For a general ${\mathfrak{D}}$ containing $Y^{0}$, pick an ordering
$X_{1},\dots,X_{r}$ of the elements of
${\mathfrak{F}}_{em}(Y)\backslash{\mathfrak{D}}$ such that their ranks are
nondecreasing, and let
${\mathfrak{D}}_{\ell}={\mathfrak{D}}\cup\\{X_{1},\dots,X_{\ell}\\}$. Then for
$1\leq\ell\leq r$, we have
${\mathfrak{F}}_{em}^{\mathcal{F}}(X_{\ell})\backslash\\{X_{\ell}\\}\subset{\mathfrak{D}}_{\ell-1}$,
so an identical argument shows that

$\displaystyle\varprojlim_{X\in{\mathfrak{D}}_{\ell}}\widetilde{R}_{X}\rightarrow\varprojlim_{X\in{\mathfrak{D}}_{\ell-1}}\widetilde{R}_{X}\quad\text{and}\quad\varprojlim_{X\in{\mathfrak{D}}_{\ell}}R_{X}\rightarrow\varprojlim_{X\in{\mathfrak{D}}_{\ell-1}}R_{X}$

are isomorphisms. ∎

###### Proof of Theorem 6.1.

Put the equations (6.2) and (6.3) together with Lemma 6.9 to get

$\displaystyle
Z(\widetilde{B}_{\mathrm{ext}})\cong\varprojlim_{X\in{\mathcal{A}}}\widetilde{R}_{X}\cong\varprojlim_{X\in{\mathcal{A}}^{\mathcal{P}}}\widetilde{R}_{X}\cong
Z(\widetilde{B})$

and

$\displaystyle
Z(B_{\mathrm{ext}})\cong\varprojlim_{X\in{\mathcal{A}}}R_{X}\cong\varprojlim_{X\in{\mathcal{A}}^{\mathcal{P}}}R_{X}\cong
Z(B).$

All of these isomorphisms are compatible with
$\overline{\zeta}_{\mathrm{ext}}$, $\overline{\zeta}$ and the natural
quotients $\widetilde{B}_{\mathrm{ext}}\rightarrow B_{\mathrm{ext}}$ and
$\widetilde{B}\rightarrow B$, so we are done. ∎

##  7\. The module category of $A$

In this section, we study the simple modules for $A=A({\mathbf{P}},U)$ and
their projective covers using a class of standard modules.

######  Definition 7.1.

For any $\alpha\in{\mathcal{P}}$, let

$L_{\alpha}:=A/(e_{\beta}\mid\beta\neq\alpha).$

Then $L_{\alpha}$ is the simple one-dimensional $A$-module supported at
$\alpha$, and let

$P_{\alpha}:=e_{\alpha}A$

be its projective cover. We also define $V_{\alpha}$ to be
$P_{\alpha}/K_{\alpha}$, where

$K_{\alpha}:=\sum_{i\in b}p(\alpha,\alpha^{i})\cdot A\subset P_{\alpha}$

and $b$ is the basis of $M$ such that $\mu(b)=\alpha$ under the bijection of
Corollary 2.15. We refer to $V_{\alpha}$ as the _standard module_ and
$L_{\alpha}$ as the _simple module_ associated to $\alpha$.

######  Lemma 7.2.

Let $\alpha\in{\mathcal{P}}$. The standard module $V_{\alpha}$ has a basis
consisting of a taut path from $\alpha$ to each $\beta\preceq\alpha$.

###### Proof.

We simply copy the argument from [BLPW10, Lemma 5.21]. It is clear that the
collection of such taut paths is linearly independent. We now show that the
image of any other path is trivial in $V_{\alpha}$.

Let $b=\mu^{-1}(\alpha)\in{\mathbb{B}}$.

Suppose $p$ is a taut path from $\alpha$ to $\gamma\in{\mathcal{F}}$ and
$\gamma\not\preceq\alpha$. Then $\gamma(i)\neq\alpha(i)$ for some $i\in b$ and
$\alpha^{i}\in{\mathcal{F}}$. By Corollary 3.7 $p$ can be replaced by one of
the form $p(\alpha,\alpha^{i})\cdot x\in K_{\alpha}$. Thus $p=0$ as an element
of $V_{\alpha}$.

If $p$ is a non-taut path, then by Proposition 3.6, we can write

$\displaystyle
p=p^{\prime}\cdot\prod_{i}u_{i}^{a_{i}}=\left(\prod_{i}e_{\alpha}u_{i}^{a_{i}}\right)\cdot
p^{\prime}$

where $p^{\prime}$ is taut (with the same endpoint as $p$) and $a_{i}>0$ for
some $i$. Corollary 3.8 implies that for all $i\in E_{n}$ and some $c_{i,j}\in
k$, we have

$\displaystyle e_{\alpha}u_{i}=\sum_{j\in
b}c_{i,j}p(\alpha,\alpha^{j},\alpha)\in K_{\alpha}.$

Thus, $p\in K_{\alpha}$ and we are done. ∎

######  Corollary 7.3.

The kernel of $V_{\alpha}\twoheadrightarrow L_{\alpha}$ has a filtration with
subquotients isomorphic to $L_{\beta}$ for $\beta\prec\alpha$, each appearing
exactly once.

###  7.1. $A$ is a quasi-hereditary algebra when ${\mathbf{P}}$ is Euclidean

Recall that a finite-dimensional algebra is quasi-hereditary if its category
of finitely generated modules is highest weight in the following sense.

######  Definition 7.4.

Let ${\mathcal{C}}$ be an abelian, artinian $k$-linear category and let
${\mathcal{I}}$ be the set indexing the isomorphism classes of simple objects
$\\{L_{\alpha}\mid\alpha\in{\mathcal{I}}\\}$ and indecomposable projective
objects $\\{P_{\alpha}\mid\alpha\in{\mathcal{I}}\\}$. Then ${\mathcal{C}}$ is
a highest weight category if the set ${\mathcal{I}}$ can be endowed with a
partial order $\leq$ and there exists a collection of objects
$\\{V_{\alpha}\mid\alpha\in{\mathcal{I}}\\}$ with surjections

$P_{\alpha}\rightarrow V_{\alpha}\rightarrow L_{\alpha}$

that satisfy:

  * (i)

the kernel of $V_{\alpha}\rightarrow L_{\alpha}$ has a filtration for which
each subquotient is isomorphic to $L_{\gamma}$ for some $\gamma<\alpha$, and

  * (ii)

The kernel of $P_{\alpha}\rightarrow V_{\alpha}$ has a filtration for which
each subquotient is isomorphic to $V_{\beta}$ for some $\beta>\alpha$.

Now consider the category of finitely generated $A({\mathbf{P}},U)$-modules.
As discussed above, the isomorphism classes of simple modules are indexed by
the set ${\mathcal{P}}$ of bounded feasible topes. For $A=A({\mathbf{P}},U)$
to be a quasi-hereditary algebra, we will assume that the oriented matroid
program ${\mathbf{P}}$ is Euclidean. Recall from Section 2.3, that this
implies there is partial order $\leq$ on ${\mathcal{P}}$ defined by:
$\alpha\leq\beta$ when there exists a directed sequence of edges from
$\mu^{-1}(\alpha)$ to $\mu^{-1}(\beta)$ in the graph $G_{\mathbf{P}}$ of the
program. By Lemma 2.25, this is the same partial order as that defined by the
transitive closure of the cone relation $\preceq$.

Suppose ${\mathbf{P}}$ is Euclidean. Then the category of
$A({\mathbf{P}},U)$-modules and the poset ${\mathcal{I}}=({\mathcal{P}},\leq)$
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 $\beta\in{\mathcal{P}}$ and $i\in\mu^{-1}(\beta)$. Then the feasible
sign vector $\beta^{i}$ is either unbounded or $\beta^{i}\succeq\beta$.

###### Proof.

Suppose $\beta^{i}$ is bounded. As $\beta^{i}(j)=\beta(j)$ for all $j\neq i$,
if $i\not\in\mu^{-1}(\beta^{i})$, then for all $j\in\mu^{-1}(\beta^{i})$,
$\beta^{i}(j)=\beta(j)$. Thus $\beta\in{\mathcal{B}}_{\mu^{-1}(\beta^{i})}$ or
equivalently $\beta\preceq\beta^{i}$.

If $\beta^{i}$ is bounded and $i\in\mu^{-1}(\beta^{i})$, then the optimal
solutions of $\beta$ and ${\beta^{i}}$ are also optimal solutions of their
common subtope $Y=\beta\wedge\beta^{i}$. But $Y$ is a tope of
${\mathbf{P}}/\\{i\\}$ and so $Y$ has a unique optimal solution. Thus
$\mu^{-1}(\beta)=\mu^{-1}(\beta^{i})$, which contradicts the fact that $\mu$
is bijection. ∎

######  Theorem 7.6.

Assume ${\mathbf{P}}$ is Euclidean. Then the kernel of the quotient
homomorphism $P_{\alpha}\twoheadrightarrow V_{\alpha}$ has a filtration with
each successive subquotient isomorphic to $V_{\beta}$ for $\beta\succ\alpha$.
Each of these standard modules appears exactly once.

In particular, $A$ is quasihereditary.

###### Proof.

For any $\gamma\in{\mathcal{P}}$, we define $P_{\alpha}^{\gamma}\subset
P_{\alpha}$ to be the submodule generated by paths which pass through
$\gamma$. For any $\beta\in{\mathcal{P}}$, let

$K_{\alpha}^{\beta}:=\sum_{\gamma>\beta}P_{\alpha}^{\gamma}.$

After choosing a total order on
$\\{\beta\in{\mathcal{P}}\mid\alpha\leq\beta\\}$ refining $\leq$, the set of
submodules $P_{\alpha}^{\beta}+K_{\alpha}^{\beta}$ with $\beta\geq\alpha$
forms a filtration of $K_{\alpha}$ with successive subquotients

$M_{\alpha}^{\beta}:=P_{\alpha}^{\beta}+K_{\alpha}^{\beta}/K_{\alpha}^{\beta}.$

Our goal now is to prove that $M_{\alpha}^{\beta}$ is zero if
$\beta\not\succeq\alpha$, and is isomorphic to $V_{\beta}$ if
$\beta\succeq\alpha$. Notice that $M_{\alpha}^{\alpha}=V_{\alpha}$.

If $\beta\not\succeq\alpha$, then there is an index $i\in\mu^{-1}(\beta)$ such
that $\alpha(i)\neq\beta(i)$. By Proposition 3.6, any path starting at
$\alpha$ and passing through $\beta$ can be written as $p_{\alpha,\beta}\cdot
r$ in $P_{\alpha}^{\geq\beta}$, where $p_{\alpha,\beta}$ represents a taut
path from $\alpha$ to $\beta$ and $r$ represents a path starting at $\beta$.
We may then apply Corollary 3.7 to the taut path $p_{\alpha,\beta}$ and
$\gamma=\beta^{i}$, to show that $p_{\alpha,\beta}$ can be chosen to pass
through $\beta^{i}\in{\mathcal{F}}$. By Lemma 7.5,
$P_{\alpha}^{\beta}\subseteq P_{\alpha}^{\beta^{i}}\subseteq
K_{\alpha}^{\beta}$, so $M_{\alpha}^{\beta}=0$.

On the other hand, assume that $\beta\succ\alpha$. There is a natural map
$P_{\beta}\rightarrow P_{\alpha}^{\beta}$ given by composing any element of
$P_{\beta}$ with a fixed taut path $p_{\alpha,\beta}$ from $\alpha$ to
$\beta$. This induces a homomorphism $V_{\beta}\rightarrow M_{\alpha}^{\beta}$
that we wish to show is an isomorphism.

By Proposition 3.6, any path starting at $\alpha$ and passing through $\beta$
can be expressed as a product of an element in $P_{\beta}$ with some taut path
from $\alpha$ to $\beta$ and by Proposition 3.5 the taut path can be chosen to
be the one we have fixed. It follows that the map $P_{\beta}\rightarrow
P_{\alpha}^{\beta}$ is surjective and thus the induced map
$V_{\beta}\rightarrow M_{\alpha}^{\beta}$ is surjective as well.

Finally, we need to show that $V_{\beta}\rightarrow M_{\alpha}^{\beta}$ is
injective. We proceed by showing that they have the same dimension. The
surjectivity of the map implies

$\dim_{k}M_{\alpha}^{\beta}\leq\dim_{k}V_{\beta}=|\\{\gamma\in{\mathcal{P}}\mid\gamma\preceq\beta\\}|,$

so that

$\dim_{k}P_{\alpha}=\sum_{\beta\succeq\alpha}\dim_{k}M_{\alpha}^{\beta}\leq|\\{(\gamma,\beta)\in{\mathcal{P}}\times{\mathcal{P}}\mid\alpha,\gamma\preceq\beta\\}|.$

We’re done if we can show this is an equality. As
$A=\sum_{\alpha\in{\mathcal{P}}}P_{\alpha}$, it suffices to prove that

$\dim_{k}A=\sum_{\alpha\in{\mathcal{P}}}\dim_{k}P_{\alpha}=\\{(\alpha,\gamma,\beta)\in{\mathcal{P}}\times{\mathcal{P}}\times{\mathcal{P}}\mid\alpha,\gamma\preceq\beta\\}.$

But recall that

$A=A({\mathbf{P}},U)\simeq
B({\mathbf{P}}^{\vee},U^{\perp})=\bigoplus_{(\alpha,\gamma)\in{\mathcal{P}}^{\vee}\times{\mathcal{P}}^{\vee}}R_{\alpha\gamma}^{\vee}$

and so by Lemma 5.7

$\dim_{k}B({\mathbf{P}}^{\vee},U^{\perp})=\sum_{(\alpha,\gamma)\in{\mathcal{P}}^{\vee}\times{\mathcal{P}}^{\vee}}|\\{\text{common
feasible cocircuit faces of $\alpha$ and $\gamma$}\\}|.$

We are then reduced to showing that the number of common feasible cocircuit
faces in ${\mathbf{P}}^{\vee}$ of $\alpha$ and $\gamma$ is equal to the number
of bounded feasible topes $\beta$ of ${\mathbf{P}}$ such that
$\alpha\preceq\beta$ and $\gamma\preceq\beta$. This follows from Complementary
Slackness (Proposition 2.19). ∎

###  7.2. The structure of projectives when ${\mathbf{P}}$ is not Euclidean

Note that the definition of the standard modules makes sense for any
${\mathbf{P}}$ and Lemma 7.2 holds even in the non-Euclidean case. However,
when ${\mathbf{P}}$ is not Euclidean, the transitive closure of the cone
relation is not a poset and so the standard modules are not part of a highest
weight structure.

Nonetheless, one might still hope for a version of Theorem 7.6: that the
kernel of $P_{\alpha}\twoheadrightarrow V_{\alpha}$ has a filtration with
successive subquotients isomorphic to $V_{\beta}$ for $\beta\succ\alpha$.

In this section we observe that this is too optimistic a hope, but that it
does hold on the level of graded Grothendieck groups.

Recall from Lemma 5.7, that for any $\alpha\in{\mathcal{P}}$, the dimension
$\dim_{k}R_{\alpha}$ is equal to the number of feasible cocircuit faces of
$\alpha$. We begin with a graded refinement of this statement.

######  Lemma 7.7.

Let $(h_{0},h_{1}\ldots,h_{d-1})$ denote the $h$-vector of
$z(\Delta_{\alpha})$ or equivalently $h_{i}$ is equal to the dimension of the
graded piece of $R_{\alpha}$ of degree $2i$. Then $h_{i}$ is equal to the
number of feasible vertices of $\alpha$ with $i$ outgoing edges.

###### Proof.

We proceed by showing that $z(\Delta_{\alpha})$ is partitionable. Recall that
a pure simplicial complex $\Delta$ is partitionable, if it can be expressed as
a disjoint union of closed intervals of the form

$\Delta=[G_{1},F_{1}]\sqcup\ldots\sqcup[G_{s},F_{s}],$

where each $F_{i}$ is a facet of $\Delta$. By [Sta04, Proposition III.2.3] the
$h$-polynomial of such a simplicial complex is given by

$h_{i}=\\#\\{j:|G_{j}|=i\\}.$

Recall that $z(\alpha)=\varnothing$ and $z(\Delta_{\alpha})$ is isomorphic as
a poset to ${\mathfrak{F}}_{lv}^{\mathcal{F}}(\alpha)\backslash\\{0\\}$. Thus
we may identify a face of $\alpha$ with the faces of the abstract simplicial
complex $z(\Delta_{\alpha})$.

The facets $F_{1},\ldots,F_{s}$ of $z(\Delta_{\alpha})$ are the zero sets of
the feasible vertices (i.e., feasible cocircuit faces) of $\alpha$. If
$F_{i}=z(X_{i})$ for a feasible vertex $X_{i}$, let $G_{i}$ be (the zero set
of) the meet in $\Delta_{\alpha}$ of the incoming edges of $X_{i}$.

Recall that each face of $\alpha$ has a unique optimal solution (this follows
from Theorem 2.14). For each feasible face $Y$ of $\alpha$, the face
$z(Y)\in\Delta_{\alpha}$ is in the interval $[G_{j},F_{j}]$ if and only if
$X_{j}$ is the optimal solution of $Y$. Thus
$[G_{1},F_{1}]\sqcup\ldots\sqcup[G_{s},F_{s}]$ is a partition of
$z(\Delta_{\alpha})$. Note that

$|G_{j}|=d-\\#\\{\textrm{incoming~{}edges~{}to~{}}X_{j}\\}=\\#\\{\textrm{outgoing~{}edges~{}to~{}}X_{j}\\}.$

We conclude that

$h_{i}=\\#\\{j:|G_{j}|=i\\}=\\#\\{\textrm{feasible~{}vertices~{}of~{}}Y\textrm{~{}with~{}}i\textrm{~{}outgoing~{}edges}\\}.$

∎

######  Corollary 7.8.

Let $(h_{0},h_{1}\ldots,h_{d-1})$ denote the $h$-vector of
$z(\Delta_{\alpha\beta})$ or equivalently $h_{i}$ is equal to the dimension of
the graded piece of $R_{\alpha\beta}$ of degree $2i$. Then $h_{i}$ is equal to
the number of feasible vertices of $\alpha\wedge\beta$ with $i$ outgoing edges
of $\alpha\wedge\beta$.

###### Proof.

Let $\gamma$ be the tope in ${\mathbf{P}}/z(\alpha\wedge\beta)$ given by the
restriction of $\alpha\wedge\beta$. Then the simplicial complex
$z(\Delta_{\alpha\beta})$ is equal to the simplicial join
$z(\Delta_{\gamma})*\Gamma$ of $z(\Delta_{\gamma})$ with the
$(d_{\alpha\beta}-1)$-simplex $\Gamma$ on the set $z(\alpha\wedge\beta)$. By
standard properties of the $h$-polynomial, we have:

$h(z(\Delta_{\alpha\beta}),x)=h(z(\Delta_{\gamma})*\Gamma,x)=h(z(\Delta_{\gamma}),x)h(\Gamma,x)=h(z(\Delta_{\gamma}),x).$

We conclude that the $h$-vector of $z(\Delta_{\alpha\beta})$ is equal to that
of $z(\Delta_{\gamma})$. The result then follows from Lemma 7.7. ∎

For an $A$-module $M$, let $[M]$ denote the class of $M$ in the Grothendieck
group of $A$-modules. We will consider the Grothendieck group of the category
of graded $A$-modules as a ${\mathbb{Z}}[q,q^{-1}]$-module, where

$[M\langle-k\rangle]=q^{k}[M].$

For a graded vector space $V=\oplus_{i}V_{i}$, we denote the graded dimension
of $V$ by

${\mathrm{grdim}}V=\sum_{i}(\dim V_{i})~{}q^{i}.$

######  Theorem 7.9.

For any generic oriented matroid program ${\mathbf{P}}$ and any
$\alpha\in{\mathcal{P}}$, the class of the indecomposable projective
$P_{\alpha}$ in the Grothendieck group can be expressed as the sum:

$[P_{\alpha}]=\sum_{\gamma\succeq\alpha}q^{d_{\alpha\gamma}}[V_{\gamma}].$

###### Proof.

For any $\beta\in{\mathcal{P}}$ the graded composition series multiplicity of
the simple $L_{\beta}$ in the projective $P_{\alpha}$ is equal to the graded
dimension of the space of paths in $A$ that start at $\alpha$ and end at
$\beta$. In other words we have:

$[P_{\alpha}]=\sum_{\beta\in{\mathcal{P}}}({\mathrm{grdim}}~{}P_{\alpha}e_{\beta})\cdot[L_{\beta}]=\sum_{\beta\in{\mathcal{P}}}({\mathrm{grdim}}~{}e_{\alpha}Ae_{\beta})\cdot[L_{\beta}].$

By Theorem 5.12,

${\mathrm{grdim}}~{}e_{\alpha}Ae_{\beta}={\mathrm{grdim}}~{}R^{\vee}_{\alpha\beta}\langle-
d_{\alpha\beta}\rangle=q^{d_{\alpha\beta}}\cdot{\mathrm{grdim}}~{}R^{\vee}_{\alpha\beta}.$

By Corollary 7.8, we can express the graded dimension of
$R^{\vee}_{\alpha\beta}$ as

${\mathrm{grdim}}~{}R^{\vee}_{\alpha\beta}=\sum_{i=0}^{d}\\#\\{\mathrm{feasible~{}vertices~{}of~{}}\alpha\wedge\beta\mathrm{~{}in~{}}{\mathbf{P}}^{\vee}\mathrm{~{}with~{}}i\mathrm{~{}outgoing~{}edges}\\}\cdot
q^{2i}.$

Observe that by Proposition 2.19 the feasible vertices of $\alpha\wedge\beta$
in ${\mathbf{P}}^{\vee}$ (i.e., common feasible vertices of both $\alpha$ and
$\beta$) are in bijection with the bounded feasible topes $\delta$ of
${\mathbf{P}}$ such that $\alpha\preceq\delta$ and $\beta\preceq\delta$. We
claim that the number of outgoing edges of $\alpha\wedge\beta$ of the feasible
vertex corresponding to $\delta$ is equal to $|S^{\delta}_{\alpha\beta}|$. As
in the proof of Corollary 7.8, let $\gamma$ be the bounded feasible tope
$(\alpha\wedge\beta)|_{\underline{\alpha\wedge\beta}}$ in the contraction
${\mathbf{P}}/z(\alpha\wedge\beta)$. Then the number of outgoing edges of the
vertex of $\gamma$ corresponding to $\delta$ is equal to the distance between
$\gamma$ and the restriction $\overline{\delta}$ of $\delta$ to
$\underline{\alpha\wedge\beta}$. But $\underline{\alpha\wedge\beta}=\\{i\in
E_{n}\mid\alpha(i)=\beta(i)\\}$ and so the distance between $\gamma$ and
$\overline{\delta}$ is equal to the cardinality of the difference set

$S(\gamma,\overline{\delta})=\\{i\in
E_{n}\mid\alpha(i)=\beta(i)\neq\delta(i)\\}=S_{\alpha\beta}^{\delta}.$

Rewriting the sum over topes $\delta$ of ${\mathbf{P}}$ such that
$\alpha\preceq\delta$ and $\beta\preceq\delta$ and using the formula
$d_{\alpha\delta}+d_{\delta\beta}=d_{\alpha\beta}+2|S_{\alpha\beta}^{\delta}|$,
we find:

(7.1)
${\mathrm{grdim}}~{}e_{\alpha}Ae_{\beta}=q^{d_{\alpha\beta}}{\mathrm{grdim}}~{}R^{\vee}_{\alpha\beta}=\sum_{\delta\succeq\alpha,\beta}q^{d_{\alpha\beta}+2|S_{\alpha\beta}^{\delta}|}=\sum_{\delta\succeq\alpha,\beta}q^{d_{\alpha\delta}+d_{\delta\beta}}.$

Putting it all together,

$[P_{\alpha}]=\sum_{\beta}\sum_{\delta\succeq\alpha,\beta}q^{d_{\alpha\delta}+d_{\delta\beta}}[L_{\beta}]$
$=\sum_{\delta\succeq\alpha}q^{d_{\alpha\delta}}\sum_{\beta\preceq\delta}q^{d_{\beta\delta}}[L_{\beta}]$
$=\sum_{\gamma\succeq\alpha}q^{d_{\alpha\delta}}[V_{\gamma}],$

as we wished to show. ∎

We conclude this section with an example of a generic non-Euclidean program
and sign vector $\alpha$ for which the kernel of $P_{\alpha}\twoheadrightarrow
V_{\alpha}$ does not admit a filtration with successive standard subquotients.

######  Example 7.10.

Let ${\mathbf{P}}=(\text{EFM}(8),g,f)$ be the generic non-Euclidean program
defined in [BLVS+99, Section 10.4]. Then
$M=\underline{\text{EFM}(8)/g\backslash f}$ is the uniform matroid of rank 3
on $E_{6}$. 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
$\alpha:E_{6}\rightarrow\\{0,+,-\\}$ using the string of signs

$\alpha(1)\alpha(2)\alpha(3)\alpha(4)\alpha(5)\alpha(6).$

The bijection $\mu$ between ${\mathbb{B}}$ and ${\mathcal{P}}$ can described
as follows, where we have listed the pairs
$(b,\mu(b))\in{\mathbb{B}}\times{\mathcal{P}}$ for ${\mathbf{P}}$:

$\displaystyle(123,++++++)\ (124,+++-+-)\ (123,++-+++)\ (126,+-+++-)$
$\displaystyle(134,+-++++)\ (135,-+++-+)\ (136,++++--)\ (145,-++---)$
$\displaystyle(146,+-+-+-)\ (156,+++++-)\ (234,++--++)\ (235,+++--+)$
$\displaystyle(236,-+++++)\ (245,++---+)\ (246,+++-++)\ (256,+-+---)$
$\displaystyle(345,++++-+)\ (346,++----)\ (356,-+++--)\ (456,+++---).$

Using this table one can deduce the cone relation $\preceq$ on ${\mathcal{P}}$
from the fact that $\mu(b)\preceq\mu(b^{\prime})$ if
$\mu(b)(i)=\mu(b^{\prime})(i)$ for any $i\in b^{\prime}$. For example, if
$\mu(b)\prec\mu(456)$, then $b=346,145$ or $256$.

Let $\alpha=+++---\in{\mathcal{P}}$. Recall the notation $\alpha^{S}$ denoting
the sign vector of a tope which differs from $\alpha$ on exactly the set
$S\subset E_{6}$. Using the above list, we find that

$\displaystyle\\{\beta\in{\mathcal{P}}\mid\beta\succeq\alpha\\}=\\{\alpha,\alpha^{4},\alpha^{5},\alpha^{6},\alpha^{\\{1,4\\}},\alpha^{\\{2,5\\}},\alpha^{\\{3,6\\}},\alpha^{\\{4,5,6\\}}\\}$

Suppose there were a filtration $0\subset F_{1}\subset\ldots\subset
F_{6}\subset K_{\alpha}$ of the kernel $K_{\alpha}:=\sum_{i\in
b}p(\alpha,\alpha^{i})\cdot A$ of $P_{\alpha}\rightarrow V_{\alpha}$ with
nonzero successive standard subquotients
$\\{V_{\gamma}\mid\gamma\in{\mathcal{C}}_{\alpha}\backslash\\{\alpha\\}\\}$ as
in the proof of Theorem 7.6. Let $V_{\gamma}=K_{\alpha}/F_{6}$ be the final
standard subquotient and suppose that $\gamma=\alpha^{S}$.

Let $p$ be a taut path from $\alpha$ to $\alpha^{S}$. If $p\in F_{6}$, then
$(K_{\alpha}/F_{6})e_{\alpha^{S}}=0$, which is a contradiction. Thus we may
assume that $p\not\in F_{6}$. For any $i\in S$, $p=p(\alpha,\alpha^{i})q$
where $q$ is a taut path from $\alpha^{i}$ to $\alpha^{S}$. As $p\not\in
F_{6}$, it follows that $p(\alpha,\alpha^{i})\not\in F_{6}$ and
$L_{\alpha^{i}}$ is a quotient of $V_{\gamma}$. This is a contradiction unless
$S=\\{i\\}$. Thus $S$ is either $\\{4\\},\\{5\\}$ or $\\{6\\}$.

Suppose $S=\\{4\\}$, so $\gamma=\alpha^{4}=++++--$. Note that
$\alpha^{6}\prec\alpha^{4}$, so
$(V_{\alpha^{4}})e_{\alpha^{6}}=(K_{\alpha}/F_{6})e_{\alpha^{6}}\neq 0$ and
thus $p(\alpha,\alpha^{6})\not\in F_{6}$. But this would mean that
$L_{\alpha^{6}}$ is a quotient of $V_{\alpha^{4}}$, which is a contradiction.

After permuting indices, the same argument shows that neither $V_{\alpha^{5}}$
nor $V_{\alpha^{6}}$ is a quotient of $K_{\alpha}$. We conclude that
$P_{\alpha}$ doesn’t not have the expected filtration. More generally, we will
see below in the proof of Theorem 7.14 that the change of basis matrix between
the standard and simple bases for the Grothendieck group is invertible, so
$[P_{\alpha}]$ cannot be expressed as a different sum of standard classes.
Thus $P_{\alpha}$ does not admit a filtration by standards.

###  7.3. $A$ is a Koszul algebra when ${\mathbf{P}}$ is Euclidean

Recall the notion of a Koszul algebra:

######  Definition 7.11.

Let $M=\bigoplus_{\ell\geq 0}M_{\ell}$ be a graded $k$-algebra. A complex

$\dots\rightarrow P_{3}\rightarrow P_{2}\rightarrow P_{1}\rightarrow P_{0}$

of graded projective right $M$-modules is _linear_ if each $P_{\ell}$ is
generated in degree $\ell$. We say that $M$ is _Koszul_ if every simple right
$M$-module has a linear projective resolution.

######  Theorem 7.12.

Assume ${\mathbf{P}}$ is Euclidean. Then for all $\alpha\in{\mathcal{P}}$, the
standard module $V_{\alpha}$ has a linear projective resolution.

###### Proof.

We follow the proof of [BLPW10, Theorem 5.24].

Let $a$ be the basis corresponding to the optimal cocircuit for $\alpha$. We
will define the promised resolution as the total complex of the following
multicomplex.

For any $S\subseteq a$, let $\alpha^{S}\in\\{+,-\\}^{n}$ be the sign vector
which disagrees with $\alpha$ on exactly the entries in $S$. For example,
$\alpha^{\varnothing}=\alpha$, and $\alpha^{\\{i\\}}=\alpha^{i}$ for any $i\in
a$. Notice that if $i\in S\subseteq a$ and $\alpha^{S},\alpha^{S\backslash
i}\in{\mathcal{P}}$, then there is a degree one map

$\begin{array}[]{cccc}\phi_{S,i}\colon&P_{\alpha^{S}}&\longrightarrow&P_{\alpha^{S\backslash
i}},\\\ &q&\mapsto&p(\alpha^{S\backslash i},\alpha^{S})\cdot q.\end{array}$

We extend this to all $S\subseteq a$ and $i\in a$ by declaring that
$P_{\alpha^{S}}=0$ if $\alpha^{S}\not\in{\mathcal{P}}$ and $\phi_{S,i}=0$ if
$i\not\in S$. Consider the module

$\Pi_{\alpha}:=\bigoplus_{S\subseteq a}P_{\alpha^{S}},$

which we view as being graded by the free abelian group
${\mathbb{Z}}\\{\varepsilon_{i}\mid i\in a\\}$ where the summand
$P_{\alpha^{S}}$ is given degree $\varepsilon_{S}:=\sum_{i\in
S}\varepsilon_{i}$. For each $i\in a$, consider the differential
$\partial_{i}\colon\Pi_{\alpha}\rightarrow\Pi_{\alpha}$ of degree
$-\varepsilon_{i}$ defined as the sum

$\partial_{i}:=\sum_{S\subseteq a}\phi_{S,i}.$

Observe that $\partial_{i}\partial_{j}=\partial_{j}\partial_{i}$ for any
$i,j\in a$ by relation ($A2$) and so we can view $\Pi_{\alpha}$ as a multi-
complex with differentials $\partial_{i}$ for each $i\in a$.

Let $\mathbf{\Pi}_{\alpha}^{\bullet}$ denote the total complex of
$\Pi_{\alpha}$. Then $\mathbf{\Pi}_{\alpha}^{\bullet}$ is a linear complex of
projective modules and $H_{0}(\mathbf{\Pi}_{\alpha}^{\bullet})=V_{\alpha}$. It
remains to show that the complex $\mathbf{\Pi}_{\alpha}^{\bullet}$ is exact in
positive degrees.

To do so, we will filter the multicomplex $\Pi_{\alpha}$. For each
$\beta\in{\mathcal{P}}$, let $(\Pi_{\alpha})^{\beta}\subset\Pi_{\alpha}$ be
the submodule whose $\varepsilon_{S}$-graded part is defined as

$\sum_{\gamma\geq\beta,\alpha^{S}}P^{\gamma}_{\alpha^{S}}\subset
P_{\alpha^{S}},$

that is, the submodule consisting of all paths from $\alpha^{S}$ passing
through some $\gamma\in{\mathcal{P}}$ where $\gamma\geq\alpha^{S}$ and
$\gamma\geq\beta$. Observe that the differentials $\partial_{i}$ for $i\in a$
are compatible with the submodules $(\Pi_{\alpha})^{\beta}$ and so we have
defined a filtration of $\Pi_{\alpha}$ by the poset ${\mathcal{P}}$.

Computing the associated graded of this filtration yields a multi-complex

$\widetilde{\Pi}_{\alpha}=\bigoplus_{\beta\in{\mathcal{P}}}(\Pi_{\alpha})^{\beta}/(\Pi_{\alpha})^{>\beta}=\bigoplus_{\beta\in{\mathcal{P}}}\left(\bigoplus_{S}M^{\beta}_{\alpha^{S}}\right),$

where $M^{\beta}_{\alpha^{S}}$ is the subquotient of $P_{\alpha^{S}}$ defined
as in the proof of Theorem 7.6.

Consider the resulting quotient multi-complexes for each
$\beta\in{\mathcal{P}}$. Let $b=\mu^{-1}(\beta)$. Recall from the proof of
Theorem 7.6 that $M^{\beta}_{\alpha^{S}}$ is non-zero if and only if
$\alpha^{S}\in{\mathcal{B}}_{b}$.

If $\beta=\alpha$, then $M^{\beta}_{\alpha^{S}}=M^{\alpha}_{\alpha^{S}}=0$ for
any non-empty $S\subset a=b$. Thus the only non-zero summand of the
$\alpha$-subquotient is $M^{\alpha}_{\alpha}=V_{\alpha}$ in total degree zero.

If $\beta\neq\alpha$, choose an element $i\in a$ such that $i\not\in b$.
Consider those subsets $S\subset a$ such that $i\not\in S$. Then we have
$\alpha^{S}\in{\mathcal{B}}_{b}$ if and only if
$\alpha^{S\cup\\{i\\}}\in{\mathcal{B}}_{b}$. If
$\alpha^{S}\in{\mathcal{B}}_{b}$, then $M_{\alpha^{S\cup i}}^{\beta}\simeq
V_{\beta}\simeq M_{\alpha^{S}}^{\beta}$ and the differential induced by
$\partial_{i}$ is the isomorphism given by left multiplication with
$p(\alpha^{S},\alpha^{S\cup\\{i\\}})$. On the other hand, if
$\alpha^{S}\not\in{\mathcal{B}}_{b}$, then $M_{\alpha^{S\cup
i}}^{\beta}=0=M_{\alpha^{S}}^{\beta}$. In particular, the differential induced
by $\partial_{i}$ on the $\beta$-component of the associated graded multi-
complex is exact.

Recall that if any differential of a multi-complex is exact then the total
complex of the multi-complex is also exact. We conclude that the total complex
of the associated graded multi-complex is exact in positive degree. It then
follows that the total complex of the original multi-complex must also be
exact in positive degrees. ∎

######  Theorem 7.13.

Assume ${\mathbf{P}}$ is Euclidean. Then $A$ and $B$ are Koszul algebras and
$A$ is Koszul dual to $B$.

###### Proof.

By [ÁDL03, Theorem 1] a quasi-hereditary algebra is Koszul if the standard
modules have linear projective resolutions and such resolutions exist for $A$
by the previous theorem. Theorem 5.12 implies that $B\cong
A({\mathbf{P}}^{\vee},U^{\perp})$ must also be Koszul. Finally the Koszul
duality follows from the quadratic duality statement of Theorem 4.2. ∎

###  7.4. Numerical identity for Hilbert polynomials

We do not know whether or not the condition that ${\mathbf{P}}$ be Euclidean
is necessary for $A$ to be Koszul. In this section we prove that for any
generic oriented matroid program ${\mathbf{P}}$ the Hilbert polynomial of the
algebra $A=A({\mathbf{P}},U)$ satisfies the following numerical identity.

Let $H(A,q)$ denote the Hilbert polynomial of $A$, which is the
${\mathcal{P}}\times{\mathcal{P}}$-matrix with entries

$H(A,q)_{\alpha,\beta}={\mathrm{grdim}}~{}e_{\alpha}Ae_{\beta}.$

Recall [BGS96, Lemma 2.11.1] that if $A$ is Koszul, then there is an equality
of matrices

$H(A,q)H(A^{!},-q)^{T}=I.$

######  Theorem 7.14.

For any generic oriented matroid program ${\mathbf{P}}$, the algebra
$A=A({\mathbf{P}},U)$ satisfies the numerical identity above, that is, the
Hilbert polynomials of $A$ and its quadratic dual $A^{!}$ satisfy the matrix
equation

$H(A,q)H(A^{!},-q)^{T}=I.$

######  Remark 7.15.

This identity does not necessarily imply that $A$ is Koszul. See [Pos95] for
an example of a non-Koszul quadratic algebra whose Hilbert series satisfies
the numerical identity.

###### Proof.

Using equation (7.1) in the proof of Theorem 7.9, the $(\alpha,\beta)$-entry
of $H(A,q)$ is given by

$H(A,q)_{\alpha,\beta}={\mathrm{grdim}}~{}e_{\alpha}Ae_{\beta}=\sum_{\gamma\succeq\alpha,\beta}q^{d_{\alpha\gamma}+d_{\gamma\beta}}.$

In particular, $H(A,q)$ factors as the product $H(A,q)=XX^{T}$, where $X$ is
the ${\mathcal{P}}\times{\mathcal{P}}$-matrix with $(\alpha,\beta)$-entry
given by

$X_{\alpha,\beta}=\begin{cases}q^{d_{\alpha\beta}}&\mathrm{if~{}}\beta\succeq\alpha\\\
0&\mathrm{otherwise}.\end{cases}$

Dually, using Proposition 2.19 we find that the $(\alpha,\beta)$-entry of
$H(A^{!},-q)$ is given by

$H(A^{!},-q)_{\alpha,\beta}=\sum_{i}(-q)^{i}\dim
e_{\alpha}A^{!}_{i}e_{\beta}=\sum_{\gamma\supset
Y_{\mu^{-1}(\alpha)},Y_{\mu^{-1}(\beta)}}(-q)^{d_{\alpha\gamma}+d_{\gamma\beta}},$

in other words the sum runs over all $\gamma\in{\mathcal{P}}$ for which the
optimal solution (cocircuit) of both $\alpha$ and $\beta$ are faces of
$\gamma$. Again this factors as a product $H(A^{!},-q)=YY^{T}$, where $Y$ is
the ${\mathcal{P}}\times{\mathcal{P}}$-matrix with $(\alpha,\beta)$-entry
given by

$Y_{\alpha,\beta}=\begin{cases}(-q)^{d_{\alpha\beta}}&\mathrm{if~{}}Y_{\mu^{-1}(\beta)}\mathrm{~{}is~{}a~{}face~{}of~{}}\alpha\\\
0&\mathrm{otherwise}.\end{cases}$

We wish to show that

$H(A,q)H(A^{!},-q)^{T}=XX^{T}YY^{T}=I.$

Note that it suffices to show $X^{T}Y=I$.

Computing the product $X^{T}Y$, we find that its $(\alpha,\beta)$-entry is
given by

$(X^{T}Y)_{\alpha,\beta}=\sum_{\gamma\in
Q}q^{d_{\alpha\gamma}}(-q)^{d_{\gamma\beta}},$

where $Q$ is the set of all $\gamma\in{\mathcal{P}}$ such that
$\alpha\succeq\gamma$ and $Y_{\mu^{-1}(\beta)}$ is a face of $\gamma$. In
other words, $Q$ consists of all $\gamma\in{\mathcal{P}}$ such that

(7.2)
$\gamma(i)=\alpha(i)\mathrm{~{}if~{}}i\in\mu^{-1}(\alpha)\quad\mathrm{~{}and}\quad\gamma(i)=\beta(i)\mathrm{~{}if~{}}i\not\in\mu^{-1}(\beta).$

We wish to show that

$(X^{T}Y)_{\alpha,\beta}=\begin{cases}1&\mathrm{if}~{}\alpha=\beta\\\
0&\mathrm{otherwise}.\end{cases}$

If $\alpha=\beta$, then $Q=\\{\alpha\\}$ and the sum is equal to
$q^{d_{\alpha\alpha}}(-q)^{d_{\alpha\alpha}}=1$.

Now assume that $\alpha\neq\beta$ and let

$J:=\mu^{-1}(\beta)\backslash\mu^{-1}(\alpha)\quad\mathrm{and}\quad
J^{\prime}:=\mu^{-1}(\alpha)\backslash\mu^{-1}(\beta)$

so that

$J\sqcup
J^{\prime}=(\mu^{-1}(\alpha)\cup\mu^{-1}(\beta))\backslash(\mu^{-1}(\alpha)\cap\mu^{-1}(\beta)).$

As we have assumed that $\alpha\neq\beta$, $J$ and $J^{\prime}$ are nonempty.

Note that if $\alpha(i)\neq\beta(i)$ for some $i\in J^{\prime}$, then by the
conditions $(\ref{eq-Q})$ $Q$ is empty and $(X^{T}Y)_{\alpha,\beta}=0$ as
desired. Thus we will assume that $\alpha(i)=\beta(i)$ for all $i\in
J^{\prime}$.

Let

$K:=\\{i\in\mu^{-1}(\alpha)\cap\mu^{-1}(\beta)\mid\alpha(i)\neq\beta(i)\\}.$

For $\delta\in{\mathcal{P}}$, $Y_{\mu^{-1}(\beta)}$ is a face of $\delta$ if
and only if $\delta=\beta^{W}$ for some subset $W\subset\mu^{-1}(\beta)$. On
the other hand, $\delta=\beta^{W}\preceq\alpha$ if and only if $K\cup J\supset
W\supset K$. Thus $Q=\\{(\beta^{K})^{S}\mid S\subset J\\}$ and

$(X^{T}Y)_{\alpha,\beta}=\sum_{S\subset
J}q^{d_{\alpha,(\beta^{K})^{S}}}(-q)^{d_{(\beta^{K})^{S},\beta}}=(-1)^{|K|}\sum_{S\subset
J}(-1)^{|S|}q^{d_{\alpha,(\beta^{K})^{S}}+d_{(\beta^{K})^{S},\beta}}$
$=(-1)^{|K|}\sum_{S\subset
J}(-1)^{|S|}q^{d_{\alpha,\beta}+2|S^{(\beta^{K})^{S}}_{\alpha,\beta}|}=(-1)^{|K|}q^{d_{\alpha,\beta}}\sum_{S\subset
J}(-1)^{|S|}q^{2|S^{\beta^{S}}_{\alpha,\beta}|},$

where in the last line we have used
$S^{(\beta^{K})^{S}}_{\alpha,\beta}=S^{\beta^{S}}_{\alpha,\beta}=\\{i\in
S\mid\alpha(i)=\beta(i)\\}$. We will need the following lemma to finish this
proof.

######  Lemma 7.16.

Assume as above that $\alpha\neq\beta$ and $\alpha(i)=\beta(i)$ for all $i\in
J^{\prime}$. Then there exists an element $t\in J$ such that
$\alpha(t)\neq\beta(t)$.

###### Proof.

Suppose for the sake of contradiction that $\alpha(i)=\beta(i)$ for all $i\in
J$, then $\alpha(i)=\beta(i)$ for all $i\in J\sqcup J^{\prime}$. In the
deletion-contraction program

$\left({\mathbf{P}}/(\mu^{-1}(\alpha)\cap\mu^{-1}(\beta))\right)\setminus(\mu^{-1}(\alpha)\cup\mu^{-1}(\beta))^{c}$

defined on the set $J\sqcup J^{\prime}$, the restrictions of the sign vectors
of $\alpha$ and $\beta$ are then equal and so describe the same tope $T$. Now
$Y_{\mu^{-1}(\alpha)}$ is the optimal solution for $\alpha$ and
$Y_{\mu^{-1}(\beta)}$ is the optimal solution for $\beta$, so the restrictions
$Y_{\alpha}$ and $Y_{\beta}$ of $Y_{\mu^{-1}(\alpha)}$ and
$Y_{\mu^{-1}(\beta)}$ to $J\sqcup J^{\prime}$ should both be the unique
optimal solution of the tope $T$. But $z(Y_{\alpha})=J^{\prime}\neq
J=z(Y_{\beta})$, which is a contradiction. Thus there exists a $t\in J$ such
that $\alpha(t)\neq\beta(t)$ as desired. ∎

In particular if $S\subset J\backslash\\{t\\}$, we have
$S^{\beta^{S\sqcup\\{t\\}}}_{\alpha,\beta}=S^{\beta^{S}}_{\alpha,\beta}$.

Using this fact we rewrite the sum above:

$(X^{T}Y)_{\alpha,\beta}=(-1)^{|K|}q^{d_{\alpha,\beta}}\sum_{S\subset
J}(-1)^{|S|}q^{2|S^{\beta^{S}}_{\alpha,\beta}|},$
$=(-1)^{|K|}q^{d_{\alpha,\beta}}\sum_{S\subset
J\backslash\\{t\\}}\left((-1)^{|S|}q^{2|S^{\beta^{S}}_{\alpha,\beta}|}+(-1)^{|S\sqcup\\{t\\}|}q^{2|S^{\beta^{S\cup\\{t\\}}}_{\alpha,\beta}|}\right)$
$=(-1)^{|K|}q^{d_{\alpha,\beta}}\sum_{S\subset
J\backslash\\{t\\}}\left((-1)^{|S|}q^{2|S^{\beta^{S}}_{\alpha,\beta}|}-(-1)^{|S|}q^{2|S^{\beta^{S}}_{\alpha,\beta}|}\right)=0.$

∎

###  7.5. Self-dual projectives

Consider the duality functor

$\textrm{d}:A\text{-mod}\rightarrow A\text{-mod}$

defined by composing the equivalence $A^{\text{op}}\text{-mod}\simeq
A\text{-mod}$ induced by the isomorphism $A\cong A^{\text{op}}$ given by
reversing the arrows of the quiver $Q_{n}$ in Section 3.3 with the induced
functor $A^{\text{op}}$-mod$\rightarrow A$-mod coming from vector space
duality.

In the following result we will involve bounded feasible topes in the affine
spaces of both ${\mathbf{P}}$ and ${\mathbf{P}}^{\vee}$. To distinguish the
two topes with a fixed sign vector
$\alpha\in{\mathcal{P}}={\mathcal{P}}^{\vee}$, we write $T_{\alpha}$ to denote
the tope in ${\mathbf{P}}$ and $T_{\alpha}^{\vee}$ to denote the tope in
${\mathbf{P}}^{\vee}$.

######  Theorem 7.17.

For any generic oriented matroid program ${\mathbf{P}}$ and
$\alpha\in{\mathcal{P}}$. The following are equivalent:

  1. (1)

The projective $P_{\alpha}$ is injective.

  2. (2)

The projective $P_{\alpha}$ is self-dual.

  3. (3)

The simple $L_{\alpha}$ is contained in the socle of some standard module
$V_{\beta}$.

  4. (4)

The bounded feasible tope $T_{\alpha}$ covers an infeasible subtope $X$,
meaning $X(g)=0$.

  5. (5)

The bounded feasible tope $T_{\alpha}^{\vee}$ in the dual program
${\mathbf{P}}^{\vee}$ is in the core of the affine space for
${\mathbf{P}}^{\vee}$. In other words $(T_{\alpha}^{\vee})^{\infty}=0$ or
equivalently the cocircuit faces of the tope $T_{\alpha}^{\vee}$ are all
feasible.

When ${\mathbf{P}}$ is Euclidean, and so $A$ is quasi-hereditary by Theorem
7.6, then the statements above are also equivalent to the following:

  1. (6)

The projective $P_{\alpha}$ is tilting.

###### Proof.

The implications $(1)\Leftrightarrow(2)~{}(~{}\Leftrightarrow(6)$, if $A$ is
quasi-hereditary) are standard facts.

$(2)\implies(3):$ If $P_{\alpha}$ is self-dual, then the socle of $P_{\alpha}$
is isomorphic to the cosocle of $P_{\alpha}$, which is $L_{\alpha}$. Therefore
when expressing $[P_{\alpha}]$ as a sum of simple classes times powers of $q$
in the Grothendieck group, $L_{\alpha}$ is the only simple to appear in the
top degree. On the other hand, by Theorem 7.9,

$[P_{\alpha}]=\sum_{\beta\in C_{\alpha}}q^{d_{\alpha\beta}}[V_{\beta}],$

so the unique simple class appearing in the highest degree must also appear as
the highest degree term of some $[V_{\beta}]$. We conclude that $L_{\alpha}$
is the socle of $V_{\beta}$.

$(3)\implies(4):$ Let $b=\mu^{-1}(\beta)$. Lemma 7.2 says that $V_{\beta}$ is
spanned as a vector space by taut paths $p_{\gamma}$ from $\beta$ to
$\gamma\preceq\beta$. A taut path $p_{\alpha}$ is in the socle of $V_{\beta}$
if there does not exist a longer taut path $p_{\gamma}$ that factors through
$p_{\alpha}$. Note that this is equivalent to the condition: if $i\not\in b$
and $\alpha$ has a feasible face $Y$ such that $Y(i)=0$, then
$\alpha(i)\neq\beta(i)$.

Recall that $Y_{b}\in{\mathcal{A}}$ denotes the feasible cocircuit of
${\mathcal{N}}=\widetilde{\mathcal{M}}\backslash f$ that is the optimal
solution of the tope $T_{\beta}$. By the covector axioms of an oriented
matroid, the composition $T:=(-Y_{b})\circ T_{\alpha}$ is also a covector of
${\mathcal{N}}$ and in particular an infeasible tope such that
$T(i)=\alpha(i)$ for all $i\in b$ and for all $i$ which are zero on a feasible
face of $\alpha$. A taut path $p$ from $T_{\alpha}$ to $T$ exists in the tope
graph of $\widetilde{\mathcal{M}}\backslash f$, and this path cannot change
the sign corresponding to any feasible facet of $T_{\gamma}$. Thus the first
sign change of the path $p$ must be infeasible, which means that $T_{\gamma}$
covers an subtope $X$ with $X(g)=0$.

$(4)\implies(5)$: If the bounded tope $T_{\alpha}$ in
$\widetilde{\mathcal{M}}\backslash f$ covers a subtope $X$ with $X(g)=0$ then
$\alpha$ is a bounded feasible sign vector for both the original program
${\mathbf{P}}$ as well as the reoriented program
${}_{-g}{\mathbf{P}}=({}_{-g}\widetilde{\mathcal{M}},-g,f)$. Dually, this
means that the tope $T_{\alpha}^{\vee}$ in
$\widetilde{\mathcal{M}}^{\vee}\backslash
g=({}_{-g}\widetilde{\mathcal{M}}^{\vee})\backslash(-g)$ is bounded and
feasible in both dual programs ${\mathbf{P}}^{\vee}$ and
${}_{-g}{\mathbf{P}}^{\vee}=({}_{-g}\widetilde{\mathcal{M}}^{\vee},f,-g)$. In
particular, the tope $\alpha\in{\mathcal{P}}^{\vee}$ does not have any
cocircuit face $Y$ with $Y(f)=0$, since this would imply $T_{\alpha}^{\vee}$
was unbounded in one of these generic programs.

$(5)\implies(2)$: If $(T_{\alpha}^{\vee})^{\infty}=0$, then
$e_{\alpha}Ae_{\alpha}\simeq
R_{\alpha}^{\vee}=k[z(\Delta_{\alpha}^{\vee})]/(U^{\perp})$, where
$\Delta_{\alpha}^{\vee}={\mathfrak{F}}_{lv}(T_{\alpha}^{\vee})\backslash\\{0\\}$.
By Lemma 5.4,
$\|\Delta_{\alpha}^{\vee}\|=\|{\mathfrak{F}}_{lv}(T_{\alpha}^{\vee})\backslash\\{0\\}\|$
is homeomorphic to a $(n-d-1)$-sphere and so a result of Munkres (see [Sta04,
Theorem II.4.3]) implies that $R_{\alpha}^{\vee}$ is Gorenstein, meaning that
there is an isomorphism $\int:(e_{\alpha}Ae_{\alpha})_{n-d-1}\rightarrow k$
such that $\langle x,y\rangle=\int xy$ defines a perfect pairing on
$e_{\alpha}Ae_{\alpha}$.

We wish to produce an isomorphism of $A$-modules $\textrm{d}(P_{\alpha})\cong
P_{\alpha}$. To do so, we will show that the map

$\langle-,-\rangle:e_{\alpha}A\times Ae_{\alpha}\rightarrow k$
$\qquad\qquad\qquad(p,q)\mapsto\int pq$

defines a perfect pairing and so it will follow that
$\textrm{d}(P_{\alpha})=(e_{\alpha}A)^{*}\cong Ae_{\alpha}=P_{\alpha}$ as
right $A$-modules.

To prove that $\langle-,-\rangle$ is perfect, we first observe that it
suffices to show that the map $\cdot
p_{\beta,\alpha}:e_{\alpha}Ae_{\beta}\rightarrow e_{\alpha}Ae_{\alpha}$ is
injective for some taut path $p_{\beta\alpha}$ from $\beta$ to $\alpha$. This
is because for any nonzero $x\in e_{\beta}Ae_{\alpha}$, if $x\cdot
p_{\beta\alpha}\neq 0$, then there exists $y\in e_{\alpha}Ae_{\alpha}$ such
that $\int(xp)y=\int x(py)=\langle x,py\rangle\neq 0$. On the $B$ side, this
means showing that

$\displaystyle\cdot
u_{S_{\alpha\alpha}^{\beta}}:R_{\alpha\beta}^{\vee}\rightarrow
R_{\alpha}^{\vee}$

is injective. We proceed by showing

$\displaystyle\cdot
u_{S_{\alpha\alpha}^{\beta}}:\widetilde{R}_{\alpha\beta}^{\vee}\rightarrow\widetilde{R}_{\alpha}^{\vee}$

is the injective map in a split short exact sequence of
$\operatorname{Sym}U^{\perp}$-modules, which proves the claim by applying the
functor $-\otimes_{\operatorname{Sym}U^{\perp}}k$.

The claim is obvious if $\alpha=\beta$ or $T^{\vee}_{\alpha}\wedge
T^{\vee}_{\beta}$ is not feasible, so we assume $\alpha\neq\beta$ and
$T^{\vee}_{\alpha}\wedge T^{\vee}_{\beta}$ is a proper non-empty face of
$T_{\alpha}^{\vee}$. To see the monomial map $\cdot
u_{S_{\alpha\alpha}^{\beta}}:\widetilde{R}_{\alpha\beta}^{\vee}\rightarrow\widetilde{R}_{\alpha}^{\vee}$
is injective and to determine its cokernel, consider the image of any non-zero
monomial $m=\prod_{i\in S}u_{i}^{s_{i}}$ in
$\widetilde{R}^{\vee}_{\alpha\beta}$, where $s_{i}>0$ for any $i\in S$. As $m$
is non-zero in $\widetilde{R}^{\vee}_{\alpha\beta}$, there exists
$Y\in\Delta_{\alpha\beta}^{\vee}$ such that $S\subset z(Y)$. Note that
$S^{\beta}_{\alpha\alpha}=z(T^{\vee}_{\alpha}\wedge T^{\vee}_{\beta})\subset
z(X)$ for any $X\in\Delta^{\vee}_{\alpha\beta}$. Thus $S\cup
S^{\beta}_{\alpha\alpha}\subset z(Y)$ and the product $m\cdot
u_{S_{\alpha\alpha}^{\beta}}=\prod_{i\in S\cup
S^{\beta}_{\alpha\alpha}}u_{i}^{t_{i}}$, where $t_{i}>0$ for $i\in S\cup
S^{\beta}_{\alpha\alpha}$, is non-zero in $\widetilde{R}^{\vee}_{\alpha}$.

The computation above also shows that the cokernel of the map $\cdot
u_{S_{\alpha\alpha}^{\beta}}:\widetilde{R}^{\vee}_{\alpha\beta}\rightarrow\widetilde{R}_{\alpha}^{\vee}$
is the face ring $k[\Delta]$ of

$\Delta=\\{S\subset E_{n}\mid S\subset z(Y)\;\text{for
some}\>Y\in\Delta_{\alpha}\;\text{and}\>S^{\beta}_{\alpha\alpha}\not\subset
S\\}.$ $=z(\Delta_{\alpha})\backslash\\{S\in z(\Delta_{\alpha})\mid
S^{\beta}_{\alpha\alpha}\subset S\\}.$

Recall from Lemma 5.4 that the geometric realization of the simplicial complex
$z(\Delta_{\alpha})$ is a PL $(d-1)$-sphere. The subset of simplices $\\{S\in
z(\Delta_{\alpha})\mid S^{\beta}_{\alpha\alpha}\subset S\\}$ is the open star
of the simplex on the set $S^{\beta}_{\alpha\alpha}$ and thus its complement
$\Delta$ in $z(\Delta_{\alpha})$ is a PL $(d-1)$-ball. It follows that
$k[\Delta]$ is Cohen-Macaulay, again with parameter space $U^{\perp}$. Thus,
$k[\Delta]$ is a free $\operatorname{Sym}U^{\perp}$-module, and therefore the
exact sequence

$\displaystyle\widetilde{R}_{\alpha\beta}^{\vee}\hookrightarrow\widetilde{R}_{\alpha}^{\vee}\twoheadrightarrow
k[\Delta]$

splits. ∎

##  8\. Derived Morita equivalence

We conclude with a proof of Theorem 1.16. Recall that ${\mathcal{M}}$ is an
oriented matroid, $U$ a parameter space for $M=\underline{{\mathcal{M}}}$, and
${\mathbf{P}}_{1}=(\widetilde{\mathcal{M}}_{1},g_{1},f_{1}),{\mathbf{P}}_{2}=(\widetilde{\mathcal{M}}_{2},g_{2},f_{2})$
and
${\mathbf{P}}_{\text{mid}}=(\widetilde{\mathcal{M}}_{\text{mid}},g_{2},f_{1})$
are Euclidean generic oriented matroid programs such that

${\mathcal{M}}=(\widetilde{\mathcal{M}}_{1}/g_{1})\backslash
f_{1}=(\widetilde{\mathcal{M}}_{2}/g_{2})\backslash f_{2}$

and

$\widetilde{\mathcal{M}}_{\text{mid}}/g_{2}=\widetilde{\mathcal{M}}_{1}/g_{1},\qquad\widetilde{\mathcal{M}}_{\text{mid}}\backslash
f_{1}=\widetilde{\mathcal{M}}_{2}\backslash f_{2}.$

We wish to show there is an equivalence of categories

$D(A({\mathbf{P}}_{1},U))\cong D(A({\mathbf{P}}_{2},U)),$

where $D(A)$ denotes the bounded derived category of graded finitely generated
$A$-modules.

######  Remark 8.1.

Note that if ${\mathbf{P}}_{1}$ and ${\mathbf{P}}_{2}$ are Euclidean, it is
not automatic that ${\mathbf{P}}_{\text{mid}}$ will be Euclidean as well. For
example, one could take $\text{EFM}(8)$ (see Example 7.10) and then change the
choice of $g$ and $f$ separately to obtain two realizable (and hence
Euclidean) generic oriented matroid programs ${\mathbf{P}}_{1}$ and
${\mathbf{P}}_{2}$ such that ${\mathbf{P}}_{\text{mid}}$ is the non-Euclidean
program $\text{EFM}(8)$.

We will prove Theorem 1.16 by reducing it to the following claim.

######  Proposition 8.2.

Suppose ${\mathbf{P}}_{1}=(\widetilde{\mathcal{M}}_{1},g_{1},f)$ and
${\mathbf{P}}_{2}=(\widetilde{\mathcal{M}}_{2},g_{2},f)$ are generic Euclidean
programs extending ${\mathcal{M}}$ such that
${\mathcal{M}}_{1}/g_{1}={\mathcal{M}}_{2}/g_{2}$. Then there is an
equivalence of categories

$D(A({\mathbf{P}}_{1},U))\simeq D(A({\mathbf{P}}_{2},U)).$

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,
$\widetilde{\mathcal{M}}_{\text{mid}}/g_{2}=\widetilde{\mathcal{M}}_{1}/g_{1}$.
Then by Proposition 8.2 it follows that

(8.1) $D(A({\mathbf{P}}_{1},U))\simeq D(A({\mathbf{P}}_{\text{mid}},U).$

On the other side, duality together with the assumptions of Theorem 1.16 give:

$\widetilde{\mathcal{M}}_{\text{mid}}^{\vee}/f_{1}=(\widetilde{\mathcal{M}}_{\text{mid}}\backslash
f_{1})^{\vee}=(\widetilde{\mathcal{M}}_{2}\backslash
f_{2})^{\vee}=\widetilde{\mathcal{M}}_{2}^{\vee}/f_{2}.$

Viewing $f_{1}$ and $f_{2}$ as playing the role of $g$ in the Euclidean
programs $\widetilde{\mathcal{M}}_{\text{mid}}^{\vee}$ and
$\widetilde{\mathcal{M}}_{2}^{\vee}$ respectively, we can again apply
Proposition 8.2 to find:

(8.2) $D(A({\mathbf{P}}_{2}^{\vee},U^{\perp}))\simeq
D(A({\mathbf{P}}_{\text{mid}}^{\vee},U^{\perp})).$

Putting these equivalences (8.1) and (8.2) together with the equivalences from
Koszul duality:

$D(A({\mathbf{P}}_{\text{mid}},U))\simeq
D(A({\mathbf{P}}_{\text{mid}}^{\vee},U^{\perp}))\quad\text{and}\quad
D(A({\mathbf{P}}_{2},U))\simeq D(A({\mathbf{P}}_{2}^{\vee},U^{\perp})),$

gives the desired result:

$D(A({\mathbf{P}}_{1},U))\simeq D(A({\mathbf{P}}_{2},U)).$

∎

###  8.1. The definition and properties of the functor

It remains to prove Proposition 8.2. For the remainder of the paper we will
let

${\mathbf{P}}_{1}=(\widetilde{\mathcal{M}}_{1},g_{1},f)\quad\text{and}\quad{\mathbf{P}}_{2}=(\widetilde{\mathcal{M}}_{2},g_{2},f)$

be two Euclidean generic oriented matroid programs such that
$\widetilde{\mathcal{M}}_{1}/g_{1}=\widetilde{\mathcal{M}}_{2}/g_{2}$.

For $\ell=1,2$ let $A_{\ell}=A({\mathbf{P}}_{\ell},U)$,
$B_{\ell}=B({\mathbf{P}}_{\ell},U)$ and ${\mathcal{P}}_{\ell}$ be the set of
bounded, feasible sign vectors of ${\mathbf{P}}_{\ell}$. Note that the set of
bounded sign vectors ${\mathcal{B}}$ is the same for ${\mathbf{P}}_{1}$ and
${\mathbf{P}}_{2}$.

As in [BLPW10, Section 6], the desired equivalence will come from a derived
tensor product with a certain bimodule $N$.

It is slightly easier to define the bimodule $N$ on the $B$-side, using the
isomorphisms $A_{\ell}\simeq B_{\ell}^{\vee}$ of Theorem 5.12 for $\ell=1,2$.
Namely, let

$N=\bigoplus_{(\alpha,\beta)\in{\mathcal{P}}_{1}\times{\mathcal{P}}_{2}}R_{\alpha\beta}^{\vee}[-d_{\alpha\beta}]$

with the natural left $B_{1}^{\vee}$-action and right $B_{2}^{\vee}$-action
given by the $\star$ operation.

To translate this to the $A$-side, recall the following the alternative
definition of $A$ from Section 3.3,

$A({\mathbf{P}},U)=e_{{\mathcal{P}}}D_{n}e_{{\mathcal{P}}}/\langle
e_{f}e_{{\mathcal{P}}}\rangle+\langle\vartheta(U^{\perp})e_{{\mathcal{P}}}\rangle.$

We consider an extended version of $A$ that only depends on
$\widetilde{\mathcal{M}}/g$ by replacing $e_{{\mathcal{P}}}$ by
$e_{\mathcal{B}}=\sum_{\alpha\in{\mathcal{B}}}e_{\alpha}$. That is, let

$A_{\mathrm{ext}}({\mathbf{P}},U)=e_{\mathcal{B}}D_{n}e_{\mathcal{B}}/\langle
e_{f}e_{\mathcal{B}}\rangle+\langle\vartheta(U^{\perp})e_{\mathcal{B}}\rangle.$

As $A_{\mathrm{ext}}({\mathbf{P}},U)$ only depends on
$\widetilde{\mathcal{M}}/g$ and we have assumed that
$\widetilde{\mathcal{M}}_{1}/g_{1}=\widetilde{\mathcal{M}}_{2}/g_{2}$, let

$A_{\mathrm{ext}}:=A_{\mathrm{ext}}({\mathbf{P}}_{1},U)=A_{\mathrm{ext}}({\mathbf{P}}_{2},U).$

When viewed as an $(A_{1},A_{2})$-bimodule, $N$ can be described as

$N=e_{g_{1}}A_{\mathrm{ext}}e_{g_{2}},$

where $e_{g_{\ell}}=\sum_{\alpha\in{\mathcal{P}}_{\ell}}e_{\alpha}$ for
$\ell=1,2$.

To check that these definitions of $N$ coincide, consider the graded vector
space

$B_{\mathrm{ext}}({\mathbf{P}},U)=\bigoplus_{(\alpha,\beta)\in{\mathcal{F}}\times{\mathcal{F}}}R_{\alpha\beta}[-d_{\alpha\beta}],$

made into an algebra via $\star$, as in the definiton of $B({\mathbf{P}},U)$
from Section 5.2. Then an easy extension of the proof of Theorem 5.12 gives us
the following lemma.

######  Lemma 8.3.

There is an isomorphism $A_{\mathrm{ext}}({\mathbf{P}},U)\simeq
B_{\mathrm{ext}}({\mathbf{P}}^{\vee},U^{\perp})$. Combining this isomorphism
with the isomorphisms $A_{\ell}\simeq B_{\ell}^{\vee}$, we obtain an
equivalence between the two definitions of $N$.

We define the functor $\Phi:D(A_{1})\rightarrow D(A_{2})$ via

$\displaystyle\Phi(M)=M\overset{L}{\otimes}_{A_{1}}N.$

For $\ell=1,2$ and any $\alpha\in{\mathcal{P}}_{\ell}$, let
$P_{\alpha}^{\ell}$ and $V_{\alpha}^{\ell}$ be the corresponding projective
and standard $A_{\ell}$-modules. Define
$\nu:{\mathcal{P}}_{1}\rightarrow{\mathcal{P}}_{2}$ to be the composition

${\mathcal{P}}_{1}\overset{\mu_{1}^{-1}}{\longrightarrow}{\mathbb{B}}\overset{\mu_{2}}{\longrightarrow}{\mathcal{P}}_{2}.$

######  Proposition 8.4.

If $\alpha\in{\mathcal{P}}_{1}\cap{\mathcal{P}}_{2}$, then
$\Phi(P_{\alpha}^{1})=P_{\alpha}^{2}$.

###### Proof.

Consider the natural map

$\Gamma:P_{\alpha}^{2}=e_{\alpha}A_{2}\rightarrow
e_{\alpha}A_{1}\otimes_{A_{1}}e_{g_{1}}A_{\mathrm{ext}}e_{g_{2}}=P_{\alpha}^{1}\otimes_{A_{1}}N=\Phi(P_{\alpha}^{1})$

taking $e_{\alpha}$ to $e_{\alpha}\otimes e_{g_{1}}e_{g_{2}}$. For paths $p,q$
in the quiver $Q$ with $p$ only passing through nodes in ${\mathcal{P}}_{1}$,
the equality of the simple tensors

$e_{\alpha}p\otimes e_{g_{1}}qe_{g_{2}}=e_{\alpha}\otimes
e_{\alpha}pe_{g_{1}}qe_{g_{2}}=e_{\alpha}\otimes e_{g_{1}}e_{g_{2}}\cdot
e_{\alpha}pe_{g_{1}}qe_{g_{2}}$

implies that $\Gamma$ is an isomorphism. ∎

######  Remark 8.5.

Note that the proposition above and its proof are valid without the assumption
that ${\mathbf{P}}_{1}$ and ${\mathbf{P}}_{2}$ be Euclidean.

######  Lemma 8.6.

For any $\alpha\in{\mathcal{P}}_{1}$, the $A_{2}$-module
$\Phi(P_{\alpha}^{1})$ has a filtration with standard subquotients. For
$b\in{\mathbb{B}}$, if $\alpha\in{\mathcal{B}}_{b}$ then the standard module
$V^{2}_{\mu_{2}(b)}$ appears with multiplicity 1 in the associated graded, and
otherwise it does not appear.777Recall that the set ${\mathcal{B}}_{b}$
defined in Definition 2.18 only depends on $\widetilde{\mathcal{M}}/g$.

###### Proof.

We have

$\Phi(P_{\alpha}^{1})=P_{\alpha}^{1}\otimes_{A_{1}}N=e_{\alpha}A_{1}\otimes_{A_{1}}e_{g_{1}}A_{\mathrm{ext}}e_{g_{2}},$

so elements of $\Phi(P_{\alpha}^{1})$ can be represented as linear
combinations of paths in ${\mathcal{B}}$ which begin at $\alpha$ and end at
elements of ${\mathcal{P}}_{2}={\mathcal{B}}\cap{\mathcal{F}}_{2}$. For
$\beta\in{\mathcal{P}}_{2}$, let $\Phi(P_{\alpha}^{1})_{\beta}$ be the
submodule generated by paths $p$ such that $\beta$ is the maximal element of
${\mathcal{P}}_{2}$ (with respect to the ordering $\leq_{2}$ on
${\mathcal{P}}_{2}$ coming from our Euclidean assumption on
${\mathbf{P}}_{2}$) through which $p$ passes. Then let

$\Phi(P_{\alpha}^{1})_{>_{2}~{}\beta}=\bigcup_{\gamma>_{2}~{}\beta}\Phi(P_{\alpha}^{1})_{\gamma}\quad\text{and}\quad\Phi(P_{\alpha}^{1})_{\geq_{2}~{}\beta}=\bigcup_{\gamma\geq_{2}~{}\beta}\Phi(P_{\alpha}^{1})_{\gamma}.$

We then obtain a filtration

$\Phi(P_{\alpha}^{1})=\bigcup_{\beta\in{\mathcal{P}}_{2}}\Phi(P_{\alpha}^{1})_{\geq_{2}~{}\beta}.$

Suppose $b\in{\mathbb{B}}$ and let $\beta=\mu_{2}(b)$. It suffices to show
that the quotient
$\Phi(P_{\alpha}^{1})_{\geq_{2}~{}\beta}/\Phi(P_{\alpha}^{1})_{>_{2}~{}\beta}$
is isomorphic to $V^{2}_{\beta}$ if $\alpha\in{\mathcal{B}}_{b}$, and is zero
otherwise.

Our argument follows the proof of Theorem 7.6.

If $\alpha\not\in{\mathcal{B}}_{b}$, then for some $i\in b$,
$\alpha(i)\neq\beta(i)$. Thus any path from $\alpha$ to $\beta$ can be
represented by one passing through $\beta^{i}$. By Lemma 7.5,
$\beta^{i}>_{2}~{}\beta$ and therefore

$\Phi(P_{\alpha}^{1})_{\geq_{2}~{}\beta}/\Phi(P_{\alpha}^{1})_{>_{2}~{}\beta}\cong
0.$

Otherwise, precomposition with a taut path from $\alpha$ to $\beta$ gives a
surjective map

$V_{\beta}^{2}\twoheadrightarrow\Phi(P_{\alpha}^{1})_{\geq_{2}~{}\beta}/\Phi(P_{\alpha}^{1})_{>_{2}~{}\beta}.$

Thus

$\dim
V_{\beta}^{2}\geq\dim~{}\Phi(P_{\alpha}^{1})_{\geq_{2}~{}\beta}/\Phi(P_{\alpha}^{1})_{>_{2}~{}\beta}$

and it suffices to show equality holds. After summing over all
$\beta\in{\mathcal{P}}_{2}$:

$\begin{split}\dim(\Phi(P^{1}_{\alpha}))&=\sum_{\beta\in{\mathcal{P}}_{2}}\dim~{}~{}\Phi(P_{\alpha}^{1})_{\geq_{2}~{}\beta}/\Phi(P_{\alpha}^{1})_{>_{2}~{}\beta}\\\
&\leq\sum_{\alpha\in{\mathcal{B}}_{b}}\dim
V^{2}_{\mu_{2}(b)}=\\#\\{(\gamma,b)\in{\mathcal{P}}_{2}\times{\mathbb{B}}\mid\alpha,\gamma\in{\mathcal{B}}_{b}\\}.\end{split}$

It suffices to show that equality holds after summing over all
$\alpha\in{\mathcal{P}}_{1}$. As
$N=\oplus_{\alpha\in{\mathcal{P}}_{1}}\Phi(P_{\alpha}^{1})$, we have

$\sum_{\alpha\in{\mathcal{P}}_{1}}\dim\Phi(P_{\alpha}^{1})=\dim
N=\sum_{(\alpha,\gamma)\in{\mathcal{P}}_{1}\times{\mathcal{P}}_{2}}\dim
R_{\alpha\gamma}^{\vee}$
$=\\#\\{(\alpha,\gamma,b)\in{\mathcal{P}}_{1}\times{\mathcal{P}}_{2}\times{\mathbb{B}}\mid\alpha,\gamma\in{\mathcal{B}}_{b}\\}.$

Here we are using Lemma 5.7 and Proposition 2.19 on each
$(\alpha,\gamma)\in{\mathcal{P}}_{1}\times{\mathcal{P}}_{2}$. ∎

######  Remark 8.7.

Notice that Theorem 7.6 can be viewed as the special case
${\mathbf{P}}_{1}={\mathbf{P}}_{2}$.

######  Remark 8.8.

We note that the above proof does not use the assumption that
${\mathbf{P}}_{1}$ is Euclidean, and so for this result we need only assume
that ${\mathbf{P}}_{2}$ is Euclidean. More generally, when ${\mathbf{P}}_{2}$
is not Euclidean one can prove the result on the level of Grothendieck groups
with a nearly identical proof as was given for the analogous Theorem 7.9.

######  Proposition 8.9.

For all $\alpha\in{\mathcal{P}}_{1}$, we have
$[\Phi(V_{\alpha}^{1})]=[V_{\nu(\alpha)}^{2}]$ in the Grothendieck groups of
(ungraded) right $A_{2}$-modules. Thus $\Phi$ induces an isomorphism of
Grothendieck groups.

###### Proof.

For any $\alpha\in{\mathcal{P}}_{1}$, the equalities

$\sum[V_{\mu_{2}(b)}^{2}]=[\Phi(P_{\alpha}^{1})]=\sum[\Phi(V_{\mu_{1}(b)}^{1})]$

follow from Lemma 8.6 and its special case Theorem 7.6, where both sums are
taken over

$\\{b\in{\mathbb{B}}\mid\alpha\in{\mathcal{B}}_{b}\\}.$

The first claim then follows by induction on the poset ${\mathcal{P}}_{1}$
with base case $\alpha\in{\mathcal{P}}_{1}$ maximal, so

$P_{\alpha}^{1}=V_{\alpha}^{1}\quad\text{and}\quad\Phi(P_{\alpha}^{1})=V^{2}_{\nu(\alpha)}.$

The second statement follows from the fact that the classes of standard
modules in a highest weight category form a ${\mathbb{Z}}$-basis for the
Grothendieck group. ∎

######  Remark 8.10.

One can show that this result holds without the Euclidean condition on
${\mathbf{P}}_{1}$ and ${\mathbf{P}}_{2}$ by the second part of Remark 8.8 and
the fact that the matrix $X$ from the proof of Theorem 7.14 is invertible.

######  Proposition 8.11.

Let $\alpha\in{\mathcal{P}}_{1}$. Then $\Phi(V_{\alpha}^{1})$ is the quotient
of $\Phi(P_{\alpha}^{1})$ by the submodule generated by all paths changing
their $i$-th coordinate for some $i\in\mu^{-1}(\alpha)$. In particular,
$\operatorname{Tor}_{m}^{A_{1}}(V_{\alpha}^{1},N)=0$ for all $m>0$.

###### Proof.

Apply $\Phi$ to the linear projective resolution of $V_{\alpha}^{1}$ of
Theorem 7.12. The degree zero homology of the resulting complex is the
quotient promised. We wish to show that the resulting complex is a resolution
of $V_{\alpha}^{1}\otimes_{A_{1}}N$. This claim follows from argument is
analogous to the proof of Theorem 7.12, where for each
$S\subset\mu_{1}^{-1}(\alpha)$ we filter each $A_{2}$-module
$\Phi(P_{\alpha^{S}}^{1})=P_{\alpha}^{1}\otimes_{A_{1}}N$ by standards as in
the proof of Lemma 8.6. ∎

######  Corollary 8.12.

If a right $A_{1}$-module $M$ admits a filtration by standard modules, we have
$\operatorname{Tor}_{m}^{A_{1}}(M,N)=0$ and therefore

$\Phi(M)=M\otimes_{A_{1}}N.$

###  8.2. Ringel duality and composition of functors

Suppose that ${\mathbf{P}}=(\widetilde{\mathcal{M}},g,f)$ is a generic
Euclidean extension of ${\mathcal{M}}$. Consider the program
$\overline{{\mathbf{P}}}={}_{-g}{\mathbf{P}}=({}_{-g}\widetilde{\mathcal{M}},-g,f)$
obtained from ${\mathbf{P}}$ by reorientation of $g$. In other words
$\overline{{\mathbf{P}}}$ is the program on the oriented matroid
${\mathcal{M}}$ with feasible cocircuits equal to the negative of the feasible
cocircuits of ${\mathbf{P}}$. This program is also generic and Euclidean. We
let $\overline{A}=A(\overline{{\mathbf{P}}},U)$ and denote by

$\Phi^{-}:D(A)\rightarrow D(\overline{A}),$

the functor $\Phi$ for ${\mathbf{P}}_{1}={\mathbf{P}}$ and
${\mathbf{P}}_{2}=\overline{\mathbf{P}}$. We will prove that $\Phi^{-}$ is an
equivalence and relate it to Ringel duality.

######  Theorem 8.13.

$\Phi^{-}$ is an equivalence, the algebras $A$ and $\overline{A}$ are Ringel
dual, and the Ringel duality functor is
$\text{d}\circ\Phi^{-}=\Phi^{-}\circ\text{d}$.

###### Proof.

Notice that for any $\alpha\in{\mathcal{P}}$, we have that the $B$-side
description of $\Phi^{-}$ gives

$\Phi^{-}(P_{\alpha})=\bigoplus_{\overline{\beta}\in\overline{\mathcal{P}}}R_{\alpha\overline{\beta}}^{\vee}[-d_{\alpha\overline{\beta}}].$

Then we have that the tope corresponding to the restrictions of both $\alpha$
and $\overline{\beta}$ in the oriented matroid
$\widetilde{\mathcal{M}}^{\vee}/S_{\alpha\alpha}^{\overline{\beta}}$ on
$E_{n}\cup\\{f\\}$ has all cocircuit faces taking the value $+$ on $f$. As in
the proof of Theorem 7.17, this implies that
$R_{\alpha\overline{\beta}}^{\vee}$ is Gorenstein and $\Phi^{-}(P_{\alpha})$
is self-dual. By Lemma 8.6 it follows that $\Phi^{-}(P_{\alpha})$ is tilting.

It remains to show that $\Phi^{-}$ is an equivalence. With Proposition 8.4 and
Theorem 7.17 in hand, one can repeat the proof of [BLPW10, Theorem 6.10] word
for word. ∎

To complete the proof of Proposition 8.2 in the general case, we will need to
study the composition of functors.

Let ${\mathbf{P}}_{1}=(\widetilde{\mathcal{M}}_{1},g_{1},f)$,
${\mathbf{P}}_{2}=(\widetilde{\mathcal{M}}_{2},g_{2},f)$, and
${\mathbf{P}}_{3}=(\widetilde{\mathcal{M}}_{3},g_{3},f)$ be generic Euclidean
programs extending ${\mathcal{M}}$ for which
$\widetilde{\mathcal{M}}_{1}/g_{1}=\widetilde{\mathcal{M}}_{2}/g_{2}=\widetilde{\mathcal{M}}_{3}/g_{3}$.
We can then define the three functors

$D(A_{1})\overset{\Phi_{12}}{\longrightarrow}D(A_{2})\overset{\Phi_{23}}{\longrightarrow}D(A_{3})\quad\text{and}\quad
D(A_{1})\overset{\Phi_{13}}{\longrightarrow}D(A_{3})$

as before. We would like to compare $\Phi_{13}$ with the composition
$\Phi_{23}\circ\Phi_{12}$.

Notice that

$N_{12}=\Phi_{12}(A_{1})=\bigoplus_{\alpha\in{\mathcal{P}}_{1}}\Phi_{12}(P_{\alpha}^{1})$

has a filtration by standard modules as a right $A_{2}$-module by Lemma 8.6.
Then

$\Phi_{23}\circ\Phi_{12}(M)=(M\overset{L}{\otimes}_{A_{1}}N_{12})\overset{L}{\otimes}_{A_{2}}N_{23}=M\overset{L}{\otimes}_{A_{1}}(N_{12}\otimes_{A_{2}}N_{23})$

by Corollary 8.12. The natural map $N_{12}\otimes_{A_{2}}N_{23}\rightarrow
N_{13}$ given by concatenation of paths induces a natural transformation
$\Phi_{23}\circ\Phi_{12}\rightarrow\Phi_{13}$. We also have that
$\Phi_{23}\circ\Phi_{12}$ and $\Phi_{13}$ induce the same map on Grothendieck
groups by Proposition 8.9. This implies that

$\dim_{k}N_{12}\otimes_{A_{2}}N_{23}=\dim_{k}N_{13}$

since the classes of $N_{12}\otimes_{A_{2}}N_{23}$ and $N_{13}$ agree in the
Grothendieck group of $A_{3}$-modules.

We now combine this discussion with the equivalence we have already proved.
Let ${\mathbf{P}}_{3}=\overline{{\mathbf{P}}_{1}}$, so that

$\Phi_{13}=\Phi^{-}:D(A_{1})\rightarrow
D(\overline{A_{1}})\quad\text{and}\quad\Phi_{23}=\Phi_{2\overline{1}}:D(A_{2})\rightarrow
D(\overline{A_{1}}).$

######  Lemma 8.14.

$\Phi^{-}\cong\Phi_{2\overline{1}}\circ\Phi_{12}$.

###### Proof.

The conclusion follows from the discussion above if we can show that the map
$N_{12}\otimes_{A_{2}}N_{2\overline{1}}\rightarrow N_{1\overline{1}}$ is an
isomorphism. We have already observed that the source and target have the same
dimension, so it will suffice to show that this map is surjective. This means
showing that for any
$(\alpha,\beta)\in{\mathcal{P}}_{1}\times\overline{{\mathcal{P}}_{1}}$, every
path from $\alpha$ to $\beta$ in $e_{\alpha}A_{\mathrm{ext}}e_{\beta}$ can be
represented as a path that passes through some sign vector $\gamma$ in
${\mathcal{P}}_{2}$. It suffices to do this for a taut path from $\alpha$ to
$\beta$. Translating this to the $B$-side, we wish to show that
$1_{\alpha\beta}^{\vee}=1_{\alpha\gamma}^{\vee}\star 1_{\gamma\beta}^{\vee}$
for some $\gamma\in{\mathcal{P}}_{2}$.

Let $(\alpha,\beta)\in{\mathcal{P}}_{1}\times\overline{{\mathcal{P}}_{1}}$ and
suppose that $R_{\alpha\beta}^{\vee}$ is nonzero. This means the maximal
common covector face $T^{\vee}_{\alpha}\wedge T^{\vee}_{\beta}$ of the topes
$T^{\vee}_{\alpha}$ and $T^{\vee}_{\beta}$ in

$\widetilde{\mathcal{M}}^{\vee}:=\widetilde{\mathcal{M}}_{1}^{\vee}\backslash
g_{1}=(\widetilde{\mathcal{M}}_{1}/g_{1})^{\vee}=(\widetilde{\mathcal{M}}_{\overline{1}}/g_{\overline{1}})^{\vee}=\widetilde{\mathcal{M}}_{\overline{1}}^{\vee}\backslash
g_{\overline{1}}$

is nonzero and all of its nonzero cocircuit faces are feasible, i.e. they take
the value $+$ on $f$. Together with the fact that $z(T^{\vee}_{\alpha}\wedge
T^{\vee}_{\beta})=S^{\beta}_{\alpha\alpha}$, this implies that
$T^{\vee}_{\alpha}\wedge T^{\vee}_{\beta}$ restricts to a bounded feasible
tope in the oriented matroid program

${\mathbf{P}}_{2}^{\vee}/S_{\alpha\alpha}^{\beta}=(\widetilde{\mathcal{M}}^{\vee}/S_{\alpha\alpha}^{\beta},f,g_{2}).$

Let $Y$ be the optimal cocircuit face of $T^{\vee}_{\alpha}\wedge
T^{\vee}_{\beta}$ viewed as a covector in
${\mathbf{P}}_{2}^{\vee}/S_{\alpha\alpha}^{\beta}.$ Then $Y$ lifts to a unique
feasible cocircuit of ${\mathbf{P}}^{\vee}_{2}$ and let
$\gamma\in{\mathcal{P}}_{2}^{\vee}={\mathcal{P}}_{2}$ be the corresponding
sign vector. By construction, $T^{\vee}_{\alpha}\wedge T^{\vee}_{\beta}$ is a
face of $T^{\vee}_{\gamma}$.

Thus
$R^{\vee}_{\alpha\gamma\beta}=R^{\vee}_{\alpha\beta},S^{\gamma}_{\alpha\beta}=\varnothing$
and

$1_{\alpha\beta}^{\vee}=1_{\alpha\gamma}^{\vee}\star 1_{\gamma\beta}^{\vee}.$

∎

###### Proof of Proposition 8.2.

We can set up everything as in Lemma 8.14, and we know that $\Phi^{-}$ is an
equivalence by Theorem 8.13. This gives us that $\Phi_{12}:D(A_{1})\rightarrow
D(A_{2})$ is faithful while $\Phi_{2\overline{1}}$ is full and essentially
surjective.

Note that $\overline{{\mathbf{P}}_{2}}$ is Euclidean if ${\mathbf{P}}_{2}$ is
Euclidean. Appealing to the same arguments as before,
$\Phi_{12}\circ\Phi_{\overline{2}1}$ is an equivalence. We conclude that
$\Phi_{12}$ is also full and essentially surjective and thus an equivalence. ∎

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