gale duality and koszul duality - pages.uoregon.edupages.uoregon.edu/njp/gk.pdf · there are two...

GALE DUALITY AND KOSZUL DUALITY

TOM BRADEN, ANTHONY LICATA, NICHOLAS PROUDFOOT, AND BEN WEBSTER

ABSTRACT. Given an affine hyperplane arrangement with some additional struc-ture, we define two finite-dimensional, noncommutative algebras, both of which aremotivated by the geometry of hypertoric varieties. We show that these algebras areKoszul dual to each other, and that the roles of the two algebras are reversed byGale duality. We also study the centers, deformations, and representation categoriesof our algebras, which are in many ways analogous to integral blocks of categoryO.

1. INTRODUCTION

In this paper, we define and study a class of finite dimensional graded algebraswhich are related to the combinatorics of hyperplane arrangements and to the ge-ometry of hypertoric varieties. To define these algebras, we take as our input whatwe call a polarized arrangement, consisting of a linear subspace V of a coordinatevector space RI , a vector η ∈ RI/V , and a covector ξ ∈ V ∗. Geometrically, it is con-venient to think of this data as describing an affine space (V + η ⊆ RI) along withan affine linear functional (given by ξ) and a finite hyperplane arrangement H (therestrictions of the coordinate hyperplanes in RI). We will usually denote a polarizedarrangement by the triple V = (V, η, ξ), with the understanding that the inclusion ofV into RI is part of the data.

If V is rational, meaning that V , η, and ξ are all defined over Q, then we mayassociate to V a symplectic complex algebraic variety called a hypertoric variety,equipped with a Hamiltonian C× action. The variety itself depends only on the ar-rangementH (that is, on V and η), and is denoted MH. It is defined as a hyperkählerquotient of the quaternionic vector space Hn by an (n − dimV )-dimensional torusdetermined by V , where the quotient parameter is specified by η. This hypertoricvariety carries a natural Hamiltonian action of an algebraic torus with Lie algebraV ∗C , and ξ determines a one-dimensional subtorus. The definitions and results of thispaper do not require any knowledge of hypertoric varieties (indeed, they will holdeven if V is not rational, in which case there are no varieties in the picture). Theywill, however, be strongly motivated by hypertoric geometry, and we will take every

1

2 T. BRADEN, A. LICATA, N. PROUDFOOT, AND B. WEBSTER

opportunity to point out this motivation. The authors plan to make this connectionmore precise in future work. The interested reader can learn more about hypertoricvarieties in the survey [Pro].

Given a polarized arrangement V , we define two quadratic graded algebras, whichwe denote by A = A(V) and B = B(V). The algebra A has a presentation in termsof the combinatorics and linear algebra of V ; if V is rational, its category of mod-ules may be identified with a certain category of sheaves on MH, supported on acollection of Lagrangian subvarieties determined by the C×-action. The algebra Bis defined using a convolution product on the direct sum of the cohomology groupsof the toric varieties associated to certain faces of H, which appear as pairwise in-tersections of the aforementioned Lagrangian subvarieties of MH. This definitioncan be extended to the non-rational case using the fact that the cohomology ring ofa toric variety has a purely combinatorial1 description in terms of its polytope.

There are two forms of duality lurking in this picture, one of which comes fromlinear programming and the other from ring theory. In linear programming, we candefine the Gale dual of a polarized arrangement V = (V, η, ξ) as the triple

V∨ = (V ⊥,−ξ,−η),

where V ⊥ ⊆ (RI)∗ is the space of linear forms on RI that vanish on V . On the otherhand, to any quadratic algebra E we may associate another quadratic algebra E!

which is known as its quadratic dual. We will prove that the algebras A and B aredual to each other in both of these senses:

Theorem (A). We have isomorphisms

A(V)!∼= //

∼=��

A(V∨)∼=

��B(V) ∼=

// B(V∨)!.

The module categories of the algebras A and B are similar in many ways to inte-gral blocks of various flavors of category O, which is a category of representationsof a semi-simple Lie algebra. In particular, we prove the following three facts, all of

1Here and elsewhere in the paper we use the term "combinatorial" loosely to refer to constructionsinvolving finite operations on linear algebra data.

GALE DUALITY AND KOSZUL DUALITY 3

which are analogous to results in representation theory related to category O andthe geometry of the Springer resolution [Spa76, Irv85, Bru08, SW].

Theorem (B).

(1) The algebras A and B are quasi-hereditary and Koszul (and thus are Koszul dual).(2) If V is rational, then the center of B is canonically isomorphic to the cohomology

ring of MH.(3) There is a canonical bijection between indecomposible projective-injectiveB-modules

and compact chambers of the hyperplane arrangement H; if V is rational, these arein bijection with the set of all irreducible projective Lagrangian subvarieties of MH.

Part (2) of Theorem (B) says that the centers of our algebras are isomorphic tothe cohomology rings of hypertoric varieties, which are entirely determined by thesubspace V ⊂ RI [Kon00, HS02, Pro]. This leads us to ask to what extent the algebrasthemselves depend on η and ξ. The answer turns out two be that two polarizedarrangements with the same underlying vector space are neither isomorphic norMorita equivalent, but they are derived Morita equivalent.

Theorem (C). The bounded derived category of graded modules overA(V) orB(V) dependsonly on the subspace V ⊂ RI .

The paper is structured as follows. In Section 2, we lay out the combinatorics andlinear algebra of polarized arrangements, introducing definitions and constructionsupon which we will rely throughout the paper. Section 3 is devoted to the algebraA, and contains a proof of the isomorphism on the top edge of the square in The-orem (A). Section 4 is about the algebra B; in it we complete the proof of Theorem(A), as well as part (2) of Theorem (B). Section 5 begins with a general overview ofquasi-hereditary and Koszul algebras, culminating in the proofs of parts (1) and (3)of Theorem (B). The purpose of Section 6 is to prove Theorem (C), and along theway we study Ringel duality, Serre functors, and mutations of exceptional collec-tions in the category of right A-modules. Finally, in Section 7, we consider naturaldeformations of the algebras A and B, which we show are both special cases of acompletely general construction that may be applied to any quadratic algebra. Thisconstruction is (as far as we know) new, and will be studied in greater detail in afuture paper.

Much of this work is motivated by our belief that Gale duality for hypertoricvarieties should be thought of as a special case of a more general phenomenon called


symplectic duality between the varieties MH and MH∨ [BLP, BLPW], where H∨ isthe hyperplane arrangement associated to V∨. Other examples of symplectic dualpairs include Springer resolutions for Langlands dual groups, and certain pairs ofmoduli spaces of instantons on surfaces. These examples all appear as the Higgsbranches of the moduli space of vacua for mirror dual 3-d N = 4 super-conformalfield theories, or as the Higgs and Coulomb branches of a single such theory. Forhypertoric varieties, this was shown by Strassler and Kapustin [KS99]. We anticipatethat our results on symplectic duality will ultimately be related to the structure ofthese field theories.

Acknowledgments. The authors would like to thank Jon Brundan, Michael Falk, Da-vide Gaiotto, Sergei Gukov, Christopher Phan, Catharina Stroppel, and Edward Wit-ten for invaluable conversations.

T.B. would like to thank the Institute for Advanced Studies of the Hebrew Univer-sity, Jerusalem, where some of the initial ideas of this paper germinated. A.L. wassupported in part by a Clay Liftoff Fellowship. N.P. was supported in part by NSFgrant DMS-0738335. B.W. was supported by a Clay Liftoff Fellowship and an NSFPostdoctoral Research Fellowship.

2. LINEAR PROGRAMMING

2.1. Polarized arrangements. Let I be a finite set.

Definition 2.1. A polarized arrangement indexed by I is a triple V = (V, η, ξ) con-sisting of

• a vector subspace V ⊂ RI ,• a vector η ∈ RI/V , and• a covector ξ ∈ V ∗ = (RI)∗/V ⊥,

such that

(a) every element of V + η has at least |I| − dimV nonzero entries, and(b) every element of V ⊥ + ξ has at least dimV nonzero entries.

If V , η, and ξ are all defined over Q, then V is called rational.

Associated to a (not necessarily rational) polarized arrangement V = (V, η, ξ) isan arrangement H of |I| hyperplanes in the affine space V + η ⊂ RI , where the ith


hyperplane is given by the equation

Hi = {x ∈ V + η | xi = 0}.

For any subset S ⊂ I , we letHS =

⋂i∈S

Hi

be the flat spanned by the set S. Condition (a) implies that H is simple, meaningthat codimHS = |S| whenever HS is nonempty. Observe that ξ may be regardedas an affine-linear functional on V + η; it does not have well-defined values, but itmay be used to compare any pair of points. Condition (b) implies that ξ is genericwith respect to the arrangement, in the sense that it is not constant on any positive-dimensional HS .

Note that we do not exclude the possibility that V is contained in some hyper-plane {xi = 0}, in which case the corresponding hyperplane Hi is empty. Such anindex i is referred to as a loop of V , since it is a loop in the associated matroid.

2.2. Boundedness and feasibility. Given a sign vector α ∈ {±1}I , let

∆α = (V + η) ∩ {x ∈ RI | α(i)xi ≥ 0 for all i}

andΣα = V ∩ {x ∈ RI | α(i)xi ≥ 0 for all i}.

If ∆α is nonempty, it is the closed chamber of H obtained by setting the definingequations of the hyperplanes to inequalities according to the signs in α. The cone Σαis always nonempty, as it contains 0. If ∆α is nonempty, then Σα is simply the coneof unbounded directions in ∆α. If not, then Σα may be understood as the cone ofunbounded directions in a chamber ∆′α associated to a different polarized arrange-ment (V, η′, ξ). Put differently, we translate the hyperplanes in some generic way sothat the polyhedron ∆α becomes nonempty, and then take its cone of unboundeddirections.

We now define subsets F ,B,P ⊂ {±1}I as follows. First we let

F = {α ∈ {±1}I | ∆α 6= ∅}.

Elements of F are called feasible. It is clear that F depends only on V and η. Next,we let

B = {α ∈ {±1}I | ξ(Σα) is bounded above}.


Elements of B are called bounded, and it is clear that B depends only on V and ξ.Elements of the intersection

P := F ∩ B = {α ∈ {±1}I | ξ(∆α) is nonempty and bounded above}

are called bounded feasible; here ξ(∆α) is regarded as a subset of the affine line.Our use of these terms comes from linear programming, where we consider α asrepresenting the linear program “find the maximum of ξ on the polyhedron ∆α”.

2.3. Gale duality. Here we introduce one of the two main dualities of this paper.

Definition 2.2. The Gale dual V∨ of a polarized arrangement V is given by the triple(V ⊥,−ξ,−η). We denote by F∨, B∨, and P∨ the feasible, bounded, and boundedfeasible sign vectors for V∨.

This definition agrees with the notion of duality in linear programming: the lin-ear programs for V and V∨ and a fixed sign vector α are dual to each other. Thefollowing key result is a form of the strong duality theorem of linear programming.

Theorem 2.3. F∨ = B, B∨ = F , and therefore P∨ = P .

Proof. It is enough to show that α = (+1, . . . ,+1) is feasible for V if and only if it isbounded for V∨. The Farkas lemma [Zie95, 1.8] says that exactly one of the followingstatements holds:

(1) there exists a lift of η to RI which lies in RI≥0,(2) there exists c ∈ V ⊥ ⊂ (RI)∗ which is positive on RI≥0 and negative on η.

Statement (1) is equivalent to α ∈ F , while a vector c satisfying (2) lies in Σ∨α, soc(η) < 0 means that −η is not bounded above on Σ∨α. �

2.4. Restriction and deletion. We define two operations which reduce the numberof hyperplanes in the system as follows. First, consider a subset S ⊂ I such thatV + RIrS = RI . Since η is assumed to be generic, this condition is equivalent tosaying that HS 6= ∅. Consider the natural isomorphism

i : RIrS/(V ∩ RIrS) −→ RI/V


induced by the inclusion of RIrS into RI . We define a new polarized arrangementVS = (V S, ξS, ηS), indexed by the set I r S, as follows:

V S := V ∩ RIrS ⊂ RIrS

ξS := ξ|V S

ηS := i−1(η).

The arrangement VS is called the restriction of V to S, since the associated hyper-plane arrangement is isomorphic to the hyperplane arrangement obtained by re-stricting to the subspace HS .

Dually, suppose that S ⊂ I is a subset such that RS ∩ V = {0}, and let

π : RI → RIrS

be the coordinate projection, which restricts to an isomorphism

π|V : V −→ π(V ).

We define the another polarized arrangement VS = (VS, ηS, ξS), also indexed byI r S, as follows:

VS = π(V ) ⊂ RIrS

ηS = π(η)

ξS = ξ ◦ π|−1V .

The arrangement VS is called the deletion of S from V , since the associated hy-perplane arrangement is obtained by removing the hyperplanes {Hi}i∈S from thearrangement associated to V . The following lemma is an easy consequence of thedefinitions.

Lemma 2.4. (VS)∨ is equal to (V∨)S .

2.5. The adjacency relation. Define a relation↔ on {±1}I by setting

α↔ β

if and only if α and β differ in exactly one place; if α, β ∈ F , this means that ∆αand ∆β are obtained from each other by flipping across a single hyperplane. Wewill write α i↔ β to indicate that α and β differ in the ith component of {±1}I . Thefollowing lemma says that an infeasible neighbor of a bounded feasible sign vectoris feasible for the Gale dual system.


Lemma 2.5. Suppose that α /∈ F , β ∈ P , and α↔ β. Then α ∈ F∨.

Proof. Suppose that α i↔ β. The fact that β ∈ P ⊆ F tells us that ∆β 6= ∅, while∆α = ∅, thus

Hi ∩∆β = ∆α ∩∆β = ∅.From this we can conclude that

Σα = Σβ ∩ {xi = 0} ⊂ Σβ.

The fact that β ∈ P ⊂ B, tells us that ξ(Σβ) is bounded above, thus so is ξ(Σα). Thisin turn tells us that α ∈ B, which is equal to F∨ by Theorem 2.3. �

2.6. Bases and the partial order. Let Bas be the set of subsets b ⊂ I of order dimVsuch that Hb 6= ∅. Such a subset is called a basis for the matroid associated to V ,and in fact this property depends only on the subspace V ⊂ RI . A set b is a basis ifand only if the composition V ↪→ RI � Rb is an isomorphism, which is equivalentto saying that we have a direct sum decomposition RI = V ⊕ Rbc , where we putbc = I r b. We have a bijection

µ : Bas→ Ptaking b to the unique sign vector α such that ξ attains its maximum on ∆α at thepoint Hb.

The covector ξ induces a partial order ≤ on Bas ∼= P . It is the transitive closure ofthe relation �, where b1 ≺ b2 if |b1 ∩ b2| = |b1| − 1 = dimV − 1 and ξ(Hb1) < ξ(Hb2).The first condition means that Hb1 and Hb2 lie on the same one-dimensional flat, soξ cannot take the same value on these two points.

Let Bas∨ denote the set of bases of V∨. We have a bijection b 7→ bc from Bas toBas∨, since RI = V ⊕ Rbc if and only if RI = V ⊥ ⊕ Rb. The next result says that thisbijection is compatible with the equality P = P∨ and the bijections µ : Bas→ P andµ∨ : Bas∨ → P∨.

Lemma 2.6. For all b ∈ Bas, µ(b) = µ∨(bc).

Proof. Let b ∈ Bas. It will be enough to show that µ(b) = α if and only if(1) the projection of α to {±1}bc is feasible for the restriction Vb, and(2) the projection of α to {±1}b is bounded for the deletion Vbc ,

since these conditions are clearly interchanged by swapping b and bc, and Theorem2.3 and Lemma 2.4 tell us that they are also interchanged by Gale duality.


Note that Vb represents the restriction of the arrangement to Hb, which is a point.All of the remaining hyperplanes are therefore loops, whose positive and negativesides are either all of Hb or empty. Condition (1) just says then that Hb lies in thechamber ∆α. Given that, in order for Hb to be the ξ-maximum on ∆α, it is enoughthat when all the hyperplanes not passing through Hb are removed from the ar-rangement, the chamber containing α is bounded. But this is exactly the statementof condition (2), which completes the proof. �

The following lemma demonstrates that this bijection is compatible with our par-tial order.

Lemma 2.7. Under the bijection b 7→ bc, the partial order on Bas∨ is the opposite of thepartial order on Bas. That is, b1 ≤ b2 if and only if bc1 ≥ bc2.

Proof. It will be enough to show that the generating relations ≺ are reversed. Thismeans checking that whenever µ(b1) ↔ µ(b2), we have ξ(Hb1) < ξ(Hb2) if and onlyif ξ∨(H∨bc1) > ξ

∨(H∨bc2). The fact that µ(b1)↔ µ(b2) implies that there exists i1 ∈ b1 andi2 ∈ b2 such that b2 = b1 ∪ {i2} r {i1}. We may then reduce the lemma to the (easy)case where |I| = 2 by replacing V with the polarized arrangement Vb1∩b2(b1∪b2)c obtainedby restricting to the 1-dimensional flat spanned by Hb1 and Hb2 and then deleting allhyperplanes but Hi1 and Hi2 . �

For b ∈ Bas, let

Bb = {α ∈ {±1}I | α(i) = µ(b)(i) for all i ∈ b}.

Note that Bb ⊂ B, and Bb depends only on V and ξ. Geometrically, the feasible signvectors in Bb are those such that ∆α lies in the “negative cone” defined by ξ withvertex Hb. Dually, we define

Fb = {α ∈ {±1}I | α(i) = µ(b)(i) for all i /∈ b} = B∨bc

to be the set of sign vectors such that Hb ∈ ∆α. In particular, Fb ⊂ F . We will needthe following lemma in Section 5.

Lemma 2.8. If α ∈ Bb and µ(a) = α, then a ≤ b.

Proof. Let a = µ−1(α), and let C be the negative cone of µ(b). Then α ∈ Bb means thatHa ∈ C. Let C ′ be the smallest face of C on which Ha lives. We will prove the lemmaby induction on d = dimC ′. If d = 0, then a = b and we are done. Otherwise, thereis a 1-dimensional flat which is contained in C ′ and passes through Ha. Following it


in the ξ-positive direction, it must leave C ′, since C ′ is ξ-bounded. The point whereit exits will be Hc for some basis c, and a < c by construction. But Hc lies on a faceof C of smaller dimension, so the inductive hypothesis gives c ≤ b. �

3. THE ALGEBRA A

3.1. Definition of A. Fix a polarized arrangement V , and consider the quiver Q =Q(V) with vertex set F and arrows {(α, β) | α↔ β}. Note in particular that there isan arrow from α to β if and only if there is an arrow from β to α. Let P (Q) be thealgebra of real linear combinations of paths in the quiver Q, generated by pairwiseorthogonal idempotents {eα | α ∈ F} along with edge paths {p(α, β) | α ↔ β}. Weuse the following notation: if α1 ↔ α2 ↔ . . . ↔ αk is a path in the quiver, then wewrite2

p(α1, α2, . . . , αk) := p(α1, α2) · p(α2, α3) · . . . · p(αk−1, αk).Let ti ∈ V ∗ be the restriction to V of the ith coordinate function on RI .

Definition 3.1. We define A = A(V) to be the quotient of P (Q) ⊗R SymV ∗ by thetwo-sided ideal generated by the following relations:

A1: If α ∈ F r P , then eα = 0.

A2: If four distinct elements α, β, γ, δ ∈ F satisfy α↔ β ↔ γ ↔ δ ↔ α, then

p(α, β, γ) = p(α, δ, γ).

A3: If α, β ∈ F and α i↔ β, then

p(α, β, α) = tieα.

We put a grading on this algebra by letting deg(eα) = 0 for all α ∈ F , deg(p(α, β)) = 1for all α↔ β, and deg(ti) = 2 for all i ∈ I ..

Remark 3.2. We have made the generating set ofA considerably larger than it needsto be in order to simplify the relations. First, we note that the relations of typeA3 allow us to express any element of SymV ∗ in terms of paths, thus A really is aquotient of the path algebra P (Q). This follows from the fact that for any α ∈ F , theset {ti | i ∈ Iα} generates V ∗, where

Iα := {i ∈ I | ∆α ∩Hi 6= ∅}.2Note that with this convention, a representation of Q is a right module over P (Q).


Second, the relations of type A1 tells us that we could have written A as a quotientof P (QP), where QP is the subquiver of Q consisting only of those vertices in P . Inthis case, however, the relations of types A2 and A3 would each have bifurcated intotwo cases, depending on whether or not β ∈ P .

It is important to note that if we write A as a quotient of P (QP), the relations aregenerated in degree 2. Thus A is what is called a quadratic algebra.

Remark 3.3. In a future paper we will show that if V is rational, the category ofrepresentations of A is equivalent to a category of sheaves on the hypertoric vari-ety MH. In the special case where MH is a cotangent bundle, these sheaves willbe microlocal D-modules whose characteristic varieties are contained in the conor-mal varieties to a certain stratification of the base. More generally, the sheaves willbe supported on the relative core of MH (see Remark 4.2), a finite collection of La-grangian subvarieties of MH defined by ξ.

Remark 3.4. The quiver Q has appeared in the literature before; it is the Cayleygraph of the Deligne groupoid of H [Par00]. The precise relationship between thealgebra A and the Deligne groupoid will be explained in Remark 7.2.

3.2. Taut paths. We next establish a series of results that allow us to understandthe elements of A more explicitly. Though we do not need these results for theremainder of Section 3, they will be used in Sections 4 and 5.

Definition 3.5. We say that a path α1 ↔ α2 ↔ . . .↔ αk inQ is taut if the sign vectorsα1 and αk differ in exactly k − 1 places; in other words, each place changes sign atmost once along the path.

Proposition 3.6. Let α1 ↔ α2 ↔ . . .↔ αk be a path in Q. Then there is a taut path

α1 = β1 ↔ β2 ↔ . . .↔ βd = αk

and a polynomial f(t) ∈ SymV ∗ such that

p(α1, . . . , αk) = p(β1, . . . , βd) · f(t)

in the algebra A.

Proof. We can represent paths in the quiver geometrically by paths in the affine spaceV + η in which H lives. Let φ : [0, 1] → V + η be a piecewise linear path with theproperty that for any t ∈ [0, 1], the point φ(t) lies in at most one hyperplane Hi,


and the endpoints φ(0) and φ(1) lie in no hyperplanes. Such a path determines anelement p(φ) = p(γ1, . . . , γr) in the algebra A, where ∆γ1 , . . . ,∆γr are the successivechambers visited by φ.

We can represent our given element p(α1, . . . , αk) in this way by choosing pointsx1, . . . , xk in each chamber and letting φ be the concatenation of the line segmentsxjxj+1. By choosing the xj generically, we can also assume that (1) the line segmentx1xj only passes through one hyperplane at a time for 2 ≤ j ≤ k, and (2) for all1 < j < k, the plane containing x1, xj , and xj+1 contains no point which lies in morethan two hyperplanes.

We then contract each point xj successively to the basepoint x1 along a straightline segment. The sequence of chambers visited by the path can change in twopossible ways. First, when the line segment xjxj+1 passes through the intersectionof two hyperplanes, a sequence · · ·α ↔ β ↔ γ · · · , α 6= γ is replaced by anothersequence · · ·α ↔ δ ↔ γ · · · . The quiver relation A2 implies that the element inthe algebra A represented by the path does not change. Second, when the point xjpasses through a hyperplaneHi, either the sequence of chambers visited by the pathis left unchanged, or a loop · · ·α ↔ β ↔ α · · · is replaced by · · ·α · · · . In this case,the old path is equal to the new path times ti, by relation A3 in the definition of thealgebra A.

The final result is the product of a polynomial and a straight-line path from x1to xk, which evidently represents a taut path, since it cannot cross any hyperplanemore than once. �

Corollary 3.7. Letα = α1 ↔ α2 ↔ . . .↔ αd = β

andα = β1 ↔ β2 ↔ . . .↔ βd = β

be two taut paths between fixed elements α, β ∈ P . Then

p(α1, . . . , αk) = p(β1, . . . , βd).

Proof. The proof of Proposition 3.6 tells us that any taut path is equivalent to onewhich is represented by a single line segment joining a point of ∆α to a point of ∆β .So it is enough to show that two such line segments give rise to the same element ofA. This, too, follows from the proof of the proposition, as we can add line segments


which stay within ∆α and ∆β to the beginning and end of one of these paths withoutchanging the sequence of visited chambers. �

Corollary 3.8. Consider an element

a = p ·∏i∈I

tdii ∈ eαAeβ,

where p is a taut path from α to β. Let S ⊂ I be the union

S = {i ∈ I | p crosses Hi} ∪ {i ∈ I | di > 0}.

Then for any γ ∈ F which agrees with α at all positions in I r S, there is an expression fora which passes through γ. In particular, if γ ∈ F r P , then a = 0.

Proof. First suppose that di = 0 for all i. In this case, we can define a taut path (inF) which goes through γ by composing taut paths from α to γ and from γ to β. Theresult then follows from Corollary 3.7.

Using this special case and the centrality of the elements ti ∈ A, we can reduce tothe case in which di > 0 for every i such that Hi separates ∆β from ∆γ . In this case,we observe that eβ ·

∏i∈I t

dii can be represented by a taut path from β to γ followed

by the reverse path, which completes the proof. �

3.3. Quadratic duality. We conclude this section by establishing the first part ofTheorem (A) from the Introduction, namely that the algebras A(V) and A(V∨) arequadratic duals of each other. First a quick review of quadratic duality.

Let R = R{eα | α ∈ I} be a ring spanned by finitely many pairwise orthogonalidempotents, and let M be an R-bimodule. Let TR(M) be the tensor algebra of Mover R, and let W be an R-bimodule equipped with an inclusion3

ι : W↪→M ⊗R M = TR(M)2.

LetE = TR(M)

/TR(M) · ι(W ) · TR(M)

be the associated quadratic algebra. We will write Wαβ = eαWeβ to denote the spaceof relations between nodes α and β.

The quadratic dual E! of E is defined as the quotient

E! = TR(M∗)/TR(M

∗) · ι(W )⊥ · TR(M∗),

3In Section 7 we will need to make a distinction between W and its image in M⊗RM . For consistencywe will make the distinction here, as well, even though it is somewhat unnecessary in this context.


where M∗ is the vector space dual of M , and

ι(W )⊥ ⊂M∗ ⊗R M∗ ∼=(M ⊗R M

)∗is the space of elements that vanish on ι(W ). Note that dualizing M interchangesthe left and right R-actions, so that (Mαβ)∗ = (M∗)βα. It is clear from this definitionthat this is a true duality, meaning that there is a natural isomorphism E!! ∼= E.

In the case of the algebra A(V), we have I = P (as explained in Remark 3.2) andM = R{ p(α, β) | α ↔ β}. Relations of type A2 from Definition 3.1 come fromWαγ , while relations of type A3 come from Wαα. Since P∨ = P , the algebras A(V)and A(V∨) have the same R and the same M . Thus, in order to make sense of thestatement that they are quadratic duals, we must define a perfect pairing

M ⊗M → R

to identify M with its dual. The obvious way to do this would be to make the edgeset into an orthonormal basis, but this does not quite give us what we need; instead,we need to twist this pairing by a sign.

Choose a subset X of edges in the underlying undirected graph of Q with theproperty that for any square α ↔ β ↔ γ ↔ δ ↔ α of distinct elements, an oddnumber of the edges of the square are in X . The existence of such anX follows fromthe fact that our graph is a subgraph of the edge graph of an n-cube. Then define apairing 〈 , 〉 by putting

〈 p(α, β), p(β, α) 〉 =

−1 if {α, β} ∈ X1 if {α, β} /∈ Xand

〈 p(α, β), p(δ, γ) 〉 = 0unless α = γ and β = δ.

Remark 3.9. Note that in the pairing that we have defined, paths from a node α toanother node β pair nontrivially only with paths from β to α; this is consistent withour general treatment of quadratic duality above. We warn the reader that this willbecome a little bit confusing in the proof of Theorem 3.10 due to the fact that A(V)admits an anti-automorphism reversing the directions of paths. In other words, thepaths from α and β and the relations between them look exactly like paths from βto α and the relations between them.


Theorem 3.10. With respect to the above pairing, A(V∨) ∼= A(V)!.

Proof. We will analyze M ⊗R M piece by piece by considering every pair α, γ ∈ Pwhich admit paths of length two connecting them. If α 6= γ, then they must differin two places, and there are exactly two elements δ1, δ2 in {±1}I so that α↔ δj ↔ γ,j = 1, 2. Since we are assuming that there is a path from α to γ inP , at least one of theδj must be in P . If both δ1 and δ2 are in P , then eαM ⊗R Meγ is the two-dimensionalvector space spanned by the paths p(α, δ1, γ) and p(α, δ2, γ), while eγM⊗RMeα is thetwo-dimensional vector space spanned by the paths p(γ, δ1, α) and p(γ, δ2, α). Thealgebra A(V) has the unique relation p(α, δ1, γ) − p(α, δ2, γ) between α and γ, andthe algebra A(V∨) has the unique relation p(γ, δ1, α) − p(γ, δ2, α) between γ and α,thus the orthogonality property holds.

If δ1 ∈ P and δ2 /∈ P , then by Lemma 2.5 we must either have δ1 ∈ F r P andδ2 /∈ F∨ or vice-versa. In the first case, Wαγ = {0} and W∨γα = (eγM ⊗R Meα)2 byrelation A3, and thus are mutual annihilators. In the second, these are reversed.

We have now dealt with the case where α 6= γ, and must consider the case whereα = γ. The length two paths from α to itself can be indexed by the set

Jα := {i ∈ I | αi ∈ P},

where αi denotes the element of {±1}I which differs from α in precisely the positioni. Note that Jα is a subset of the set Iα that we defined in Remark 3.2, consistingexactly of those i for which ξ is bounded above on the chamber opposite ∆α acrossHi. Let I∨α and J∨α be the corresponding sets for V∨, and note that J∨α = Jα.

The space ι(Wαα) of relations among the loops at α can therefore be considered asa vector subspace of RJα , which can be computed as follows. The relations amongthe elements {ti}i∈I of Sym(V ∗) are given by V ⊥ ⊂ (RI)∗ ∼= RI . The relations ι(Wαα)are given by first intersecting V ⊥ with RIα and then projecting (isomorphically) ontoRJα . Alternatively, we can first project V ⊥ onto RIcα∪Jα and then intersect with RJα .

On the dual side, the space of relations among loops at α is a vector subspace ofRJ∨α = RJα , and the pairing is the standard one without any signs. Here we obtainour relations by first intersecting V = V ⊥⊥ with RI∨α and then projecting onto RJ∨α .The fact that we obtain the orthogonal complement of the relations ι(Wαα) followsfrom the fact that Iα r Jα = I r I∨α , which is a consequence of Lemma 2.5. �

4. THE ALGEBRA B


4.1. Toric varieties. Let V = (V, η, ξ) be a rational polarized arrangement. Then Vinherits an integer lattice VZ := V ∩ ZI , and the dual vector space V ∗ inherits a duallattice which is a quotient of (ZI)∗ ⊂ (RI)∗. Let

T I = (RI)∗/(ZI)∗ and T = V ∗/V ∗Z ,

so that we have a surjective homomorphism

T I � T.

Then for every α ∈ F , the polyhedron ∆α determines a toric variety with an actionof the complexification TC. The construction that we give below is originally due toCox [Cox97].

Let T IC be the complexification of the coordinate torus TI , and let CIα be the repre-

sentation of T IC in which the ith coordinate of T IC acts on the i

th coordinate of CIα withweight α(i) ∈ {±1}. Let

G = ker(T IC � TC),

and letZα = CIα r

⋃S⊂I

such thatHS∩∆α=∅

{z ∈ CI | zi = 0 for all i ∈ S},

which is acted upon by T IC and therefore by G. The toric variety Xα associated to ∆αis defined as the quotient

Xα = Zα/G,

and inherits an action of TC = T IC/G. The action of the subtorus T ⊂ TC is hamil-tonian with respect to a natural symplectic structure on X , and ∆α is the momentpolyhedron. If ∆α is compact4, then Xα is projective. More generally, Xα is projec-tive over the affine toric variety whose coordinate ring is the semi-group ring of thesemi-group Σα ∩ VZ.

Since the polyhedron ∆α is simple by our assumption on η, the toric variety Xαhas at worst finite quotient singularities. This implies that the ordinary and T -equivariant cohomology rings of Xα may be presented purely in terms of the com-binatorics and linear algebra of the polyhedron ∆α. Specifically, we have a natural

4We use the word "compact" rather than "bounded" to avoid confusion with the fact that α is calledbounded if ξ is bounded above on ∆α.


isomorphism5

(1) H∗T (Xα) ∼= R[ui]i∈I/〈uS | S ⊂ I such that HS ∩∆α = ∅〉 ,

where uS =∏

i∈S ui. Here the element ui represents α(i) times the class of the toricdivisor corresponding to the (possibly empty) facet Hi ∩ ∆α of ∆α. In particular, itlives in degree two. This isomorphism identifiesH∗T (X) not just as an R-algebra, butas an algebra over the polynomial ring SymV = H∗T (pt), where SymV acts on theright-hand side via the inclusion of V into RI = R{ui | i ∈ I}. We then obtain

(2) H∗(Xα) ∼= H∗T (Xα)⊗Sym V R

by killing the positive degree elements of SymV .

Remark 4.1. Note that we can use the above presentations to make sense of therings H∗T (Xα) and H

∗(Xα) even when V is not rational, despite the fact that there isno toric variety Xα in this situation.

4.2. The relative core. Again suppose that V is rational, and let

X̃ =∐α∈P

Xα

be the disjoint union of the toric varieties associated to the bounded feasible signvectors, that is, to the chambers of H on which ξ is bounded above. We will de-fine a quotient X of X̃ which, informally, is obtained by gluing the components of X̃together along the toric subvarieties corresponding to the faces at which the corre-sponding polyhedra intersect.

More precisely, let CI be the standard coordinate representation of T IC, and let

HI = CI × (CI)∗

be the product of CI with its dual. For every α ∈ {±1}I , CIα can be found in a uniqueway as a subrepresentation of HI . Then

X :=

( ⋃α∈P

Zα

)/G,

where the union is taken inside of HI . Then TC acts on X, and we have a TC-equivariant projection

π : X̃→ X.5All cohomology groups in this paper will be taken with coefficients in R.


The restriction to this map to each Xα is obviously an embedding. As a result, anyface of a polyhedron ∆α corresponds to a T -invariant subvariety which is itself atoric variety for a subtorus of TC.

Remark 4.2. The singular variety X sits naturally as a closed subvariety of the hy-pertoric variety MH, which is defined as an algebraic symplectic quotient of HI byG (or, equivalently, as a hyperkähler quotient of HI by the compact form of G). See[Pro] for more details. The subvariety X is Lagrangian, and is closely related to twoother Lagrangian subvarieties of MH which have appeared before in the literature.The projective components of X (meaning the components whose correspondingpolyhedra are compact) form a complete list of irreducible projective Lagrangiansubvarieties of MH, and their union is called the core of MH. On the other hand, ifwe had defined X as a union over all α ∈ F rather than restricting to those α ∈ P ,we would have obtained a larger subvariety of MH called the extended core, whichwe will denote Xη. The subvariety Xη can also be described as the zero level of themoment map for the Hamiltonian TC-action on MH.

Our variety X, which sits in between the core and the extended core, will be re-ferred to as the relative core of MH with respect to the C×-action defined by ξ. Itmay be characterized as the set of points x ∈MH such that limλ→∞ λ · x exists.

Example 4.3. Suppose that H consists of n points in a line. Then MH is isomorphicto the minimal resolution of C2/Zn, and its core is equal to the exceptional fiberof this resolution, which is a chain of n − 1 projective lines. The extended core islarger; it includes two affine lines attached to the projective lines at either end of thechain. The relative core lies half-way in between, containing exactly one of the twoaffine lines. This reflects the fact that ξ is bounded above on exactly one of the twounbounded chambers ofH.

The core, relative core, and extended core of MH are all T -equivariant deforma-tion retracts of MH, which allows us to give a combinatorial description of theirordinary and equivariant cohomology rings [Pro, 3.2.2].

Theorem 4.4. We have natural isomorphisms

H∗T (X)∼= H∗T (MH) ∼= R[ui]i∈I

/〈uS | S ⊂ I such that HS = ∅〉

andH∗(X) ∼= H∗(MH) ∼= H∗T (MH)⊗Sym V R.


Remark 4.5. As in Remark 4.1, we can use Theorem 4.4 to make sense of the ringsH∗T (X) and H

∗(X) even if V is not rational, in which case the variety X does not itselfexist.

4.3. Definition of B. For (α, β) ∈ P × P and (α, β, γ) ∈ P × P × P , let

Xαβ = Xα ∩Xβ and Xαβγ = Xα ∩Xβ ∩Xγ,

where the intersections are taken inside of X. Then

X̃×π X̃ ∼=∐

(α,β)∈P×P

Xαβ

andX̃×π X̃×π X̃ ∼=

∐(α,β,γ)∈P×P×P

Xαβγ.

Letp12, p13, p23 : X̃×π X̃×π X̃ −→ X̃×π X̃

denote the natural projections. Note that these maps are proper; they are finitedisjoint unions of closed immersions of toric subvarieties.

Definition 4.6. Let dαβ denote the complex codimension of Xαβ in X (equivalentlythe real codimension of ∆α ∩∆β in V ), and consider the graded vector space

B = B(V) =⊕

(α,β)∈P×P

H∗(Xαβ)[−dαβ].

If we ignore the grading, then B may be identified with the cohomology ring ofX̃×π X̃. We may therefore define a multiplication ? : B ⊗B → B by the formula

a ? b = p13∗(p∗12(a) ∪ p∗13(b)

).

Proposition 4.7. B is a graded ring with respect to the above convolution product.

Proof. Suppose that a ∈ Hk(Xαβ)[−dαβ] and b ∈ H`(Xβγ)[−dβγ], so that

deg(a) = k + dαβ and deg(b) = `+ dβγ.

Let dαβγ denote the complex codimension of Xαβγ in X (equivalently the real codi-mension of ∆α ∩∆β ∩∆γ in V ). The push-forward p13∗ raises cohomological degreeby twice the complex codimension of Xαβγ in Xαγ , that is, by 2(dαβγ − dαγ). Thus

deg(a ? b) = k + `+ 2(dαβγ − dαγ) + dαγ = k + `+ 2dαβγ − dαγ.


It is an easy exercise in linear algebra to check that

2dαβγ = dαβ + dβγ + dαγ,

thereforedeg(a ? b) = k + `+ dαβ + dβγ = deg(a) + deg(b).

�

Remark 4.8. As in Remarks 4.1 and 4.5, we can make sense of the definition ofB regardless of whether or not V is rational. Indeed, the cohomology of Xαβ isexpressed combinatorially as

(3) H∗(Xαβ) ∼= R[ui]i∈I/〈uS | S ⊂ I such that HS ∩∆α ∩∆β = ∅〉 ⊗Sym V R.

Denote this ring by Rαβ . The cohomology ring Rαβγ of the triple intersections has asimilar expression. Then the pullback map Rαβ → Rαβγ is induced by the identityon R[ui]i∈I , while the pushforward Rαβγ → Rαγ is induced by multiplication by uS ,where

S = {i ∈ I | α(i) = γ(i) 6= β(i)}.This allows us to put a ring structure on

⊕(α,β)Rαβ[−dαβ]. Although we will use

geometric arguments from toric geometry in a few places, they can all be reformu-lated purely combinatorially, and all of our results about the ring B hold for arbi-trary arrangements unless otherwise noted (as in Theorem 4.11).

Remark 4.9. Suppose that V is rational. As we have observed in Remark 4.2, eachcomponent Xα of X can be thought of as an irreducible Lagrangian subvariety ofthe hypertoric variety MH, which allows us to interpret the cohomology groups oftheir intersections as Floer cohomology groups. With this identification, B can bethought of an Ext-algebra in the Fukaya category of MH. This description should berelated to the description in Remark 3.3 by taking homomorphisms to the canonicalcoisotropic brane, as described by Kapustin and Witten [KW07].

4.4. A and B. We now state and prove the main theorem of this section, which,along with Theorem 3.10, comprises Theorem (A) from the Introduction.

Theorem 4.10. There is a natural isomorphism A(V∨) ∼= B(V).

Proof. We define a map φ : A(V∨)→ B(V) by• sending the idempotent eα to the unit element 1αα ∈ H0(Xαα) for all α ∈ P ,


• sending p(α, β) to the unit 1αβ ∈ H0(Xαβ) for all α, β ∈ P with α↔ β, and• sending ti ∈ (V ⊥)∗ ∼= RI/V to the sum of its images in each H2(Xαα), which

by Equations (1) and (2) is a quotient of RI/V .

To show that this is a homomorphism, we need to check that these elements satisfythe relations A2 and A3 from Definition 3.1 (for V∨). In order to check that relationA2 holds, suppose that α, β, γ, δ ∈ F∨ satisfy α↔ β ↔ γ ↔ δ ↔ α, and α, γ ∈ P∨ =P . There are two possibilities: first, if β and δ also lie in P∨, then

1αβ ? 1βγ = 1αγ = 1αδ ? 1δγ,

so relation A2 is satisfied in B. The other possibility is that only one of β and δ liesin P∨, and the other is in F∨ r P∨ = B r P . Suppose β ∈ P and δ ∈ B r P . Theneδ = 0 in A(V∨), hence the relation A2 tells us that

p(α, β)p(β, γ) = p(α, δ)p(δ, γ) = 0.

On the other hand, the fact that δ ∈ BrP implies that Xαγ = ∅, hence 1α,β ? 1β,γ = 0in B(V).

Now suppose that α ∈ P∨, β ∈ F∨, and α i↔ β. Relation A3 breaks into twocases, depending on whether or not β ∈ P∨. If β ∈ P∨, then we have the relationp(α, β, α) = tieα in A(V∨). The left-hand side p(α, β, α) maps to 1αβ ? 1βα ∈ B(V),which corresponds to the class [Xαβ] ∈ H2(Xαα). This is the same class to which timaps via the isomorphism of Equation 2, thus the relation holds in B(V). On theother hand, if β ∈ F∨ r P∨ = B r P , then we have the relation tieα = 0 in A(V∨). Inthis case Hi ∩∆α = ∅, so φ(tieα) = 0 in B(V), as well.

The fact that B(V) is generated by the classes 1αβ for (α, β) ∈ P × P followsfrom Equations 1 and 2 and the fact that for all α, H2(Xαα) is spanned by the set{ui | i ∈ Iα}. Thus we have defined a surjective graded ring homomorphism

φ : A(V∨)→ B(V).

To show that φ is injective, we show that each block eαA(V∨)eβ has dimension nolarger than the total dimension of the ringH∗(Xαβ) (or the ringRαβ from Remark 4.8in the non-rational case). By Proposition 3.6 and Corollary 3.7, we have a surjectivemap

χ : R[ui]i∈I ⊗Sym V R = Sym RI/V = Sym(V ⊥)∗ → eαA(V∨)eβgiven by χ(f) = p ·f , where p is any taut path from α to β. It will be enough to showthat the monomials uS of Equation (3) are in the kernel of χ.


The condition HS ∩ ∆α ∩ ∆β = ∅ can be rephrased as HS′ ∩ ∆α = ∅, whereS ′ = S ∪ {i ∈ I | αi 6= βi}. This is equivalent to saying that the projection ᾱ ofα to {±1}IrS′ gives an infeasible sign vector for VS′ . By Theorem 2.3 and Lemma2.4, this is equivalent to saying that ᾱ is unbounded for the Gale dual arrangement(V∨)S′ (note that ᾱ cannot be infeasible for (V∨)S′ , since ∆∨ᾱ ⊃ ∆∨α and α ∈ P = P∨).The vanishing of χ(uS) in A(V∨) then follows from Corollary 3.8. �

4.5. The center. We now state and prove part (2) of Theorem (B) from the Introduc-tion, which gives a cohomological interpretation of the center of B.

Theorem 4.11. If V is rational, we have natural graded isomorphisms

Z(B) ∼= H∗(X) ∼= H∗(MH).

Proof. Let z ∈ Z(B). Since z commutes with the idempotents 1αα for all α ∈ P , wemust have z =

∑α∈P zα, where zα ∈ H∗(Xαα) = H∗(Xα). Also, since z commutes

with every element 1αβ , we see that zα and zβ will restrict to the same element ofH∗(Xαβ) for all α, β ∈ P . On the other hand, since the elements 1αβ generate B as aring, these two conditions completely characterize Z(B). That is, we have

(4) Z(B) ∼=

{z ∈

⊕α∈P

H∗(Xα)∣∣∣ for all α, β ∈ P , (zα − zβ)|Xαβ = 0

}.

Thus we want to show that the pullback

π∗ : H∗(X)→ H∗(X̃) ∼=⊕α∈P

H∗(Xα)

is injective, with image equal to the right hand side of Equation (4).Fix a total ordering6 α1, . . . , αr of P which is a refinement of the partial order

described in Section 2.6. For 1 ≤ j, k ≤ r let

Xk = Xαk ,

Xjk = Xαjαk ,

Ck = Xk r⋃j


and

X≤k :=k⋃

i=1

Ci =k⋃

i=1

Xi ⊂ X.

Then X≤k is the set of all points x ∈ X for which the limit limλ→∞ λ · x under the C×

action defined by ξ is a T -fixed point corresponding to a 0-flat Hb where b ∈ Bas andµ(b) ≤ k. This implies that Ci is a rational homology cell for all i. We will prove byinduction on k that the chain complex

(5) H∗(X≤k)→⊕

1≤i≤k

H∗(Xi)→⊕

1≤i j. The case k = 1 is trivial, since X≤1 = X1. Thecase k = r gives our theorem.

Let C0i , X0≤k, X0k , and X

0jk denote the intersections of Ci, X≤k, Xk, and Xjk with the

core of MH (see Remark 4.2), that is with the union of all of the compact Xi. Thenthe inclusions

C0i ↪→ Ci, X0≤k ↪→ X≤k, X0k ↪→ Xk, and X0jk ↪→ Xjk

are all homotopy equivalences, and the C0i decompose X0≤k into cells. It follows thatthe long exact sequence of the pair (X≤k,X≤k−1) splits into short exact sequences

0→ H∗(X≤k,X≤k−1)→ H∗(X≤k)→ H∗(X≤k−1)→ 0.

This then implies that the chain complex (5) surjects onto the same complex with kreplaced by k − 1, with kernel the complex

H∗(X0≤k,X0≤k−1)→ H∗(X0k)→

⊕1≤j


variety X, but we believe that it would be possible to give a purely combinatorialproof.

5. THE REPRESENTATION CATEGORY

We begin with a general discussion of highest weight categories, quasi-hereditaryalgebras, self-dual projectives, and Koszul algebras. With the background in place,we analyze our algebras A(V) and B(V) in light of these definitions.

5.1. Highest weight categories. Let C be an abelian, Artinian category enrichedover R with simple objects {Lα | α ∈ I}, projective covers {Pα | α ∈ I}, and in-jective hulls {Iα | α ∈ I}. Let ≤ be a partial order on the index set I.

Definition 5.1. We call C highest weight with respect to this partial order if there isa collection of objects {Vα | α ∈ I} and epimorphisms Pα

Πα→ Vαπα→ Lα such that for

each α ∈ I, the following conditions hold:(1) The object kerπα has a filtration such that each successive quotient is isomor-

phic to Lβ for some β < α.(2) The object ker Πα has a filtration such that each successive quotient is isomor-

phic to Vγ for some γ > α.

The objects Vα are called standard objects. Classic examples of highest weight cate-gories in representation theory include the various integral blocks of parabolic cate-gory O [FM99, 5.1].

Suppose that C is highest weight with respect to a given partial order on I. Tosimplify the discussion, we will assume that the endomorphism algebras of everysimple object in C is just the scalar ring R; this will hold for the categories we con-sider. For all α ∈ I, let C6>α be the subcategory of objects whose composition seriescontain no simple objects Lβ with β > α.

Proposition 5.2. The standard object Vα of Definition 5.1 is isomorphic to the projectivecover of Lα in the subcategory C6>α of C.

Proof. We must show that

Exti (Vα, Lβ) ∼=

R if α = β and i = 0,

0 if α = β and i > 0,

0 if α � β.


By definition, we have

Exti (Pα, Lβ) ∼=

R if α = β and i = 0,0 otherwise.If α 6< β, then ker Πα has a filtration by Vγ with γ > α 6< β, and therefore with γ � β.Thus Exti(ker Πα, Lβ) = 0 for all i. Then by the long exact sequence for Ext, we haveExti(Vα, Lβ) ∼= Exti(Pα, Lβ) whenever α 6< β. �

Dually, we define the costandard object Λα to be the the injective hull of Lα in thesame subcategory C6>α.

Definition 5.3. An object of C is called tilting if it admits a filtration by standardmodules and one by costandard modules. An equivalent condition is that T is tilt-ing if and only if Exti(T,Λα) = 0 = Exti(Vα, T ) for all i > 0 and α ∈ I. (The firstcondition is equivalent to the existence of a standard filtration, and the second to theexistence of a costandard filtration.) For each α ∈ I, there is a unique indecomposi-ble tilting module Tα with Vα as its largest standard submodule and Λα as its largestcostandard quotient [Rin91].

We now have six important sets of objects of C, all indexed by the set I:• the simples {Lα}• the indecomposable projectives {Pα}• The indecomposable injectives {Iα}• the standard objects {Vα}• the costandard objects {Λα}• the tilting objects {Tα}.

Each of these six sets forms a basis for the Grothendieck group K(C), and thus eachis a minimal set of generators of the bounded derived category Db(C). In particular,any exact functor from Db(C) to any other triangulated category is determined bythe images of these objects and the morphisms between them and their shifts.

Let {Mα | α ∈ I} and {Nα | α ∈ I} be two sets of objects that form bases for K(C).We say that the second set is left dual to the first set (and that the first set is rightdual to the second) if

Exti (Nα,Mβ) ∼=

R if α = β and i = 0,0 otherwise.


It is an easy exercise to check that if a dual set to {Mα} exists, then it is unique upto isomorphism. Note that dual sets descend to dual bases for K(C) under the Eulerform 〈

[M ], [N ]〉

:=∞∑i=0

(−1)i dim Exti(M,N).

Proposition 5.4. The sets {Pα} and {Iα} are left and right (respectively) dual to {Lα}, andthe set {Λα} is right dual to {Vα}.

Proof. The first statement follows immediately from the definition of projective cov-ers and injective hulls. For the second statement, let α, β ∈ I, and suppose first thatα 6< β. If α is maximal in I, then Vα = Pα and we get that Exti(Vα,Λβ) is R if i = 0and α = β and is 0 otherwise, as required. If α is not maximal, we can assume byinduction that the statement holds for all larger α. Apply the cohomological functorExt•(−,Λβ) to the short exact sequence

0→ Ker → Pα → Vα → 0;

since Ker has a filtration by standards Vα′ with α′ > α, the result follows by induc-tion on the resulting long exact sequence.

On the other hand, if α < β then we apply the dual argument using the exactsequence 0 → Λβ → Iβ → Coker → 0, where Coker has a filtration by costandardsΛβ′ with β′ > b. �

5.2. Quasi-hereditary algebras. We now study those algebras whose module cate-gories are highest weight.

Definition 5.5. An algebra is quasi-hereditary if its category C(E) of finitely gen-erated right modules is highest weight with respect to some partial ordering of itssimple modules.

Let E be a finite-dimensional, quasi-hereditary R-algebra with respect to a fixedpartial order on the indexing set I of its simple modules. Let

P∗ =⊕α∈I

Pα, I∗ =⊕α∈I

Iα, and T∗ =⊕α∈I

Tα

be the sums of the indecomposible projectives, injectives, and tilting modules, re-spectively. Let Db(E) = Db(C(E)) be the bounded derived category of finitely gen-erated right E-modules.


Definition 5.6. We say that E is basic if the simple module Lα is one dimensionalfor all α ∈ I. This is equivalent to requiring that the canonical homomorphism

E → End(P∗) ∼= End(I∗)op

is an isomorphism.

Definition 5.7. The endomorphism algebra R(E) := End(T∗) is called the Ringeldual of E. It has simple modules indexed by I, and it is quasi-hereditary withrespect to the partial order on I opposite to the given one. If E is basic, then thecanonical homomorphism E → R(R(E)) is an isomorphism [Rin91, Theorems 6,7].The functor R = RHom•(−, T∗) from Db(E) to Db(R(E)) is called the Ringel dualityfunctor.

Proposition 5.8. Suppose that E is basic. Up to automorphisms of E and R(E), R is theunique contravariant equivalence that satisfies any of the following conditions:

(1) R sends tilting modules to projective modules,(2) R sends projective modules to tilting modules,(3) R sends standard modules to standard modules.

Proof. We first prove statement (1). The Ringel duality functor is an equivalence be-cause Db(E) is generated by T∗. Since R(T∗) is equal to R(E) as a right module overitself, it is clear that R takes tilting modules to projective modules. Suppose that R′

is another such equivalence. Since the indecomposable tilting modules {Tα} gen-erate K(E) and R′ induces an isomorphism on Grothendieck groups, R′ must takethe tilting modules to the complete set of indecomposable projectiveR(E)-modules.Thus R′(T∗) is isomorphic to the direct sum of all indecomposable projective R(E)-modules, which is isomorphic to R(E) as a right module over itself. Since any exactfunctor is determined by its values on sends a generator and on the endomorphismsof that generator, R′ can only differ from R in its isomorphism between End(T∗) andR(E). This is precisely the uniqueness statement we have claimed for (1).

Statement (2) follows by applying statement (1) to the adjoint functor.As for Statement (3), it was shown in [Rin91, Theorem 6] that R takes standard

modules to standard modules. Suppose that R′ is another such equivalence. Forany α ∈ I, the projective module Pα has a standard filtration, therefore so doesR′(Pα). Furthermore, we have Exti(R′(Vβ),R′(Pα)) = Exti(Pα, Vβ) = 0 for all i > 0


and β ∈ I, thus R′(Pα) has a costandard filtration as well, and is therefore tilting.Then part (2) tells us that R′ is the Ringel duality functor. �

5.3. Self-dual projectives and the double centralizer property. Suppose that ouralgebraE is basic and quasi-hereditary, and that it is endowed with an anti-involutionψ, inducing an equivalence of categories C(E) ' C(Eop). We have another suchequivalence given by taking the dual of the underlying vector space, and these twoequivalences compose to a contravariant auto-involution d of C(E).

We will assume for simplicity that ψ fixes all idempotents of E. The case whereit gives a non-trivial involution on idempotents is also interesting, but requires a bitmore care in the statements below, and will not be relevant to this paper. The fol-lowing proposition follows easily from the fact that any contravariant equivalencetakes projectives to injectives.

Proposition 5.9. For all α ∈ I,

dLα ∼= Lα, dPα ∼= Iα, dVα ∼= Λα, and dTα ∼= Tα.

Remark 5.10. Proposition 5.9 has two important consequences. First, since d pre-serves simples, it acts trivially on the Grothendieck group of C(E). In particular, wehave [Vα] = [Λα], so by Proposition 5.4, the classes [Vα] are an orthonormal basis ofthe Grothendieck group.

Second, the isomorphism T∗ ∼= dT∗ induces an anti-automorphism of R(E) thatfixes idempotents, and thus a duality functor on the Ringel dual category C(R(E)).

Proposition 5.11. For all α ∈ I, the following are equivalent:

(1) The projective module Pα is injective.(2) The projective module Pα is tilting.(3) The projective module Pα is self-dual, that is, d(Pα) = Pα.

Furthermore, these conditions imply that the simple module Lα is contained in the socle ofsome standard module Vβ .

Proof. (1) ⇒ (2): Projectives all carry standard filtrations, and injectives carry co-standard ones. Thus, any module which is both must be tilting.


(2) ⇒ (3): All tilting modules are self-dual, since indecomposible tiltings are de-termined by the highest simple module appearing in their composition series, whichis unchanged by duality.

(3) ⇒ (1): The dual of a projective is injective, so any self-dual projective is alsoinjective.

Suppose that the projective module Pα is self-dual. Since Pα is indecomposible,it is the injective hull of its socle. Since Pα is self-dual, its socle is isomorphic to itscosocle Lα. Since Pα has a standard filtration, it has at least one standard module Vβas a submodule. Since socle is a left exact functor and Vβ is finite-dimensional, thesocle of Vβ is a non-trivial submodule of the socle Lα of Pα. Since Lα is simple, thesocle of Vβ must be isomorphic to Lα. �

Let Id = {α ∈ I | d(Pα) ∼= Pα}. Let

P =⊕α∈Id

Pα ⊂ P∗

be the direct sum of all of the self-dual projective right E-modules, and consider itsendomorphism algebra

(6) S := End(P) ⊂ End(P∗) ∼= E.

Definition 5.12. An algebra is said to be symmetric if it is isomorphic to its vectorspace dual as a bimodule over itself. It is immediate from the definition that S issymmetric.

For Theorem 5.13 we make one additional assumption, which is that the last im-plication of Proposition 5.11 can be reversed. More precisely, we assume that if thesimple module Lα includes into the socle of some standard module, then α ∈ Id.

Theorem 5.13. With the above hypothesis, the functor from right E-modules to right S-modules taking a module M to HomE(P,M) is fully faithful on projectives.

Proof. Fix an index α ∈ Id, and let L be the socle of the standard module Vα. ThenL is a direct sum of simple modules, and the assumption above implies that theinjective hull of L is also projective; we denote this hull by P . Since P is an injectivemodule, the inclusion L ↪→ P extends to Vα, and since the map is injective on thesocle L, it must be injective on all of Vα. Thus P is the injective hull of Vα. Anapplication of [MS, 2.6] gives the desired result. �


Remark 5.14. The property attributed to the S-E-bimodule P in Theorem 5.13 isknown as the double centralizer property. See [MS] for a more detailed treatmentof this phenomenon.

5.4. Koszul algebras. To discuss the notion of Koszulity, we must begin to workwith graded algebras and graded modules. LetE =

⊕k≥0Ek be a graded R-algebra,

and let R = E0.

Definition 5.15. A complex

. . .→ Pk → Pk−1 → . . .→ P1 → P0

of graded projective right E-modules is called linear if Pk is generated in degree k.

Definition 5.16. The algebra E is called Koszul if each every simple rightE-moduleadmits a linear projective resolution.

The notion of Koszulity gives us a second interpretation of quadratic duality.

Theorem 5.17. [BGS96, 2.3.3, 2.9.1, 2.10.1] If E is Koszul, then E is the quotient ofthe tensor algebra TRE1 by a quadratic ideal, and the quadratic dual E! is isomorphic toExtE(R,R)

op. Furthermore, E! is also Koszul.

Remark 5.18. In this case E! is also known as the Koszul dual of E.

Let D(E) be the bounded derived category of graded right E-modules.

Theorem 5.19. [BGS96, 1.2.6] If E is Koszul, we have an equivalence of derived categories

D(E) ∼= D(E!).

Remark 5.20. There are two other triangulated categories naturally associated toE, namely the bounded derived category Dug(E) of ungraded E-modules and thebounded derived category Ddg(E) of dg-E-modules, where E is thought of a dg-algebra with trivial differential. Given a complex of graded E-modules, one caneither forget the grading on the modules to obtain a complex of ungraded modules,or add the module grading to the grading on the complex to get a dg-module overE. This gives us forgetful functors

εu : D(E)→ Dug(E) and εd : D(E)→ Ddg(E).


both of which are degradings (in the sense of [BGS96]) with respect to different shiftfunctors on D(E). Using the theory of derived Morita equivalence (for example,see [Sch04, Proposition 3.20]), one can construct an equivalence of categories fromDug(E) to Ddg(E!) that commutes with the functors εu and εd.

We conclude this section with a discussion of Koszulity for quasi-hereditary alge-bras. Suppose that our graded algebra E is quasi-hereditary. For all α ∈ I, thereexists an idempotent eα ∈ R such that Pα = eαE, thus each projective module Pαinherits a natural grading. Let us assume that the grading of E is compatible withthe quasi-hereditary structure. In other words, we suppose that for all α ∈ I, thestandard module Vα admits a grading that is compatible with the map Πα : Pα → Vαof Definition 5.1. It is not hard to check that each of Lα, Vα,Λα, Pα, and Iα inherits agrading as a quotient of E, and that Tα admits a unique grading that is compatiblewith the inclusion of Vα. Thus R(E) = End(T∗) inherits a grading as well, and thisgrading is compatible with the quasi-hereditary structure [Zhu04].

Theorem 5.21. [ÁDL03, Theorem 1] Let E be a finite-dimensional graded algebra with agraded anti-automorphism that preserves idempotents. If E is graded quasi-hereditary andeach standard module admits a linear projective resolution, then E is Koszul.

5.5. The algebra A(V). Let V = (V, η, ξ) be a polarized arrangement, and let A =A(V) be the associated quiver algebra. A has a canonical anti-automorphism tak-ing p(α, β) to p(β, α) for all α ↔ β in P . Under the identification A(V) ∼= B(V∨),this corresponds to swapping the left and right factors of X̃∨ ×X∨ X̃∨. This anti-automorphism fixes the idempotents, and thus gives rise to a contravariant involu-tion d of C(A) as in Section 5.3.

For all α ∈ P , letLα = A/〈eβ | β 6= α〉

be the one-dimensional simple right A-module supported at the node α, and letPα = eαA denote the projective cover of Lα. Since Lα is one dimensional for each α,A is basic. Let a = µ−1(α) be the basis corresponding to the sign vector α, and let

K>α =∑i∈a

p(α, αi) · A ⊂ Pα

be the right-submodule of Pα generated by paths that begin at the node α and moveto a node that is higher in the partial order given in Section 2.6. (Recall that αi is the


unique sign vector such that α i↔ αi.) Let

Vα = Pα/K>α,

and letPα → Vα → Lα

be the natural projections.

Lemma 5.22. The module Vα has a vector space basis consisting of a taut path from α toeach element of F ∩ Ba.

Proof. Corollary 3.7 implies that this collection of paths is linearly independent. Anytaut path which terminates outside of F ∩ Ba must cross a hyperplane Hi for somei ∈ a, and by Corollary 3.8 it can be replaced by a path which crosses this hyperplanefirst, thus it lies in K>α. It will therefore suffice to show that any path which isnot taut will also have trivial image in Vα. By Proposition 3.6, this is equivalent toshowing that the positive degree part of SymV acts trivially on Vα, which followsfrom the fact that V is spanned by {ti | i ∈ a}. �

When V is rational, the modules Pα, Vα, Lα acquire natural geometric interpreta-tions via the isomorphism A ∼= B(V∨) of Theorem 4.10. Let X∨ be the relative coreof the hypertoric variety MH∨ associated to V∨. For each α ∈ P = P∨, we have acomponent X∨α ⊆ X. Let yα ∈ X∨α be an arbitrary element of the dense toric stratum(in other words, an element whose image under the moment map lies in the interiorof the polyhedron ∆∨α), and let xα = limλ→∞ λ·yα ∈ Xα be the toric fixed point whoseimage under the moment map is the ξ∨-maximum point of ∆∨α.

Proposition 5.23. If V is rational, then we have module isomorphisms

Pα ∼= H∗(Xα ×X∨ X̃∨), Vα ∼= H∗({xα} ×X∨ X̃∨), and Lα ∼= H∗({yα} ×X∨ X̃∨),

where A(V) ∼= B(V∨) = H∗(X̃∨ ×X∨ X̃∨) acts on the right by convolution.

Proof. The first isomorphism is immediate from the definitions.Restriction to the point xα defines a B(V∨)-module surjection from

Pα ∼= H∗(X∨α ×X∨ X̃∨)→ H∗({xα} ×X∨ X̃∨).

Note that xα ∈ X∨β if and only if α(i) = β(i) for all i /∈ a, or in other words, if andonly if β ∈ Fa ∩ B = B∨a ∩ F∨. The second isomorphism now follows from Lemma


5.22, using the fact that a taut path from α to β in the algebra A(V) gives rise to theunit class 1αβ ∈ H0(Xαβ) ⊂ B(V∨).

The third isomorphism follows from the fact that H∗({yα} ×X∨ X̃∨) ∼= H∗({yα}) isone-dimensional, eα acts by the identity, and eβ acts by zero for all β 6= α. �

Theorem 5.24. The algebra A is quasi-hereditary with respect to the partial order on Pgiven in Section 2.6, with the modules {Vα} as the standard modules. This structure iscompatible with the grading on A.

Proof. We must show that the modules {Vα | α ∈ I} satisfy the conditions of Defini-tion 5.1. Condition (1) follows from Lemmas 5.22 and 2.8.

For condition (2), we define a filtration of K>α = ker Πα as follows. For γ ∈ P , letP γα ⊂ Pα be the submodule generated by paths that pass through the node γ, andfor any γ ∈ P let

P≥γα =∑δ≥γ

P δα, P>γα =

∑δ>γ

P δα.

Note that P≥αα = Pα and P>αα = K>α.Then P γα ⊂ K>α for all γ > α, and these submodules form a filtration with succes-

sive quotientsMγα := P

≥γα /P

>γα .

Let g = µ−1(γ) ∈ Bas. If α is not in the negative cone Bg, then there exists i ∈ g suchthat α(i) 6= γ(i). It follows from Corollary 3.7 that P γα = P>γα , hence Mγα = 0.

If α ∈ Bg, then composition with a taut path p from α to γ defines a map Pγ →P≥γα which induces a map Vγ → Mγα . We will show that this induced map is anisomorphism. First, note that Mγα is spanned by the classes of paths which passthrough γ. Using Proposition 3.6, such a path is equivalent to one which beginswith a taut path from α to γ, and by Corollary 3.7 implies that this taut path can betaken to be p, so our map is surjective.

To see that it is injective, it will be enough to show that dimR Pα is bounded belowby ∑

α∈Bg

dimR Vµ(g) = |{(δ, g) ∈ P × Bas | α, δ ∈ Bg}|,

or, adding over all α, that

dimRA ≥ |{(α, δ, g) ∈ P × P × Bas | α, δ ∈ Bg}|.


Assume for the moment that V is rational. By Theorem 4.10 we have

dimRA = dimRH•(X̃∨ ×X∨ X̃∨)

=∑

α,δ∈P∨dimRH

•(X∨α ∩X∨δ )

= |{(α, δ, b) ∈ P∨ × P∨ × Bas∨ | H∨b ⊂ ∆∨α ∩∆∨δ }|.

Here the last equality uses the fact that the variety X∨α ∩X∨δ is a semi-projective toricvariety, so the total dimension of its cohomology is equal to the number of torusfixed points, which is the same as the number of vertices H∨b , b ∈ Bas contained inthe moment polytope ∆∨α ∩∆∨δ .

Now our result follows from the fact that for any basis b ∈ Bas,

H∨b ⊂ ∆∨α ⇔ α ∈ F∨b ⇔ α ∈ Bbc .

If V is not rational, the same argument works, replacing the cohomology ring of X∨αδby the appropriate combinatorial ring R∨αδ as in Remark 4.8. �

Theorem 5.25. Let V be a polarized arrangement. The algebrasA(V) andB(V) are Koszul,and Koszul dual to each other.

Proof. By Theorems 3.10, 4.10, and 5.17, it is enough to prove that A = A(V) isKoszul. By Theorem 5.21, it is enough to show that each standard module Vα has alinear projective resolution.

Let a = µ−1(α) be the basis associated with the sign vector α. For any subsetS ⊂ a, let αS be the sign vector that differs from α in exactly the indices in S. Thus,for example, α∅ = α, and α{i} = αi for all i ∈ a. (Note that the sign vectors that arisethis way are exactly those in the set Fa.) If S = S ′ t {i} ⊂ a, then we have a mapϕS,i : PαS → PαS′ given by left multiplication by the element p(αS′ , αS). We adoptthe convention that PαS = 0 if αS /∈ P and ϕS,i = 0 if i /∈ S.

LetΠα =

⊕S⊂a

PαS

be the sum of all of the projective modules associated to the sign vectors αS . Thismodule is multi-graded by the abelian group Za = Z{�i | i ∈ a}, with the summand


PαS sitting in multi-degree �S :=∑

i∈S �i. For each i ∈ a, we define a differential

∂i =∑

i∈S⊂a

ϕS,i

of degree −�i. These differentials commute because of the relation (A2), and thusdefine a multi-complex structure on Πα. The total complex Π•α of this multi-complexis linear and projective; we claim that it is a resolution of the standard module Vα.It is clear from the definition that H0(Π•α) ∼= Vα, thus we need only show that ourcomplex is exact in positive degrees.

We will use two important facts about filtered chain complexes and multi-complexes.Both are manifest from the theory of spectral sequences, but could also easily beproven by hand by any interested reader.

(*) If any one of the differentials in a multi-complex is exact, then the total com-plex is exact as well.

(**) If a chain complex C• has a filtration such that the associated graded C̃• isexact at an index i, then C• is also exact at i.

As in the proof of Theorem 5.24, we may filter each projective module PαS bysubmodules of the form P βαS for β ≥ αS , which consists of paths from αS that passthrough the node β. We extend this filtration to all β by defining P βαS to be the sumof P β′αS over all β

′ ∈ P for which β′ ≥ αS and β′ ≥ β. It is easy to see that thisfiltration is compatible with the differentials, hence we obtain an associated gradedmulti-complex

Π̃•α :=⊕

β

(Π•α)β/(Π•α)

>β =⊕β,S

MβαS .

Take β ∈ P , and let b = µ−1(β). We showed in the proof of Theorem 5.24 thatMβαS is nonzero if and only if αS ∈ Bb, in which case it is isomorphic to Vβ . If β = α,then we have a nonzero summand only when S = ∅, so that summand sits in totaldegree zero. For β 6= α, choose an element i ∈ a∩bc. This ensures that if S = S ′t{i},then αS′ ∈ Bb if and only if αS ∈ Bb. For such a pair S and S ′, we have

MβαS∼= Vβ ∼= MβαS′ ,

and ∂̃βi is the isomorphism given by left-composition with p(αS′ , αS). Thus ∂̃βi is

exact in nonzero degree. By (*) we can conclude that the total complex Π̃•α is exactin nonzero degree, and thus by (**) so is Π•α. �

We next determine which projective A-modules are self-dual.


Theorem 5.26. For all α ∈ P , the following are equivalent:(1) The projective Pα is injective.(2) The projective Pα is tilting.(3) The projective Pα is self-dual, that is, d(Pα) = Pα.(4) The simple Lα is contained in the socle of some standard module Vβ .(5) The cone Σα ⊂ V has non-trivial interior.(6) The chamber ∆∨α ⊂ V ⊥ − ξ is compact.

Proof. The implications (3)⇒ (2)⇒ (1)⇒ (4) were proved in Proposition 5.11.(4) ⇒ (5): Let b = µ−1(β) ∈ Bas. By Lemma 5.22, Vβ is spanned as a vector space

by taut paths pγ from β to nodes γ ∈ F ∩ Bb. The socle of Vβ is spanned by thosepγ for which γ is as far away from β as possible; i.e. any hyperplane Hi, i /∈ b whichmeets ∆γ must separate ∆γ and ∆β . This implies that any ray starting at the pointHb and passing through an interior point q of ∆γ will not leave this chamber once itenters. It follows that the direction vector of this ray lies in Σγ . Since this holds forany q, Σγ has nonempty interior.

(5) ⇒ (6): The fact that Σα has non-empty interior implies that α is feasible forthe polarized arrangement (V, η′, ξ) for any η′ ∈ RI/V . Dually, α is bounded for(V ⊥,−ξ,−η′) for any covector η′, and and thus ∆∨α is compact.

(6) ⇒ (3): If V is rational, then Proposition 5.23 tells us that Pα is isomorphic tothe vector space H∗(X∨α ×X∨ X̃∨). Since ∆∨α is compact, so is X∨α , and the Poincarépairing provides an isomorphism between Pα and its vector space dual d(Pα). In thenon-rational case, the same argument goes through using the fact that the rings Rαβof Remark 4.8 are Gorenstein [Sta83, II.5.2]. �

Remark 5.27. The equivalence (1) ⇔ (6) is part (3) of Theorem (B) from the Intro-duction, keeping in mind that A = A(V) ∼= B(V∨). If V is rational, then the set ofα ∈ P for which ∆∨α is compact indexes the components of the core of the hyper-toric variety MH∨ (see Remark 4.2), which may also be described as the set of allirreducible projective Lagrangian subvarieties of MH∨ .

Remark 5.28. Theorems 5.24, 5.25, and 5.26 are all analogous to theorems that arisein the study of parabolic category O and other important categories in representa-tion theory [MS].


6. DERIVED EQUIVALENCES

The purpose of this section is to show that the dependence of A(V) on the param-eters ξ and η is relatively minor. Indeed, suppose that

V1 = (V, η1, ξ1) and V2 = (V, η2, ξ2)

are polarized arrangements with the same underlying linear subspace V ⊂ RI . ThusV1 and V2 are related by translations of the hyperplanes and a change of affine-linear functional on the affine space in which the hyperplanes live. The associatedquiver algebras A1 = A(V1) and A2 = A(V2) are not necessarily isomorphic, noreven Morita equivalent. They are, however, derived Morita equivalent, as stated inTheorem (C) of the Introduction and proved in Theorem 6.13 of this section. That is,the triangulated category D(V) defined in Section 5.4 is an invariant of the subspaceV ⊂ RI . The corresponding results forDug(V) andDdg(V) can be obtained by similarreasoning.

6.1. Definition of the functors. We begin by restricting our attention to the specialcase in which ξ1 = ξ2 = ξ for some ξ ∈ V ∗. On the dual side, this means thatη∨1 = η

∨2 = η

∨ = −ξ, and therefore that H∨1 = H∨2 = H∨. Our first task will be todefine an A1 − A2 bimodule N . We will give two equivalent descriptions of N , oneon the A-side and one on the B-side, exploiting the isomorphism A(Vj) ∼= B(V∨j ) ofTheorem 4.10. We begin with the B-side description, as it is the easier of the two tomotivate. Let Aj = A(Vj) and B∨j = B(V∨j ). For simplicity, we will assume that eachVj is rational.

Recall that the relative coreX∨j :=

⋃α∈Pj

X∨α

is defined as a union of toric varieties indexed by the chambers of H∨ which arebounded above with respect to the affine linear function ξ∨j . This sits inside of theextended core (Remark 4.2)

X∨ext :=⋃

α∈F∨X∨α ,

in which we use all of the chambers of H∨; the extended core depends only on H∨

and is therefore the same for V∨1 and V∨2 . We then define N to be the graded vectorspace

H∗(X̃∨1 ×X∨ext X̃∨2 ) =

⊕(α,β)∈P1×P2

H∗(X∨αβ)[−dαβ],


which has a natural left B∨1 -action and right B∨2 -action via convolution.To formulate this definition on the A-side, rather than considering all feasible

sign vectors we must consider all bounded sign vectors. Let Aext(V) be the algebradefined by the same relations as A(V), but without the feasibility restrictions. Thatis, we begin with a quiver Qext whose nodes are indexed by the set {±1}I of allsign vectors, and let Aext(V) be the quotient of P (Qext) ⊗R SymV ∗ by the followingrelations:

Aext1: If α ∈ {±1}I r B, then eα = 0.

Aext2: If four distinct elements α, β, γ, δ ∈ {±1}I satisfy α↔ β ↔ γ ↔ δ ↔ α, then

p(α, β, γ) = p(α, δ, γ).

Aext3: If α, β ∈ {±1}I and αi↔ β, then

p(α, β, α) = tieα.

We note that since B1 = B2, we have Aext(V1) = Aext(V2) = Aext. Let

eηj =∑α∈Pj

eα ∈ Aext.

Then Aj is isomorphic to the subalgebra eηjAext eηj of Aext. Consider the gradedvector space

N := eη1Aext eη2 ,

which is a left A1-module and a right A2-module in the obvious way.The following proposition is an easy extension of Theorem 4.10; its proof will be

left to the reader.

Proposition 6.1. The quiver algebra Aext is isomorphic to the extended convolution algebraH∗(X̃∨ext ×X∨ext X̃

∨ext). Furthermore, the isomorphisms Aj ∼= B∨j induce a graded bimodule

isomorphism between N and H∗(X̃∨1 ×X∨ext X̃∨2 ).

We define a functor Φ : D(V1)→ D(V2) by the formula

Φ(M) := ML⊗A1 N.

For α ∈ Pj , let P jα and V jα denote the corresponding projective module and standardmodule for Aj .

Proposition 6.2. If α ∈ P1 ∩ P2, then Φ(P 1α) = P 2α .


Proof. The natural map

Γ : P 2α = eαA2 → eαA1 ⊗A1 eη1Aext eη2 = P 1α ⊗A1 N = Φ(P 1α)

taking eα to eα ⊗ eη1eη2 , is an isomorphism, where surjectivity follows from an ana-logue of Proposition 3.7 for the module N . �

Remark 6.3. Note that by Proposition 6.2 and the equivalence (3)⇔ (6) of Theorem5.26, Φ takes self-dual projectives to self-dual projectives.

Consider a basis b ∈ Bas(V1) = Bas(V2), and recall that we have bijections

P1µ1←− Bas(V1) = Bas(V2)

µ2−→ P2.

Let ν : P1 → P2 denote the composition. Recall also that the sets Bb ⊂ B, defined inSection 2.6, do not depend on η.

Lemma 6.4. For any α ∈ P1, the A2-module Φ(P 1α) has a filtration with standard sub-quotients. If α ∈ Bb then the standard module V 2µ2(b) appears with multiplicity 1 in theassociated graded, and otherwise it does not appear.

Proof. As in the proof of Proposition 6.2, we have Φ(P 1α) = eαA1 ⊗A1 eη1Aext eη2 , thuswe may represent an element of Φ(P 1α) by a path in B that begins at α and ends atan element of P2 = B ∩ F2. For β ∈ P2, let Φ(P 1α)β be the submodule generated bythose paths p such that β is the maximal element of P2 through which p passes, andlet

Φ(P 1α)>β =⋃γ>β

Φ(P 1α)γ and Φ(P1α)≥β =

⋃γ≥β

Φ(P 1α)γ.

We then obtain a filtrationΦ(P 1α) =

⋃β

Φ(P 1α)≥β.

Suppose that β = µ2(b); we claim that the quotient Φ(P 1α)≥β/Φ(P 1α)>β is isomorphic

to V 2β if α ∈ Bb, and is trivial otherwise.If α ∈ Bb, then we have a map

V 2β → Φ(P 1α)≥β/Φ(P 1α)>β

given by precomposition with any taut path from α to β, and an adaptation of theproof of Theorem 5.24 shows that it is an isomorphism. If α /∈ Bb, then there existsi ∈ b such that αi 6= β(i), and any path from α to β will be equivalent to one thatpasses through βi > β. Thus in this case the quotient is trivial. �


Proposition 6.5. For all α ∈ P1, we have [Φ(V 1α )] = [V 2ν(α)] in the Grothendieck group of(ungraded) right A2-modules. Thus Φ induces an isomorphism of Grothendieck groups.

Proof. For all b ∈ Bas, we have∑α∈Bb

[Φ(V 1α )] = [Φ(P1µ1(b)

)] =∑α∈Bb

[V 2ν(α)],

where the first equality follows from the proof of Theorem 5.24 and the second fol-lows from Lemma 6.4. The first statement of the theorem then follows from induc-tion on b. The second statement follows from the fact that the Grothendieck groupof modules over a quasi-hereditary algebra is freely generated by the classes of thestandard modules. �

Remark 6.6. We emphasize that Φ(V 1β ) and V2ν(β) are not isomorphic as modules;

Proposition 6.5 says only that they have the same class in the Grothendieck group.In fact, the next proposition provides an explicit description of Φ(V 1β ) as a module.

Proposition 6.7. Φ(V 1α ) is the quotient of Φ(P 1α) by the submodule generated by all pathswhich cross the hyperplane Hi for some i ∈ µ−11 (α). In particular, TorA1k (V 1α , N) = 0 for allk > 0.

Proof. It is clear that if we take a projective resolution of V 1α and tensor it with N ,the degree zero cohomology of the resulting complex will be this quotient. Thuswe need only show that the complex is exact in positive degree, that is, that it is aresolution of V 1α ⊗N . The proof of this fact is identical to the proof of Lemma 6.4. �

Corollary 6.8. If a right A1-module M admits a filtration by standard modules, thenTorA1k (M,N) = 0 for all k > 0, and thus Φ(M) = M ⊗A1 N .

Remark 6.9. Though we will not need this fact, it is interesting to note that Φ takesthe exceptional collection {V 1α } to the mutation of {V 2ν(α)} with respect to a linearrefinement of our partial order. (See [Bez06] for definitions of exceptional collectionsand mutations.) We leave the proof as an exercise to the reader.


6.2. Ringel duality and Serre functors. In this section we pass to an even furtherspecial case; we still require that ξ1 = ξ2, and we will now assume in addition thatη1 = −η2. Rather than referring to V1 and V2, we will write

V = (V, η, ξ) and V̄ = (V,−η, ξ),

and we will refer to V̄ as the reverse of V . Let

Φ− : D(V)→ D(V̄) and Φ+ : D(V̄)→ D(V)

be the functors constructed above.

Theorem 6.10. The algebras A and Ā are Ringel dual, and the Ringel duality functor isd ◦Φ− = Φ−◦ d. In particular, Φ− sends projectives to tiltings, tiltings to injectives, andstandards to costandards.

Proof. Using the B-side description of the functor Φ−, we find that, for any α ∈ P ,

Φ−(Pα) =⊕β̄∈P̄

H∗(Xαβ̄)[−dαβ̄].

The polyhedron ∆αβ̄ is always compact, thus H∗(Xαβ̄) is Gorenstein and Φ−(Pα) isself-dual. We showed in Lemma 6.4 that Φ−(Pα) admits a filtration with standardsubquotients, with V̄ν(α) as its largest standard submodule, from which we can con-clude that Φ−(Pα) is isomorphic to T̄ν(α). Thus d ◦Φ− is a contravariant functor thatsends projective modules to tilting modules; by Proposition 5.8, it will now be suffi-cient to show that Φ− is an equivalence.

For all α, β ∈ P , the functor Φ− induces a map Hom(Pα, Pβ) → Hom(T̄ν(α), T̄ν(β)).We will show that this map is an isomorphism by first showing it to be injectiveand then comparing dimensions. By the double centralizer property (Remark 5.14),there exists a self-dual projective Pα′ and a map Pα′ → Pα such that compositionwith this map defines an injection from Hom(Pα, Pβ) to Hom(Pα′ , Pβ). On the otherhand, the injective hull of Pβ is the same as the injective hull of its socle. Since Pβhas a standard filtration, each simple summand of this socle lies in the socle of somestandard module. Then the implication (4) ⇒ (3) of Theorem 5.26 tells us that theinjective hull of Pβ is isomorphic to some self-dual projective Pβ′ .

Now consider the commutative diagram below, in which the vertical arrow onthe left is injective. To prove injectivity of the top horizontal arrow, it is enough toshow injectivity of the bottom horizontal arrow, which follows from Proposition 6.2.


Hom(Pα, Pβ) //

��

Hom(T̄ν(α), T̄ν(β))

��Hom(Pα′ , Pβ′) // Hom(T̄ν(α′), T̄ν(β′))

Next we need to prove that the two Hom-spaces on the top of the diagram havethe same dimension. Since standards and costandards are dual sequences (Proposi-tion 5.4), we have

Exti(Tα, Tβ) = 0 for all i > 0.

By Lemma 6.4, we have the decomposition

[T̄ν(α)] = [Φ−(Pα)] =

∑α∈Bb

[Vµ2(b)]

in the Grothendieck group of Ā-modules. From this statement and Proposition 6.5,we may deduce that

[Pα] =∑α∈Bb

[Vµ1(b)]

in the Grothendieck group ofA-modules. The standard classes form an orthonormalbasis with respect to the Euler form (Remark 5.10), thus

dim Hom(T̄ν(α), T̄ν(β)) =〈[T̄ν(α)], [T̄ν(β)]

〉= #{b ∈ Bas | α, β ∈ Bb}

=〈[Pα], [Pβ]

〉= dim Hom(Pα, P̄β).

Thus Φ− is an equivalence of categories.By Propositions 5.8 and 5.9, it is now sufficient to show that R(A) is isomorphic to

Ā. To this end, consider the equivalence Φ+ from Ā modules to A modules, whichtakes P̄ν(α) to Tα for all α ∈ P . From this we find that

R(A) = EndA(⊕Tα) ∼= EndĀ(⊕P̄ν(α)) = Ā.

The last statement follows from Proposition 5.9. �

The functors Φ± are not mutually inverse (we will see this explicitly in Theo-rem 6.11), but their composition is interesting and natural from a categorical per-spective. For any graded algebra E, an auto-equivalence S : D(E)→ D(E) is called


a Serre functor7 if we

gale duality and koszul duality - pages.uoregon.edupages.uoregon.edu/njp/gk.pdf · there are two...

Documents