# Category \mathcal{O} for Oriented Matroids

aa r X i v : . [ m a t h . R T ] F e b A CATEGORY O FOR ORIENTED MATROIDS

ETHAN KOWALENKO AND CARL MAUTNERA

BSTRACT . We associate to a sufﬁciently generic oriented matroid program andchoice of linear system of parameters a ﬁnite dimensional algebra, whose repre-sentation theory is analogous to blocks of Bernstein–Gelfand–Gelfand category O .When the data above comes from a generic linear program for a hyperplane ar-rangement, we recover the algebra deﬁned by Braden–Licata–Proudfoot–Webster.

1. I

NTRODUCTION

In [BLPW10, BLPW12], Braden–Licata–Proudfoot–Webster introduced a classof ﬁnite-dimensional algebras related to the combinatorics of hyperplane arrange-ments, whose representation theory is closely analogous to the integral blocks ofBernstein–Gelfand–Gelfand (BGG) category O . Recall that BGG category O playsan important role in Lie theory and can be described using the geometry of theSpringer resolution. Braden–Licata–Proudfoot–Webster discovered their algebrasby analogy, motivated by the geometry of toric hyperk¨ahler (or hypertoric) vari-eties, but the algebras can be deﬁned from basic linear algebra data. The input fortheir deﬁnition was the data of a polarized arrangement V = ( V, η, ξ ) , where V ⊂ R n is a d -dimensional linear subspace, η ∈ R n /V is a (generic) vector and ξ ∈ V ∗ is a (generic) covector. Braden–Licata–Proudfoot–Webster [BLPW16] and others(e.g., [Los17]) have since introduced and studied other such geometric categories O associated to conical symplectic resolutions.In this paper we extend the deﬁnition of Braden–Licata–Proudfoot–Webster ina different, more combinatorial direction: from the setting of polarized arrange-ments to the combinatorics of oriented matroids. More precisely, the role of V ⊂ R n is replaced by a rank d orientable matroid M and parameter space U and therole of η and ξ by an oriented matroid program ( f M , g, f ) that extends and liftsan orientation of M . One motivation for our work was a desire to categorify andbetter understand the matroidal Schur algebras of [BM17, BM].To explain our results and motivation, we ﬁrst recall in more detail the resultsof Braden–Licata–Proudfoot–Webster.1.1. Hypertoric category O . In [BLPW10], Braden–Licata–Proudfoot–Webster de-ﬁned a quadratic algebra A ( V ) . One motivation was a description of a regu-lar block of category O as arising from a quantization of the structure sheaf of T ∗ ( G/B ) , the cotangent bundle of a ﬂag variety. When V is rational (meaningthat V, η, and ξ are deﬁned over Q ), one may associate to V a hyperk¨ahler variety M , sometimes called a hypertoric variety, which behaves in various ways like thecotangent bundle of a ﬂag variety. Braden–Licata–Proudfoot–Webster show that More precisely, the subspace V and vector η alone determine the variety M . The covector ξ can beused to endow M with a C ∗ -action. ξ ǫ δγ βα (a) f g ǫ δγ βα (b)F IGURE

1. Hyperplane arrangement and corresponding pseudo-sphere arrangement.in this case the category of representations of A ( V ) is equivalent to that obtainedby applying the same sort of quantization construction to the hypertoric varietyfor V . Moreover they show: Theorem 1.1 (Braden–Licata–Proudfoot–Webster) . Let V = ( V, η, ξ ) be a polarizedarrangement where η and ξ are generic. (1) The algebra A ( V ) is quadratic with quadratic dual A ( V ∨ ) , where V ∨ = ( V ⊥ , − ξ, − η ) denotes the Gale dual polarized arrangement. (2) The algebra A ( V ) is quasi-hereditary. (3) The algebra A ( V ) is Koszul (and thus by the ﬁrst result, Koszul dual to A ( V ∨ ) ). (4) Up to derived Morita equivalence, the algebra A ( V ) depends only on V ⊂ R n andnot on η or ξ . To give a feeling for the representation theory of these algebras, we will de-scribe a labelling of the simple modules for A ( V ) . It is convenient to consider thefollowing hyperplane arrangement deﬁned by V = ( V, η, ξ ) . Note that η ∈ R n /V can be viewed as the afﬁne subspace η + V ⊂ R n , and we consider the arrange-ment of hyperplanes in η + V cut out by the coordinate hyperplanes of R n . Thegenericity condition on η is the requirement that the resulting arrangement be sim-ple , meaning that the nonempty intersection of m hyperplanes has codimension m .The covector ξ ∈ V ∗ lifts to an afﬁne linear functional on η + V . The genericity con-dition on ξ is the requirement that ξ be nonconstant on any positive dimensionalintersection of V and a coordinate subspace. Example 1.2.

The polarized arrangement from Example 2.2 of [BLPW10] consistsof a two-dimensional subspace V ⊂ R and the choices of η and ξ there producesthe hyperplane arrangement depicted in Figure 1.1(a).The set P of chambers of the hyperplane arrangement in η + V that are boundedwith respect to ξ parametrize the simple modules { L α } α ∈P for A ( V ) . In Exam-ple 1.2, we can label these chambers α, β, γ, δ, ǫ as in Figure 1.1(a). See the following paragraph for the meaning of the word generic used here.

CATEGORY O FOR ORIENTED MATROIDS 3

For each bounded chamber α ∈ P , let β (cid:22) α if β is contained in the cone gen-erated by α originating from its maximal vertex. The transitive closure of thisrelation gives the highest weight partial order on simple objects for the quasi-hereditary structure in the theorem. In the example above, this produces the posetdescribed by the following Hasse diagram: αβγ δǫ More precisely, Braden–Licata–Proudfoot–Webster deﬁne standard modules V α for every α ∈ P and prove (see the proof of [BLPW10, Theorem 5.23]): Theorem 1.3 (Braden–Licata–Proudfoot–Webster) . For any α ∈ P the indecompos-able projective cover P α of L α has a ﬁltration with successive subquotients isomorphic to V β for each β (cid:23) α and each such standard module appears exactly once. Matroidal setting.

Fix a ﬁeld k and a ﬁnite index set I . In this paper we willbegin with an orientable matroid M of rank d and a choice of parameter space U ⊂ k I for M . By parameter space, we mean a subspace U ⊂ k I such that thecomposition U ֒ → k n ։ Span { t i | i ∈ b } is an isomorphism for any basis b of M . Example 1.4.

Note that the subspace V ⊂ R n in a polarized arrangement ofBraden–Licata–Proudfoot–Webster provides such a pair for k = R : let M be thematroid on the index set I = { , . . . , n } represented by the coordinate functions of R n restricted to V , viewed as vectors x , . . . , x n ∈ V ∗ , and let U = V .Let M be an orientation of M , meaning an oriented matroid M such that M = M , where M denotes the underlying unoriented matroid. (In the polarized ar-rangement example, there is a natural choice for M , as M is represented by vectorsin a real vector space.)The remaining input data we need is the structure of a oriented matroid pro-gram P = ( f M , g, f ) , meaning f M is an oriented matroid on the underlying set I ⊔ { g, f } such that g is not a loop, f is not a coloop, and ( f M\ f ) /g = M . Like wedid for η and ξ , we ask that g and f be sufﬁciently generic (see Deﬁniton 2.2).The matroid M is determined by P , so we can and will omit it from our notationand consider pairs ( P , U ) where P = ( f M , g, f ) is a sufﬁciently generic orientedmatroid program and U ⊂ k I is a parameter space for the underlying (unoriented)matroid M = ( f M\ f ) /g . Example 1.5.

Polarized arrangements give a natural class of examples. For a d -dimensional polarized arrangement ( V, η, ξ ) , consider the d + 1 -dimensional sub-space e V of R n × R g × R f spanned by the graph of ξ : V → R f and the vector ( η, , ∈ R n × R g × R f , where η is any representative of the coset η ∈ R n /V . Let f M be the oriented matroid on the set { , . . . , n } ⊔ { g, f } deﬁned by the coordinatefunctions x , . . . , x n , x g , x f ∈ e V ∗ . ETHAN KOWALENKO AND CARL MAUTNER f gF

IGURE

2. Ringel exampleNot every oriented matroid program P comes from a polarized arrangement,but by the Topological Realization Theorem of Folkman–Lawrence, every loop-free program P can be expressed a pseudosphere arrangement - a topological repre-sentation generalizing the notion of a hyperplane arrangement. Example 1.6.

Figure 1.1(b) shows the feasible region of the pseudosphere arrange-ment corresponding to the polarized arrangement from Example 1.2.

Example 1.7.

Figure 1.2 depicts the feasible part of a pseudosphere arrangement,where | I | = 8 and M is the uniform rank 2 matroid on 8 points, that deﬁnes anon-realizable oriented matroid program P = ( f M , g, f ) . Here the oriented sub-matroid f M\ f , a rank 3 oriented matroid on 9 points, is the non-stretchable simplearrangement of 9 pseudolines deﬁned by Ringel [Rin55] as a perturbation the Pap-pus matroid. Remark . Every oriented matroid program P where d = 2 and | I | ≤ is realiz-able, so the program described in Example 1.7 is a minimal non-realizable exam-ple.For a pair ( P = ( f M , g, f ) , U ) as above we deﬁne the dual pair ( P ∨ , U ⊥ ) , where P ∨ = ( f M ∨ , f, g ) is the dual oriented matroid program (here the roles of f and g are swapped), and U ⊥ ⊂ k I is the orthogonal complement. Remark . It is an exercise in linear algebra to check that when the oriented ma-troid program P = ( f M , g, f ) comes from a polarized arrangement V ∨ = ( V, η, ξ ) as in Example 1.5, this duality agrees with the standard Gale duality of linearprogramming. In other words, the dual program P ∨ = ( f M ∨ , f, g ) is the ori-ented matroid program associated to the Gale dual polarized arrangement V ∨ =( V ⊥ , − ξ, − η ) .1.3. Main results.

As above, let the pair ( P , U ) consist of a sufﬁciently genericoriented matroid program P = ( f M , g, f ) together with a parameter space U ⊂ k I CATEGORY O FOR ORIENTED MATROIDS 5 for the (unoriented) matroid M = ( f M\ f ) /g . Modifying the deﬁnition of Braden–Licata–Proudfoot–Webster to this setting, we introduce a ﬁnite-dimensional alge-bra A ( P , U ) over k . In particular, in the realizable case of Example 1.4 and 1.5, onerecovers the original algebra A ( P , V ) = A ( V ) .In the more general setting, we observe that part (1) of Theorem 1.1 extendswithout modiﬁcation: Theorem 1.10.

Let ( P , U ) be a pair as above. The algebra A ( P , U ) is quadratic withquadratic dual A ( P ∨ , U ⊥ ) corresponding to the dual pair. Similarly to the realizable case, the simple modules for A ( P , U ) are labelled bythe set P of bounded, feasible topes. For example, in Example 1.7 the bounded,feasible topes correspond to shaded regions in Figure 1.2. Again one can deﬁne acone relation on P and standard modules V α for each α ∈ P .However, unlike in the realizable case, the transitive closure of the cone relationneed not deﬁne a poset. An oriented matroid program P = ( f M , g, f ) is said to be Euclidean , if the transitive closure of the cone relation on bounded, feasible topesof P is a poset.Using this condition, we obtain the following analogue of Theorems 1.1(2),1.1(3) and 1.3. Theorem 1.11.

For a pair ( P , U ) be as above with the additional assumption that theprogram P is Euclidean , the algebra A ( P , U ) is quasi-hereditary and Koszul.Moreover, for any α ∈ P the indecomposable projective cover P α of L α has a ﬁltrationwith successive subquotients isomorphic to V β for each β (cid:23) α and each such standardmodule appears exactly once.Remark . While oriented matroid programs are not always Euclidean, everyoriented matroid program of rank at most 3 (equivalently d at most 2) is Euclidean.Thus there are plenty of Euclidean, non-realizable programs, such as Example 1.7.We do not know whether or not every non-realizable oriented matroid M ad-mits a Euclidean program P = ( f M , g, f ) such that f M /g \ f = M . For connectionsto a well-known conjecture of Las Vergnas, see the discussion surrounding Propo-sition 2.27.We observe in Example 7.10 that in the non-Euclidean case, A ( P , U ) need not bequasi-hereditary. In particular, we give an example of a non-Euclidean program P and projective A ( P , U ) -module which does not admit a standard ﬁltration.However, in Theorem 7.9 we do prove that for any oriented matroid program P the following analogue of Theorem 1.3 holds on the level of the Grothendieckgroup of graded A ( P , U ) -modules. Theorem 1.13.

For any generic oriented matroid program P and any α ∈ P , the class ofthe indecomposable projective P α in the Grothendieck group can be expressed as the sum: [ P α ] = X γ (cid:23) α q d αγ [ V γ ] , where d αγ denotes the distance between the topes α and γ . While our proof of Koszulity in the Euclidean case relies on A ( P , U ) beingquasi-hereditary, it is conceivable that A ( P , U ) is Koszul more generally. As ev-idence in this direction, in Theorem 7.14 we prove the Hilbert series of A ( P , U ) and A ( P ∨ , U ⊥ ) satisfy the numerical identity discussed in [BGS96, Lemma 2.11.1]. ETHAN KOWALENKO AND CARL MAUTNER

Derived Morita equivalence.

In light of Theorem 1.1(4) it seems natural toask:

Question . Let M be an orientable matroid and U a choice of parameter spacefor M . For any two orientations M , M of M and generic oriented matroid pro-grams P = ( f M , g , f ) , P = ( f M , g , f ) such that f M i /g i \ f i = M i , i = 1 , ,are the algebras A ( P , U ) and A ( P , U ) derived Morita equivalent?If the answer to this question is yes, it would appear to give a rather interest-ing algebraic invariant of the matroid M . Or weaker, one might still hope for anafﬁrmative answer under the assumption that M = M : Question . Let M be an oriented matroid and U a choice of parameter space for M = M . For any two generic oriented matroid programs P = ( f M , g , f ) , P =( f M , g , f ) such that M = f M /g \ f = f M /g \ f , are the algebras A ( P , U ) and A ( P , U ) derived Morita equivalent?If the answer to one or both of these questions is no, the number of derivedMorita equivalence classes could also provide a interesting invariant of M or M .As a partial result in this direction, following the strategy of Braden–Licata–Proudfoot–Webster, we prove the following theorem in Section 8. Theorem 1.16.

Fix M and let P = ( f M , g , f ) and P = ( f M , g , f ) be Euclideansuch that f M i /g i \ f i = M for i = 1 , . Suppose in addition that the oriented matroidprogram P mid = ( f M mid , g , f ) is also Euclidean, where P mid is a generic oriented matroidprogram such that f M mid /g = f M /g , f M mid \ f = f M \ f . Then the bounded derived categories of graded ﬁnitely generated A ( P , U ) - and A ( P , U ) -modules are equivalent. This allow us to answer Questions 1.14 and 1.15 in some simple cases.

Corollary 1.17.

Question 1.15 has an afﬁrmative answer for any oriented matroid M ofrank 2.Proof. Recall from Remark 1.12, that any oriented matroid program of rank 3 isEuclidean, so for any P and P , the three oriented matroid programs P , P and P mid are all Euclidean and the result follows from Theorem 1.16. (cid:3) Corollary 1.18.

Question 1.14 has an afﬁrmative answer for M = U ,n , the uniformmatroid of rank 2 deﬁned on a set E n of n elements.Proof. By Corollary 1.17, it sufﬁces to show that for any two orientations M and M of U ,n , there are generic oriented matroid programs P = ( f M , g , f ) , P =( f M , g , f ) such that M = f M /g \ f and M = f M /g \ f , for which A ( P , U ) and A ( P , U ) are derived equivalent.But any two orientations of U ,n are related by a relabeling of E n and reorienta-tion. Note that relabelling and reorientation each induce a canonical isomorphismbetween the associated algebras. (cid:3) Note that such an oriented matroid program ( f M mid , g , f ) always exists and there will typi-cally be many such oriented matroid programs. However the particular choice will not matter forus, because, as mentioned in Remark 2.1, all of our constructions depend only on the contraction andrestriction oriented matroids f M mid /g and f M mid \ f . CATEGORY O FOR ORIENTED MATROIDS 7

The same sort of argument gives a handful of similar examples.1.5.

Matroidal Schur algebras.

Motivated in part by [BLPW10], Braden and thesecond author deﬁned a hypertoric Schur algebra [BM] - an analogue of the Schuralgebra associated to afﬁne hypertoric varieties. Recall that one can construct anafﬁne hypertoric variety M with the input of a rational subspace V ⊂ R n . In thissetting the resulting hypertoric Schur algebra R ( V ) can be interpreted as a con-volution algebra for a union of resolutions of stratum slices of M . In particular,for a rational polarized arrangement ( V, η, ξ ) with the same underlying subspace V ⊂ R n , the convolution algebra for the resolution M → M is a subalgebraof the associated hypertoric Schur algebra. Braden–Proudfoot–Webster showedin [BPW16, Proposition 6.16, Example 6.18] that the convolution algebra of theresolution M → M is categoriﬁed by Harish-Chandra bimodules for hypertoriccategory O . One expects the entire hypertoric Schur algebra to be similarly cate-goriﬁed by Harish-Chandra bimodules with more general support and similarlyto obtain natural a q -deformation of the hypertoric Schur algebra, or q -hypertoricSchur algebra.In [BM17], Braden and the second author observed that the hypertoric Schuralgebras studied in [BM] can be deﬁned in terms of the underlying matroid. Fol-lowing this observation, they deﬁned a matroidal Schur algebra R ( M ) associated toany matroid M .One motivation for deﬁning the category O for oriented matroid programs inthe present paper was to provide the foundation to categorify and ﬁnd natural q -deformations of matroidal Schur algebras for orientable, but non-realizable ma-troids using an appropriate category of Harish-Chandra bimodules.1.6. Outline of paper.

In Section 2 we describe the combinatorial set-up of ori-ented matroid programs and parameter spaces, as well as some useful topology.In Section 3 we deﬁne the algebra A ( P , U ) and in Section 4 we study its quadraticdual and prove Theorem 4.2. In Section 5 we deﬁne the algebra B ( P , U ) and proveTheorem 5.12. Section 6 develops more topology, resulting in a nice description ofthe center of B ( P , U ) .In Section 7, we study the category of ﬁnitely-generated (right) A ( P , U ) -modules.Under the Euclidean assumption, we prove that A ( P , U ) is quasihereditary (The-orem 7.6) and Koszul (Theorem 7.13). In the non-Euclidean setting, we describethe Grothendieck class of the graded indecomposable projective modules (Theo-rem 7.9), prove the Koszulity condition on Hilbert series (Theorem 7.14) and showthat Theorem 7.6 requires the Euclidean assumption (Example 7.10). In Section 8,we study the derived categories of graded ﬁnitely-generated A ( P , U ) -modules forvarying Euclidean P and a ﬁxed M and prove Theorem 1.16.1.7. Acknowledgements.

We would especially like to thank Tom, Tony, Nick andBen, whose paper [BLPW10] was a major source of inspiration for us and formsthe foundation of this paper.The second author is grateful to Jens Eberhardt for introducing him to the worldof oriented matroids. Thanks to Federico Ardila who put us in touch with JimLawrence and thanks to Jim for answering our silly questions. The second au-thor is also grateful to Catharina Stroppel for helpful conversations about ﬁnite-dimensional algebras.

ETHAN KOWALENKO AND CARL MAUTNER

The second author was supported in part by a Simons Foundation Collabora-tion Grant and NSF grant DMS-1802299. He would also like to thank the Mathe-matics department at Dartmouth College for its hospitality.2. C

OMBINATORIAL S ETUP

In this section we brieﬂy introduce the notation we will need to work with ori-ented matroids, but assume some familiarity with the basic notions. To the uniniti-ated reader, we recommend [BLVS +

99] (particularly the ﬁrst chapter) for a surveyand as a useful reference.2.1.

Generic oriented matroid programs.

For an index set I and any function Z : I → { , + , −} , let Z := { i | Z ( i ) = 0 } ⊂ I be the support of Z and z ( Z ) := I \ Z and be the zero set of Z .Let M be an orientable matroid of rank d on the set I . Let M be an orientedmatroid such that M = M is its underlying unoriented matroid.Let B denote the set of bases for M . We let C = C ( M ) denote the set of signedcircuits and C ∗ = C ∗ ( M ) the set of signed cocircuits, both regarded as subsetsof the set of functions I → { , + , −} . Note the unoriented matroid M = M hascircuits { X | X ∈ C } and cocircuits { Y | Y ∈ C ∗ } . The dual oriented matroid M ∨ is given by switching the roles of the circuits and cocircuits (i.e. C ( M ∨ ) = C ∗ ( M ) and C ∗ ( M ∨ ) = C ( M ) ), while the bases B ∨ of the underlying matroid M ∨ = M ∨ are the complements in I of the elements of B .Let S ⊂ I . Then the set { X ∈ C ( M ) | X ⊂ I \ S } is the set of circuits of an oriented matroid M\ S on I \ S , called the deletion of S from M . The set { X | S : S → { , + , −} | X ∈ C ( M ) and X ∩ S is inclusion minimal and nonempty } gives the set of circuits of an oriented matroid M / ( I \ S ) on S , called the contraction of M to S . Duality exchanges contraction and deletion: ( M /S ) ∨ = M ∨ \ S and ( M\ S ) ∨ = M ∨ /S. An element i ∈ I is a loop of M if { i } is the support of a circuit of M . Dually, i ∈ I is a coloop of M if i is not contained in the support of any circuit of M .Let I be the set E n = { , . . . , n } . An oriented matroid program P = ( f M , g, f ) is anoriented matroid f M on the set E n ⊔ { f, g } such that ( f M\ f ) /g = M , g is not a loopand f is not a coloop. In particular, the rank of f M is d + 1 , and this is deﬁned to bethe rank of P .The deletion N = f M\ f of f from f M is called the corresponding afﬁne orientedmatroid . Remark . Our constructions will only depend on the contraction f M /g , whichis a single element extension of M on E n ⊔ { f } , and the deletion f M\ f , which isa single element lift of M on E n ⊔ { g } . Thus for our purposes it would be morenatural to deﬁne an oriented matroid program as a pair, which we have taken toaffectionately calling an eft , of a single element extension and single element lift of M . We will refrain from doing so in this paper as the original notion appears tobe standard in the ﬁeld. CATEGORY O FOR ORIENTED MATROIDS 9

Deﬁnition 2.2.

We say that the oriented matroid program P = ( f M , g, f ) is generic for M if(1) for any cocircuit Y of N = f M\ f , if | z ( Y ) | > d , then Y ( g ) = 0 , and(2) for any circuit X of f M /g , if | z ( X ) | > n − d , then X ( f ) = 0 . Remark . As the rank of the oriented matroid f M /g on E n ⊔ { f } is d , for anycircuit X of f M /g , | X | ≤ d + 1 and so the X has at least n − d zero entries. Inthe case of equality, the X contains a basis of f M /g and so z ( X ) is independent in ( f M /g ) ∨ .Dually, for any cocircuit Y of N , | z ( Y ) | ≥ d and if equality holds z ( Y ) is inde-pendent in N . Example 2.4.

As explained in Example 1.5, every polarized arrangement V =( V, η, ξ ) naturally gives rise to an oriented matroid program P . If η and ξ aregeneric in the sense of Theorem 1.1, then P is generic as well.We now deduce some simple consequences of genericity. Lemma 2.5.

Suppose P = ( f M , g, f ) is generic. Then N = f M\ f has no loops and f M /g has no coloops.Proof. We prove the ﬁrst statement and the second follows by duality. By ourassumption in the deﬁnition of an oriented matroid program, g is not a loop of N , so g ∈ Y for some cocircuit Y of N . By Deﬁnition 2.2(1), | z ( Y ) | = d and soRemark 2.3 implies z ( Y ) is independent. If there were a loop i of N , then i ∈ z ( Y ) contradicting the fact that z ( Y ) is independent. (cid:3) Lemma 2.6.

Assume P is generic. Then there is a natural bijection between the set B ofbases for M and the set of feasible cocircuits for N = f M\ f .Proof. Consider the map that takes a feasible cocircuit Y for N to its zero set b := z ( Y ) . As Y is feasible, Y ( g ) = + and so by condition (1), Y must have d zeroentries. Then Y has n + 1 − d non-zero entries and is a circuit of N ∨ (which hasrank n − d ), so any subset of Y of size n − d is a basis for N ∨ . In particular, Y \{ g } is a basis for N ∨ , so its complement b ⊔ { g } is basis for N . Thus b is a basis for M = N /g .To show that this is a bijection, suppose b is a basis for M . Then b ⊔ { g } is abasis for N , its complement E n \ b is a basis for N ∨ and so ( E n \ b ) ∪ g must containa cocircuit Y for N . By condition (1), either Y = ( E n \ b ) ∪ g or Y ⊂ E n \ b . But thelatter is not possible as E n \ b is a basis for N ∨ . We conclude that there is a uniquechoice feasible cocircuit Y with support Y = ( E n \ b ) ∪ g . (cid:3) For b ∈ B , we let Y b be denote the corresponding feasible cocircuit.We will often use three constructions to obtain new generic oriented matroidprograms from a generic oriented matroid program P : duality, deletion, and con-traction. Recall that duality for oriented matroid programs takes the program P = ( f M , g, f ) to the program P ∨ = ( f M ∨ , f, g ) with underlying oriented matroid ( f M ∨ \ g ) /f = (( f M /g ) \ f ) ∨ = M ∨ . For any S ⊂ E n , we denote the contraction and deletion of S by P /S := ( f M /S, g, f ) and P \ S := ( f M\ S, g, f ) , respectively.Note that P is generic if and only of P ∨ is generic. If P is generic and S ⊂ b for some b ∈ B , then P /S is generic and has rank d + 1 − | S | . If P is generic and S ∩ b = ∅ for some b ∈ B , then P \ S is also generic of rank d + 1 . Lemma 2.7.

For any oriented matroid M there exists a generic oriented matroid program P = ( f M , g, f ) such that ( f M /g ) \ f = M .Proof. For example, for any order on E n , consider the lexicographic extension M ′ = M [ E n ] by a point f with respect to this order (Note that this is the sameas taking the extension M [ I min ] where I min is a lexigraphically minimal basis of M ). By a Lemma of Todd [BLVS +

99, Lemma 7.2.6], any circuit X of M ′ with morethan n − d zero entries satisﬁes X ( f ) = 0 . It then remains to deﬁne f M as a sin-gle element of lifting of M ′ by a point g , such that N = f M\ f satisﬁes property(1) above. This can be done via the dual construction: consider the colocalization τ : C ( M ′ ) → { + , − , } deﬁned for any X ∈ C ( M ′ ) by τ ( X ) = X i , where i is theminimal element of E n such that X i = 0 . Let f M be the lexicographic lifting of M ′ deﬁned by τ (in other words the dual of the lexicographic extension of ( M ′ ) ∨ associated to τ ). (cid:3) For the rest of the paper we assume that P is generic.2.2. Bounded feasible topes and sign vectors.

In this section we recall the notionsof bounded and feasible topes and show in Corollary 2.15 that when P is genericthere is a natural bijection between bases B of M and bounded feasible topes for P . For any functions Z, Z ′ : I → { , + , −} , their composition Z ◦ Z ′ : I → { , + , −} is deﬁned by Z ◦ Z ′ ( i ) = ( Z ( i ) if Z ( i ) = 0 Z ′ ( i ) otherwiseWe say that Z is a face of Z ′ if Z ◦ Z ′ = Z ′ .The nonzero covectors of an oriented matroid on the set I are the functions I → { , + , −} which can be written as the composition of cocircuits. The set ofcovectors of M is denoted by L ( M ) , and includes the zero function . It has anatural poset structure deﬁned by Y ≤ X if Y is a face of X . The poset L ( M ) ispure with minimal element , the zero function. The maximal elements of L ( M ) are called topes , while the minimal elements of L ( M ) \{ } are the covectors of M .The rank of Y ∈ L ( M ) is given as ρ ( Y ) = d − r ( z ( Y )) , where r is the rankfunction of the underlying matroid M . For Y , Y ∈ L ( M ) , the join Y ∨ Y is theminimal covector that has both Y and Y as faces, which only exists if there is atope T with both Y and Y as faces. The meet Y ∧ Y is the maximal covector faceof both Y and Y . Note that the meet of Y , Y ∈ L ( M ) always exists, but is thezero function when Y and Y do not have a common cocircuit face. Deﬁnition 2.8. (Feasible covectors and afﬁne space) Let P = ( f M , g, f ) be a genericprogram and let L = L ( N ) denote the set of covectors for N = f M\ f . The afﬁne CATEGORY O FOR ORIENTED MATROIDS 11 space of P is: A = { Y ∈ L | Y ( g ) = + } . We call elements of A feasible covectors.We say that the boundary of afﬁne space is: A ∞ = { Y ∈ L | Y ( g ) = 0 } Notice that A ∞ deﬁnes an oriented matroid on E n ∪ { g } which is equal to M with g adjoined as a loop. Also, notice that the join of covectors in A is also in A ifit exists, while their meet is in A if and only if they share a common cocircuit facein A ; Otherwise, their meet is in A ∞ . Deﬁnition 2.9. (Feasible topes) Let T ( L ) denote the set of topes of N . We let F = A ∩ T ( L ) denote the set of feasible topes.Notice that the deﬁnition of feasible topes does not depend on f . Remark . By Lemma 2.5 the topes of N are the covectors T such that z ( T ) = ∅ .A sign vector is a function α : E n → { + , −} , usually written as α ∈ { + , −} n .There is an obvious injective map from F to { + , −} n given by forgetting the valueat g (which is always + ). We may refer to the sign vectors in the image as feasible sign vectors, and in a slight abuse of notation we identify feasible topes with thecorresponding sign vectors. When there is a risk of confusion, we will write T α todenote the feasible tope of N corresponding to a sign vector α . Deﬁnition 2.11. (Directions and optimality)(1) We refer to covectors in the boundary of afﬁne space Z ∈ A ∞ as directions in A . We say that a direction is increasing (resp. decreasing or constant)with respect to f if Z ( f ) = + (resp. Z ( f ) = − or Z ( f ) = 0 ).(2) For a feasible tope T ∈ F and a feasible covector face Y of T , we say thatthe direction Z ∈ A ∞ is feasible for Y in T if Z ◦ Y is a face of T .(3) A feasible covector Y that is a face of T ∈ F is an optimal solution for T ifthere is no feasible increasing direction for Y in T . Deﬁnition 2.12. (Bounded sign vectors) A sign vector α ∈ { + , −} n is unbounded ifthere exists a increasing direction Z ∈ A ∞ such that Z | E n ◦ α = α . If no such Z exists, we say α is bounded . Similarly, a tope T is bounded if the sign vector T | E n is bounded.Let B denote the set of bounded sign vectors and P = F ∩ B denote the set ofbounded and feasible sign vectors.

Remark . Let Y be a bounded, feasible tope. Note that if Y , Y are optimalsolutions for Y , then so is Y ◦ Y . If Y is an optimal solution and Y ∈ A is a faceof Y , then Y is also optimal. It follows that if Y has an optimal solution, then ithas an optimal cocircuit. Theorem 2.14.

Assume P is generic. Then every feasible, bounded tope Y has a uniqueoptimal solution (cocircuit) and the resulting map from P to the set of feasible cocircuitsfor N is a bijection. Proof.

Let Y ∈ P . Recall that an optimal solution exists, without any conditionon P , by Bland and Lawrence’s Main Theorem of Oriented Matroid Programming(see [BLVS +

99, 10.1.13] for a survey).We ﬁrst show that if P is generic, then such a solution is unique.Suppose that Y has two distinct optimal cocircuits Y and Y . Then Y ◦ Y ∈ A must also be an optimal solution. Replacing Y by another cocircuit face of Y ◦ Y ifnecessary, without loss of generality, we may assume that Y and Y are joined byan edge (i.e., the rank of Y ◦ Y is 2). There then exist two cocircuits at the boundary ± Z ∈ A ∞ on the pseudoline Y Y such that Y ◦ Y = Y ◦ Z and Y ◦ Y = Y ◦ − Z .(The cocircuit Z can by obtained via elimination of g from the pair Y , Y and thiselimination is unique up to sign as Y , Y form a modular pair). By optimality of Y and Y , we conclude that ± Z must be constant directions. Note that z ( Y ◦ Y ) isan independent set in M of cardinality d − . The contraction f M / ( { g } ∪ z ( Y ◦ Y )) is a rank 1 oriented matroid where f is a loop since the cocircuits are both zero on f . Thus z ( Y ◦ Y ) ∪ { f } contains a circuit X of f M /g such that X ( f ) = 0 . Thiscontradicts condition (2) since X is zero on at least n − d + 1 entries.It remains to show that the map from P to feasible cocircuits is a bijection. Givena cocircuit Y ∈ A , we would like to show that Y is the optimal solution for aunique tope T ∈ P . We construct such a T as follows. For any i ∈ z ( Y ) , weknow that P i := P / ( z ( Y ) \{ i } ) is a generic program whose afﬁne space is one-dimensional. There is a unique cocircuit Z i in P i such that Z i ( f ) = − , and thiscocircuit is the restriction of a cocircuit Z i in f M /g . Then Z i ( j ) = 0 for j ∈ z ( Y ) \{ i } ,while Z i ( i ) = 0 and Z i ( f ) = − . Then T is deﬁned to be the composition Z ◦ Y ,where Z is the composition of all Z i , i ∈ z ( Y ) , taken in any order.This T is feasible since T ( g ) = Y ( g ) = + , and unique since it agrees on z ( Y ) with the unique bounded feasible tope of P \ ( E n \ z ( Y )) . To show that T is bounded,recall that an equivalent deﬁnition for a feasible tope to be bounded is that it mustbe in the bounded cone of some b ∈ B (see [BLVS +

99, Deﬁnition 10.1.8.ii, Corollary10.1.10.ii]), meaning that it agrees on b with a bounded tope of P \ ( E n \ b ) (see alsoDeﬁnition 2.18). Since T is in the bounded cone of z ( Y ) ∈ B , we have that T isbounded. (cid:3) Combining the bijections of Lemma 2.6 and Theorem 2.14 we obtain our desiredcorrespondence.

Corollary 2.15.

There is a natural bijection between the set B of bases for M and the set P of bounded feasible topes: µ : B → P , which takes a basis b to the tope whose optimal cocircuit is Y b . (Recall that Y b is the feasiblecocircuit with z ( Y b ) = b .) We conclude this section with a discussion of the effect of duality on the bijec-tion µ .Recall the following result about duality for oriented matroid programs. Proposition 2.16. [BLVS +

99, Corollary 10.1.11]

Let F ∨ , B ∨ , and P ∨ respectively de-note the sets of feasible, bounded, and bounded feasible sign vectors for the dual program P ∨ = ( f M ∨ , f, g ) . Then F ∨ = B and B ∨ = F , and so P ∨ = P . CATEGORY O FOR ORIENTED MATROIDS 13

Let B ∨ denote the set of bases of M ∨ . Then b b c := E n \ b deﬁnes a bijection B → B ∨ . Let µ ∨ : B ∨ → P ∨ be the bijection for the dual program P ∨ deﬁned asabove. Recall that P ∨ = P . Proposition 2.17 (Complementary Slackness) . For any b ∈ B , µ ( b ) = µ ∨ ( b c ) .Proof. Recall that P is generic if and only if P ∨ is generic. This is then the “Comple-mentary Slackness” theorem of Bland applied to generic programs (cf. [BLVS + (cid:3) Cone relation and Euclidean oriented matroid programs.

In this section weconsider a binary relation on the set P of bounded feasible topes (or via µ equiva-lently on the set of bases B ). Deﬁnition 2.18.

For any basis b ∈ B we deﬁne the negative cone as B b = { β ∈ { + , −} n | µ ( b )( i ) = β ( i ) for all i ∈ b } . Notice that this set of sign vectors depends on M and f but does not depend on g ,in the sense that the signs µ ( b )( i ) for i ∈ b only depend on the cocircuits of f M /g . Proposition 2.19 (Complementary Slackness) . For any b ∈ B , let X b c be the feasiblecocircuit of P ∨ with z ( X b c ) = b c = E n \ b . Then for any α ∈ P = P ∨ , α ∈ B b if andonly if X b c is a face of α .Proof. By deﬁnition, the tope α is in the cone B b if and only if α ( i ) = µ ( b )( i ) for all i ∈ b. By Proposition 2.17, we can rewrite the above condition as α ( i ) = µ ∨ ( b c )( i ) for all i ∈ b. Under the bijection µ ∨ from Corollary 2.15 for the dual program P ∨ , we have that µ ∨ ( b c )( i ) = X b c ( i ) for all i ∈ b . Thus the previous condition becomes α ( i ) = X b c ( i ) for all i ∈ b, which in turn is equivalent to X b c is a face of α in the afﬁne space associated to P ∨ . (cid:3) Deﬁnition 2.20 (Cone relation) . For α, β ∈ P , we write β (cid:22) α whenever β ∈B µ − ( α ) .The binary relation (cid:22) on P is reﬂexive and anti-symmetric, but not necessarilytransitive. Let ≤ denote the transitive closure of (cid:22) . In general, the binary relation ≤ on P does not deﬁne a poset, as the closure may no longer be anti-symmetric.In the following sections we will deﬁne algebras associated to the program P .To ensure that these algebras are quasi-hereditary, we will need the relation ≤ todeﬁne a poset. It turns out that this is equivalent to a well-known condition onthe oriented matroid program P , namely we ask that P be Euclidean . To recall thedeﬁnition, we ﬁrst deﬁne the following graph associated to P . Deﬁnition 2.21.

Let G P be the graph whose vertices are vertices in A (i.e., thefeasible cocircuits for N ) and whose edges are the edges in A (i.e., the feasible cov-ectors of N of rank 2). By our genericity condition, the graph is naturally directedby orienting each edge in increasing direction with respect to f . Deﬁnition 2.22.

For b, b ′ ∈ B , we write b ≤ b ′ if there is a directed path from Y b to Y b ′ in the graph G P . Deﬁnition 2.23.

The program P is Euclidean if the directed graph G P containsno directed cycles. Equivalently, P is Euclidean if the binary relation on B fromDeﬁnition 2.22 is anti-symmetric.By a result of Edmonds and Fukuda, the Euclidean property is well-behavedunder duality: Proposition 2.24. [BLVS +

99, Corollary 10.5.9]

An oriented matroid program P =( f M , g, f ) is Euclidean if and only if its dual program P ∨ = ( f M ∨ , f, g ) is Euclidean. Importantly for us, for Euclidean programs, the transitive closure of the conerelations is a poset. In fact these two conditions are equivalent:

Lemma 2.25.

The oriented matroid program P = ( f M , g, f ) is Euclidean if and only ifthe transitive closure of the cone relation is anti-symmetric.Proof. By Proposition 2.24, it sufﬁces to show that the dual program P ∨ is Eu-clidean if and only if the transitive closure on the cone relation is anti-symmetric.By deﬁnition, P ∨ is Euclidean if and only if the binary relation ( B ∨ , ≤ ) is anti-symmetric. Thus it sufﬁces to show that the bijection µ ∨ : B ∨ → P ∨ = P is order reversing.In other words, we wish to show that for b , b ∈ B , µ ( b ) ≥ µ ( b ) if and only if b c ≤ b c . As ( P , ≤ ) is the transitive closure of (cid:22) , without loss of generality we may sup-pose µ ( b ) (cid:23) µ ( b ) . Then µ ( b ) ∈ B b and by Proposition 2.19, X b c is a face of µ ( b ) = µ ∨ ( b c ) . As X b c is the (unique) optimal cocircuit of the tope µ ∨ ( b c ) , there isa directed path from X b c to X b c . Thus b c ≤ b c as desired.For the other direction, it sufﬁces to consider the case where there is a directededge from X b c to X b c . Then X b c is a cocircuit face of µ ∨ ( b c ) = µ ( b ) . By Propo-sition 2.19 this implies µ ( b ) ∈ B b and so µ ( b ) ≤ µ ( b ) , which completes theproof. (cid:3) On the existence of Euclidean generic programs.

Unlike genericity, it is notclear (at least to the authors) that any oriented matroid M can be extended to aEuclidean generic oriented matroid program f M .On the other hand, we have already seen in Example 1.7 a non-realizable Eu-clidean program ( f M , g, f ) for which M = f M /g \ f is realizable. There also appearto be many examples for which the oriented matroid M is non-realizable, and soour setting signiﬁcantly generalizes that of [BLPW10].In this section we ﬁrst give such an example and then give a criterion on M forthe existence of a Euclidean generic oriented matroid program lifting and extend-ing M . Example 2.26.

The V´amos matroid V (rank 4 on 8 elements) is not representableover any ﬁeld, but is orientable by [BV78]. We now deﬁne a Euclidean genericprogram e V by adjoining elements to V as labelled and orientated in [BV78]. Let Here by ( B ∨ , ≤ ) we are referring to the binary relation on B ∨ coming from the directed graph G P ∨ . CATEGORY O FOR ORIENTED MATROIDS 15 ( e V , g, f ) be the generic oriented matroid program obtained from V by adjoining g using the lexicographic one-element lift deﬁned by the cobasis { , , , } , and ad-joining f by a lexicographic one-element extension deﬁned by the basis { , , , } .One can verify that the graph G P of this program has no directed cycles, so thisprogram is Euclidean and generic.We are grateful to Jim Lawrence for suggesting that we consider the followingstatement. Proposition 2.27.

Let M be an oriented matroid. If one can adjoin elements g and f to M to obtain a Euclidean generic oriented matroid program ( f M , g, f ) , then M has asimplicial tope. Before beginning the proof, we note that it is an open conjecture of Las Vergnas [LV80](see also [BLVS +

99, 7.3.10]) that any oriented matroid M should have a simplicialtope. Thus, the proposition implies that if for any oriented matroid M there is aEuclidean generic oriented matroid program ( f M , g, f ) for which f M /g \ f = M ,then Las Vergnas’ conjecture holds. Proof.

Suppose ( f M , g, f ) is a Euclidean generic oriented matroid program suchthat ( f M /g ) \ f = M .As P is Euclidean, there exists a minimal vertex of G P . Let Y min be the feasiblecocircuit of N corresponding to such a minimal vertex. Let T min be the tope whoseoptimal cocircuit is Y min , which exists by Theorem 2.14. Then Y min is the onlyfeasible cocircuit face of T min , so for all other cocircuit faces Y , Y ( g ) = 0 . Considerthe subtopes covered by T min . There are d such subtopes that have Y min as a face,all of which are feasible. Any other subtope Z covered by T min cannot be feasibleand as all non-feasible cocircuit faces Y of T satisfy Y ( g ) = 0 , then z ( Z ) = g .There can be at most one such subtope covered by T min . By [BLVS +

99, Exercise4.4], any tope of f M covers at least d + 1 subtopes, so there does exist a subtope Z and T min covers exactly d + 1 subtopes. By the same exercise, it follows that T min is simplicial, hence the subtope Z is as well, which is in turn a tope of M . (cid:3) Linear systems of parameters.

Recall that M = M denotes the underlyingmatroid of M . Consider the matroid complex ∆ of M - the simplicial complex ofindependent sets of M .Fix a ﬁeld k . Let k n be the standard n -dimensional vector space with basis t , . . . , t n . Deﬁnition 2.28.

For any ﬁeld k , the face ring of the matroid complex ∆ of M isdeﬁned as k [ M ] : = k [ t , . . . , t n ] / ( t S | S ∆)= k [ t , . . . , t n ] / ( t X | X ∈ C ) where t S := Q i ∈ S t i for any S ⊂ E n . We give k [ M ] a Z ≥ -grading by setting deg( t i ) = 2 for all i . Deﬁnition 2.29.

Recall d denotes the rank of M . A linear system of parameters (l.s.o.p.) for k [ M ] is a set Θ = { θ , θ , . . . , θ d } ⊂ k n such that k [ M ] is a ﬁnitely-generated k [ θ , θ , . . . , θ d ] -module. Equivalently k [ M ] / (Θ) is a zero-dimensional ring, where Θ is the image of Θ in k [ M ] . Remark . Stanley [Sta04] deﬁnes an l.s.o.p. as a subset of k [ M ] ⊂ k [ M ] . Theset Θ is a l.s.o.p. in the sense we deﬁne above if its image in k [ M ] is an l.s.o.p. inthe sense used by Stanley.We introduce the following perhaps non-standard deﬁnition: Deﬁnition 2.31.

We call a subspace U ⊂ k n a parameter space for k [ M ] if the com-position U ֒ → k n ։ Span { t i | i ∈ b } is an isomorphism for any basis b ∈ B . Example 2.32. If M is realizable as a hyperplane arrangement coming from a k -vector subspace V ⊂ k n , we can choose U to be V . Lemma 2.33. If U ⊂ k n is a parameter space for k [ M ] , then any basis of U is a l.s.o.p.for k [ M ] . If Θ is an l.s.o.p. for k [ M ] , then its span Span (Θ) ⊂ k n is a parameter space for k [ M ] .Proof. Suppose U is a parameter space. By deﬁnition dim U = dim Span { t i | i ∈ b } = d. Suppose θ , . . . , θ d is a basis of U . By [Sta04, Lemma 2.4(a)], θ , . . . , θ d is an l.s.o.p.for k [ M ] if and only if for every facet of ∆ , that is basis b ∈ B , the list θ , . . . , θ d ∈ k n projects to a spanning set of Span { t i | i ∈ b } . This is true by the deﬁnition of U .Similarly, if θ , . . . , θ d ∈ k n is an l.s.o.p., then for any basis b ∈ B , the projectionsto Span { t i | i ∈ b } of θ , . . . , θ d are a spanning set. Thus the projection fromSpan (Θ) to Span { t i | i ∈ b } is an isomorphism and Span (Θ) is a parameter space. (cid:3) By the Noether Normalization Lemma, if k is an inﬁnite ﬁeld, then a l.s.o.p. for k [ M ] exists. From now on, we assume that an l.s.o.p. exists and ﬁx a choice ofl.s.o.p. Θ and its span U .2.6. Linear systems of parameters and duality.

Let u , . . . , u n be the basis of ( k n ) ∗ dual to t , . . . , t n . It will be convenient for us to view the matroid complex k [ M ∨ ] of the dual matroid M ∨ as the appropriate quotient of k [ u , . . . , u n ] .Let U ⊥ be the kernel of the natural map ( k n ) ∗ → U ∗ . Lemma 2.34. U ⊥ is a parameter space for the face ring of the dual matroid k [ M ∨ ] .Proof. For any b ∈ B a basis for M , let b c := E n \ b be its complement, which is abasis for M ∨ . It sufﬁces to show that the projection U ⊥ → Span { u i | i ∈ b c } isan isomorphism. As the two vector spaces have the same dimension, it sufﬁces toshow the null space is trivial. Suppose x is in the null space, so x ∈ U ⊥ ∩ Span { u i | i ∈ b } . Then for any u ∈ U , h x, u i = h x, pr b u i , where pr b : U → Span { t i | i ∈ b } is the projection. As U is a parameter space for k [ M ] , the projection pr b is an isomorphism and so h x, w i = 0 for any w ∈ Span { t i | i ∈ b } . Thus x = 0 and we may conclude that the projection U ⊥ → Span { t i | i ∈ b c } is an isomorphism. (cid:3) CATEGORY O FOR ORIENTED MATROIDS 17

We deﬁne the dual of the pair ( P , U ) to be ( P , U ) ∗ = ( P ∨ , U ⊥ ) . Proposition 2.35. k [ M ] is a free algebra over k [ θ , . . . , θ d ] = Sym U ⊂ k [ t , . . . , t n ] ,while k [ M ∨ ] is a free algebra over Sym U ⊥ ⊂ k [ u , . . . , u n ] . Both have rank | B | .Proof. Recall the face ring of a matroid complex is shellable (e.g., [Sta04, Proposi-tion III.3.1]) and so the result follows [Sta04, Theorem III.2.5]. (cid:3)

3. T

HE ALGEBRA A The deﬁnition of A . Recall that P = ( f M , g, f ) is a generic oriented matroidprogram and U ⊂ k n is a parameter space for M = M .Let Q denote the quiver with vertex set F , the set of feasible topes, and arrowsbetween topes that differ by exactly one sign. We say that two topes α, β that differby exactly one sign are adjacent and write α ↔ β . If α ↔ β and differ in the i -thcomponent, we write β = α i .Let P ( Q ) denote the path algebra for Q , which is generated by orthogonal idem-potents { e α | α ∈ F} and edge paths { p ( α, β ) } where α and β are adjacent and p ( α, β ) is the path from α to β . We write p ( α , . . . , α k ) for the element in the quiveralgebra obtained as the composition p ( α , α ) · . . . · p ( α k − , α k ) . Deﬁnition 3.1.

Let e A = e A ( P ) = e A ( f M , g, f ) be the quotient of P ( Q ) ⊗ k Sym( k n ) ∗ = P ( Q ) ⊗ k k [ u , . . . , u n ] by the two-sided ideal generated by the following relations: ( A e α = 0 for α

6∈ P , ( A p ( α, γ, β ) = p ( α, δ, β ) for any four distinct topes α, β, γ, δ ∈ F where α and β are each connected to γ and δ by an edge, and( A ) p ( α, α i , α ) = e α u i whenever α, α i ∈ F differ only in the sign of i ∈ E n .We let A = A ( P , U ) = A ( f M , g, f, U ) be A := e A ⊗ Sym( k n ) ∗ Sym(( k n ) ∗ / ( U ⊥ )) = e A ⊗ Sym( k n ) ∗ Sym U ∗ , or equivalently, the quotient of e A by the additional relations( A ) x = 0 for any x ∈ U ⊥ ⊂ Sym( k n ) ∗ . Remark . When the pair ( P , U ) comes from a polarized arrangement as in Ex-amples 1.4 and 1.5, there is an equality A ( P , U ) = A ( V, η, ξ ) .As in 2.6, for bookkeeping, we will use the dual coordinates for the dual matroidprogram, so we view e A ( P ∨ ) and A ( P ∨ , U ⊥ ) as the analogous quotients of P ( Q ) ⊗ k Sym( k n ) = P ( Q ) ⊗ k k [ t , . . . , t n ] .3.2. Expressions for elements of A . We ﬁrst introduce some terminology for pathsin the quiver Q . To distinguish between paths in Q and their images in A , we willuse the notation α → α → · · · → α r for a path of length r in the quiver Q .A path α → · · · → α r in Q is taut if α and α r differ in exactly r coordinates.To relate paths, we use the following notion. Deﬁnition 3.3.

Two paths

P, P ′ in Q are related by an elementary homotopy (a sym-metric relation) if(i) P is the path α → · · · → α j → α j +1 → · · · → α r of length r while P ′ isthe path α → · · · → α j → β → α j → α j +1 → · · · → α r of length r + 2 forsome β adjacent to α j , or (ii) P is the path α → · · · → α j − → α j → α j +1 → · · · → α r while P ′ isthe path α → · · · → α j − → α ′ j → α j +1 → · · · → α r of the same length,where we assume α j − and α j +1 differ in exactly 2 coordinates. Remark . Under our assumption that ( f M , g, f ) is generic, any feasible covectorfor N = f M\ f of rank d − has exactly two zero coordinates. Thus, in this setting,this deﬁnition of elementary homotopy coincides with that given in [BLVS + +

99] between feasible paths is of the formabove.We will use the following result:

Proposition 3.5.

Let P and P ′ be any two taut paths in Q with the same start and endpoints. Then P and P ′ are related by a sequence of elementary homotopies of type (ii) suchthat every intermediate path is also taut.Proof. If we consider instead paths in the entire tope graph and the more generalnotion of elementary homotopy, this is a result of Cordovil-Moreira [CM93] (seealso [BLVS +

99, Proposition 4.4.7]). Recall that a subset

R ⊂ T is T -convex ifit contains every shortest path between any two of its members and the set offeasible topes is T -convex (see [BLVS +

99, Deﬁnition 4.2.5] and the discussion thatfollows). The result then follows for paths in Q . (cid:3) Proposition 3.6.

Given a path P = ( α → · · · → α s ) in Q , let d i be the number of timesthe i -th coordinate changes twice. Then for any taut path P ′ = ( α = β → · · · → β r = α s ) , we have p ( α , . . . , α s ) = p ( β , · · · , β r ) · Y i ∈ E n u d i i in e A .Proof. Note that s − r ≥ with equality if and only if P is a taut path. We provethe proposition by induction on s − r . If s = r , then both paths are taut, and so byProposition 3.5 they are related by a sequence of elementary homotopies of type(ii). But an elementary homotopy of type (ii) descends to an equality in the pathalgebra by deﬁnition, so p ( α , . . . , α r ) = p ( β , . . . , β r ) , as desired.Assume that the statement holds whenever s − r < k for some positive integer k . Suppose that s − r = k . There exists a minimal ℓ such that α → · · · → α ℓ is taut,while α → · · · → α ℓ +1 is not taut. Then α and α ℓ +1 differ in ℓ − coordinatesand for some i , α ( i ) = α ℓ +1 ( i ) = α ℓ ( i ) .Notice that any taut path between α and α ℓ will have length exactly one morethan the length of a taut path from α to α ℓ +1 . Therefore, by Proposition 3.5, usinga sequence of elementary homotopies of type (ii) we can replace α → · · · → α ℓ − → α ℓ with a taut path α → α ′ → · · · → α ′ ℓ − → α ℓ +1 → α ℓ .This gives an equality in the algebra A : p ( α , α , . . . , α ℓ − , α ℓ − , α ℓ , α ℓ +1 ) = p ( α , α ′ , . . . , α ′ ℓ − , α ℓ +1 , α ℓ , α ℓ +1 )= p ( α , α ′ , . . . , α ′ ℓ − , α ℓ +1 ) u i . CATEGORY O FOR ORIENTED MATROIDS 19

We are then reduced to considering the path α → α ′ → · · · → α ′ ℓ − → α ℓ +1 → α ℓ +2 → · · · → α s of length s − and the number of times the i -th coordinate changes twice is d i − ,while the number of times every j -th coordinate changes twice remains d j for all j = i . We can then invoke the induction hypothesis to complete the proof. (cid:3) The following two corollaries are analogous to [BLPW10, Corollary 3.10].

Corollary 3.7.

Consider an element a = p · Y i ∈ E n u d i i ∈ e A, where p is a taut path in Q from α to β . Suppose γ denotes a feasible tope such that if α ( i ) = β ( i ) and d i = 0 , then γ ( i ) = α ( i ) = β ( i ) . Then a = a ′ · m, where a ′ is the concatenation of a taut path from α to γ with a taut path from γ to β and m is a product of u i ’s.In particular, if γ is not bounded, then a = 0 in both e A and A .Proof. For all j ∈ E n , either:(1) α ( i ) = β ( j ) and the j -th coordinate changes in the concatenation a ′ exactlyonce.(2) γ ( j ) = α ( j ) = β ( j ) and the j -th coordinate does not change in the concate-nation a ′ .(3) γ ( j ) = α ( j ) = β ( j ) and so the j -th coordinate changes exactly twice in theconcatenation a ′ .Proposition 3.6 then says: a ′ = p · Y i ∈ E n u d ′ i i , where d ′ i ∈ { , } and by our assumption on γ , we have d i ≥ d ′ i . Thus a = a ′ · Y i ∈ E n u d i − d ′ i i , as desired. (cid:3) Corollary 3.8.

Let b be the zero set of any feasible cocircuit face of α ∈ P . For any j ∈ E n , e α u j ∈ A can be written as a k -linear combination of paths { p ( α, α i , α ) | i ∈ b } .In particular, the image in A of the element a ∈ e A described in Corollary 3.7 can beexpressed a linear combination of paths in Q that pass through γ .Proof. For j ∈ b , the tope α j is feasible, so e α u j = p ( α, α j , α ) . On the other hand,as U is a parameter space for k [ M ] , the set { u i | i ∈ b } restricts to a basis of U ∗ . Thusfor any j b , u j ∈ A can be expressed as a linear combination of { u i | i ∈ b } . (cid:3) Alternative description of A . We conclude this section with a slightly dif-ferent description of our algebra A , which will make it easier to describe how A changes when we modify the choice of generic oriented matroid program.Let D be the path algebra over k of the quiver with two vertices labelled by + and − and an arrow in each direction. Let D n = D ⊗ D ⊗ . . . ⊗ D denote the n -foldtensor product of D with itself. In particular, D n is the path algebra on the quiverwith vertices labelled by the set { + , −} n of sign vectors, or equivalently verticesof an n -cube, and edges connecting any two sign vectors that differ in exactly oneposition, modulo the relations that whenever α, β ∈ { + , −} n differ in exactly twopositions i and j , we have an equivalence of paths in D n : p ( α, α i , β ) = p ( α, α j , β ) . (As before, α i ∈ { + , −} n denotes the sign vector that differs from α in exactly the i -th coordinate.)For any sign vector α , we again let e α denote the idempotent deﬁned as thetrivial path at the vertex labelled by α . Let e P = P α ∈P e α be the sum of idempo-tents corresponding to bounded, feasible topes and e f = P α e α be the sum ofidempotents corresponding to unbounded sign vectors.For each i ∈ E n , we consider the element θ i ∈ D n deﬁned as the sum θ i = X α ∈{ + , −} n p ( α, α i , α ) . Note that the center Z ( D n ) of D n is a polynomial algebra with generators θ i . Let ϑ : ( k n ) ∗ → k { θ i | i ∈ E n } be the isomorphism sending u i to θ i . Lemma 3.9.

The algebra e A from Deﬁnition 3.1 is isomorphic to the quotient of e P D n e P by the relation: ( A ′ ) e α = 0 for α

6∈ B ,and the algebra A is obtained by adding the additional relation: ( A ′ ) ϑ ( x ) = 0 for x ∈ U ⊥ .Equivalently, there are isomorphisms e A ∼ = e P D n e P / h e f e P i , A ∼ = e P D n e P / h e f e P i + h ϑ ( U ⊥ ) e P i . Proof.

To distinguish between the two deﬁnitions, let e A and A denote the orig-inal algebras and e A and A be the algebras deﬁned as in the lemma. Note thatthere is an injective homomorphism from e A to e A uniquely deﬁned by sendinga path in Q to the corresponding path in the cube quiver { + , −} n and sending u i to θ i . To see that it is surjective, it sufﬁces to observe that for any two topes α, β ∈ P , there exists a taut path in Q from α to β , which follows from [Cor82,Lemme 3.7]. (cid:3)

4. T

HE QUADRATIC DUAL OF A In this section we observe that A ( P , U ) is a quadratic algebra and that its qua-dratic dual is isomorphic to A ( P ∨ , U ⊥ ) . To ease notational clutter, in this sectionwe will write A for A ( P , U ) and A ∨ for A ( P ∨ , U ⊥ ) .To state and prove the results of this section, we will need some additionalnotation. Let Q P ⊂ Q be the full subquiver with vertices P ⊂ F . For α ∈ P , we let J α := { i | α i ∈ P} , CATEGORY O FOR ORIENTED MATROIDS 21 I α := { i | α i ∈ F} ,K α := { i | α i ∈ B} . Lemma 4.1.

The algebra A = A ( P , U ) is the quotient of the path algebra P ( Q P ) by thefollowing relations: ( A ′′ ) For distinct topes α, β, γ ∈ P , δ ∈ F , where α and β are each connected to γ and δ by an edge, p ( α, γ, β ) = ( p ( α, δ, β ) if δ ∈ P ( A / ′′ ) For any α ∈ P and w ∈ U ⊥ ∩ Span { u i | i ∈ I α } , if w = P i ∈ I α w i u i for some w i ∈ k , then X i ∈ J α w i p ( α, α i , α ) = 0 . In particular, it follows that A is a quadratic algebra.Proof. Note that there is a surjection from P ( Q ) to P ( Q P ) (by setting e α = 0 for all α

6∈ P ) and that this map factors through the natural map from P ( Q ) to A .We ﬁrst show that the map from P ( Q ) to A (and hence from P ( Q P ) to A ) issurjective. To do so, it sufﬁces to show that the image in A of any element of w ∈ U ∗ is in the image of the map from P ( Q ) to A . In A , we can express w as thesum w = P α ∈F we α , so it is enough to show that for any α ∈ F , we α is in theimage.Because α is feasible, there exists a feasible cocircuit face Y of α and z ( Y ) isa basis of M . Thus z ( Y ) ⊂ I α and so by our assumption that U is a parameterspace, the image of { u i | i ∈ I α } ⊃ { u i | i ∈ z ( Y ) } is a spanning set of U ∗ . For any w ∈ U ∗ , we can therefore write we α as a linear combination of elements of theform p ( α, α i , α ) where α i ∈ F . We conclude that map from P ( Q ) to A is surjective.To identify the kernel, note that ( A ′′ ) is simply the image of the relation ( A in P ( Q P ) and similarly the relations ( A and ( A combine to give ( A / ′′ ) .As P ( Q P ) is generated over its degree zero component by its degree one com-ponent and the relations above are quadric, we conclude that A is a quadraticalgebra. (cid:3) Recall that A is an A -bimodule. Since A is quadratic, A is a quotient of thetensor algebra T ( A ) over A of the form A = T ( A ) /T ( A ) · W · T ( A ) where W ⊆ N := ( T ( A ) ⊗ A T ( A )) is the space of quadratic relations in A . The quadratic dual of A is deﬁned to be A ! = T ( A ∗ ) /T ( A ∗ ) · W ⊥ · T ( A ∗ ) where W ⊥ ⊆ N ∗ := ( T ( A ∗ ) ⊗ A T ( A ∗ )) is the set of elements orthogonal to W . Theorem 4.2.

There is an isomorphism A ( P ∨ , U ⊥ ) ≃ A ( P , U ) ! .Proof. Mimicking the proof of [BLPW10, Theorem 3.11], we deﬁne an isomor-phism between ( A ∨ ) and ( A ) ∗ and show that space W ∨ of quadratic relationsin A ∨ coincides with the space W ⊥ of quadratic relations in A ! under this identiﬁ-cation. In degree zero, we have a canonical identiﬁcation A = k { e α | α ∈ P = P ∨ } = ( A ∨ ) . In degree one, { p ( α, β ) | α, β ∈ P such that α ↔ β } ⊂ A is a natural basis for A .As P = P ∨ , to distinguish elements of A and A ∨ , we let p ∨ ( α, β ) denote theelement of ( A ∨ ) associated to the arrow α → β in P ∨ .We now identify ( A ∨ ) with ( A ) ∗ as follows. First we attach a sign ε ( α ↔ β ) toeach pair α ↔ β of adjacent topes in P such that for distinct topes α, α i , α j , ( α i ) j ,an odd number of the edges of the square α o o / / O O (cid:15) (cid:15) α i O O (cid:15) (cid:15) α j o o / / ( α i ) j are attached a negative sign. We then identify ( A ∨ ) with ( A ) ∗ via the perfect pairing ( A ∨ ) × A → k h p ∨ ( α, β ) , p ( γ, δ ) i = ( ε ( α ↔ β ) if α = δ, β = γ, . For the remainder of the proof, we let N : = A ⊗ A A = T ( A ) ,N ∨ : = ( A ∨ ) ⊗ A ( A ∨ ) = T (( A ∨ ) ) . We wish to show that our choice of perfect pairing induces an isomorphism be-tween W ∨ ⊆ N ∨ and W ⊥ ⊆ N ∗ .Note that the relations of type ( A ′′ ) lie in e α W e β and the relations of type( A / ′′ ) lie in e α W e α . As these relations are homogeneous with respect to theidempotents e α for α ∈ P , we have a direct sum decomposition W = M α,β ∈P e α W e β . Moreover, e α W e β and e γ W ∨ e δ are orthogonal unless α = δ and β = γ . Thus itis enough to check that e β W ∨ e α ⊂ e β N ∨ e α is the perpendicular complement to e α W e β ⊂ e α N e β for any α, β ∈ P .Note that e α N e β , e β N ∨ e α are zero unless α = β or α and β differ in exactly twopositions and there is a path from α to β in P .We ﬁrst deal with the latter case. Suppose the two elements of E n where α and β differ are i = j . We must have that at least one of α i or α j is in P by assumption.We can assume that a j ∈ P . This can be done for the edges of the n-cube { + , −} n by identifying its vertices with monomialsin the exterior algebra Λ k { e , . . . , e n } and then using the standard differential to attach signs to edges.Restricting to P then gives a collection of signs as desired. CATEGORY O FOR ORIENTED MATROIDS 23 If a i ∈ P , then we have that e α N e β is a two dimensional k -vector space withbasis { p ( α, α i ) ⊗ p ( α i , β ) , p ( α, α j ) ⊗ p ( α j , β ) } and e α W e β = k { p ( α, α i ) ⊗ p ( α i , β ) − p ( α, α j ) ⊗ p ( α j , β ) } ⊂ e α N e β , , and e β W ∨ e α = k { p ∨ ( β, α i ) ⊗ p ∨ ( α i , α ) − p ∨ ( β, α j ) ⊗ p ∨ ( α j , α ) } ⊂ e β N ∨ e α . Pairing the two basis vectors together using the form deﬁned above we get ε ( α ↔ α i ) ε ( α i ↔ β ) + ε ( α ↔ α j ) ε ( α j ↔ β ) . By our choice of signs, the terms cancel and we conclude that e α W ⊥ e β = e β W ∨ e α . If α j

6∈ P , then either α j ∈ F\P and α j

6∈ B = F ∨ , or α j ∈ B\P = F ∨ \P ∨ and α j

6∈ F = B ∨ . Assume that α j ∈ F\P and α j

6∈ B = F ∨ . Then we have that e α N e β = k { p ( α, α i ) ⊗ p ( α i , β ) } = e α W e β since p ( α, α i , β ) = p ( α, α j , β ) = 0 in A . On the other hand, α j

6∈ F ∨ means that e β N ∨ e α = k { p ∨ ( β, α i ) ⊗ p ∨ ( α i , α ) } and ( A ) does not impose any relations, so e β W ∨ e α = 0 . Therefore e β W ⊥ e α = 0 = e β W ∨ e α . The case α j ∈ B\P = F ∨ \P ∨ and α j

6∈ F = B ∨ follows from the same argument on the dual side.Finally, consider the case where α = β . Note that e α N e α = k { p ( α, α i ) ⊗ p ( α i , a ) | i ∈ J α } . We identify e α N e α with k J α by regarding p ( α, α i ) ⊗ p ( α i , a ) as the standard basiselement labelled by i ∈ J α . We also use the standard pairing on k n to view U ⊥ as a subspace of k n . From the relations ( A / ′′ ), we ﬁnd that e α W e α is givenby pr J α ( U ⊥ ∩ k I α ) or equivalently by ( pr K α U ⊥ ) ∩ k J α , where pr S denotes theorthogonal projection from k n to k S for any S ⊂ { , . . . , n } .Taking the orthogonal complement of e α W e α ⊂ e α N e α using the ﬁrst descrip-tion gives: ( e α W e α ) ⊥ = ( pr J α ( U ⊥ ∩ k I α )) ⊥ = ( pr I α U ) ∩ k J α = ( pr K ∨ α U ) ∩ k J ∨ α = e α W ∨ e α . Here the second equality follows from the fact that ( pr S V ) ⊥ = V ⊥ ∩ k S , the thirdequality uses the fact that I α = K ∨ α and J α = J ∨ α (see Proposition 2.16) and theﬁnal equality follows from the second description of W above. (cid:3)

5. T

HE ALGEBRA B Following Braden–Licata–Proudfoot–Webster, we deﬁne in this section anotheralgebra B = B ( P , U ) associated to the pair ( P , U ) and prove that B is isomorphicto the quadratic dual A ! of A ( P , U ) . We also consider a deformed version e B ( P ) such that e B ( P ) ∼ = e A ( P ∨ ) . The algebras B and e B deﬁned in [BLPW10, Section4.1] coincide with those deﬁned here in the special case when ( P , U ) comes froma linear subspace as in Example 2.4.5.1. A topological lemma.

In this section we assume some familiarity with reg-ular cell complexes, posets and their geometric realizations and refer the readerto [BLVS +

99, Section 4.7] for more on these topics.Recall that L = L ( N ) denotes the poset of covectors for N = f M\ f . This posetis pure with a unique minimal element and a rank function ρ . Every covector isuniquely determined by its cocircuit faces. We will deﬁne the algebra B ( P , U ) from the poset structure of L , using an afﬁneversion of the following notion [BLVS +

99, Deﬁnition 4.1.2(ii)]:

Deﬁnition 5.1.

For any Y ∈ L ( N ) , the Edmonds-Mandel face lattice of Y , denoted F em ( Y ) , is the set of all faces of Y in L ( N ) . The opposite poset F lv ( Y ) := F em ( Y ) op is called the Las Vergnas face lattice of Y .Both F em ( Y ) and F lv ( Y ) are graded posets and by theorems of Folkman–Lawrenceand Edmonds–Mandel, they have the following topological interpretation [BLVS + Theorem 5.2.

The lattices F em ( Y ) and F lv ( Y ) are each isomorphic to the face lattices (oraugmented face posets) of PL regular cell decompositions of the ( ρ ( Y ) − -sphere. For any Y ∈ A , let Y ∞ be the unique maximal face of Y such that Y ∞ ∈ A ∞ .The face Y ∞ is equal to the composition of all cocircuit faces of Y in A ∞ . Deﬁnition 5.3.

Let Y ∈ A . The feasible Edmonds-Mandel face lattice of Y is F F em ( Y ) := ( F em ( Y ) ∩ A ) ∪ { } , while the feasible Las-Vergnas face lattice of Y is F F lv ( Y ) := F F em ( Y ) op . The core of A is A := { Y ∈ A | Y ∞ = 0 } .Note that A does not have a cellular interpretation, since the faces of a feasiblecovector need not be feasible. The same is true of F F em ( Y ) when Y ∞ = 0 . On theother hand, F F lv ( Y ) will always have a cellular interpretation, even if the program P is not generic. The assumption that g is generic implies F F lv ( Y ) is the face latticeof a pure simplicial complex, whose vertices correspond to the feasible facets of Y and whose maximal simplices correspond to the feasible cocircuits faces of Y .We will use the following lemma on the topology of F F lv ( Y ) to show that the al-gebra B ( P , U ) we deﬁne below is ﬁnite-dimensional (which will imply that A ( P , U ) is ﬁnite dimensional as well). Lemma 5.4.

Let Y ∈ A . • If Y ∞ = 0 , then the geometric realization k F F lv ( Y ) \{ , Y }k of the proper part ofthe feasible Las Vergnas face lattice of Y is a PL ( ρ ( Y ) − -sphere. • If Y ∞ = 0 , then k F F lv ( Y ) \{ , Y }k is a PL ( ρ ( Y ) − -ball.Proof. If Y ∞ = 0 , then F F lv ( Y ) = F lv ( Y ) and so the statement reduces to Theo-rem 5.2.Assume that Y ∞ = 0 , and let ∆ := F F lv ( Y ) \{ , Y } denote the proper part of the feasible Las Vergnas face lattice of Y .The geometric realization k ∆ k of ∆ is homeomorphic to the geometric realiza-tion k ∆ ord (∆) k of the order complex ∆ ord (∆) of ∆ (which is a subdivision of theformer). Further, using the canonical identiﬁcation ∆ ord (∆) ∼ = ∆ ord (∆ op ) , we ﬁndthat k ∆ k is homeomorphic to k ∆ ord (∆ op ) k .It will thus sufﬁce to prove that k ∆ ord (∆ op ) k is a PL ball. By restricting to P /z ( Y ) if necessary, we may assume that Y is a tope, so ρ ( Y ) = d + 1 and ∆ ord (∆ op ) is a ( d − -dimensional simplicial complex. CATEGORY O FOR ORIENTED MATROIDS 25

We ﬁrst note that ∆ ord (∆ op ) ∼ = ∆ ord (∆) is pure because ∆ is the face posetof a pure simplicial complex. By [BLVS +

99, Proposition 4.5.4], the graded poset F F em ( Y ) admits a recursive coatom ordering, which implies by [BLVS +

99, Lemma4.7.19] that its open interval ∆ op = (0 , Y ) is shellable (meaning that the ordercomplex ∆ ord (∆ op ) is shellable).To conclude that the shellable ( d − -dimensional simplicial complex ∆ ord (∆ op ) is a PL ( d − -ball we use the criterion of [BLVS +

99, Proposition 4.7.22(ii)]: namelywe must show that every ( d − -simplex is the face of one or two ( d − -simplicesand at least one ( d − -simplex is the face of exactly one ( d − -simplex.Note that a ( d − -simplex of ∆ ord (∆ op ) is a maximal chain x < x < · · ·

For α , . . . , α r ∈ P , let e R α ··· α r := k [ z (∆ α ··· α r )] . Remark . When ( P , U ) comes from a linear subspace as in Example 2.4, theneach feasible tope α corresponds to a bounded feasible chamber in the the corre-sponding hyperplane arrangement and the ring e R α ··· α r deﬁned here agrees withthe corresponding ones deﬁned in [BLPW10, Deﬁnition 4.1]. For any feasible covector Y , the zero set z ( Y ) ⊂ E n is an independent set of M , so there are natural quotient maps k [ M ] → e R α ··· α r . Notice that for any β ∈ F there is also a natural quotient map e R α ··· α r → e R α ··· α r β compatible with the mapsfrom k [ M ] . Furthermore, the quotient k [ M ] → e R α ··· α r makes e R α ··· α r a Sym U -module. Lemma 5.7.

For α , . . . , α r ∈ P , let Y = α ∧ · · · ∧ α r . The ring e R α ··· α r is a free Sym U -module whose rank is equal to the number of feasible cocircuit faces of Y .Proof. Lemma 5.4 tells us that k F F lv ( Y ) \{ , Y }k = k ∆ α ...α r k is a ( ρ ( Y ) − -sphereor ( ρ ( Y ) − -ball. If z ( Y ) = ∅ , then the posets z (∆ α ··· α r ) and ∆ α ··· α r are isomor-phic. More generally, the geometric realization k z (∆ α ··· α r ) k is the | z ( Y ) | -fold coneover k ∆ α ··· α r k . In any case, k z (∆ α ··· α r ) k is either a ( d − -ball or ( d − -sphere.By results of Hochster, Reisner, and Munkres (cf [Sta04, Section II.4]) it followsthat e R α ··· α r = k [ z (∆ α ··· α r )] is a Cohen-Macaulay ring. Hence [Sta04, TheoremI.5.10] implies that e R α ··· α r is a free module of ﬁnite rank over the symmetric al-gebra of any parameter space. Here U is a parameter space for e R α ··· α r by [Sta04,Lemma III.2.4] because the composition U ֒ → k I ։ k z ( X ) is an isomorphism forany cocircuit X ∈ F F em ( Y ) . Also by [Sta04, Lemma III.2.4], the rank is equal tothe number of maximal simplices of z (∆ α ··· α r ) , which are in bijection with thecocircuits of F F em ( Y ) . (cid:3) Remark . While we will not need it in what follows, we note that if P is Eu-clidean one can prove that z (∆ α ··· α r ) is in fact shellable and then [Sta04, TheoremIII.2.5] gives an explicit basis of e R α ··· α r as a free Sym U -module.For any α, β ∈ P , we let d αβ := |{ i | α ( i ) = β ( i ) }| , which coincides with the length of any taut path from α to β . For α, β, γ , we let S βαγ := { i | α ( i ) = γ ( i ) = β ( i ) } , which is the set of i ∈ E n such that the concatenation of a taut path α to β and ataut path β to γ changes the i -th coordinate exactly twice.For a graded vector space (or module) M and integer k , we write M h k i to de-note the graded vector space shifted down by k , that is ( M h k i ) i = M i + k . Deﬁnition 5.9.

Let e B = e B ( P ) , as a graded vector space in non-negative degree,be deﬁned as e B := M ( α,β ) ∈P×P e R αβ h− d αβ i where the variables t i are given degree 2.Following [BLPW10] we deﬁne a multiplication ⋆ : e B ⊗ e B → e B as zero on e R αβ ⊗ e R δγ if β = δ and for α, β, γ ∈ P , by the composition e R αβ ⊗ e R βγ → e R αβγ ⊗ e R αβγ → e R αβγ f αβγ → e R αγ where the ﬁrst map is the product of the restrictions, the second is multiplicationin e R αβγ , and the third map f αβγ is induced by multiplication by t S βαγ . Lemma 5.10.

The map f αβγ is well-deﬁned. CATEGORY O FOR ORIENTED MATROIDS 27

Proof.

It sufﬁces to check that if t S = 0 in e R αβγ , then t S βαγ t S = 0 in e R αγ . In otherwords, we want to show if S z ( Y ) for all Y ∈ ∆ αβγ , then S ∪ S βαγ z ( Y ′ ) for all Y ′ ∈ ∆ αγ . Suppose Y ′ ∈ ∆ αγ and either: (1) Y ′ ∆ β or (2) Y ′ ∈ ∆ β .In the ﬁrst case, Y ′ ∆ β means that there exists an i such that Y ′ ( i ) = − β ( i ) ,in particular Y ′ ( i ) = 0 and i z ( Y ′ ) . As Y ′ is a face of α and γ , it follows that Y ′ ( i ) = α ( i ) = γ ( i ) and so i ∈ S βαγ . Thus S ∪ S βαγ z ( Y ′ ) .In the second case, we have Y ′ ∈ ∆ αβγ hence by assumption S z ( Y ′ ) and soagain S ∪ S βαγ z ( Y ′ ) . (cid:3) Proposition 5.11.

The multiplication ∗ gives e B the structure of a graded ring.Proof. We need to check associativity and compatibility with grading. For associa-tivity, the map e R αβ ⊗ e R βγ ⊗ e R γδ → e R αδ given by x ⊗ y ⊗ z ( x ⋆ y ) ⋆ z is equalto the map given by restricting each of the components to e R αβγδ , multiplying inorder, and then multiplying by t S βαγ · t S γαδ to get back into R αδ . For the x ⋆ ( y ⋆ z ) ,the only change is that we multiply by t S γβδ · t S βαδ . To show t S βαγ · t S γαδ = t S γβδ · t S βαδ , note that the power of t i appearing on each side is equal to the number of timesa path given by the concatenation of taut paths from α to β , β to γ , and γ to δ changes the i -th coordinate twice.For the compatibility of gradings, note that d αβ + d βγ − d αγ = 2 | S βαγ | . It follows that multiplication ⋆ gives a graded preserving map e R αβ h− d αβ i ⊗ k e R βγ h− d βγ i → e R αγ h− d αγ i . (Recall that deg( t i ) = 2 for all i .) (cid:3) Note that the map ζ : Sym( k n ) = k [ t , . . . , t n ] → e B given by the composition k [ t , . . . , t n ] ֒ → M α ∈P k [ t , . . . , t n ] ։ M α ∈P e R αα ֒ → e B makes e B into a graded Sym( k n ) -algebra. Moreover, this map factors through theprojection k [ t , . . . , t n ] → k [ M ] and so we may also view e B as a graded k [ M ] -algebra.Let R α ··· α r := e R α ··· α r ⊗ Sym U k = e R α ··· α r ⊗ k [ M ] k [ M ] / ( U ) . We deﬁne B = B ( P , U ) via B := e B ⊗ Sym U k = e B ⊗ k [ M ] k [ M ] / ( U ) . Note, B is itself a graded ring whose multiplication we will also denote by ⋆ . Theorem 5.12.

There is a natural isomorphism e A ( P ∨ ) → e B ( P ) as graded Sym( k n ) -algebras, and this induces an isomorphism A ( P ∨ , U ⊥ ) ≃ B ( P , U ) as graded rings. In particular the theorem implies e A ( P ∨ ) is in fact a k [ M ] -module. Proof.

We deﬁne a map φ : e A ( P ∨ ) → e B ( P ) by(1) e α αα ∈ e R αα for all α ∈ P , (2) p ∨ ( α, β ) αβ ∈ e R αβ h− i for adjacent α, β ∈ P , and(3) f ζ ( f ) for all f ∈ Sym( k n ) = k [ t , . . . , t n ] .We will ﬁrst show that this gives a well-deﬁned homomorphism. We then provesurjectivity and conclude with injectivity.We must check that the image of the relations ( A ) and ( A ) for e A ( f M ∨ ) hold in e B ( f M ) .For ( A ), we consider α ∈ P ∨ = P and i = j in E n such that γ = ( α i ) j ∈ P ∨ and α i , α j ∈ F ∨ . This means that S α i αγ = ∅ = S α j αγ , so that t S αiαγ = 1 .If α i and α j are both in P ∨ = P , then we have αα i ⋆ α i γ = 1 αγ = 1 αα j ⋆ α j γ . Otherwise, by relabelling i and j if necessary, we may assume without loss ofgenerality that α i

6∈ P ∨ . This means that in e A ( P ∨ ) , we have p ∨ ( α, α i , γ ) = p ∨ ( α, α j , γ ) . On the other hand it also means that α i ∈ F ∨ \P ∨ = B\F . But any common faceof α and γ must also be a face of α i , which is infeasible, hence α and γ have nocommon feasible faces and so e R αγ = 0 , which means both products αα i ⋆ α i γ and αα j ⋆ α j γ must be zero.We now check the relation ( A ). Let α ∈ P ∨ and α i ∈ F ∨ .If α i ∈ P ∨ , we have that φ ( e α t i ) = 1 αα ⋆ ζ ( t i ) = t i ∈ e R αα = t S ( α ( α i ) α ) ∈ e R αα = 1 α ( α i ) ⋆ ( a i ) α = φ ( p ( α, α i , α )) . If α i

6∈ P ∨ , then α i ∈ F ∨ \P ∨ and so e α t i = e α p ∨ ( α, α i , α ) = 0 in e A ( P ∨ ) . On theother hand, α i ∈ B\P implies that i is not in the zero set of any feasible face of thefeasible tope α of N , so t i = 0 ∈ e R αα . This completes the proof that the relation( A ) is satisﬁed.Thus, the homomorphism φ is well-deﬁned.To see that φ is surjective, note that φ ( e α t i ) = 1 αα t i for all i ∈ E n and α ∈ P ∨ = P . This means that L α e R αα ⊆ e B is contained in the image of φ . Since the naturalquotient e R αα → e R ααβ = e R αβ is given by multiplication by αβ = φ ( p ∨ α,β ) for any β ∈ P and p ∨ α,β representing a taut path from α to β in the quiver Q ∨ associated to P ∨ , we have that φ is surjective.Finally, we must prove that φ is injective. It sufﬁces to show that the dimensionof e R αβ in each degree is at least the dimension of the corresponding graded partof e α e A ( P ∨ ) e β . To do so, we construct a surjection of graded Sym( k n ) -modules e R αβ → e α e A ( P ∨ ) e β .Let χ : Sym( k n ) = k [ t , . . . , t n ] → e α e A ( P ∨ ) e β be the map that takes to ataut path p from α to β in the quiver Q ∨ . By Proposition 3.6 and Proposition 3.5this map is surjective. It remains to show that χ factors through e R αβ , which isequivalent to showing that for any S z ( Y ) for all Y ∈ ∆ αβ , we have χ ( t S ) = p · t S = 0 . We reduce to the case of α = β by replacing S by S ∪ S βαα . CATEGORY O FOR ORIENTED MATROIDS 29

By Corollary 3.7 it sufﬁces to prove the existence of γ ∈ F ∨ \P ∨ = B\P suchthat γ ( i ) = α ( i ) for all i S . Since S z ( Y ) for any Y ∈ ∆ α , the image α of α in P /S = ( f M /S, g, f ) is not feasible. However, α is bounded in P /S since anycocircuit face X ∈ A ∞ ( P /S ) of α comes from a cocircuit face X ∈ A ∞ ( P ) of α ,and therefore has X ( f ) = X ( f ) = − . Hence α is a feasible but unbounded signvector in P ∨ \ S , and therefore allows us to ﬁnd γ ∈ F ∨ \P ∨ with γ ( i ) = α ( i ) for all i S . (cid:3)

6. T

HE CENTER OF B We continue to assume P = ( f M , g, f ) is a generic oriented matroid programand U ⊂ k n is a parameter space for M = M . In this section we compute thecenters of e B := e B ( P ) and B := B ( P , U ) .Let ζ be the composition k [ M ] ֒ → M α ∈P k [ M ] → M α ∈P e R αα ֒ → e B and recall that this map makes e B a graded k [ M ] -algebra. Theorem 6.1.

The map ζ : k [ M ] → e B ( P ) is injective, and its image is the center of e B ( P ) . Furthermore, the quotient e B ( P ) → B ( P , U ) induces a surjection of centers andthe center of B ( P , U ) is isomorphic to k [ M ] / ( U ) . This is the natural generalization of [BLPW10, Theorem 4.16] and we imitatethe structure of their proof, which makes use of the extended algebras : e B ext = e B ext ( P ) := M ( α,β ) ∈F×F e R αβ h− d αβ i and B ext = B ext ( P , U ) := e B ext ⊗ Sym U k, where the product ⋆ is deﬁned as before but we use all feasible topes, not just thebounded feasible topes. Braden–Licata–Proudfoot–Webster ﬁrst prove their resultis true when e B and B are replaced by the extended algebras e B ext and B ext . Thisis done by studying a chain complex whose homology homology is the center Z ( e B ext ) . To get the theorem for e B and B , they then use a categorical limit argu-ment.This section is split into two subsections. In the ﬁrst subsection we deﬁne thenecessary notation and lay the topological foundation for the proof. In the secondsubsection we adapt the arguments of [BLPW10] to our setting.6.1. The topology of afﬁne space and feasible Edmonds Mandel face lattices.

When P realizable by a polarized arrangement ( V, η, ξ ) , it is possible to view A asthe cells of the coordinate hyperplane arrangement in η + V ⊂ R n . This allowsone to ﬁnd tubular neighborhoods of intersections of hyperplanes, and computethe relative cellular Borel-Moore homology of these tubular neighborhoods usingthe decomposition by cells. In this section we recall deﬁnitions and notation togeneralize these notions to our setting.For any i ∈ E n and S ⊆ E n , let H F i := { X ∈ A | X ( i ) = 0 } and H F S := \ i ∈ S H F i . g F IGURE kA ∪ A ∞ k , k ∆ ord ( A ) k , k Ξ k Our genericity assumption on g implies that H F S = ∅ if and only if S is indepen-dent in the underlying matroid M of M , in which case any maximal covector in H F S has rank d + 1 − | S | . Note that H F S does not have a cellular interpretation unless S is a basis of M , in which case H F S consists of a single feasible cocircuit.It is known [BLVS +

99, Theorem 4.5.7.i] that k ∆ ord ( A ) k is a shellable d -ball. Thusits boundary k ∆ ord ( A\A ) k is a PL ( d − -sphere. In k ∆ ord ( A ) k , we have that k ∆ ord ( H F S ) k is a ( d − | S | ) -ball (when nonempty), with boundary k ∆ ord ( H F S \A ) k .For any Y ∈ A , let σ Y := k ∆ ord ( F F em ( Y ) \{ } ) k ⊂ k ∆ ord ( A ) k . Lemma 6.2.

For any Y ∈ A , the order complex ∆ ord ( F F em ( Y ) \{ } ) is a shellable ( ρ ( Y ) − -ball.Proof. Recalling the notation from the proof of Lemma 5.4, note that ∆ ord ( F F em ( Y ) \{ } ) is the cone on ∆ ord (∆ op ) with vertex Y . If Y ∞ = 0 , then ∆ ord (∆ op ) is a shellablesphere (see [BLVS +

99, Theorem 4.3.5(i)]) and if Y ∞ = 0 , then in the proof ofLemma 5.4 we showed that ∆ ord (∆ op ) is a shellable ball. In both cases, the cone isa shellable ball as claimed. (cid:3) The boundary of each σ Y is the union of cells σ X for the proper faces X of Y and the geometric simplices in ∆ ord ( F F em ( Y ) \{ } ) corresponding to chains that donot begin with a cocircuit. Let Ξ to be the regular cell complex of cells { σ Y } Y ∈A to-gether with the set of (geometric) simplices { σ } σ ∈ ∆ ord ( A\A ) . In particular, ∆ ord ( A ) is a subdivision of Ξ . The cells σ Y for Y ∈ A deﬁne a subcomplex of Ξ , as do thosefor Y ∈ H F S ∩ A when S is independent in M . Remark . Figure 3 shows an example to illustrate some of these deﬁnitions. Thereason we consider ∆ ord ( A ) and Ξ is because A does not have a cellular interpre-tation unless we include a boundary. The natural boundary would be A ∞ , but thisgives us undesirable topology at the boundary. By introducing ∆ ord ( A ) and Ξ weresolve this issue. Deﬁnition 6.4.

For an independent set S ⊂ E n of M , we deﬁne Σ S := { A ∈ ∆ ord ( A ) | A ⊂ ( x < x < · · · < x i ) ∈ ∆ ord ( A ) and x j ∈ H F S for some j } and let N S := k Σ S k ◦ be the interior of k Σ S k . In particular: N S := { A ∈ ∆ ord ( A ) | A = ( x < x < · · · < x i ) and x j ∈ H F S ∩ A for some j } As was discussed in the case of H F ∅ = A following Deﬁnition 5.3. CATEGORY O FOR ORIENTED MATROIDS 31

Proposition 6.5.

The Borel-Moore homology of N S is H BMm ( N S ; Z ) = ( Z m = d m = d and can be computed in k Ξ k using the relative cellular Borel-Moore homology of N S viathe decomposition by cells N S ∩ σ Y for Y ∈ A . These N S ∩ σ Y are nonempty exactlywhen F F em ( Y ) ∩ H F S = ∅ .Proof. We ﬁrst show that N S is a d -ball, from which the ﬁrst statement follows.By [BLVS +

99, Theorem 4.5.7(i)], k ∆ ord ( H F S ) k is a shellable ball. In particular, k ∆ ord ( H F S ) k is collapsible (i.e., it collapses to a point).Now N S is a regular neighborhood of k ∆ ord ( H F S ) k in the d -ball k ∆ ord ( A ) k andso by [RS72, Corollary 3.27], N S of k ∆ ord ( H F S ) k is a d-ball.To see that each intersection N S ∩ σ Y for Y ∈ A is a cell, note that N S ∩ σ Y is aregular neighborhood of the shellable ball H S ∩ σ Y in σ Y . The same collapsibilityargument from above then implies N S ∩ σ Y is a ball.The cells N S ∩ σ Y provide a cellular decomposition of N S modulo its boundaryand the space N S ∩ σ Y is nonempty if and only if a face of Y is contained in H F S orequivalently, F F lv ( Y ) ∩ H F S = ∅ . (cid:3) Proposition 6.6.

Let Y ∈ A\A such that Y has a face in H F S ∩ A . Then H BMm ( N S ∩ σ Y ; Z ) = 0 for all m . This can be computed in k Ξ k using the decomposition of N S ∩ σ Y by cells N S ∩ σ X for X ∈ F F em ( Y ) , which are nonempty exactly for X which have a face in H F S .Proof. The idea of this proof is to again use relative homology, this time on thepair ( σ Y , σ Y \ N S ) . Recall that σ Y is a PL ( ρ ( Y ) − -ball by Lemma 6.2. Then σ Y iscontractible, so we are done if we can prove the same for σ Y \ N S .There is a unique maximal element of H F S ∩ A ∩ F F em ( Y ) , which we call X . Thecomplement of N S in σ Y is k ∆ k , where ∆ := ord( F F em ( Y ) \ ( H F S ∩ A )) ⊂ ord( F F em ( Y )) . Notice that ∆ is nonempty since Y ∞ = 0 , and ∆ is equivalent to the cone over ord( F F em ( Y ) \ F F em ( X )) with cone point Y . Thus k ∆ k is contractible, and we aredone. (cid:3) The proof of Theorem 6.1.

We will construct a chain complex with homologyisomorphic to the center Z ( e B ext ) .Let A r = { Y ∈ A | ρ ( Y ) − r } = { Y ∈ A | dim( σ Y ) = r } , where ρ is the poset rank of Y in L ( N ) , N = f M\ f , and σ Y = k ord( F F em ( Y )) k ⊂k Ξ k is a ball by Lemma 6.2. For example, A d is the set of feasible topes F and A is the set of feasible cocircuits.For all Y ∈ A r , the space of orientations of σ Y is a one-dimensional vector space or ( Y ) := H BMr ( σ ◦ Y ; k ) , where σ ◦ Y denotes the interior of σ Y . There is a natural boundary map ∂ Y : or ( Y ) → M X ∈A r − ∩ F F em ( Y ) or ( X ) . Assembling all such maps, we obtain a chain complex on L Y ∈A or ( Y ) , graded by dim( σ Y ) , which computes the cellular Borel-Moore homology of k Ξ k ◦ . As k Ξ k isa closed PL d -ball, this homology is one-dimensional in degree d and zero in allother degrees.For Y ∈ A r , let e R Y := k [ z ( F F lv ( Y ) \{ } )] = e R α ··· α ℓ , for any choice of α , . . . , α ℓ ∈ F such that Y = α ∧ · · · ∧ α ℓ . We deﬁne a chaincomplex C • such that C r = M Y ∈A r e R Y ⊗ k or ( Y ) with differentials for each Y ∈ A r e R Y ⊗ k or ( Y ) → M X ∈A r − ∩ F F em ( Y ) e R X ⊗ k or ( X ) induced by the natural boundary maps or ( Y ) → or ( X ) and the quotients e R Y → e R X for each facet X of Y . Lemma 6.7.

Fix an orientation class Ω ∈ H BMd ( k Ξ k ◦ ; k ) , and let Ω α ∈ or ( α ) be therestriction of Ω for any α ∈ F . Let ψ α : k [ M ] → e R α denote the natural quotient map.Then the homology of C • is zero outside of degree d and k [ M ] ≃ H d ( C • ) via the map x X α ∈F ψ α ( x ) ⊗ Ω α . Proof.

Following the proof of [BLPW10, Lemma 4.17], for a monomial m = Q i t s i i ,let S = { i | s i > } and C m • ⊂ C • be the subcomplex consisting of all images of m ,namely C mr = M Y ∈ A r H F S ∩ F F em ( Y ) = ∅ or ( Y ) . Note that the complex C • decomposes as a direct sum ⊕ m C m • of subcomplexesbecause the terms of C • are direct sums of quotients of k [ x , . . . , x n ] by monomialideals, while the differentials are induced by the identity map on k [ x , . . . , x n ] , upto sign.If the set S = { i | s i > } is dependent in M , then H S = ∅ and so C m • = 0 . If S is independent in M , then C m • is the cellular Borel-Moore complex of the neigh-borhood N S ⊂ k Ξ k , the homology of which is one-dimensional and concentratedin degree d by Proposition 6.5. (cid:3) Proposition 6.8.

The obvious map ζ ext : k [ M ] → e B ext is injective, and its image is thecenter of e B ext . The quotient homomorphism e B ext → B ext induces a surjection of centers,and yields an isomorphism Z ( B ext ) ∼ = k [ M ] / ( U ) .Proof. As in the proof of [BLPW10, Proposition 4.18], for an element z ∈ e B ext to bein the center, it must commute with each idempotent αα and thus be of the form z = P α ∈F z α where z α ∈ R α . Similarly, using the fact that z must commute with αβ for α and β adjacent, we ﬁnd that ψ αβ ( z α ) = ψ βα ( z β ) CATEGORY O FOR ORIENTED MATROIDS 33 where ψ αβ : e R α → e R αβ and ψ βα : e R β → e R αβ denote the canonical quotienthomomorphisms. As e B ext is generated by these element and the image of ζ ext , itfollows that: Z ( e B ext ) ∼ = ( ( z α ) ∈ M α ∈F e R α | ψ αβ ( z α ) = ψ βα ( z β ) for all α ↔ β ∈ F ) . (6.1)On the other hand, H d ( C • ) is equal to the set of cycles y = P α ∈F y α ⊗ Ω α ∈ C d ,which is equivalent to the condition that ψ αβ ( y α ) = ψ βα ( y β ) for all α ↔ β ∈ F .We conclude that ζ ext induces an isomorphism k [ M ] ∼ = H d ( C • ) ∼ = Z ( e B ext ) . Finally, we can deﬁne a chain complex ˆ C • of free Sym U -modules with ˆ C m = C m for ≤ m ≤ d and ˆ C d +1 = ker( ∂ d ) ∼ = k [ M ] . This is acyclic, and thus so is ˆ C • ⊗ Sym U k . Arguments analogous to above prove that k [ M ] / ( U ) ∼ = H d ( C • ⊗ Sym U k ) ∼ = Z ( B ext ) through an isomorphism compatible with the quotient e B ext → B ext . (cid:3) We now include boundedness into our considerations. Deﬁne A P := [ Y ∈P⊂A d F F em ( Y ) , and notice that we have a chain of proper inclusions A ⊂ A P ⊂ A . For any Y ∈ A P , let Y be the (unique) maximal face of F F em ( Y ) ∩ A .Note that the description of Z ( e B ext ) in (6.1) can be rewritten as asking that ψ αβ ( z α ) = ψ βα ( z β ) for all α, β ∈ F , not necessarily adjacent. This can be rephrasedas a limit: Z ( e B ext ) ∼ = lim ←− X ∈A e R X . (6.2)By the same sort of argument, we ﬁnd Z ( B ext ) ∼ = lim ←− X ∈A R X , Z ( e B ) ∼ = lim ←− X ∈A P e R X , and Z ( B ) ∼ = lim ←− X ∈A P R X . (6.3)The next lemma allows us to conclude that these centers only depend on A . Lemma 6.9.

For any Y ∈ A P and any subcomplex D ⊂ F F em ( Y ) with Y ∈ D , therestrictions lim ←− X ∈ F F em ( Y ) e R X → lim ←− X ∈ D e R X and lim ←− X ∈ F F em ( Y ) R X → lim ←− X ∈ D R X (6.4) are isomorphisms.Proof. This proof is simply a rephrasing of [BLPW10, Lemma 4.20] in our setting.This is trivial if Y = Y , which includes the case where Y is a feasible cocircuit.So we may assume Y = Y (equivalently, Y ∞ = 0 ), and we may also inductivelyassume the statement is true for all X ∈ F F em ( Y ) \{ Y } .We will prove the statement ﬁrst for D = F F em ( Y ) \{ Y } . Let C Y • be the sub-complex of C • consisting only of the summands e R X ⊗ k or ( X ) for X ∈ F F em ( Y ) .As in the proof of Lemma 6.7, this complex splits into a direct sum of complexes C Y,m • = C Y • ∩ C m • for each monomial m = Q i x s i . The summand C Y,m • computes the cellular Borel-Moore homology of N S ∩ σ Y ⊂ σ Y , for S = { i | s i > } . ByProposition 6.6, this Borel-Moore homology is trivial, so every C Y,m • is acyclic andso is C Y • .We have that lim ←− X ∈ D e R X is isomorphic to the kernel of the boundary map C Yd − → C Yd − . Since C Y • is acyclic, the ﬁrst map of (6.4) is therefore an isomor-phism. Similarly, the second map is an isomorphism since C Y • is an acyclic com-plex of free Sym U -modules, which implies C Y • ⊗ Sym U k is an acyclic complex ofvector spaces.For a general D containing Y , pick an ordering X , . . . , X r of the elements of F em ( Y ) \ D such that their ranks are nondecreasing, and let D ℓ = D ∪{ X , . . . , X ℓ } .Then for ≤ ℓ ≤ r , we have F F em ( X ℓ ) \{ X ℓ } ⊂ D ℓ − , so an identical argumentshows that lim ←− X ∈ D ℓ e R X → lim ←− X ∈ D ℓ − e R X and lim ←− X ∈ D ℓ R X → lim ←− X ∈ D ℓ − R X are isomorphisms. (cid:3) Proof of Theorem 6.1.

Put the equations (6.2) and (6.3) together with Lemma 6.9 toget Z ( e B ext ) ∼ = lim ←− X ∈A e R X ∼ = lim ←− X ∈A P e R X ∼ = Z ( e B ) and Z ( B ext ) ∼ = lim ←− X ∈A R X ∼ = lim ←− X ∈A P R X ∼ = Z ( B ) . All of these isomorphisms are compatible with ζ ext , ζ and the natural quotients e B ext → B ext and e B → B , so we are done. (cid:3)

7. T

HE MODULE CATEGORY OF A In this section, we study the simple modules for A = A ( P , U ) and their projec-tive covers using a class of standard modules. Deﬁnition 7.1.

For any α ∈ P , let L α := A/ ( e β | β = α ) . Then L α is the simple one-dimensional A -module supported at α , and let P α := e α A be its projective cover. We also deﬁne V α to be P α /K α , where K α := X i ∈ b p ( α, α i ) · A ⊂ P α and b is the basis of M such that µ ( b ) = α under the bijection of Corollary 2.15. Werefer to V α as the standard module and L α as the simple module associated to α . Lemma 7.2.

Let α ∈ P . The standard module V α has a basis consisting of a taut pathfrom α to each β (cid:22) α . CATEGORY O FOR ORIENTED MATROIDS 35

Proof.

We simply copy the argument from [BLPW10, Lemma 5.21]. It is clear thatthe collection of such taut paths is linearly independent. We now show that theimage of any other path is trivial in V α .Let b = µ − ( α ) ∈ B .Suppose p is a taut path from α to γ ∈ F and γ α . Then γ ( i ) = α ( i ) forsome i ∈ b and α i ∈ F . By Corollary 3.7 p can be replaced by one of the form p ( α, α i ) · x ∈ K α . Thus p = 0 as an element of V α .If p is a non-taut path, then by Proposition 3.6, we can write p = p ′ · Y i u a i i = Y i e α u a i i ! · p ′ where p ′ is taut (with the same endpoint as p ) and a i > for some i . Corollary 3.8implies that for all i ∈ E n and some c i,j ∈ k , we have e α u i = X j ∈ b c i,j p ( α, α j , α ) ∈ K α . Thus, p ∈ K α and we are done. (cid:3) Corollary 7.3.

The kernel of V α ։ L α has a ﬁltration with subquotients isomorphic to L β for β ≺ α , each appearing exactly once. A is a quasi-hereditary algebra when P is Euclidean. Recall that a ﬁnite-dimensional algebra is quasi-hereditary if its category of ﬁnitely generated modulesis highest weight in the following sense.

Deﬁnition 7.4.

Let C be an abelian, artinian k -linear category and let I be the setindexing the isomorphism classes of simple objects { L α | α ∈ I} and indecom-posable projective objects { P α | α ∈ I} . Then C is a highest weight category if theset I can be endowed with a partial order ≤ and there exists a collection of objects { V α | α ∈ I} with surjections P α → V α → L α that satisfy:(i) the kernel of V α → L α has a ﬁltration for which each subquotient is iso-morphic to L γ for some γ < α , and(ii) The kernel of P α → V α has a ﬁltration for which each subquotient is iso-morphic to V β for some β > α .Now consider the category of ﬁnitely generated A ( P , U ) -modules. As dis-cussed above, the isomorphism classes of simple modules are indexed by the set P of bounded feasible topes. For A = A ( P , U ) to be a quasi-hereditary algebra,we will assume that the oriented matroid program P is Euclidean. Recall fromSection 2.3, that this implies there is partial order ≤ on P deﬁned by: α ≤ β whenthere exists a directed sequence of edges from µ − ( α ) to µ − ( β ) in the graph G P of the program. By Lemma 2.25, this is the same partial order as that deﬁned bythe transitive closure of the cone relation (cid:22) .Suppose P is Euclidean. Then the category of A ( P , U ) -modules and the poset I = ( P , ≤ ) satisﬁes condition (i) of Deﬁnition 7.4 by Corollary 7.3. To show that A is quasi-hereditary it remains to show condition (ii).We will use the following simple lemma. Lemma 7.5.

Suppose β ∈ P and i ∈ µ − ( β ) . Then the feasible sign vector β i is eitherunbounded or β i (cid:23) β .Proof. Suppose β i is bounded. As β i ( j ) = β ( j ) for all j = i , if i µ − ( β i ) , then forall j ∈ µ − ( β i ) , β i ( j ) = β ( j ) . Thus β ∈ B µ − ( β i ) or equivalently β (cid:22) β i .If β i is bounded and i ∈ µ − ( β i ) , then the optimal solutions of β and β i arealso optimal solutions of their common subtope Y = β ∧ β i . But Y is a tope of P / { i } and so Y has a unique optimal solution. Thus µ − ( β ) = µ − ( β i ) , whichcontradicts the fact that µ is bijection. (cid:3) Theorem 7.6.

Assume P is Euclidean. Then the kernel of the quotient homomorphism P α ։ V α has a ﬁltration with each successive subquotient isomorphic to V β for β ≻ α .Each of these standard modules appears exactly once.In particular, A is quasihereditary.Proof. For any γ ∈ P , we deﬁne P γα ⊂ P α to be the submodule generated by pathswhich pass through γ . For any β ∈ P , let K βα := X γ>β P γα . After choosing a total order on { β ∈ P | α ≤ β } reﬁning ≤ , the set of submodules P βα + K βα with β ≥ α forms a ﬁltration of K α with successive subquotients M βα := P βα + K βα /K βα . Our goal now is to prove that M βα is zero if β α , and is isomorphic to V β if β (cid:23) α .Notice that M αα = V α .If β α , then there is an index i ∈ µ − ( β ) such that α ( i ) = β ( i ) . By Proposition3.6, any path starting at α and passing through β can be written as p α,β · r in P ≥ βα ,where p α,β represents a taut path from α to β and r represents a path starting at β . We may then apply Corollary 3.7 to the taut path p α,β and γ = β i , to show that p α,β can be chosen to pass through β i ∈ F . By Lemma 7.5, P βα ⊆ P β i α ⊆ K βα , so M βα = 0 .On the other hand, assume that β ≻ α . There is a natural map P β → P βα givenby composing any element of P β with a ﬁxed taut path p α,β from α to β . Thisinduces a homomorphism V β → M βα that we wish to show is an isomorphism.By Proposition 3.6, any path starting at α and passing through β can be ex-pressed as a product of an element in P β with some taut path from α to β and byProposition 3.5 the taut path can be chosen to be the one we have ﬁxed. It fol-lows that the map P β → P βα is surjective and thus the induced map V β → M βα issurjective as well.Finally, we need to show that V β → M βα is injective. We proceed by showingthat they have the same dimension. The surjectivity of the map implies dim k M βα ≤ dim k V β = |{ γ ∈ P | γ (cid:22) β }| , so that dim k P α = X β (cid:23) α dim k M βα ≤ |{ ( γ, β ) ∈ P × P | α, γ (cid:22) β }| . We’re done if we can show this is an equality. As A = P α ∈P P α , it sufﬁces to provethat dim k A = X α ∈P dim k P α = { ( α, γ, β ) ∈ P × P × P | α, γ (cid:22) β } . CATEGORY O FOR ORIENTED MATROIDS 37

But recall that A = A ( P , U ) ≃ B ( P ∨ , U ⊥ ) = M ( α,γ ) ∈P ∨ ×P ∨ R ∨ αγ and so by Lemma 5.7 dim k B ( P ∨ , U ⊥ ) = X ( α,γ ) ∈P ∨ ×P ∨ |{ common feasible cocircuit faces of α and γ }| . We are then reduced to showing that the number of common feasible cocircuitfaces in P ∨ of α and γ is equal to the number of bounded feasible topes β of P such that α (cid:22) β and γ (cid:22) β . This follows from Complementary Slackness (Propo-sition 2.19). (cid:3) The structure of projectives when P is not Euclidean. Note that the deﬁni-tion of the standard modules makes sense for any P and Lemma 7.2 holds even inthe non-Euclidean case. However, when P is not Euclidean, the transitive closureof the cone relation is not a poset and so the standard modules are not part of ahighest weight structure.Nonetheless, one might still hope for a version of Theorem 7.6: that the kernel of P α ։ V α has a ﬁltration with successive subquotients isomorphic to V β for β ≻ α .In this section we observe that this is too optimistic a hope, but that it does holdon the level of graded Grothendieck groups.Recall from Lemma 5.7, that for any α ∈ P , the dimension dim k R α is equal tothe number of feasible cocircuit faces of α . We begin with a graded reﬁnement ofthis statement. Lemma 7.7.

Let ( h , h . . . , h d − ) denote the h -vector of z (∆ α ) or equivalently h i isequal to the dimension of the graded piece of R α of degree i . Then h i is equal to thenumber of feasible vertices of α with i outgoing edges.Proof. We proceed by showing that z (∆ α ) is partitionable. Recall that a pure sim-plicial complex ∆ is partitionable , if it can be expressed as a disjoint union of closedintervals of the form ∆ = [ G , F ] ⊔ . . . ⊔ [ G s , F s ] , where each F i is a facet of ∆ . By [Sta04, Proposition III.2.3] the h -polynomial ofsuch a simplicial complex is given by h i = { j : | G j | = i } . Recall that z ( α ) = ∅ and z (∆ α ) is isomorphic as a poset to F F lv ( α ) \{ } . Thus wemay identify a face of α with the faces of the abstract simplicial complex z (∆ α ) .The facets F , . . . , F s of z (∆ α ) are the zero sets of the feasible vertices (i.e., fea-sible cocircuit faces) of α . If F i = z ( X i ) for a feasible vertex X i , let G i be (the zeroset of) the meet in ∆ α of the incoming edges of X i .Recall that each face of α has a unique optimal solution (this follows from The-orem 2.14). For each feasible face Y of α , the face z ( Y ) ∈ ∆ α is in the interval [ G j , F j ] if and only if X j is the optimal solution of Y . Thus [ G , F ] ⊔ . . . ⊔ [ G s , F s ] is a partition of z (∆ α ) . Note that | G j | = d − { incoming edges to X j } = { outgoing edges to X j } . We conclude that h i = { j : | G j | = i } = { feasible vertices of Y with i outgoing edges } . (cid:3) Corollary 7.8.

Let ( h , h . . . , h d − ) denote the h -vector of z (∆ αβ ) or equivalently h i is equal to the dimension of the graded piece of R αβ of degree i . Then h i is equal to thenumber of feasible vertices of α ∧ β with i outgoing edges of α ∧ β .Proof. Let γ be the tope in P /z ( α ∧ β ) given by the restriction of α ∧ β . Then thesimplicial complex z (∆ αβ ) is equal to the simplicial join z (∆ γ ) ∗ Γ of z (∆ γ ) with the ( d αβ − -simplex Γ on the set z ( α ∧ β ) . By standard properties of the h -polynomial,we have: h ( z (∆ αβ ) , x ) = h ( z (∆ γ ) ∗ Γ , x ) = h ( z (∆ γ ) , x ) h (Γ , x ) = h ( z (∆ γ ) , x ) . We conclude that the h -vector of z (∆ αβ ) is equal to that of z (∆ γ ) . The result thenfollows from Lemma 7.7. (cid:3) For an A -module M , let [ M ] denote the class of M in the Grothendieck groupof A -modules. We will consider the Grothendieck group of the category of graded A -modules as a Z [ q, q − ] -module, where [ M h− k i ] = q k [ M ] . For a graded vector space V = ⊕ i V i , we denote the graded dimension of V by grdim V = X i (dim V i ) q i . Theorem 7.9.

For any generic oriented matroid program P and any α ∈ P , the class ofthe indecomposable projective P α in the Grothendieck group can be expressed as the sum: [ P α ] = X γ (cid:23) α q d αγ [ V γ ] . Proof.

For any β ∈ P the graded composition series multiplicity of the simple L β in the projective P α is equal to the graded dimension of the space of paths in A that start at α and end at β . In other words we have: [ P α ] = X β ∈P (grdim P α e β ) · [ L β ] = X β ∈P (grdim e α Ae β ) · [ L β ] . By Theorem 5.12, grdim e α Ae β = grdim R ∨ αβ h− d αβ i = q d αβ · grdim R ∨ αβ . By Corollary 7.8, we can express the graded dimension of R ∨ αβ as grdim R ∨ αβ = d X i =0 { feasible vertices of α ∧ β in P ∨ with i outgoing edges } · q i . Observe that by Proposition 2.19 the feasible vertices of α ∧ β in P ∨ (i.e., com-mon feasible vertices of both α and β ) are in bijection with the bounded feasibletopes δ of P such that α (cid:22) δ and β (cid:22) δ . We claim that the number of outgoingedges of α ∧ β of the feasible vertex corresponding to δ is equal to | S δαβ | . As inthe proof of Corollary 7.8, let γ be the bounded feasible tope ( α ∧ β ) | α ∧ β in thecontraction P /z ( α ∧ β ) . Then the number of outgoing edges of the vertex of γ CATEGORY O FOR ORIENTED MATROIDS 39 corresponding to δ is equal to the distance between γ and the restriction δ of δ to α ∧ β . But α ∧ β = { i ∈ E n | α ( i ) = β ( i ) } and so the distance between γ and δ isequal to the cardinality of the difference set S ( γ, δ ) = { i ∈ E n | α ( i ) = β ( i ) = δ ( i ) } = S δαβ . Rewriting the sum over topes δ of P such that α (cid:22) δ and β (cid:22) δ and using theformula d αδ + d δβ = d αβ + 2 | S δαβ | , we ﬁnd:(7.1) grdim e α Ae β = q d αβ grdim R ∨ αβ = X δ (cid:23) α,β q d αβ +2 | S δαβ | = X δ (cid:23) α,β q d αδ + d δβ . Putting it all together, [ P α ] = X β X δ (cid:23) α,β q d αδ + d δβ [ L β ]= X δ (cid:23) α q d αδ X β (cid:22) δ q d βδ [ L β ]= X γ (cid:23) α q d αδ [ V γ ] , as we wished to show. (cid:3) We conclude this section with an example of a generic non-Euclidean programand sign vector α for which the kernel of P α ։ V α does not admit a ﬁltration withsuccessive standard subquotients. Example 7.10.

Let P = ( EFM (8) , g, f ) be the generic non-Euclidean program de-ﬁned in [BLVS +

99, Section 10.4]. Then M = EFM (8) /g \ f is the uniform matroidof rank 3 on E . As short hand, we simply write ijk for the basis { i, j, k } of M .We denote the sign vector of a bounded feasible tope α : E → { , + , −} using thestring of signs α (1) α (2) α (3) α (4) α (5) α (6) . The bijection µ between B and P can described as follows, where we have listedthe pairs ( b, µ ( b )) ∈ B × P for P : (123 , + + + + ++) (124 , + + + − + − ) (123 , + + − + ++) (126 , + − + + + − )(134 , + − + + ++) (135 , − + + + − +) (136 , + + + + −− ) (145 , − + + − −− )(146 , + − + − + − ) (156 , + + + + + − ) (234 , + + − − ++) (235 , + + + − − +)(236 , − + + + ++) (245 , + + − − − +) (246 , + + + − ++) (256 , + − + − −− )(345 , + + + + − +) (346 , + + − − −− ) (356 , − + + + −− ) (456 , + + + − −− ) . Using this table one can deduce the cone relation (cid:22) on P from the fact that µ ( b ) (cid:22) µ ( b ′ ) if µ ( b )( i ) = µ ( b ′ )( i ) for any i ∈ b ′ . For example, if µ ( b ) ≺ µ (456) , then b = 346 , or .Let α = + + + − −− ∈ P . Recall the notation α S denoting the sign vector of atope which differs from α on exactly the set S ⊂ E . Using the above list, we ﬁndthat { β ∈ P | β (cid:23) α } = { α, α , α , α , α { , } , α { , } , α { , } , α { , , } } Suppose there were a ﬁltration ⊂ F ⊂ . . . ⊂ F ⊂ K α of the kernel K α := P i ∈ b p ( α, α i ) · A of P α → V α with nonzero successive standard subquotients { V γ | γ ∈ C α \{ α }} as in the proof of Theorem 7.6. Let V γ = K α /F be the ﬁnal standardsubquotient and suppose that γ = α S .Let p be a taut path from α to α S . If p ∈ F , then ( K α /F ) e α S = 0 , which isa contradiction. Thus we may assume that p F . For any i ∈ S , p = p ( α, α i ) q where q is a taut path from α i to α S . As p F , it follows that p ( α, α i ) F and L α i is a quotient of V γ . This is a contradiction unless S = { i } . Thus S is either { } , { } or { } .Suppose S = { } , so γ = α = + + + + −− . Note that α ≺ α , so ( V α ) e α =( K α /F ) e α = 0 and thus p ( α, α ) F . But this would mean that L α is a quotientof V α , which is a contradiction.After permuting indices, the same argument shows that neither V α nor V α isa quotient of K α . We conclude that P α doesn’t not have the expected ﬁltration.More generally, we will see below in the proof of Theorem 7.14 that the change ofbasis matrix between the standard and simple bases for the Grothendieck group isinvertible, so [ P α ] cannot be expressed as a different sum of standard classes. Thus P α does not admit a ﬁltration by standards.7.3. A is a Koszul algebra when P is Euclidean. Recall the notion of a Koszulalgebra:

Deﬁnition 7.11.

Let M = L ℓ ≥ M ℓ be a graded k -algebra. A complex . . . → P → P → P → P of graded projective right M -modules is linear if each P ℓ is generated in degree ℓ . We say that M is Koszul if every simple right M -module has a linear projectiveresolution. Theorem 7.12.

Assume P is Euclidean. Then for all α ∈ P , the standard module V α hasa linear projective resolution.Proof. We follow the proof of [BLPW10, Theorem 5.24].Let a be the basis corresponding to the optimal cocircuit for α . We will deﬁnethe promised resolution as the total complex of the following multicomplex.For any S ⊆ a , let α S ∈ { + , −} n be the sign vector which disagrees with α onexactly the entries in S . For example, α ∅ = α , and α { i } = α i for any i ∈ a . Noticethat if i ∈ S ⊆ a and α S , α S \ i ∈ P , then there is a degree one map φ S,i : P α S −→ P α S \ i ,q p ( α S \ i , α S ) · q. We extend this to all S ⊆ a and i ∈ a by declaring that P α S = 0 if α S

6∈ P and φ S,i = 0 if i S . Consider the module Π α := M S ⊆ a P α S , which we view as being graded by the free abelian group Z { ε i | i ∈ a } wherethe summand P α S is given degree ε S := P i ∈ S ε i . For each i ∈ a , consider thedifferential ∂ i : Π α → Π α of degree − ε i deﬁned as the sum ∂ i := X S ⊆ a φ S,i . CATEGORY O FOR ORIENTED MATROIDS 41

Observe that ∂ i ∂ j = ∂ j ∂ i for any i, j ∈ a by relation ( A ) and so we can view Π α as a multi-complex with differentials ∂ i for each i ∈ a .Let Π • α denote the total complex of Π α . Then Π • α is a linear complex of projec-tive modules and H ( Π • α ) = V α . It remains to show that the complex Π • α is exactin positive degrees.To do so, we will ﬁlter the multicomplex Π α . For each β ∈ P , let (Π α ) β ⊂ Π α bethe submodule whose ε S -graded part is deﬁned as X γ ≥ β,α S P γα S ⊂ P α S , that is, the submodule consisting of all paths from α S passing through some γ ∈ P where γ ≥ α S and γ ≥ β . Observe that the differentials ∂ i for i ∈ a are compatiblewith the submodules (Π α ) β and so we have deﬁned a ﬁltration of Π α by the poset P . Computing the associated graded of this ﬁltration yields a multi-complex e Π α = M β ∈P (Π α ) β / (Π α ) >β = M β ∈P M S M βα S ! , where M βα S is the subquotient of P α S deﬁned as in the proof of Theorem 7.6.Consider the resulting quotient multi-complexes for each β ∈ P . Let b = µ − ( β ) . Recall from the proof of Theorem 7.6 that M βα S is non-zero if and onlyif α S ∈ B b .If β = α , then M βα S = M αα S = 0 for any non-empty S ⊂ a = b . Thus the onlynon-zero summand of the α -subquotient is M αα = V α in total degree zero.If β = α , choose an element i ∈ a such that i b . Consider those subsets S ⊂ a such that i S . Then we have α S ∈ B b if and only if α S ∪{ i } ∈ B b . If α S ∈ B b ,then M βα S ∪ i ≃ V β ≃ M βα S and the differential induced by ∂ i is the isomorphismgiven by left multiplication with p ( α S , α S ∪{ i } ) . On the other hand, if α S

6∈ B b ,then M βα S ∪ i = 0 = M βα S . In particular, the differential induced by ∂ i on the β -component of the associated graded multi-complex is exact.Recall that if any differential of a multi-complex is exact then the total com-plex of the multi-complex is also exact. We conclude that the total complex ofthe associated graded multi-complex is exact in positive degree. It then followsthat the total complex of the original multi-complex must also be exact in positivedegrees. (cid:3) Theorem 7.13.

Assume P is Euclidean. Then A and B are Koszul algebras and A isKoszul dual to B .Proof. By [ ´ADL03, Theorem 1] a quasi-hereditary algebra is Koszul if the standardmodules have linear projective resolutions and such resolutions exist for A by theprevious theorem. Theorem 5.12 implies that B ∼ = A ( P ∨ , U ⊥ ) must also be Koszul.Finally the Koszul duality follows from the quadratic duality statement of Theo-rem 4.2. (cid:3) Numerical identity for Hilbert polynomials.

We do not know whether ornot the condition that P be Euclidean is necessary for A to be Koszul. In thissection we prove that for any generic oriented matroid program P the Hilbertpolynomial of the algebra A = A ( P , U ) satisﬁes the following numerical identity. Let H ( A, q ) denote the Hilbert polynomial of A , which is the P × P -matrix withentries H ( A, q ) α,β = grdim e α Ae β . Recall [BGS96, Lemma 2.11.1] that if A is Koszul, then there is an equality of ma-trices H ( A, q ) H ( A ! , − q ) T = I. Theorem 7.14.

For any generic oriented matroid program P , the algebra A = A ( P , U ) satisﬁes the numerical identity above, that is, the Hilbert polynomials of A and its qua-dratic dual A ! satisfy the matrix equation H ( A, q ) H ( A ! , − q ) T = I. Remark . This identity does not necessarily imply that A is Koszul. See [Pos95]for an example of a non-Koszul quadratic algebra whose Hilbert series satisﬁesthe numerical identity. Proof.

Using equation (7.1) in the proof of Theorem 7.9, the ( α, β ) -entry of H ( A, q ) is given by H ( A, q ) α,β = grdim e α Ae β = X γ (cid:23) α,β q d αγ + d γβ . In particular, H ( A, q ) factors as the product H ( A, q ) = XX T , where X is the P ×P -matrix with ( α, β ) -entry given by X α,β = ( q d αβ if β (cid:23) α . Dually, using Proposition 2.19 we ﬁnd that the ( α, β ) -entry of H ( A ! , − q ) is givenby H ( A ! , − q ) α,β = X i ( − q ) i dim e α A ! i e β = X γ ⊃ Y µ − α ) ,Y µ − β ) ( − q ) d αγ + d γβ , in other words the sum runs over all γ ∈ P for which the optimal solution (cocir-cuit) of both α and β are faces of γ . Again this factors as a product H ( A ! , − q ) = Y Y T , where Y is the P × P -matrix with ( α, β ) -entry given by Y α,β = ( ( − q ) d αβ if Y µ − ( β ) is a face of α . We wish to show that H ( A, q ) H ( A ! , − q ) T = XX T Y Y T = I. Note that it sufﬁces to show X T Y = I .Computing the product X T Y , we ﬁnd that its ( α, β ) -entry is given by ( X T Y ) α,β = X γ ∈ Q q d αγ ( − q ) d γβ , where Q is the set of all γ ∈ P such that α (cid:23) γ and Y µ − ( β ) is a face of γ . In otherwords, Q consists of all γ ∈ P such that(7.2) γ ( i ) = α ( i ) if i ∈ µ − ( α ) and γ ( i ) = β ( i ) if i µ − ( β ) . CATEGORY O FOR ORIENTED MATROIDS 43

We wish to show that ( X T Y ) α,β = ( α = β . If α = β , then Q = { α } and the sum is equal to q d αα ( − q ) d αα = 1 .Now assume that α = β and let J := µ − ( β ) \ µ − ( α ) and J ′ := µ − ( α ) \ µ − ( β ) so that J ⊔ J ′ = ( µ − ( α ) ∪ µ − ( β )) \ ( µ − ( α ) ∩ µ − ( β )) . As we have assumed that α = β , J and J ′ are nonempty.Note that if α ( i ) = β ( i ) for some i ∈ J ′ , then by the conditions (7 . Q is emptyand ( X T Y ) α,β = 0 as desired. Thus we will assume that α ( i ) = β ( i ) for all i ∈ J ′ .Let K := { i ∈ µ − ( α ) ∩ µ − ( β ) | α ( i ) = β ( i ) } . For δ ∈ P , Y µ − ( β ) is a face of δ if and only if δ = β W for some subset W ⊂ µ − ( β ) . On the other hand, δ = β W (cid:22) α if and only if K ∪ J ⊃ W ⊃ K . Thus Q = { ( β K ) S | S ⊂ J } and ( X T Y ) α,β = X S ⊂ J q d α, ( βK ) S ( − q ) d ( βK ) S,β = ( − | K | X S ⊂ J ( − | S | q d α, ( βK ) S + d ( βK ) S,β = ( − | K | X S ⊂ J ( − | S | q d α,β +2 | S ( βK ) Sα,β | = ( − | K | q d α,β X S ⊂ J ( − | S | q | S βSα,β | , where in the last line we have used S ( β K ) S α,β = S β S α,β = { i ∈ S | α ( i ) = β ( i ) } . We willneed the following lemma to ﬁnish this proof. Lemma 7.16.

Assume as above that α = β and α ( i ) = β ( i ) for all i ∈ J ′ . Then thereexists an element t ∈ J such that α ( t ) = β ( t ) .Proof. Suppose for the sake of contradiction that α ( i ) = β ( i ) for all i ∈ J , then α ( i ) = β ( i ) for all i ∈ J ⊔ J ′ . In the deletion-contraction program (cid:0) P / ( µ − ( α ) ∩ µ − ( β )) (cid:1) \ ( µ − ( α ) ∪ µ − ( β )) c deﬁned on the set J ⊔ J ′ , the restrictions of the sign vectors of α and β are thenequal and so describe the same tope T . Now Y µ − ( α ) is the optimal solution for α and Y µ − ( β ) is the optimal solution for β , so the restrictions Y α and Y β of Y µ − ( α ) and Y µ − ( β ) to J ⊔ J ′ should both be the unique optimal solution of the tope T . But z ( Y α ) = J ′ = J = z ( Y β ) , which is a contradiction. Thus there exists a t ∈ J suchthat α ( t ) = β ( t ) as desired. (cid:3) In particular if S ⊂ J \{ t } , we have S β S ⊔{ t } α,β = S β S α,β .Using this fact we rewrite the sum above: ( X T Y ) α,β = ( − | K | q d α,β X S ⊂ J ( − | S | q | S βSα,β | , = ( − | K | q d α,β X S ⊂ J \{ t } (cid:18) ( − | S | q | S βSα,β | + ( − | S ⊔{ t }| q | S βS ∪{ t } α,β | (cid:19) = ( − | K | q d α,β X S ⊂ J \{ t } (cid:18) ( − | S | q | S βSα,β | − ( − | S | q | S βSα,β | (cid:19) = 0 . (cid:3) Self-dual projectives.

Consider the duality functord : A -mod → A -moddeﬁned by composing the equivalence A op -mod ≃ A -mod induced by the isomor-phism A ∼ = A op given by reversing the arrows of the quiver Q n in Section 3.3 withthe induced functor A op -mod → A -mod coming from vector space duality.In the following result we will involve bounded feasible topes in the afﬁnespaces of both P and P ∨ . To distinguish the two topes with a ﬁxed sign vector α ∈ P = P ∨ , we write T α to denote the tope in P and T ∨ α to denote the tope in P ∨ . Theorem 7.17.

For any generic oriented matroid program P and α ∈ P . The followingare equivalent: (1) The projective P α is injective. (2) The projective P α is self-dual. (3) The simple L α is contained in the socle of some standard module V β . (4) The bounded feasible tope T α covers an infeasible subtope X , meaning X ( g ) = 0 . (5) The bounded feasible tope T ∨ α in the dual program P ∨ is in the core of the afﬁnespace for P ∨ . In other words ( T ∨ α ) ∞ = 0 or equivalently the cocircuit faces of thetope T ∨ α are all feasible.When P is Euclidean, and so A is quasi-hereditary by Theorem 7.6, then the statementsabove are also equivalent to the following: (6) The projective P α is tilting.Proof. The implications (1) ⇔ (2) ( ⇔ (6) , if A is quasi-hereditary) are standardfacts. (2) = ⇒ (3) : If P α is self-dual, then the socle of P α is isomorphic to the cosocleof P α , which is L α . Therefore when expressing [ P α ] as a sum of simple classestimes powers of q in the Grothendieck group, L α is the only simple to appear inthe top degree. On the other hand, by Theorem 7.9, [ P α ] = X β ∈ C α q d αβ [ V β ] , so the unique simple class appearing in the highest degree must also appear as thehighest degree term of some [ V β ] . We conclude that L α is the socle of V β . (3) = ⇒ (4) : Let b = µ − ( β ) . Lemma 7.2 says that V β is spanned as a vectorspace by taut paths p γ from β to γ (cid:22) β . A taut path p α is in the socle of V β ifthere does not exist a longer taut path p γ that factors through p α . Note that this isequivalent to the condition: if i b and α has a feasible face Y such that Y ( i ) = 0 ,then α ( i ) = β ( i ) .Recall that Y b ∈ A denotes the feasible cocircuit of N = f M\ f that is the opti-mal solution of the tope T β . By the covector axioms of an oriented matroid, thecomposition T := ( − Y b ) ◦ T α is also a covector of N and in particular an infeasibletope such that T ( i ) = α ( i ) for all i ∈ b and for all i which are zero on a feasibleface of α . A taut path p from T α to T exists in the tope graph of f M\ f , and thispath cannot change the sign corresponding to any feasible facet of T γ . Thus the CATEGORY O FOR ORIENTED MATROIDS 45 ﬁrst sign change of the path p must be infeasible, which means that T γ covers ansubtope X with X ( g ) = 0 . (4) = ⇒ (5) : If the bounded tope T α in f M\ f covers a subtope X with X ( g ) = 0 then α is a bounded feasible sign vector for both the original program P as well asthe reoriented program − g P = ( − g f M , − g, f ) . Dually, this means that the tope T ∨ α in f M ∨ \ g = ( − g f M ∨ ) \ ( − g ) is bounded and feasible in both dual programs P ∨ and − g P ∨ = ( − g f M ∨ , f, − g ) . In particular, the tope α ∈ P ∨ does not have any cocircuitface Y with Y ( f ) = 0 , since this would imply T ∨ α was unbounded in one of thesegeneric programs. (5) = ⇒ (2) : If ( T ∨ α ) ∞ = 0 , then e α Ae α ≃ R ∨ α = k [ z (∆ ∨ α )] / ( U ⊥ ) , where ∆ ∨ α = F lv ( T ∨ α ) \{ } . By Lemma 5.4, k ∆ ∨ α k = k F lv ( T ∨ α ) \{ }k is homeomorphic to a ( n − d − -sphere and so a result of Munkres (see [Sta04, Theorem II.4.3]) implies that R ∨ α is Gorenstein, meaning that there is an isomorphism R : ( e α Ae α ) n − d − → k such that h x, y i = R xy deﬁnes a perfect pairing on e α Ae α .We wish to produce an isomorphism of A -modules d ( P α ) ∼ = P α . To do so, wewill show that the map h− , −i : e α A × Ae α → k ( p, q ) Z pq deﬁnes a perfect pairing and so it will follow that d ( P α ) = ( e α A ) ∗ ∼ = Ae α = P α asright A -modules.To prove that h− , −i is perfect, we ﬁrst observe that it sufﬁces to show that themap · p β,α : e α Ae β → e α Ae α is injective for some taut path p βα from β to α . Thisis because for any nonzero x ∈ e β Ae α , if x · p βα = 0 , then there exists y ∈ e α Ae α such that R ( xp ) y = R x ( py ) = h x, py i 6 = 0 . On the B side, this means showing that · u S βαα : R ∨ αβ → R ∨ α is injective. We proceed by showing · u S βαα : e R ∨ αβ → e R ∨ α is the injective map in a split short exact sequence of Sym U ⊥ -modules, whichproves the claim by applying the functor − ⊗ Sym U ⊥ k .The claim is obvious if α = β or T ∨ α ∧ T ∨ β is not feasible, so we assume α = β and T ∨ α ∧ T ∨ β is a proper non-empty face of T ∨ α . To see the monomial map · u S βαα : e R ∨ αβ → e R ∨ α is injective and to determine its cokernel, consider the image of any non-zeromonomial m = Q i ∈ S u s i i in e R ∨ αβ , where s i > for any i ∈ S . As m is non-zero in e R ∨ αβ , there exists Y ∈ ∆ ∨ αβ such that S ⊂ z ( Y ) . Note that S βαα = z ( T ∨ α ∧ T ∨ β ) ⊂ z ( X ) for any X ∈ ∆ ∨ αβ . Thus S ∪ S βαα ⊂ z ( Y ) and the product m · u S βαα = Q i ∈ S ∪ S βαα u t i i ,where t i > for i ∈ S ∪ S βαα , is non-zero in e R ∨ α .The computation above also shows that the cokernel of the map · u S βαα : e R ∨ αβ → e R ∨ α is the face ring k [∆] of ∆ = { S ⊂ E n | S ⊂ z ( Y ) for some Y ∈ ∆ α and S βαα S } . = z (∆ α ) \{ S ∈ z (∆ α ) | S βαα ⊂ S } . Recall from Lemma 5.4 that the geometric realization of the simplicial complex z (∆ α ) is a PL ( d − -sphere. The subset of simplices { S ∈ z (∆ α ) | S βαα ⊂ S } is the open star of the simplex on the set S βαα and thus its complement ∆ in z (∆ α ) is a PL ( d − -ball. It follows that k [∆] is Cohen-Macaulay, again with parameterspace U ⊥ . Thus, k [∆] is a free Sym U ⊥ -module, and therefore the exact sequence e R ∨ αβ ֒ → e R ∨ α ։ k [∆] splits. (cid:3)

8. D

ERIVED M ORITA EQUIVALENCE

We conclude with a proof of Theorem 1.16. Recall that M is an oriented ma-troid, U a parameter space for M = M , and P = ( f M , g , f ) , P = ( f M , g , f ) and P mid = ( f M mid , g , f ) are Euclidean generic oriented matroid programs suchthat M = ( f M /g ) \ f = ( f M /g ) \ f and f M mid /g = f M /g , f M mid \ f = f M \ f . We wish to show there is an equivalence of categories D ( A ( P , U )) ∼ = D ( A ( P , U )) , where D ( A ) denotes the bounded derived category of graded ﬁnitely generated A -modules. Remark . Note that if P and P are Euclidean, it is not automatic that P mid willbe Euclidean as well. For example, one could take EFM (8) (see Example 7.10) andthen change the choice of g and f separately to obtain two realizable (and henceEuclidean) generic oriented matroid programs P and P such that P mid is thenon-Euclidean program EFM (8) .We will prove Theorem 1.16 by reducing it to the following claim. Proposition 8.2.

Suppose P = ( f M , g , f ) and P = ( f M , g , f ) are generic Eu-clidean programs extending M such that M /g = M /g . Then there is an equivalenceof categories D ( A ( P , U )) ≃ D ( A ( P , U )) . Before giving a proof of this Proposition, we will use it to deduce Theorem 1.16.

Proof that Proposition 8.2 implies Theorem 1.16.

Under the assumptions of Theorem 1.16, f M mid /g = f M /g . Then by Proposition 8.2 it follows that(8.1) D ( A ( P , U )) ≃ D ( A ( P mid , U ) . On the other side, duality together with the assumptions of Theorem 1.16 give: f M ∨ mid /f = ( f M mid \ f ) ∨ = ( f M \ f ) ∨ = f M ∨ /f . Viewing f and f as playing the role of g in the Euclidean programs f M ∨ mid and f M ∨ respectively, we can again apply Proposition 8.2 to ﬁnd:(8.2) D ( A ( P ∨ , U ⊥ )) ≃ D ( A ( P ∨ mid , U ⊥ )) . Putting these equivalences (8.1) and (8.2) together with the equivalences fromKoszul duality: D ( A ( P mid , U )) ≃ D ( A ( P ∨ mid , U ⊥ )) and D ( A ( P , U )) ≃ D ( A ( P ∨ , U ⊥ )) , CATEGORY O FOR ORIENTED MATROIDS 47 gives the desired result: D ( A ( P , U )) ≃ D ( A ( P , U )) . (cid:3) The deﬁnition and properties of the functor.

It remains to prove Proposition8.2. For the remainder of the paper we will let P = ( f M , g , f ) and P = ( f M , g , f ) be two Euclidean generic oriented matroid programs such that f M /g = f M /g .For ℓ = 1 , let A ℓ = A ( P ℓ , U ) , B ℓ = B ( P ℓ , U ) and P ℓ be the set of bounded,feasible sign vectors of P ℓ . Note that the set of bounded sign vectors B is the samefor P and P .As in [BLPW10, Section 6], the desired equivalence will come from a derivedtensor product with a certain bimodule N .It is slightly easier to deﬁne the bimodule N on the B -side, using the isomor-phisms A ℓ ≃ B ∨ ℓ of Theorem 5.12 for ℓ = 1 , . Namely, let N = M ( α,β ) ∈P ×P R ∨ αβ [ − d αβ ] with the natural left B ∨ -action and right B ∨ -action given by the ⋆ operation.To translate this to the A -side, recall the following the alternative deﬁnition of A from Section 3.3, A ( P , U ) = e P D n e P / h e f e P i + h ϑ ( U ⊥ ) e P i . We consider an extended version of A that only depends on f M /g by replacing e P by e B = P α ∈B e α . That is, let A ext ( P , U ) = e B D n e B / h e f e B i + h ϑ ( U ⊥ ) e B i . As A ext ( P , U ) only depends on f M /g and we have assumed that f M /g = f M /g , let A ext := A ext ( P , U ) = A ext ( P , U ) . When viewed as an ( A , A ) -bimodule, N can be described as N = e g A ext e g , where e g ℓ = P α ∈P ℓ e α for ℓ = 1 , .To check that these deﬁnitions of N coincide, consider the graded vector space B ext ( P , U ) = M ( α,β ) ∈F×F R αβ [ − d αβ ] , made into an algebra via ⋆ , as in the deﬁniton of B ( P , U ) from Section 5.2. Thenan easy extension of the proof of Theorem 5.12 gives us the following lemma. Lemma 8.3.

There is an isomorphism A ext ( P , U ) ≃ B ext ( P ∨ , U ⊥ ) . Combining thisisomorphism with the isomorphisms A ℓ ≃ B ∨ ℓ , we obtain an equivalence between the twodeﬁnitions of N . We deﬁne the functor

Φ : D ( A ) → D ( A ) via Φ( M ) = M L ⊗ A N. For ℓ = 1 , and any α ∈ P ℓ , let P ℓα and V ℓα be the corresponding projective andstandard A ℓ -modules. Deﬁne ν : P → P to be the composition P µ − −→ B µ −→ P . Proposition 8.4. If α ∈ P ∩ P , then Φ( P α ) = P α .Proof. Consider the natural map

Γ : P α = e α A → e α A ⊗ A e g A ext e g = P α ⊗ A N = Φ( P α ) taking e α to e α ⊗ e g e g . For paths p, q in the quiver Q with p only passing throughnodes in P , the equality of the simple tensors e α p ⊗ e g qe g = e α ⊗ e α pe g qe g = e α ⊗ e g e g · e α pe g qe g implies that Γ is an isomorphism. (cid:3) Remark . Note that the proposition above and its proof are valid without theassumption that P and P be Euclidean. Lemma 8.6.

For any α ∈ P , the A -module Φ( P α ) has a ﬁltration with standard sub-quotients. For b ∈ B , if α ∈ B b then the standard module V µ ( b ) appears with multiplicity1 in the associated graded, and otherwise it does not appear. Proof.

We have Φ( P α ) = P α ⊗ A N = e α A ⊗ A e g A ext e g , so elements of Φ( P α ) can be represented as linear combinations of paths in B whichbegin at α and end at elements of P = B ∩ F . For β ∈ P , let Φ( P α ) β be thesubmodule generated by paths p such that β is the maximal element of P (withrespect to the ordering ≤ on P coming from our Euclidean assumption on P )through which p passes. Then let Φ( P α ) > β = [ γ> β Φ( P α ) γ and Φ( P α ) ≥ β = [ γ ≥ β Φ( P α ) γ . We then obtain a ﬁltration Φ( P α ) = [ β ∈P Φ( P α ) ≥ β . Suppose b ∈ B and let β = µ ( b ) . It sufﬁces to show that the quotient Φ( P α ) ≥ β / Φ( P α ) > β is isomorphic to V β if α ∈ B b , and is zero otherwise.Our argument follows the proof of Theorem 7.6.If α

6∈ B b , then for some i ∈ b , α ( i ) = β ( i ) . Thus any path from α to β can berepresented by one passing through β i . By Lemma 7.5, β i > β and therefore Φ( P α ) ≥ β / Φ( P α ) > β ∼ = 0 . Otherwise, precomposition with a taut path from α to β gives a surjective map V β ։ Φ( P α ) ≥ β / Φ( P α ) > β . Recall that the set B b deﬁned in Deﬁnition 2.18 only depends on f M /g . CATEGORY O FOR ORIENTED MATROIDS 49

Thus dim V β ≥ dim Φ( P α ) ≥ β / Φ( P α ) > β and it sufﬁces to show equality holds. After summing over all β ∈ P : dim(Φ( P α )) = X β ∈P dim Φ( P α ) ≥ β / Φ( P α ) > β ≤ X α ∈B b dim V µ ( b ) = { ( γ, b ) ∈ P × B | α, γ ∈ B b } . It sufﬁces to show that equality holds after summing over all α ∈ P . As N = ⊕ α ∈P Φ( P α ) , we have X α ∈P dim Φ( P α ) = dim N = X ( α,γ ) ∈P ×P dim R ∨ αγ = { ( α, γ, b ) ∈ P × P × B | α, γ ∈ B b } . Here we are using Lemma 5.7 and Proposition 2.19 on each ( α, γ ) ∈ P × P . (cid:3) Remark . Notice that Theorem 7.6 can be viewed as the special case P = P . Remark . We note that the above proof does not use the assumption that P is Euclidean, and so for this result we need only assume that P is Euclidean.More generally, when P is not Euclidean one can prove the result on the level ofGrothendieck groups with a nearly identical proof as was given for the analogousTheorem 7.9. Proposition 8.9.

For all α ∈ P , we have [Φ( V α )] = [ V ν ( α ) ] in the Grothendieck groupsof (ungraded) right A -modules. Thus Φ induces an isomorphism of Grothendieck groups.Proof. For any α ∈ P , the equalities X [ V µ ( b ) ] = [Φ( P α )] = X [Φ( V µ ( b ) )] follow from Lemma 8.6 and its special case Theorem 7.6, where both sums aretaken over { b ∈ B | α ∈ B b } . The ﬁrst claim then follows by induction on the poset P with base case α ∈ P maximal, so P α = V α and Φ( P α ) = V ν ( α ) . The second statement follows from the fact that the classes of standard modulesin a highest weight category form a Z -basis for the Grothendieck group. (cid:3) Remark . One can show that this result holds without the Euclidean conditionon P and P by the second part of Remark 8.8 and the fact that the matrix X fromthe proof of Theorem 7.14 is invertible. Proposition 8.11.

Let α ∈ P . Then Φ( V α ) is the quotient of Φ( P α ) by the submodulegenerated by all paths changing their i -th coordinate for some i ∈ µ − ( α ) . In particular, Tor A m ( V α , N ) = 0 for all m > . Proof.

Apply Φ to the linear projective resolution of V α of Theorem 7.12. The de-gree zero homology of the resulting complex is the quotient promised. We wishto show that the resulting complex is a resolution of V α ⊗ A N . This claim fol-lows from argument is analogous to the proof of Theorem 7.12, where for each S ⊂ µ − ( α ) we ﬁlter each A -module Φ( P α S ) = P α ⊗ A N by standards as in theproof of Lemma 8.6. (cid:3) Corollary 8.12.

If a right A -module M admits a ﬁltration by standard modules, we have Tor A m ( M, N ) = 0 and therefore Φ( M ) = M ⊗ A N. Ringel duality and composition of functors.

Suppose that P = ( f M , g, f ) is ageneric Euclidean extension of M . Consider the program P = − g P = ( − g f M , − g, f ) obtained from P by reorientation of g . In other words P is the program on theoriented matroid M with feasible cocircuits equal to the negative of the feasiblecocircuits of P . This program is also generic and Euclidean. We let A = A ( P , U ) and denote by Φ − : D ( A ) → D ( A ) , the functor Φ for P = P and P = P . We will prove that Φ − is an equivalenceand relate it to Ringel duality. Theorem 8.13. Φ − is an equivalence, the algebras A and A are Ringel dual, and theRingel duality functor is d ◦ Φ − = Φ − ◦ d.Proof. Notice that for any α ∈ P , we have that the B -side description of Φ − gives Φ − ( P α ) = M β ∈P R ∨ αβ [ − d αβ ] . Then we have that the tope corresponding to the restrictions of both α and β inthe oriented matroid f M ∨ /S βαα on E n ∪ { f } has all cocircuit faces taking the value + on f . As in the proof of Theorem 7.17, this implies that R ∨ αβ is Gorenstein and Φ − ( P α ) is self-dual. By Lemma 8.6 it follows that Φ − ( P α ) is tilting.It remains to show that Φ − is an equivalence. With Proposition 8.4 and Theorem7.17 in hand, one can repeat the proof of [BLPW10, Theorem 6.10] word for word. (cid:3) To complete the proof of Proposition 8.2 in the general case, we will need tostudy the composition of functors.Let P = ( f M , g , f ) , P = ( f M , g , f ) , and P = ( f M , g , f ) be generic Eu-clidean programs extending M for which f M /g = f M /g = f M /g . We canthen deﬁne the three functors D ( A ) Φ −→ D ( A ) Φ −→ D ( A ) and D ( A ) Φ −→ D ( A ) as before. We would like to compare Φ with the composition Φ ◦ Φ .Notice that N = Φ ( A ) = M α ∈P Φ ( P α ) has a ﬁltration by standard modules as a right A -module by Lemma 8.6. Then Φ ◦ Φ ( M ) = ( M L ⊗ A N ) L ⊗ A N = M L ⊗ A ( N ⊗ A N ) CATEGORY O FOR ORIENTED MATROIDS 51 by Corollary 8.12. The natural map N ⊗ A N → N given by concatenationof paths induces a natural transformation Φ ◦ Φ → Φ . We also have that Φ ◦ Φ and Φ induce the same map on Grothendieck groups by Proposition8.9. This implies that dim k N ⊗ A N = dim k N since the classes of N ⊗ A N and N agree in the Grothendieck group of A -modules.We now combine this discussion with the equivalence we have already proved.Let P = P , so that Φ = Φ − : D ( A ) → D ( A ) and Φ = Φ : D ( A ) → D ( A ) . Lemma 8.14. Φ − ∼ = Φ ◦ Φ .Proof. The conclusion follows from the discussion above if we can show that themap N ⊗ A N → N is an isomorphism. We have already observed that thesource and target have the same dimension, so it will sufﬁce to show that this mapis surjective. This means showing that for any ( α, β ) ∈ P × P , every path from α to β in e α A ext e β can be represented as a path that passes through some sign vector γ in P . It sufﬁces to do this for a taut path from α to β . Translating this to the B -side, we wish to show that ∨ αβ = 1 ∨ αγ ⋆ ∨ γβ for some γ ∈ P .Let ( α, β ) ∈ P × P and suppose that R ∨ αβ is nonzero. This means the maximalcommon covector face T ∨ α ∧ T ∨ β of the topes T ∨ α and T ∨ β in f M ∨ := f M ∨ \ g = ( f M /g ) ∨ = ( f M /g ) ∨ = f M ∨ \ g is nonzero and all of its nonzero cocircuit faces are feasible, i.e. they take the value + on f . Together with the fact that z ( T ∨ α ∧ T ∨ β ) = S βαα , this implies that T ∨ α ∧ T ∨ β restricts to a bounded feasible tope in the oriented matroid program P ∨ /S βαα = ( f M ∨ /S βαα , f, g ) . Let Y be the optimal cocircuit face of T ∨ α ∧ T ∨ β viewed as a covector in P ∨ /S βαα . Then Y lifts to a unique feasible cocircuit of P ∨ and let γ ∈ P ∨ = P be thecorresponding sign vector. By construction, T ∨ α ∧ T ∨ β is a face of T ∨ γ .Thus R ∨ αγβ = R ∨ αβ , S γαβ = ∅ and ∨ αβ = 1 ∨ αγ ⋆ ∨ γβ . (cid:3) Proof of Proposition 8.2.

We can set up everything as in Lemma 8.14, and we knowthat Φ − is an equivalence by Theorem 8.13. This gives us that Φ : D ( A ) → D ( A ) is faithful while Φ is full and essentially surjective.Note that P is Euclidean if P is Euclidean. Appealing to the same argumentsas before, Φ ◦ Φ is an equivalence. We conclude that Φ is also full and essen-tially surjective and thus an equivalence. (cid:3) R EFERENCES[ ´ADL03] I. ´Agoston, V. Dlab, and E. Luk´acs. Quasi-hereditary extension algebras.

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