , then p ≡ q.
Henceforth we will assume ψ is adequate unless stated otherwise.
Question 3.12. Do all non-cancellative brane tilings adequately contract to cancellative brane tilings whose algebras admit impressions?
Remark 3.13. We do not know of an example where condition 1 is not satisfied, though it is a non-trivial condition (see Proposition 3.20).
Lemma 3.14. If ψ : A → A ′ is a contraction with A ′ cancellative and c ∈ A is a cycle of positive length, then ψ(c) is not a vertex.
Proof. Suppose c is a cycle of positive length that contracts to a vertex. Then the lift c + must be a cycle: otherwise the underlying graph of ψ(Q) = Q ′ could not embed into a two-torus since we are assuming Q ′ is non-degenerate. Furthermore, the unit cycle u t(c) must also contract to a vertex, for otherwise again the underlying graph of Q ′ could not embed into a two-torus. Let i ∈ Q 0 be a vertex that is not the head or tail of any arrow in Q * 1 , and let p be a path from i to t(c). Then by Lemma 3.8, Since Q ′ is cancellative, ψ(u n i ) = e ψ(i) . This contradicts our choice of i. Lemma 3.15. Let p, q ∈ e j kQe i be paths such that ψ(p) ≡ ψ(q). Then their lifts p + and q + bound a compact region R in the covering quiver Q ⊂ R 2 of Q. Furthermore, if the interior of R does not contain the lift of any arrow in Q * 1 , then p ≡ q. Proof. Suppose ψ(p) ≡ ψ(q). Then their lifts ψ(p) + and ψ(q) + have coincident heads and tails. By Lemma 3.14, there is no cycle c ∈ e i kQe i of positive length satisfying ψ(c) = e ψ(i) . Therefore p + and q + have coincident heads and tails as well, so p + and q + bound a compact region R.
Suppose p ≡ q. Since ψ(p) ≡ ψ(q) and the relations (10) are 'homotopy relations', the lift of some δ ∈ Q * 1 must lie in the interior of R as an obstruction.
Proposition 3.16. Let γ ∈ B be a monomial. Suppose that for each i ∈ Q 0 there is a cycle c i ∈ e i kQe i such thatτ (c i ) = γ. Then i∈Q 0 c i + I ∈ Z. Furthermore, if d ∈ e i kQe i is another cycle satisfyingτ (d) = γ then d ≡ c i .
Proof. For the first claim, it suffices to show that for each arrow a ∈ Q 1 we have ac t(a) ≡ c h(a) a. Fix a ∈ Q 1 . First suppose (ac t(a) ) + and (c h(a) a) + do not contain cyclic proper subpaths. Then we may write R = ∪ j P j ⊂ R 2 , where each P j is a closed region bounded by two paths p + j and q + j with no cyclic proper subpaths; the intersection of interiors is empty, P • j ∩ P • k = ∅ for j = k; and the lift of any arrow in Q * 1 that lies in R lies on the boundary of some P j . Therefore, by condition 1 in Definition 3.11, p j ≡ q j for each j, whence ac t(a) ≡ c h(a) a. Now suppose (ac t(a) ) + or (c h(a) a) + contains a cyclic proper subpath, say ac t(a) = p 2 qp 1 , where q + is a cycle. By Lemma 3.8, q ≡ u m h(p 1 ) for some m ≥ 1. Thus, since i∈Q 0 u i + I is in the center of A, ac t(a) ≡ p 2 p 1 u m t(a) . Similarly c h(a) a ≡ ru n t(a) for some path r and n ≥ 0. Therefore the previous argument with p 2 p 1 and r in place of ac t(a) and c h(a) a implies that p 2 p 1 ≡ r. Butτ (p 2 p 1 )σ m =τ (ac t(a) ) =τ (c h(a) a) =τ (r)σ n . Thus m = n since B is a polynomial ring andτ (r) =τ (p 2 p 1 ) = 0. This yields ac t(a) ≡ c h(a) a.
The second claim follows from the same argument with c i and d in place of ac t(a) and c h(a) a.
Proposition 3.17. The subalgebras S and S ′ of B are equal.
Proof. Here we use condition 2 in Definition 3.11.
It is clear that S ⊆ S ′ . To show the converse, suppose that γ ∈ S ′ . Since A ′ is cancellative, S ′ is generated by monomials in B (S ′ is toric), and therefore we may suppose γ is a monomial. Furthermore, A ′ cancellative implies S ′ = R ′ . Therefore for each i ∈ Q ′ 0 there is a cycle c i ∈ e i kQ ′ e i such thatτ ′ (c i ) = γ. If γ = αβ with α, β ∈ S ′ , then either γ is in S, or α or β is not in S. Therefore we loose no generality in assuming γ is irreducible in S ′ . In particular σ ∤ γ in S ′ since σ ∈ S ′ .
Theorem 3.18. Let ψ : A → A ′ be an adequate contraction between superpotential algebras of brane tilings, where A ′ is cancellative and A is not. Further suppose A ′ admits an impression (τ ′ , B), with B a polynomial ring andτ ′ (a) ∈ B a monomial for each a ∈ Q ′ 1 . Define the k-homomorphism τ : A → M |Q 0 | (B) by (11), and let R, S and R ′ , S ′ be as in (8), (9) with τ and τ ′ respectively. Then Furthermore, R is depicted by S. In particular, the 'mesonic chiral ring' of A, namely S, is a depiction of its center Z.
Proof. Denote by 1 the identity matrix in M |Q 0 | (B). Recall that the k-homomorphism τ : A → M |Q 0 | (B) is an algebra homomorphism on each e i Ae i , i ∈ Q 0 . We will show that the restriction is an algebra isomorphism.
(i) The map (12) is well-defined, i.e., τ (Z) ⊆ R1: Suppose c ∈ Z. Since c commutes with the vertex idempotents, c must be a sum of cycles: c = i∈Q 0 c i with each c i ∈ e i Ae i . Let p be a path. Since τ is an algebra homomorphism on each e i Ae i , we have Furthermore,τ (p) =τ ′ (ψ(p)) is nonzero since τ ′ is an impression of A ′ . Thus, since B is an integral domain, (13) impliesτ (c h(p) ) =τ (c t(p) ). But there is a path between each pair of vertices in Q that is nonzero modulo I, and therefore τ (c) ∈ R1.
(ii) The map (12) is surjective, i.e., R1 ⊆ τ (Z): In the following, by monomial, path, or cycle, we mean a scalar multiple thereof. We first show that R is generated by a set of monomials in B. Suppose m j=1 β j ∈ R, with each β j a monomial. By the definition of R, for each i ∈ Q 0 there exists a b ∈ e i kQe i such thatτ (b) = j β j . Suppose b = n ℓ=1 c ℓ for some cycles c ℓ ∈ e i kQe i . By assumption, theτ -image of any arrow is a monomial in B. Thus theτ -image of any path is a monomial in B since B is a polynomial ring. Therefore, by k-linearity ofτ ,τ where each γ ℓ :=τ (c ℓ ) is a monomial since c ℓ is a path. But then Since B is a polynomial ring, n = m and (possibly re-indexing) γ j = β j for 1 ≤ j ≤ m. β j is therefore theτ -image of a cycle in e i kQe i . Since i ∈ Q 0 was arbitrary, β j ∈ R, proving our claim. Now let γ be a monomial in R. As we have just shown, γ is theτ -image of an element c = i∈Q 0 c i with each c i ∈ e i kQe i a cycle whoseτ -image is γ. By Proposition 3.16, c+I ∈ Z, so γ1 ∈ τ (Z). Since R is generated by a set of monomials in B, we have R1 ⊆ τ (Z).
(iii) The map (12) is injective: If b + I and c + I are in Z and satisfy τ (b) = τ (c) then b ≡ c by Proposition 3.16. [B,Lemma 2.1], and R ′ = S ′ by [B,Theorem 2.11]. Furthermore, since (τ ′ , B) is an impression of A ′ , the map Corollary 3.19. Suppose the hypotheses of Theorem 3.18 hold. Then the center Z of A is birational to the normal toric Gorenstein singularity S = S ′ and contains a positive dimensional subvariety that is identified as a single (closed) point.
The following proposition gives an infinite class of examples for which Theorem 3.18 is applicable. A notable example is the quiver for the cone over Q 111 , given in Example 3.3.ii below, which has been studied in the context of Chern-Simons quiver gauge theories in string theory.
Proof. Condition 1 is only non-trivial in the following case: Suppose p and q = r 2 sr 1 are paths whose lifts contain no cyclic proper subpaths modulo I, bound a compact region whose interior does not contain the lift of any arrow in Q * 1 , and such that sδ is a unit cycle for some δ ∈ Q * 1 . For condition 1 to be satisfied we must show that p ≡ q, which is non-trivial since it implies that σ divides theτ -image of every such path in Q.
Otherwise h(p) = h(q) = t(δ) has indegree 1, so p and q have the same leftmost arrow subpath a, say p = ap ′ and q = aq ′ . The paths p ′ and q ′ then bound a compact region whose interior does not contain the lift of any arrow in Q * 1 , and δ only meets q ′ at its head since a is not a loop. Therefore p ′ ≡ q ′ , whence p ≡ q.
The case where h(δ) has indegree and outdegree 1 is similar.
3.3. Examples. We now consider some 'nonnoetherian deformations' of square superpotential algebras. A superpotential algebra A = kQ/I of a brane tiling is square if the underlying graph of Q is a square grid graph with vertex set Z × Z, and with at most one diagonal edge in each unit square.
x 1 x 2 y 1 y 2 x 1 y 1 x 2 y 2 x 1 y 2 x 2 y 1 Figure 2. A labeling of arrows in the quiver of a square superpotential algebra that specifies an impression.
Any square superpotential algebra A admits an impression (τ, B = k[x 1 , x 2 , y 1 , y 2 ]), where for each arrow a in the covering quiver Q,τ (a) is the monomial corresponding to the orientation of a given in figure 2 [B,Theorem 3.7]. Specifically, τ : A → M |Q 0 | (B) is the k-algebra homomorphism defined on the generating set Q 0 ∪ Q 1 by e i → E ii for each i ∈ Q 0 and a →τ (a)E h(a),t(a) for each a ∈ Q 1 .
If Q only possesses three arrow orientations, say up, left, and right-down, then we may label the respective arrows by x, y, and z, and obtain an impression (τ, k[x, y, z]). In either case, A is cancellative by Remark 3.7.
Consider the four examples of nonnoetherian deformations of square superpotential algebras given in figure 3. In each example, the quiver labeled (a) is non-cancellative and contracts to the cancellative quiver Q ′ labeled (b). In examples (iii) and (iv), quiver (c) is obtained from (b) by removing all 2-cycles. The superpotential algebras corresponding to (b) and (c) are equal, although their path algebras (without relations) are not. The non-cancellative quivers (a) first appeared in [DHP, [FHPR,Section 4], [FKR]; [Bo,Example 3.2]; and [DHP, Table 6, 2.6], respectively, each in a different form from what is shown here.
In example (iii), if both of the upward pointing arrows are contracted, then the resulting quiver Q ′ consists of one vertex and 3 loops-the standard C 3 brane tilingand is therefore cancellative. However, for this choice of Q * 1 , condition 2 in Definition 3.11 is not satisfied, and the conclusions of Theorem 3.18 do not hold since S = k[x, y, xz, yz] = k[x, y, z] = S ′ .
In a forthcoming paper [B2], some questions regarding the representation theory of non-cancellative brane tilings will be addressed.
Douglas, Peng Gao, Mauricio Romo, and James Sparks for discussions on the physics of non-cancellative brane tilings.