Lower bound cluster algebras: presentations, Cohen–Macaulayness, and normality

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ALGEBRAIC COMBINATORICS

Greg Muller, Jenna Rajchgot & Bradley Zykoski

Lower bound cluster algebras: presentations, Cohen–Macaulayness, and normality Volume 1, issue 1 (2018), p. 95-114.

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https://doi.org/10.5802/alco.2

Lower bound cluster algebras:

presentations, Cohen–Macaulayness, and normality

Greg Muller, Jenna Rajchgot & Bradley Zykoski

Abstract We give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Gröbner degenerates to the Stanley–

Reisner scheme of a vertex-decomposable ball or sphere, and is thus Cohen–Macaulay. Finally, we use Stanley–Reisner combinatorics and a result of Knutson–Lam–Speyer to show that all lower bound algebras are normal.

1. Introduction and statement of results

Cluster algebras are a family of combinatorially-defined commutative algebras which were introduced by Fomin and Zelevinsky at the turn of the millennium to axiomatize and generalize patterns appearing in the study of dual canonical bases in Lie theory [8]. Since their introduction, cluster algebras have been discovered in the rings of functions on many important spaces, such as semisimple Lie groups, Grassmannians, flag varieties, and Teichmüller spaces [2, 20, 10, 11].⁽¹⁾

In each of these examples, the cluster algebra is realized as the coordinate ring of a smooth variety. This makes it all the more surprising that the varieties associated to general cluster algebras can be singular; in fact, they can possess such nightmar- ish pathologies as a non-Noetherian singularity [16]. Various approaches have been introduced to mitigate this.

• Restricting to a subclass of cluster algebras with potentially better behavior:

acyclic cluster algebras [2], locally acyclic cluster algebras [16, 1], or cluster algebras with a maximal green sequence [6, 18].

• Replacing the cluster algebra by a closely-related algebra with potentially better behavior: upper cluster algebras [2, 1], the span of convergent theta functions [12], orlower bound algebras [2].

In this note, we study the algebraic and geometric behavior of lower bound algebras.⁽²⁾

Manuscript received 2nd August 2017, accepted 18th August 2017.

Keywords. Cluster algebras, lower bound cluster algebras, combinatorial commutative algebra, Stanley–Reisner complexes.

Acknowledgements. The third author was supported by NSF grant DMS-1001764, as an REU project at the University of Michigan in 2015.

(1)A more interesting and morally correct statement is that each of these spaces possesses a stratification such that each stratum naturally has a cluster algebra in its ring of functions.

(2)More specifically, we consider lower bound algebras defined by a quiver in the body of the paper, and consider the more general context ofgeometric typein Appendix B.

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1.1. Lower bound algebras. Lower bound algebras were introduced in [2] as a kind of “lazy approximation” of a cluster algebra, in the following sense. A cluster algebra is defined to be the subalgebra of a field of rational functions generated by a (usually infinite) set ofcluster variables, produced by a recursive procedure called mutation. A lower bound algebra is defined to be the subalgebra generated by trun- cating this process at a specific finite set of steps. The resulting algebra is contained in the associated cluster algebra and is manifestly finitely generated.

A lower bound algebra is constructed from an ice quiver Q: this is a quiver (i.e.

a finite directed graph) without loops or directed 2-cycles, in which each vertex is designated unfrozen or frozen. As a matter of convenience, we assume the vertices ofQ have been indexed by the numbers {1,2, . . . , n}. To each unfrozen vertex i, we associate a pair of monomialsp⁺_i , p⁻_i ∈Z[x1, x2, . . . , xn] as follows.

(1.1) p⁺_i := Y

arrowsa∈Q

source(a)=i

x_target(a), p⁻_i := Y

arrowsa∈Q

target(a)=i

x_source(a)

Each vertex then determines a Laurent polynomial x⁰_i, called the adjacent cluster variable ati, which is defined by the following formula.⁽³⁾

(1.2) x⁰_i:=

x⁻¹_i (p⁺_i +p⁻_i ) ifiis unfrozen x⁻¹_i ifiis frozen

The lower bound algebra L(Q) defined by Q is the subalgebra of Z[x^±1₁ , . . . , x^±1_n ] generated by the variablesx1, . . . , xn and the adjacent cluster variablesx⁰₁, . . . , x⁰_n.

1 2

3

4

Figure 1. An ice quiver (the unique frozen vertex is depicted as a square)

Example 1.1. Consider the ice quiver Q in Figure 1. The four adjacent cluster variables are below.

x⁰₁= x²₂+ 1 x1

, x⁰₂=x₃x₄+x²₁ x2

, x⁰₃=1 +x₂ x3

, x⁰₄= 1 x4

1.2. Relations in L(Q). We first consider the problem of finding relations among the generators of L(Q). Each adjacent cluster variable satisfies a defining relation immediately from its definition.

(1.3) ∀iunfrozen, x⁰_ixi= (p⁺_i +p⁻_i )

(1.4) ∀ifrozen, x⁰_ixi= 1

A more interesting class of relations is given by the following proposition.

(3)This is an abuse of terminology. Technically speaking, a frozen vertex ishould not have an adjacent cluster variablex⁰_i, and instead we should include x⁻¹_i as a generator (though this latter step is a matter of some debate). We are streamlining the process by calling the inversex⁻¹_i “the adjacent cluster variable ati”.

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Proposition 1.2 (The cycle relations).For each directed cycle of unfrozen vertices v₁→v₂→ · · · →v_k →v_k+1=v₁ inQ, the following cycle relation holds.

(1.5) X

S⊂{1,2,...,k}

S∩(S+1)=∅

(−1)^|S| Y

i∈S

p⁺_v_ip⁻_v_i+1 xv_ixv_i+1

!

 Y

i6∈S∪(S+1)

x⁰_v_i



=

k

Y

i=1

p⁺_v_i xv_i

+

k

Y

i=1

p⁻_v_i xv_i

We note that the expressions on either side reduce to polynomials in the generators, despite the presence of fractions. Also note that choosing a different initial vertexv₁ in the same directed cycle does not change the corresponding cycle relation.

1

2 3

Figure 2. An ice quiver (no frozen vertices)

Remark 1.3. A quiver Q is called acyclic if it has no directed cycles of unfrozen vertices. The unifying theme of this paper is the use of the cycle relations to generalize results aboutL(Q) which were previously known only whenQis acyclic (that is, when there are no cycle relations).

Example1.4. LetQbe the ice quiver in Figure 2. The three adjacent cluster variables are

x⁰₁= x₂+x₃ x1

, x⁰₂= x₃+x₁ x2

, x⁰₃=x₁+x₂ x3

The defining relations here are obtained by clearing the denominators above. A nontrivial directed 3-cycle starting at any vertex determines the cycle relation

x⁰₁x⁰₂x⁰₃−x⁰₁−x⁰₂−x⁰₃= 2 which may be verified by direct computation.

1.3. A presentation of L(Q). We may ask whether there are other relations in L(Q) that are not an immediate consequence of the preceding relations; or more concretely, whether the defining relations and the cycle relations generate the entire ideal of relations among the generators.

Explicitly, we consider the homomorphism of rings⁽⁴⁾

π:Z[x1, x2, . . . , xn, y1, y2, . . . , yn]−→Z[x^±1₁ , x^±1₂ , . . . , x^±1_n ]

∀i, π(x_i) =x_i, π(y_i) =x⁰_i

The image of this homomorphism isL(Q), and soKQ:= ker(π) is theideal of relations among the generators of L(Q), where each adjacent cluster variable x⁰_i has been re- placed by the abstract variableyi. The homomorphismπdescends to an isomorphism

Z[x₁, x₂, . . . , x_n, y₁, y₂, . . . , y_n]/K_Q−→ L(Q)^∼

We will say a directed cyclev₁→v₂→ · · · →v_k →v_k+1=v₁isvertex-minimal if no vertex appears more than once and there is no directed cycle whose vertex set is a proper subset of{v₁, v₂. . . , vk}.

(4)The y-variables introduced here have no relation to they-variables or coefficient variables introduced in [9].

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Theorem 1.5.The ideal of relationsK_Q is generated by the following elements.

• For each unfrozen vertex i,

(1.6) y_ix_i−p⁺_i −p⁻_i

• For each frozen vertexi,

(1.7) y_ix_i−1

• For each vertex-minimal directed cycle of unfrozen verticesv₁→v₂→ · · · → vk→vk+1=v1,

(1.8) X

S⊂{1,2,...,k}

S∩(S+1)=∅

(−1)^|S| Y

i∈S

p⁺_v

ip⁻_v

i+1

xv_ixv_i+1

!

 Y

i6∈S∪(S+1)

y_v_i



−

k

Y

i=1

p⁺_v

i

xv_i

−

k

Y

i=1

p⁻_v

i

xv_i

which simplifies to a polynomial.

The theorem is true without the vertex-minimal condition, which is used here to reduce the set of relations.

Example 1.6. Let Q be the ice quiver in Figure 2. By Theorem 1.5, L(Q) is iso- morphic to the quotient of Z[x₁, x₂, x₃, y₁, y₂, y₃] by the ideal K_Q generated by the following 4 relations.

KQ=hy1x1−x2−x3, y2x2−x3−x1, y3x3−x1−x2, y1y2y3−y1−y2−y3−2i 1.4. A Gröbner basis for KQ. We prove Theorem 1.5 by means of a stronger result, that the given generators are a Gröbner basis for the ideal of relations KQ. Recall that, given a polynomial ring with a monomial order<, aGröbner basisof an idealI is a generating set{g1, g2, .., gk} ofI satisfying the additional condition that {in<(g1),in<(g2), . . . ,in<(gk)}is a generating set of in<(I).

The monomial orders relevant to us are those in which the y-variables are much more expensive than the x-variables, that isx^αy^β >x^γy^δ wheneverP

iβi is larger thanP

iδi. An example of such a monomial order is the lexicographical order where the variables are ordered by

y₁> y₂>· · ·> y_n > x₁> x₂>· · ·> x_n.

Theorem1.7.For any monomial order ofZ[x₁, x₂, . . . , x_n, y₁, y₂, . . . , y_n]in which all of they-variables are much more expensive than all of thex-variables, the polynomials given in Theorem 1.5 are a Gröbner basis for K_Q. Consequently, the initial ideal in_<K_Q is squarefree monomial ideal with generating set

{xiyi |16i6n} ∪ {yv₁yv₂· · ·yv_k

v1→v2→ · · · →

→v_k →v_k+1=v₁ is a vertex-minimal cycle inQ}.

Remark 1.8. When Q is acyclic, Theorem 1.7 specializes to Corollary 1.17 in [2].

The proof of Theorem 1.7 given in Section 2.2 uses [2, Corollary 1.17] in an essential way, so our proof is not independent of the original result.

1.5. Simplicial complexes and Cohen–Macaulayness of lower bound algebras. From here on, we work over a fieldK, and consider theK-algebraL(Q)⊗_ZK. We will still refer to this algebra as a lower bound algebra and, though a slight abuse of notation, will simply denote it byL(Q). Similarly, we letKQdenote the associated lower bound ideal, so that it is the kernel of the mapπ:K[x1, . . . , xn, y1. . . , yn]→ L(Q).

To a squarefree monomial ideal I ⊆ K[z1, . . . , zn], one can associate a simplicial complex ∆ on the vertex set{z1, . . . , zn}. This simplicial complex is called the Stanley–Reisner complex and is defined as follows:{zi₁, . . . zi_r} is a face of ∆ if and

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x₁

x2 x3

y1 y2

y₃

Figure 3. The Stanley–Reisner complex forhx₁y₁, x₂y₂, x₃y₃, y₁y₂y₃i only if the monomialzi₁· · ·zir ∈/ I. Observe that the minimal non-faces of ∆ are in one-to-one correspondence with a minimal generating set ofI. For further information on Stanley–Reisner complexes, see the textbook [15, Chapter 1].

Whenever the Stanley–Reisner complex is a simplicial ball or a sphere,⁽⁵⁾the corresponding face ringK[z₁, . . . , z_n]/Iis a Cohen–Macaulay ring [19]. Furthermore, when I = in_<J for some ideal J ⊆ K[z₁, . . . , z_n], we may also conclude that J itself is Cohen–Macaulay (see eg. [5, Proposition 3.1]).

Example 1.9. We continue Example 1.6 and observe that the initial ideal in<KQ

(for a term order< as in Theorem 1.7) is hx1y1, x2y2, x3y3, y1y2y3i. The facets (i.e.

the maximal faces) of the associated Stanley–Reisner complex are precisely those {z1, z2, z3} where zi is either xi or yi and at least one of z1, z2, z3 is an xi. This Stanley–Reisner complex is readily seen to be a simplicial ball, and is pictured in Figure 3.

This example generalizes, and we are able to conclude that all lower bound algebras are Cohen–Macaulay.

Theorem 1.10.Let Q be a quiver with nvertices, letK_Q ⊆K[x₁, . . . , x_n, y₁, . . . , y_n] be its ideal of relations, and let < be any monomial order where the y-variables are much more expensive than thex-variables. Let∆_Q be the Stanley–Reisner complex of the squarefree monomial ideal in_<K_Q.

(i) If Qis acyclic, then∆Q is a simplicial sphere.

(ii) If Qis not acyclic, then ∆Q is a simplicial ball.

We show additional properties of ∆Q. If Q is acyclic, then ∆Q is the boundary of a cross-polytope. In both cases, ∆Q satisfies the stronger condition of vertex- decomposibility. Details are in Section 3.

Corollary 1.11.For anyQ the lower bound algebra L(Q)over a field Kis Cohen–

Macaulay.

Remarks 1.12.

(i) We prove Theorem 1.10 by way of a more general result, which gives a larger class of simplicial complexes which are automatically vertex-decomposable balls (see Theorem 3.3).

(5)More precisely, we mean the geometric realization of the simplicial complex is homeomorphic to a ball or sphere, respectively. Whenever we refer to a simplicial complex as a topological object, we more precisely mean its geometric realization.

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(ii) WhenQis acyclic,L(Q) was already known to be Cohen–Macaulay; specifically, [2, Corollary 1.17] implies that L(Q) is a complete intersection, and, consequently, that it is Cohen–Macaulay.

1.6. Normality of lower bound algebras. Our last main result is the normality of theK-algebra L(Q).

Theorem 1.13.Every lower bound cluster algebra defined by a quiver is normal.

Since L(Q) = K[x₁, . . . , x_n, y₁, . . . , y_n]/K_Q is finitely-generated, Serre’s Criterion reduces normality to a pair of geometric conditions on the varietyV(KQ).

• (R1) The variety V(KQ) has no codimension-1 singularities.

• (S2) Any regular function on an open subset in V(KQ) with codimension-2 complement extends to a regular function on all of V(KQ).

The Cohen–Macaulay property implies the S2 condition, and so normality of L(Q) reduces to proving there are no codimension-1 singularities.

As with Cohen–Macaulayness, this geometric question will be reduced to Stanley–

Reisner combinatorics, along with a result of Knutson–Lam–Speyer [14, Proposition 8.1]. See Section 4 for further information.

Remark 1.14. Like our other results, Theorem 1.13 is only new in the case of non- acyclicQ. In the acyclic setting,L(Q) is equal to its upper cluster algebra⁽⁶⁾, which is a normal domain by [16, Prop. 2.1].

Structure of paper. Section 2 considers relations in L(Q) and proves the associated results: Proposition 1.2, Theorem 1.5 and Theorem 1.7. Section 3 introduces the relevant combinatorial tools, leading to the proof of Theorem 1.10. Section 4 addresses normality, proving Theorem 1.13.

The paper concludes with a pair of appendices which frame the scope of the paper.

Appendix A considers the singularities of lower bound algebras directly, and provides an example to suggest this is a difficult problem. Appendix B explains how the results of the paper can be extended toskew-symmetrizablelower bound algebras, which are more general but also somewhat less intuitive.

2. Presentations and Gröbner Bases

2.1. Choice graphs and cycle relations. To every cycle of Q there exists a corresponding relation inL(Q). LetQhave a directed cyclev1→v2→ · · · →v_k−1→ vk →v1. By giving an alternate presentation for the product

(2.1)

k

Y

i=1

x⁰_v_i=

k

Y

i=1

x⁻¹_v_i (p⁺_v_i+p⁻_v_i),

we acquire a nontrivial relation that holds inL(Q). It is our goal to expand the right- hand product as a sum, and from this, Proposition 1.2 will follow. Each term of the expansion of this product represents a choice, for each i, of either p⁺_v_i or p⁻_v_i from x⁻¹_v_i (p⁺_v_i+p⁻_v_i). Therefore, to each term, we associate a directed graph withZ/kZas its vertex set and{(i, i±1)}ⁿ_i=1 as its set of arrows, where the sign of±corresponds to the abovementioned choice ofp⁺_v_i (corresponding to the positive sign becausexv_i+1

dividesp⁺_v

i) orp⁻_v

i (negative sign, sincex_v_i−1 dividesp⁻_v

i). Call these graphs thechoice graphs of the terms of the expansion of (2.1), and letC denote the set of all choice

(6)This was proven with an additional assumption in [2, Thm. 1.18], and without said assumption in [17].

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1 2 3 4 5 6

Figure 4. The choice graph for (x⁻¹₁ p⁺₁)(x⁻¹₂ p⁺₂)(x⁻¹₃ p⁻₃)(x⁻¹₄ p⁻₄)(x⁻¹₅ p⁻₅)(x⁻¹₆ p⁻₆)

graphs. Formally, we write the correspondence between choice graphs and terms as a function

(2.2) M :C→ {monomials in Z[x^±1₁ , . . . , x^±1_k ]}, M(g) =

k

Y

i=1

x⁻¹_v_i p^signv_i ^g⁽ⁱ⁾,

where

(2.3) sign_g(i) =

(+ if (i, i+ 1)∈g,

− if (i, i−1)∈g..

Example 2.1. LetQ be the quiver onZ/6Zwhose set of arrows is {(j, j+ 1)}⁶_j=1. The choice graph in Figure 4 represents the term

(x⁻¹₁ p⁺₁)(x⁻¹₂ p⁺₂)(x⁻¹₃ p⁻₃)(x⁻¹₄ p⁻₄)(x⁻¹₅ p⁻₅)(x⁻¹₆ p⁻₆) =x⁻¹₁ p⁺₁ p⁺₂p⁻₃ x3x2

p⁺₄ p⁻₅ x4

p⁻₆ x5

x⁻¹₆ .

By our construction ofCandM, we have that (2.1) may be written as (2.4)

k

Y

i=1

x⁻¹_v

i (p⁺_v

i+p⁻_v

i) =X

g∈C

M(g),

and so it is our goal to expand the sum on the right of (2.4). The following proposition gives us such an expansion.

Proposition 2.2.We may expand (2.4)as follows:

k

Y

i=1

x⁻¹_v_i (p⁺_v_i+p⁻_v_i) =

k

Y

i=1

p⁺_v_i xvi−1

+

k

Y

i=1

p⁻_v_i xv_i+1

+ X

∅6=S⊂{1,2,...,k}

S∩(S+1)=∅

(−1)^|S|+1 Y

i∈S

p⁺_v

i

xv_i+1

p⁻_v_i+1 xv_i

!

 Y

i /∈S∪(S+1)

x⁰_v

i

 (2.5) 

Proof. Note that a choice graph has a directed 2-cycle if and only if its associated term has a factor of the form (x⁻¹_v_i p⁺_v_i)(x⁻¹_v_i+1p⁻_v_i+1), which is a monomial ^p

+ vi

x_vi+1 p⁻_vi+1

x_vi

in the variablesxv₁, . . . , xvn sincexvi+1 dividesp⁺_v_i andxvi dividesp⁻_v_i+1. Also notice that no two 2-cycles may share vertices, since each vertex meets the tail of one and only one arrow. It follows that a choice graph may have at mostb^k₂c2-cycles. Finally notice that a 2-cycle may only exist on adjacent vertices. Now, let Sj denote the collection of subsetsS of{1, . . . , k}of sizej >1 such thatS∩(S+ 1) =∅, and let S=Sb^k₂c

j=1Sj. These subsetsS correspond to the “left endpoints” of the 2-cycles in choice graphs withj 2-cycles.

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LetC(S) be the set of all choice graphs with a 2-cycle on every pair {i, i+ 1} for i∈S ∈S, and letC₀ be the set of choice graphs with no 2-cycles, then we have

(2.6) C=C₀∪ S

S∈S₁

C(S)

!

∪ S

S∈S₂

C(S)

!

∪ · · · ∪



 S

S∈S_bn 2c

C(S)



.

Since everyg∈C(S) has a pair of arrows (i, i+ 1),(i+ 1, i) for everyi∈S, we have

(2.7) X

g∈C(S)

M(g) = Y

i∈S

p⁺_v_i xv_i+1

p⁻_v

i+1

xv_i

!

 Y

i /∈S∪(S+1)

x⁻¹_v_i p⁺_v_i+p⁻_v_i



.

We also have

X

g∈C0

M(g) =

k

Y

i=1

p⁺_v_i xvi−1

+

k

Y

i=1

p⁻_v_i xv_i+1

,

since there are precisely two choice graphs with no 2-cycles, corresponding to a consis- tent choice of either + or−. We wish use (2.6) to write (2.4) as a sum with summands of the form (2.7). Such a sum must have precisely one term corresponding to each member of C. However, there is a certain amount of overcounting involved in (2.6), since for anyS∈S, we have

(2.8) C(S)⊆C(Sr{i}) for everyi∈S.

We therefore proceed iteratively by way of the inclusion-exclusion principle. We first include a summand corresponding toC(S) for everyS∈S1:

T₁= X

S∈S1

X

g∈C(S)

M(g).

As (2.8) shows, for everyS∈S2, the summandT1contains two terms corresponding to each element ofC(S). Therefore, we now exclude a summand for eachC(S),S ∈ S2:

T2=T1− X

S∈S₂

X

g∈C(S)

M(g).

Again, (2.8) shows us that, for everyS∈S₃, the summandT₂excludes one term too many for each element ofC(S). We therefore defineT₃ accordingly, and continue the process of inclusion and exclusion until we obtain

T_bk 2c =

b^k₂c

X

j=1



(−1)^j+1 X

S∈Sj

X

g∈C(S)

M(g)



.

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Finally, we must include a term ofP

g∈C0M(g) corresponding toC₀, and we conclude that

X

g∈C

M(g) =X

g∈C0

M(g) +T_bk 2c

k

Y

i=1

x⁻¹_v

i (p⁺_v

i+p⁻_v

i) =

k

Y

i=1

p⁺_v

i

xvi−1

+

k

Y

i=1

p⁻_v

i

xv_i+1

+

b^k₂c

X

j=1



(−1)^j+1 X

S∈Sj

Y

i∈S

p⁺_v

i

x_v_i+1 p⁻_v_i+1

x_v_i

!

 Y

i /∈S∪(S+1)

x⁰_v

i









=

k

Y

i=1

p⁺_v_i xv_i−1

+

k

Y

i=1

p⁻_v_i xv_i+1

+

X

∅6=S⊂{1,2,...,k}

S∩(S+1)=∅



(−1)^|S|+1 Y

i∈S

p⁺_v

i

xv_i+1

p⁻_v

i+1

xv_i

!

 Y

i /∈S∪(S+1)

x⁰_v

i







.

Proposition 1.2 follows, since we now have

k

Y

i=1

x⁻¹_v_i (p⁺_v_i+p⁻_v_i)− X

∅6=S⊂{1,2,...,k}

S∩(S+1)=∅



(−1)^|S|+1 Y

i∈S

p⁺_v_i xv_i+1

p⁻_v_i+1 xv_i

!

 Y

i /∈S∪(S+1)

x⁰_v_i







=

k

Y

i=1

p⁺_v_i xvi−1

+

k

Y

i=1

p⁻_v_i xv_i+1

X

S⊂{1,2,...,k}

S∩(S+1)=∅



(−1)^|S| Y

i∈S

p⁺_v_i xv_i+1

p⁻_v

i+1

xv_i

!

 Y

i /∈S∪(S+1)

x⁰_v_i







=

k

Y

i=1

p⁺_v

i

xvi−1

+

k

Y

i=1

p⁻_v

i

xv_i+1

.

2.2. Generators forK_Q. Now, in addition to thedefining polynomialsy_ix_i−(p⁺_i + p⁻_i ),yixi−1 given by the defining relations (1.6) and (1.7), we have by Proposition 2.2 that the ideal of relationsKQalso contains thecycle polynomials. We define the cycle polynomials in Z[x1, x2, . . . , xn, y1, y2, . . . , yn] to be those polynomials coming from vertex-minimal directed cycles whose images underπvanish by virtue of (2.5). That is, for every vertex-minimal directed cycle of unfrozen verticesv1→v2→ · · · →vk →v1

inQ, we have the cycle polynomial X

S⊂{1,2,...,k}

S∩(S+1)=∅

(−1)^|S| Y

i∈S

p⁺_v_i xv_i+1

p⁻_v

i+1

xv_i

!

 Y

i6∈S∪(S+1)

yv_i



−

k

Y

i=1

p⁺_v_i xvi−1

−

k

Y

i=1

p⁻_v_i xv_i+1

.

Note again that this expression reduces to a polynomial in the x- and y-variables because eachxv_i+1 dividesp⁺_v_i and eachxv_i dividesp⁻_v_i+1. In Table 1, we present the cycle polynomials given by some basic ice quivers.

We now obtain a presentation for the ideal of relations KQ and for the initial ideal in<KQ, where< is a monomial order in which they-variables are much more expensive than the x-variables. This presentation will suffice to prove Theorems 1.5 and 1.7. We first require the following standard lemma.

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Table 1. Cycle polynomials in several examples 1

2 3

y₁y₂y₃−y₁−y₂−y₃−2

1 2

3 4

y1y2y3y4−y1y2−y1y4−y2y3−y3y4

1 2 3 4

5 y₁y₂y₃y₄y₅−y₁y₂y₃−y₁y₂y₅− y₂y₃y₄−y₃y₄y₅+y₁+y₂+y₃+

y₄+y₅−2

2

1 4

3 5

y1y2y5−y1x3−y2−y5x4−x3−x4

and

y₂y₃y₄−y2−y3x₁−y4x₅−x1−x5

Lemma 2.3.Let J and L be ideals in a polynomial ring. Suppose that J ⊆ L and in<J = in<L. ThenJ =L.

Proof. LetG be a Gröbner basis forJ and let f ∈L. Since in<G generates in<L, dividingf byGgives a remainder of 0, and sof ∈J. Theorem2.4.Given an ice quiverQonnvertices, the defining polynomials together with the cycle polynomials form a Gröbner basis forKQ= kerπ, where

π:Z[x1, x2, . . . , xn, y1, y2, . . . , yn]−→Z[x^±1₁ , x^±1₂ , . . . , x^±1_n ]

∀i, π(x_i) =x_i, π(y_i) =x⁰_i.

Proof. LetJ be the ideal of Z[x₁, x₂, . . . , xn, y₁, y₂, . . . , yn] that is generated by the set Gof defining and cycle polynomials. We know thatJ ⊆KQ, and therefore that in<J ⊆in<KQ. LetM be the monomial ideal generated by the initial terms of the polynomials inG. We know thatM ⊆in<J ⊆in<KQ, so we would like to show that in<KQ⊆M.

Assume (for the purpose of contradiction) that there is some f ∈ KQ such that in<(f)6∈M. We may write (assume alla,b andλnon-zero for simplicity)

(2.9) in_<(f) =λx^a_iⁱ¹

1 x^a_iⁱ²

2 · · ·x^a_i^ik

k y^b_j^j¹

1 y^b_j^j²

2 · · ·y^b_j^j`

`

Note that{j₁, j₂, . . . , j`}cannot contain the indices of a directed cycle of unfrozen vertices. Otherwise, it would also contain the indices of a vertex-minimal directed cycle, and in<(f) would be a multiple of the initial term of a cycle polynomial, contradicting the assumption that in<(f)6∈M.

Let Y ⊂ [n] be the indices of unfrozen vertices which are not in {j1, j2, . . . , j`}, and letQ⁰ be the ice quiver obtained by freezing the vertices in Qindexed byY. By

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the preceding observation,Q⁰ is an acyclic quiver. There is a natural inclusion L(Q),→ L(Q⁰)

induced by inclusions into Z[x^±1₁ , . . . , x^±1_n ]. This inclusion may be lifted to a ring homomorphism

µ:Z[x1, . . . , xn, y1, . . . , yn]→Z[x1, . . . , xn, y1, . . . , yn] µ(xi) =xi, µ(yi) =

yi(p⁺_i +p⁻_i ) ifi∈Y yi otherwise

with the property thatπ⁰◦µ=π, whereπ⁰ is the map

Z[x₁, x₂, . . . , x_n, y₁, y₂, . . . , y_n]−→Z[x^±1₁ , x^±1₂ , . . . , x^±1_n ] defined byQ⁰ instead ofQ.

Each of the variables appearing in the initial term off are fixed by µ. In lower- order terms off, µmay introduce monomials inx; however, this will never create a term greater than in<(f). Hence,

in<(f) = in<(µ(f))∈in<(KQ⁰)

SinceQ⁰ is acyclic, it was shown in [2, Corollary 1.17] that in_<(K_Q⁰) is generated by {x_iy_i | i ∈ [n]}. Hence, in_<(f) is a multiple of x_iy_i for some i. However, this implies that in_<(f) is a multiple of the initial term of theith defining polynomial in KQ, contradicting the assumption that in<(f)6∈M.

It follows that in<(KQ)⊂M. This consequently implies that in<(J) = in<(KQ) and, by the preceding lemma, that J = KQ. Furthermore, since in<(G) generates in<(KQ), the setGis a Gröbner basis forKQ.

3. Simplicial Complexes and Cohen–Macaulayness

Now that we have obtained a generating set for in<KQ, we can explicitly construct the Stanley–Reisner complex of in<KQ. We first consider a larger class of simplicial complexes, defined as follows:

Definition 3.1.Let S = {1, . . . , n}, let C be a collection {C1, . . . , Ck} of subsets of S, and let Y ⊆ S. Define the simplicial complex ∆(S,C, Y) on the set {xi

i ∈ S} ∪ {yi

i∈Y}by the rule⁽⁷⁾

F ∈∆(S,C, Y) ⇐⇒ ∀i{xi, yi} 6⊆F and∀j{yi}i∈C_j 6⊆F.

Since every facet of ∆(S,C, Y) is of the form {z1, . . . , z_n}, where z_i is either x_i or y_i, we see that ∆(S,C, Y) is always a pure simplicial complex. Note that for any quiverQon vertex setS, whereC is the collection of sets of vertices of vertex-minimal directed cycles onQ, we have by Theorem 2.4 that

(3.1) in<KQ=

*

x1y1, . . . , xnyn, Y

i∈C1

yi, . . . , Y

i∈Ck

yi

+

, Cj⊆S,

and the Stanley–Reisner complex of in<KQ is precisely ∆(S,C, S).

Remark 3.2. Whenever{i} ∈C, there is no vertex of the formyi in the simplicial complex ∆(S,C, Y). Such confusing notation is necessary for later induction. A vertex in ∆(S,C, Y) of the formyi will be called ay-vertex.

(7)If a subsetCjis not contained inY, we may simply ignore the condition{yi}i∈C_j 6⊆S, which is vacuously true becauseyiis not defined fori6∈Y.

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We now recall some definitions. Given a simplicial complex ∆ and a vertexv of ∆, thelinkofv is the set

link_∆(v) :={F∈∆

F 63v andF∪ {v} ∈∆}, and thedeletionofv is the set

del∆(v) :={F∈∆

F∪ {v}∈/ ∆},

where the bar denotes closure, so that del∆(v) is a simplicial complex. We call a vertex v of a simplicial complex ∆ ashedding vertex of ∆ if no face of link∆(v) is a facet of del∆(v). Finally, we recall the (recursive) notion of vertex-decomposability: a simplicial complex ∆ isvertex-decomposableif it is a simplex, or if it has a shedding vertex v such that both link∆(v) and del∆(v) are vertex-decomposable (see [3], also [4]).

It is our goal to prove the following theorem, from which Theorem 1.10 will follow.

Theorem 3.3.The complex ∆(S,C, Y) is always homeomorphic to a vertex-decom- posable (n−1)-ball, except when C = ∅ and Y = S, in which case ∆(S,C, Y) is homeomorphic to a vertex-decomposable(n−1)-sphere.

Note that the case whereC =∅andY =S is precisely the case in which{y_j}_j∈S is a face of ∆(S,C, Y). We first characterize the link and the deletion in ∆(S,C, Y) for any vertex of the formyi fori∈Y.

Proposition 3.4.Fory-vertexyi in∆(S,C, Y), we have link∆(S,C,Y)(yi) = ∆(Sⁱ,Cⁱ, Yⁱ), whereSⁱ:=Sr{i},Cⁱ:={C_j∩S_i

C_j ∈C}, andYⁱ:=Y ∩Sⁱ.

Proof. We first showLink := link∆(S,C,Y)(y_i)⊆∆(Sⁱ,Cⁱ, Yⁱ). SinceLink is a subcomplex of ∆(S,C, Y), no face ofLinkcontains{xj, y_j}for anyj, since ∆(S,C, Y) is defined so as never to contain any such face. Since no face ofLinkmay contain either x_i or y_i, we see that Link is a simplicial complex on {x_j

j ∈Sⁱ} ∪ {y_j

j ∈Yⁱ}.

Finally, were some F ∈ Link to contain {y_j}_j∈Ci

` for some `, then F ∪ {y_i} would contain {y_j}_j∈C_`, contradicting F ∪ {y_i} ∈ ∆(S,C, Y). We now have that Link ⊆

∆(Sⁱ,Cⁱ, Yⁱ).

We now show ∆(Sⁱ,Cⁱ, Yⁱ) ⊆ Link. Consider some F ∈ ∆(Sⁱ,Cⁱ, Yⁱ). Clearly F 63 y_i, so suppose that F∪ {y_i} ∈/ ∆(S,C, Y). Then either {x_i, y_i} ⊆F∪ {y_i} or {y_j}_j∈C_` ⊆F∪{y_i}for some`. The former case is impossible sincex_i∈/Sⁱ. The latter case implies {y_j}_j∈Ci

` ⊆ F, contradicting the definition of ∆(Sⁱ,Cⁱ, Yⁱ). Therefore we must haveF∪ {yi} ∈∆(S,C, Y) for every faceF of ∆(Sⁱ,Cⁱ, Yⁱ), from which it follows that ∆(Sⁱ,Cⁱ, Yⁱ)⊆Link. We conclude that ∆(Sⁱ,Cⁱ, Yⁱ) =Link.

Proposition 3.5.Fory-vertexy_i in∆(S,C, Y), we have del_∆(S,_C_,Y₎(yi) = ∆(S,C, Yⁱ), whereYⁱ is defined as above.

Proof. We first showDel:= del∆(S,C,Y)(y_i)⊆∆(S,C, Yⁱ). Since no face of Delmay containyi, we see that Delis a simplicial complex on{xj

j ∈S} ∪ {yj

j ∈Yⁱ}.

Since Delis a subcomplex of ∆(S,C, Y), no face of Delcontains either{xj, yj} for anyj or{yj}_j∈C_` for any`. ThereforeDel⊆∆(S,C, Yⁱ).

Since ∆(S,C, Yⁱ) has xi as a vertex but not yi, we have by the definition of

∆(S,C, Yⁱ) that every facet of ∆(S,C, Yⁱ) containsxi. Consider some arbitrary facet Fof ∆(S,C, Yⁱ). Sincexi ∈F, we cannot haveF∪{yi} ∈∆(S,C, Y), and soF ∈Del.

ThereforeDel⊆∆(S,C, Yⁱ), and so we conclude that Del= ∆(S,C, Yⁱ).