{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{129.51611pt}{-49.05363pt}\pgfsys@lineto{211.56966pt}{-8.03075pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.89444}{0.44717}{-0.44717}{0.89444}{211.74854pt}{-7.94131pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}}
{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@rect{164.7395pt}{-33.34048pt}{11.96451pt}{9.77545pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{168.07251pt}{-29.05081pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\tiny$\alpha_{\mathcal{N}}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right).$

We observe that $B_{\bullet}$ is a convolution with six parts, three of which
are shifts of $M_{1}\sqcup M_{2}$. The leftmost terms live in homological
degree $d_{1}+d_{2}+2$, and the other copy of $M_{1}\sqcup M_{2}$ is in
homological degree $d_{1}+d_{2}+1$. The remaining terms are complexes that
begin in the homological degree indicated in the diagram. The terms appearing
in red make up the termwise-split quotient complex
$\operatorname{Null}(M_{1}\sqcup M_{2},d_{1}+d_{2}+1)[1]$.

Let us draw $B_{\bullet}$ again, regrouping the six parts into three groups of
two which we emphasize with color:

$B_{\bullet}=\left(\leavevmode\hbox to267.43pt{\vbox
to72.21pt{\pgfpicture\makeatletter\hbox{\hskip 21.3691pt\lower-36.10464pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.03609pt}{25.93832pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$M_{1}\sqcup
M_{2}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.03609pt}{-30.9672pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{{\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>M_{2}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{95.77495pt}{-30.9672pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}$M_{1}\sqcup
M_{2}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{97.23953pt}{25.93832pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$M_{1}\sqcup
C_{2}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{97.23953pt}{-2.51443pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{{\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color<EMAIL_ADDRESS>M_{2}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{212.51515pt}{-2.51443pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}$C_{1}\sqcup
C_{2}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}}{}{ {}{}{}}{}{
{}{}{}}{}\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{21.56909pt}{28.45276pt}\pgfsys@lineto{93.30653pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{93.50653pt}{28.45276pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{21.56909pt}{23.06181pt}\pgfsys@lineto{93.3185pt}{5.1158pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.97011}{-0.24265}{0.24265}{0.97011}{93.51251pt}{5.06728pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{133.91554pt}{23.4297pt}\pgfsys@lineto{208.59412pt}{4.75021pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.97011}{-0.24266}{0.24266}{0.97011}{208.78813pt}{4.70169pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{133.91554pt}{0.0pt}\pgfsys@lineto{208.58215pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{208.78215pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{21.56909pt}{-28.45276pt}\pgfsys@lineto{91.84195pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{92.04195pt}{-28.45276pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{21.56909pt}{-23.06181pt}\pgfsys@lineto{93.3185pt}{-5.1158pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.97011}{0.24265}{-0.24265}{0.97011}{93.51251pt}{-5.06728pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{}{}}{}{
{}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{}{}{{}}\pgfsys@moveto{135.38013pt}{-23.06378pt}\pgfsys@lineto{208.59412pt}{-4.75021pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.97011}{0.24266}{-0.24266}{0.97011}{208.78813pt}{-4.70169pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right).$ (20)

The red terms $M_{1}\sqcup M_{2}\to C_{1}\sqcup C_{2}$ form a termwise-split
subcomplex isomorphic to $C_{\bullet}(\mathcal{N})$. The green terms
$M_{1}\sqcup M_{2}\to C_{1}\sqcup M_{2}$ form a termwise-split subquotient
complex isomorphic to a shift of $m(C_{\bullet}(\mathcal{N}_{1})\boxtimes
M_{2})$. The blue terms $M_{1}\sqcup M_{2}\to M_{1}\sqcup C_{2}$ form a
termwise-split quotient complex isomorphic to a shift of $m(M_{1}\boxtimes
C_{\bullet}(\mathcal{N}_{2}))$. This demonstrates that the complex
$B_{\bullet}$ may be constructed as a convolution with three parts as just
described.

Now we conclude. The complex $m(D_{\bullet})$ clearly lives in
$\mathcal{I}(E_{1},E_{2})$. So does the contractible complex
$\operatorname{Null}(M_{1}\sqcup M_{2},d_{1}+d_{2}+1)$. By Lemma 2.9,
$B_{\bullet}=\operatorname{Cone}(\beta)$ is also in
$\mathcal{I}(E_{1},E_{2})$. Two of the three parts in a convolution describing
$B_{\bullet}$ are $m(C_{\bullet}(\mathcal{N}_{1})\boxtimes M_{2})$ and
$m(M_{1}\boxtimes C_{\bullet}(\mathcal{N}_{2}))$, both of which live in
$\mathcal{I}(E_{1},E_{2})$. Applying Lemma 2.9 again, the remaining part of
this convolution also lies in $\mathcal{I}(E_{1},E_{2})$. This remaining part
is $C_{\bullet}^{\Omega}(\mathcal{N})$, which completes the proof. ∎

###### Proof of Theorem 7.3.

We have already established Equation (17). Since $m$ is an equivalence from
$\mathcal{M}_{\operatorname{id}}^{+}(E_{1})\boxtimes\mathcal{M}_{\operatorname{id}}^{+}(E_{2})$
to $\mathcal{M}_{\operatorname{id}}^{+}(E)_{\sqcup}$ and
$\Delta=m^{-1}\circ\Delta_{\sqcup}$, Equation (18) is equivalent to the
statement that we have
$\Delta_{\sqcup}(\mathcal{I}(E))\subset\mathcal{I}(E_{1},E_{2})$. This follows
from Lemmas 7.4 and 7.5. ∎

###  7.3 Convolution of valuative functors

Suppose that $\mathcal{A}$ is a monoidal additive category, and let $\otimes$
denote the tensor product in $\mathcal{A}$. The example to have in mind is the
category of finite dimensional graded vector spaces. Let
$\Phi\colon\mathcal{M}_{\operatorname{id}}^{+}(E_{1})\to\mathcal{A}$ and
$\Psi\colon\mathcal{M}_{\operatorname{id}}^{+}(E_{2})\to\mathcal{A}$ be
additive functors. We define the additive functor

$\Phi\boxtimes\Psi\colon\mathcal{M}_{\operatorname{id}}^{+}(E_{1})\boxtimes\mathcal{M}_{\operatorname{id}}^{+}(E_{2})\to\mathcal{A}$

by putting

$\Phi\boxtimes\Psi(M_{1}\boxtimes
M_{2})=\Phi(M_{1})\otimes\Phi(M_{2})\qquad\text{and}\qquad\Phi\boxtimes\Psi(f\boxtimes
g)=\Phi(f)\otimes\Psi(g).$

We then define the convolution

$\Phi*\Psi:=(\Phi\boxtimes\Psi)\circ\Delta\colon\mathcal{M}_{\operatorname{id}}^{+}(E)\to\mathcal{A}.$

In particular, we have $\Phi*\Psi(M)=\Phi(M^{E_{1}})\otimes\Psi(M_{E_{1}})$
for any matroid $M$ on $E$.

######  Remark 7.6.

If $\Phi$ and $\Psi$ categorify matroid invariants $f$ and $g$, then the
convolution $\Phi*\Psi$ categorifies the convolution $f*g$ as defined in
[AS22, Definition 6.2]. There is no relationship between this use of the word
convolution and the notion of convolution of complexes discussed in Section
2.2.

Theorem 7.3 has the following corollary.

######  Corollary 7.7.

If $\Phi$ and $\Psi$ are valuative, then so is $\Phi*\Psi$.

###### Proof.

Theorem 7.3 tells us that $\Delta$ takes $\mathcal{I}(E)$ to
$\big{\langle}\mathcal{I}(E_{1})\boxtimes\mathcal{M}_{\operatorname{id}}^{+}(E_{2}),\mathcal{M}_{\operatorname{id}}^{+}(E_{1})\boxtimes\mathcal{I}(E_{2})\big{\rangle}$.
Since $\Phi$ and $\Psi$ are both valuative, $\Phi$ kills $\mathcal{I}(E_{1})$
and $\Psi$ kills $\mathcal{I}(E_{2})$. Therefore $\Phi*\Psi$ kills
$\mathcal{I}(E)$. ∎

##  8 Examples of valuative categorical invariants

In this section, we use Corollary 7.7 to derive new examples of valuative
categorical invariants of matroids.

###  8.1 Whitney functors

We begin with a simple lemma. Let
$\Phi:\mathcal{M}_{\operatorname{id}}(E)\to\mathcal{A}$ be a valuative
functor, and let $k$ be a natural number. Define a new functor
$[\Phi]_{k}:\mathcal{M}_{\operatorname{id}}(E)\to\mathcal{A}$ by putting
$[\Phi]_{k}(M)=\Phi(M)$ if $\operatorname{rk}M=k$ and 0 otherwise. Since all
morphisms in $\mathcal{M}_{\operatorname{id}}(E)$ relate matroids of the same
rank, $\Phi$ is naturally isomorphic to the direct sum over all $k$ of
$[\Phi]_{k}$. This immediately implies the following result.

######  Lemma 8.1.

If $\Phi$ is valuative, then so is $[\Phi]_{k}$.

Fix a natural number $r$ and an increasing $r$-tuple of natural numbers
$\textbf{k}=(k_{1},\ldots,k_{r})$. For any matroid $M$, let

$\mathcal{L}_{\textbf{k}}(M):=\\{(F_{1},\ldots,F_{r})\mid\text{$F_{i}$ is a
flat of rank $k_{i}$ and $F_{1}\subset\cdots\subset F_{r}$}\\}.$

We define the Whitney functor

$\Phi_{\textbf{k}}:\mathcal{M}\to\operatorname{Vec}_{\mathbb{Q}}$

on objects by taking $\Phi_{\textbf{k}}(M)$ to be a vector space with basis
$\mathcal{L}_{\textbf{k}}(M)$. If $\varphi:(E,M)\to(E^{\prime},M^{\prime})$ is
a morphism and $(F_{1},\ldots,F_{r})\in\mathcal{L}_{\textbf{k}}(M)$, then we
define

$\Phi_{\textbf{k}}(\varphi)(F_{1},\ldots,F_{r})=\begin{cases}\left(\overline{\varphi(F_{1})},\ldots,\overline{\varphi(F_{r})}\right)&\text{if
$\operatorname{rk}_{M^{\prime}}(\varphi(F_{i}))=k_{i}$ for all $i$}\\\
0&\text{otherwise.}\end{cases}$

The main result of this section is that the functor $\Phi_{\textbf{k}}$ is
valuative.

######  Proposition 8.2.

For any $r$ and $\mathbb{k}$, the functor $\Phi_{\mathbb{k}}$ is valuative.

To prove Proposition 8.2, we first consider the related functors introduced in
Example 4.3.

######  Lemma 8.3.

The functor $\Psi_{E,\mathbb{k},\textbf{S}}$ from Example 4.3 is valuative.

###### Proof.

We proceed by induction on $r$. The base case $r=0$ is Proposition 4.11. For
the inductive step, let $r\geq 1$ be given and assume that the lemma holds for
$r-1$. Fix a set $E$, an increasing $r$-tuple
$\textbf{k}=(k_{1},\ldots,k_{r})$ of natural numbers, and an increasing
$r$-tuple $\textbf{S}=(S_{1},\ldots,S_{r})$ of subsets of $E$. By our
inductive hypothesis, the functor

$\Psi_{S_{r},(k_{1},\ldots,k_{r-1}),(S_{1},\ldots,S_{r-1})}:\mathcal{M}_{\operatorname{id}}(S_{r})\to\operatorname{Vec}_{\mathbb{Q}}$

is valuative. By Lemma 8.1, so is the functor

$\left[\Psi_{S_{r},(k_{1},\ldots,k_{r-1}),(S_{1},\ldots,S_{r-1})}\right]_{k_{r}}:\mathcal{M}_{\operatorname{id}}(S_{r})\to\operatorname{Vec}_{\mathbb{Q}}.$

By Theorem 5.5, the functor
$\operatorname{OS}^{0}:\mathcal{M}_{\operatorname{id}}(E\setminus
S_{r})\to\operatorname{Vec}_{\mathbb{Q}}$ is valuative. Note that the degree
zero part of the Orlik–Solomon algebra of a matroid is equal to $\mathbb{Q}$
if that matroid is loopless and to 0 otherwise, thus for any matroid $M$ on
$E$, $\operatorname{OS}^{0}(M_{S_{r}})$ is equal to $\mathbb{Q}$ if $S_{r}$ is
a flat and 0 otherwise. It follows that

$\Psi_{E,\textbf{k},\textbf{S}}=\left[\Psi_{S_{r},(k_{1},\ldots,k_{r-1}),(S_{1},\ldots,S_{r-1})}\right]_{k_{r}}*\operatorname{OS}^{0},$

and therefore $\Psi_{E,\textbf{k},\textbf{S}}$ is valuative by Corollary 7.7.∎

###### Proof of Proposition 8.2..

We need to show that, for any decomposition $\mathcal{N}$ of a matroid $M$ on
the ground set $E$, and any orientation $\Omega$ of $\mathcal{N}$, the complex
$\Phi_{\textbf{k}}(C_{\bullet}^{\Omega}(\mathcal{N}))$ is exact.

For any matroid $N$, define a filtration on $\Phi_{\textbf{k}}(N)$ by taking
the $i^{\text{th}}$ filtered piece to be spanned by those tuples
$(F_{1},\ldots,F_{r})\in\mathcal{L}_{\textbf{k}}(N)$ such that
$|F_{1}|+\cdots+|F_{r}|\geq i$. The linear map of vector spaces induced by a
weak map $\iota_{N,N^{\prime}}:N\to N^{\prime}$ takes the $i^{\text{th}}$
filtered piece of $\Phi_{\textbf{k}}(N)$ to the $i^{\text{th}}$ filtered piece
of $\Phi_{\textbf{k}}(N^{\prime})$, hence we obtain a filtered complex
$\Phi_{\textbf{k}}(C_{\bullet}^{\Omega}(\mathcal{N}))$. The associated graded
of this complex is isomorphic to the complex

$\Psi_{E,\textbf{k}}(C_{\bullet}^{\Omega}(\mathcal{N}))=\bigoplus_{\textbf{S}}\Psi_{E,\textbf{k},\textbf{S}}(C_{\bullet}^{\Omega}(\mathcal{N})),$

where the sum is over all increasing $r$-tuples
$\textbf{S}=(S_{1},\ldots,S_{r})$ of subsets of $E$. By Lemma 8.3, this
associated graded complex is exact. By considering the spectral sequence
associated with the filtered complex (as in the proof of Theorem 5.5), we may
conclude that the filtered complex
$\Phi_{\textbf{k}}(C_{\bullet}^{\Omega}(\mathcal{N}))$ is exact, as well. ∎

###  8.2 Chow functors

Let $M$ be a matroid on the ground set $E$. The augmented Chow ring
$\operatorname{CH}(M)$ is defined as the quotient of the polynomial ring

$\mathbb{Q}[x_{F}\mid\text{$F$ a flat}]\otimes\mathbb{Q}[y_{e}\mid e\in E]$

by the ideal

$\left\langle\sum_{F}x_{F}\right\rangle+\left\langle y_{e}-\sum_{e\notin
F}x_{F}\;\Big{|}\;e\in E\right\rangle+\left\langle
y_{e}x_{F}\;\Big{|}\;e\notin F\right\rangle+\left\langle
x_{F}x_{G}\;\Big{|}\;\text{$F,G$ incomparable}\right\rangle.$

If $M$ has no loops, the Chow ring $\underline{\operatorname{CH}}(M)$ is
defined as the quotient of $\operatorname{CH}(M)$ by the ideal generated by
$\\{y_{e}\mid e\in E\\}$. If $M$ has loops, then
$\underline{\operatorname{CH}}(M)$ is by definition 0.

We would like to promote these rings to categorical invariants. Our first tool
will be the following theorem of Feichtner and Yusvinsky [FY04, Corollary 1].

######  Theorem 8.4.

If $M$ has no loops, then $\underline{\operatorname{CH}}(M)$ has a basis
consisting of monomials of the form $x_{F_{1}}^{m_{1}}\cdots
x_{F_{r}}^{m_{r}}$, where $r\in\mathbb{N}$, $\varnothing=F_{0}\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{r}$, and
$0<m_{i}<\operatorname{rk}F_{i}-\operatorname{rk}F_{i-1}$ for all
$i\in\\{1,\ldots,r\\}$.

For any positive integer $k$, consider the graded vector space

$Q_{k}:=\mathbb{Q}(-1)\oplus\cdots\oplus\mathbb{Q}(1-k)$

of total dimension $k-1$, with a piece in every positive degree less than $k$.
Theorem 8.4 has the following corollary.

######  Corollary 8.5.

If $M$ has no loops, then there is a canonical isomorphism of graded vector
spaces

$\underline{\operatorname{CH}}(M)\;\cong\;\bigoplus_{r\geq
0}\;\bigoplus_{\begin{subarray}{c}\mathbb{k}\,=(k_{1},\ldots,k_{r})\\\
0<k_{1}<\cdots<k_{r}\end{subarray}}\Phi_{\mathbb{k}}(M)\otimes
Q_{k_{1}}\otimes Q_{k_{2}-k_{1}}\otimes\cdots\otimes Q_{k_{r}-k_{r-1}}.$

Based on Corollary 8.5, we define a functor $\underline{\operatorname{CH}}$
from $\mathcal{M}$ to the category of finite dimensional graded vector spaces
over $\mathbb{Q}$ by setting

$\underline{\operatorname{CH}}:=\bigoplus_{r\geq
0}\;\bigoplus_{\begin{subarray}{c}\textbf{k}\,=(k_{1},\ldots,k_{r})\\\
0<k_{1}<\cdots<k_{r}\end{subarray}}\Phi_{\textbf{k}}\otimes Q_{k_{1}}\otimes
Q_{k_{2}-k_{1}}\otimes\cdots\otimes Q_{k_{r}-k_{r-1}}$

on the full subcategory spanned by matroids without loops, and
$\underline{\operatorname{CH}}=0$ on the full subcategory spanned by matroids
with loops. We observe that there are no morphisms in $\mathcal{M}$ from a
matroid with loops to a matroid without loops, so these conditions uniquely
characterize a well-defined functor $\underline{\operatorname{CH}}$.

######  Corollary 8.6.

The categorical invariant $\underline{\operatorname{CH}}$ is valuative.

###### Proof.

Corollary 8.5 tells us that $\underline{\operatorname{CH}}$ is a direct sum of
shifts of functors of the form $\Phi_{\mathbb{k}}$, which is valuative by
Proposition 8.2. ∎

Our next task is to construct an analogous basis for the augmented Chow ring
$\operatorname{CH}(M)$. Given a flat $F$ of $M$, choose any maximal
independent set $I\subset F$, and let $y_{F}:=\prod_{e\in
I}y_{e}\in\operatorname{CH}(M)$. The element $y_{F}$ does not depend on the
choice of $I$ [BHM+22, Lemma 2.11(2)].

######  Proposition 8.7.

For any matroid $M$, the augmented Chow ring $\operatorname{CH}(M)$ has a
basis consisting of monomials of the form $y_{F_{0}}x_{F_{1}}^{m_{1}}\cdots
x_{F_{r}}^{m_{r}}$, where $r\in\mathbb{N}$, $F_{0}\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{r}$, and
$0<m_{i}<\operatorname{rk}F_{i}-\operatorname{rk}F_{i-1}$ for all
$i\in\\{1,\ldots,r\\}$.

###### Proof.

Let $\mathfrak{m}\subset\operatorname{CH}(M)$ be the ideal generated by
$\\{y_{e}\mid e\in E\\}$. The argument in the proof of [BHM+, Proposition 1.8]
shows that we have an isomorphism

$\operatorname{gr}\operatorname{CH}(M):=\bigoplus_{p\geq
0}\frac{\mathfrak{m}^{p}\operatorname{CH}(M)}{\mathfrak{m}^{p+1}\operatorname{CH}(M)}\cong\bigoplus_{F}\underline{\operatorname{CH}}(M_{F})(-\operatorname{rk}F),$
(21)

where $\underline{\operatorname{CH}}(M_{F})(-\operatorname{rk}F)$ embeds into
$\operatorname{gr}\operatorname{CH}(M)$ by sending a polynomial $\eta$ in
$\\{x_{G}\mid F\subset G\subset E\\}$ to the polynomial $y_{F}\eta$. We may
therefore use the basis for each $\underline{\operatorname{CH}}(M_{F})$ from
Theorem 8.4 to construct a basis for $\operatorname{gr}\operatorname{CH}(M)$,
and this lifts to a basis for $\operatorname{CH}(M)$. ∎

######  Remark 8.8.

The decategorified version of Equation (21), which says that

$H_{M}(t)=\sum_{F}t^{\operatorname{rk}F}\underline{H}_{M_{F}}(t),$

appears in [FMSV, Theorems 1.3, 1.4, and 1.5].

Proposition 8.7 has the following corollary.

######  Corollary 8.9.

There is a canonical isomorphism of graded vector spaces

$\operatorname{CH}(M)\;\cong\;\bigoplus_{r\geq
0}\;\bigoplus_{\begin{subarray}{c}\mathbb{k}\,=(k_{0},k_{1},\ldots,k_{r})\\\
k_{0}<k_{1}<\cdots<k_{r}\end{subarray}}\Phi_{\mathbb{k}}(M)\otimes
Q_{k_{1}-k_{0}}\otimes Q_{k_{2}-k_{1}}\otimes\cdots\otimes
Q_{k_{r}-k_{r-1}}(-k_{0}).$

Motivated by Corollary 8.9, we define a functor $\operatorname{CH}$ from
$\mathcal{M}$ to the category of finite dimensional graded vector spaces over
$\mathbb{Q}$ by putting

$\operatorname{CH}\;:=\;\bigoplus_{r\geq
0}\;\bigoplus_{\begin{subarray}{c}\textbf{k}\,=(k_{0},k_{1},\ldots,k_{r})\\\
k_{0}<k_{1}<\cdots<k_{r}\end{subarray}}\Phi_{\textbf{k}}(M)\otimes
Q_{k_{1}-k_{0}}\otimes Q_{k_{2}-k_{1}}\otimes\cdots\otimes
Q_{k_{r}-k_{r-1}}(-k_{0}).$

Corollary 8.6 has the following augmented analogue.

######  Corollary 8.10.

The categorical invariant $\operatorname{CH}$ is valuative.

######  Remark 8.11.

Corollaries 8.6 and 8.10 categorify [FS, Theorem 8.14] and [FMSV, Theorem
1.11], which say that the Chow polynomial $\underline{H}_{M}(t)$ and the
augmented Chow polynomial $H_{M}(t)$ are valuative invariants.

###  8.3 Kazhdan–Lusztig functors

Given positive integers $j$ and $r$ along with a subset $R\subset[r]$, let

$s_{j}(R):=\min\\!\big{(}\mathbb{Z}_{\geq j}\setminus
R\big{)}\in\\{1,\ldots,r+1\\}.$

The following theorem is proved in [PXY18, Theorems 6.1].

######  Theorem 8.12.

Let $M$ be a loopless matroid of rank $k$ on the ground set $E$. The
Kazhdan–Lusztig polynomial $P_{M}(t)$ is equal to

$1+\sum_{i\geq
0}\sum_{r=1}^{i}t^{i}\sum_{R\subset[r]}(-1)^{|R|}\sum_{\begin{subarray}{c}a_{0}<a_{1}<\cdots<a_{r}<a_{r+1}\\\
a_{0}=0\\\ a_{r}=i\\\
a_{r+1}=k-i\end{subarray}}\operatorname{dim}\Phi_{\mathbb{k}}(M),$

where $\mathbb{k}=(k_{1},\ldots,k_{r})$ and
$k_{j}=k-a_{s_{r+1-j}(R)}-a_{r-j}$.

######  Remark 8.13.

The polynomial $P_{M}(t)$ is equal to zero for any matroid with loops, thus
Theorem 8.12 provides a full description of the Kazhdan–Lusztig polynomials of
all matroids.

Based on Theorem 8.12, we define a functor $\operatorname{KL}$ from
$\mathcal{M}$ to the category of finite dimensional bigraded vector spaces
over $\mathbb{Q}$ as follows. On the full subcategory of $\mathcal{M}$
consisting of loopless matroids of rank $k$, we define the functor

$\operatorname{KL}:=\tau\oplus\bigoplus_{1\leq r\leq
i}\;\bigoplus_{R\subset[r]}\;\bigoplus_{\begin{subarray}{c}a_{0}<a_{1}<\cdots<a_{r}<a_{r+1}\\\
a_{0}=0\\\ a_{r}=i\\\ a_{r+1}=k-i\end{subarray}}\Phi_{\textbf{k}}(-i,-|R|),$

where $\textbf{k}=(k_{1},\ldots,k_{r})$ and
$k_{j}=k-a_{s_{r+1-j}(R)}-a_{r-j}$. Here the notation means that the summand
$\Phi_{\textbf{k}}$ appears in bidegree $(i,|R|)$. We define
$\operatorname{KL}=0$ on the full subcategory of $\mathcal{M}$ consisting of
matroids with loops. By definition, the functor $\operatorname{KL}$
categorifies the matroid invariant

$\tilde{P}_{M}(t,u):=\sum_{i,j}\operatorname{dim}\operatorname{KL}^{i,j}(M)\,t^{i}u^{j},$

and Theorem 8.12 and Remark 8.13 imply that $\tilde{P}_{M}(t,-1)$ is equal to
the Kazhdan–Lusztig polynomial $P_{M}(t)$. The following result follows
immediately from Propositions 4.11 and 8.2.

######  Corollary 8.14.

The categorical invariant $\operatorname{KL}$ is valuative.

The $Z$-polynomial relates to the Kazhdan–Lusztig polynomial in the same way
that the augmented Chow polynomial relates to the Chow polynomial. That is, we
have

$Z_{M}(t):=\sum_{F}t^{\operatorname{rk}F}P_{M_{F}}(t)=\sum_{S\subset
E}t^{\operatorname{rk}S}P_{M_{S}}(t),$ (22)

where the second equality comes from the fact that, whenever $S$ is not a
flat, $M_{S}$ has a loop, and therefore $P_{M_{S}}(t)=0$. We therefore define
the functor

$\Sigma:=\bigoplus_{k\geq 0}\bigoplus_{S\subset
E}\Big{(}[\tau]_{k}*\operatorname{KL}\Big{)}(-k,0),$

where the summand indexed by $S$ is understood to be the convolution of the
functor $[\tau]_{k}$ on $\mathcal{M}_{\operatorname{id}}(S)$ with the functor
$\operatorname{KL}$ on $\mathcal{M}_{\operatorname{id}}(E\setminus S)$. Each
summand of this functor is defined only on the category
$\mathcal{M}_{\operatorname{id}}(E)$, but the direct sum extends to the entire
category $\mathcal{M}$. By definition, the functor $\Sigma$ categorifies the
polynomial

$\tilde{Z}_{M}(t,u):=\sum_{i,j}\operatorname{dim}\Sigma^{i,j}(M)\,t^{i}u^{j},$

and Equation (22) implies that $\tilde{Z}_{M}(t,-1)=Z_{M}(t)$. We obtain the
following corollary from Proposition 4.11, Corollary 7.7, Lemma 8.1, and
Corollary 8.14.

######  Corollary 8.15.

The categorical invariant $\Sigma$ is valuative.

######  Remark 8.16.

Corollaries 8.14 and 8.15 categorify [AS22, Theorem 8.9] and [FS, Theorem
9.3], which say that the Kazhdan–Lusztig polynomial $P_{M}(t)$ and the
$Z$-polynomial $Z_{M}(t)$ are valuative invariants.

######  Remark 8.17.

In light of our definition of the functor $\Sigma$, one might ask if we should
have defined the functor $\operatorname{CH}$ in an analogous way. More
precisely, consider the functor

$\Theta:=\bigoplus_{k\geq 0}\bigoplus_{S\subset
E}\Big{(}[\tau]_{k}*\underline{\operatorname{CH}}\Big{)}(-k).$

As in the definition of $\Sigma$, each summand is defined only on the category
$\mathcal{M}_{\operatorname{id}}(E)$, but the direct sum extends to the entire
category $\mathcal{M}$. By Remark 8.8 and the fact that
$\underline{\operatorname{CH}}$ vanishes on matroids with loops, $\Theta$
categorifies the augmented Chow polynomial $H_{M}(t)$. Moreover, a slight
modification of Proposition 8.7 implies that $\Theta(M)$ is canonically
isomorphic to $\operatorname{CH}(M)$ for any matroid $M$. In addition,
Proposition 4.11, Corollary 7.7, Lemma 8.1, and Corollary 8.6 imply that
$\Theta$ is valuative.

However, $\Theta$ is not naturally isomorphic to the functor
$\operatorname{CH}$ because it behaves differently on weak maps that are not
isomorphisms. For example, the degree zero part of $\Theta$ is isomorphic to
$\Psi_{(0)}$, whereas the degree zero part of $\operatorname{CH}$ is
isomorphic to $\Phi_{(0)}$. That is, if $\varphi:M\to M^{\prime}$ is a
morphism in $\mathcal{M}$ and $M^{\prime}$ has strictly more loops than $M$,
then $\operatorname{CH}(\varphi)$ will be an isomorphism but $\Theta(\varphi)$
will be zero. In particular, Corollary 8.19 (which will be stated and proved
in the next section) would fail for the functor $\Theta$. This is why we
regard the functor $\operatorname{CH}$ as a “better” categorification of the
augmented Chow polynomial than the functor $\Theta$.

Similarly, the functor $\Sigma$ is not the only valuative categorical
invariant that categorifies the $Z$-polynomial. However, since there is no
analogue of Corollary 8.19 for the $Z$-polynomial, we know of no reason to
regard one categorification as better than another. See Remark 8.21 for more
on this topic.

###  8.4 Monotonicity

If $\varphi:M\to M^{\prime}$ is a morphism in $\mathcal{M}$, it is clear from
the definition of the functor $\operatorname{OS}$ that the ring homomorphism
$\operatorname{OS}(\varphi):\operatorname{OS}(M)\to\operatorname{OS}(M^{\prime})$
is surjective, and therefore that the polynomial
$\pi_{M}(t)-\pi_{M^{\prime}}(t)$ has non-negative coefficients. We express
this statement by saying that the Poincaré polynomial is monotonic with
respect to weak maps. The aim of this section is to prove a similar result for
the Chow and augmented Chow polynomials, and to discuss the possibility of
extending it to the Kazhdan–Lusztig and $Z$-polynomials.

Fix a natural number $r$ and an increasing $r$-tuple of natural numbers
$\textbf{k}=(k_{1},\ldots,k_{r})$. The following result is equivalent to a
statement conjectured in [FMSV, Conjecture 3.38].

######  Proposition 8.18.

For any morphism $\varphi:M\to M^{\prime}$ in $\mathcal{M}$, the linear map

$\Phi_{\mathbb{k}}(\varphi):\Phi_{\mathbb{k}}(M)\to\Phi_{\mathbb{k}}(M^{\prime})$

is surjective.999We thank George Nasr for his help with this result.

###### Proof.

We may use $\varphi$ to identify the ground sets of $M$ and $M^{\prime}$, and
thus reduce to the case of a morphism $\iota_{M,M^{\prime}}:M\to M^{\prime}$
in $\mathcal{M}_{\operatorname{id}}(E)$. Suppose we are given
$(F^{\prime}_{1},\ldots,F^{\prime}_{r})\in\mathcal{L}_{\textbf{k}}(M^{\prime})$.
We need to find $(F_{1},\ldots,F_{r})\in\mathcal{L}_{\textbf{k}}(M)$ such that
$\overline{F_{i}}=F_{i}^{\prime}$ for all $1\leq i\leq r$.

We proceed by induction on $r$. When $r=0$, there is nothing to prove. The
$r=1$ case is proved in [Luc75, Proposition 5.12]. For the inductive step, we
may assume that we have flats $F_{2}\subset\cdots\subset F_{r}$ with
$\operatorname{rk}_{M}(F_{i})=k_{i}$ and $\overline{F_{i}}=F_{i}^{\prime}$ for
all $2\leq i\leq r$.

Since $\overline{F_{2}}=F^{\prime}_{2}$, we have

$\operatorname{rk}_{M^{\prime}}(F_{2})=\operatorname{rk}_{M^{\prime}}(F^{\prime}_{2})=k_{2}=\operatorname{rk}_{M}(F_{2}),$

thus we have a morphism $M^{F_{2}}\to(M^{\prime})^{F_{2}}$ in
$\mathcal{M}_{\operatorname{id}}(F_{2})$, and $F^{\prime}_{1}\cap F_{2}$ is a
flat of rank $k_{1}$ in $(M^{\prime})^{F_{2}}$. We may therefore apply [Luc75,
Proposition 5.12] again to find a flat $F_{1}$ of $M^{F_{2}}$ of rank $k_{1}$
whose closure in $(M^{\prime})^{F_{2}}$ is equal to $F^{\prime}_{1}\cap
F_{2}$.

We claim that the closure of $F_{1}$ in $M^{\prime}$ is equal to
$F^{\prime}_{1}$. Indeed, the rank of $F_{1}$ in $M^{\prime}$ is the same as
its rank in $(M^{\prime})^{F_{2}}$, which is equal to $k_{1}$, thus its
closure is a flat of rank $k_{1}$. Since $F_{1}\subset F_{1}^{\prime}\cap
F_{2}\subset F_{1}^{\prime}$, the closure is contained in $F_{1}^{\prime}$,
and must be equal by comparison of the ranks. ∎

As a consequence, we obtain a strengthening of the numerical monotonicity
result in [FMSV, Theorem 1.13].

######  Corollary 8.19.

For any morphism $\varphi:M\to M^{\prime}$ in $\mathcal{M}$, the linear maps

$\underline{\operatorname{CH}}(\varphi):\underline{\operatorname{CH}}(M)\to\underline{\operatorname{CH}}(M^{\prime})\qquad\text{and}\qquad\operatorname{CH}(\varphi):\operatorname{CH}(M)\to\operatorname{CH}(M^{\prime})$

are both surjective. In particular, the polynomials

$\underline{H}_{M}(t)-\underline{H}_{M^{\prime}}(t)\qquad\text{and}\qquad
H_{M}(t)-H_{M^{\prime}}(t)$

have non-negative coefficients.

###### Proof.

The maps $\underline{\operatorname{CH}}(\varphi)$ and
$\operatorname{CH}(\varphi)$ are surjective by Proposition 8.18 and the
definitions of $\underline{\operatorname{CH}}$ and $\operatorname{CH}$ as
direct sums of shifts of functors of the form $\Phi_{\textbf{k}}$. ∎

The Kazhdan–Lusztig polynomial and $Z$-polynomial are conjectured to have the
same monotonicity property as the Chow polynomial and augmented Chow
polynomial. The following conjecture was first formulated by Nasr,
generalizing an unpublished conjecture of Gedeon in the case where $M$ is
uniform.

######  Conjecture 8.20.

For any morphism $\varphi:M\to M^{\prime}$ in $\mathcal{M}$, the polynomials

$P_{M}(t)-P_{M^{\prime}}(t)\qquad\text{and}\qquad Z_{M}(t)-Z_{M^{\prime}}(t)$

have non-negative coefficients.

######  Remark 8.21.

There exist graded vector subspaces
$\operatorname{IH}(M)\subset\operatorname{CH}(M)$ and
$\operatorname{IH}(M)_{\varnothing}\subset\underline{\operatorname{CH}}(M)$
with Poincaré polynomials $Z_{M}(t)$ and $P_{M}(t)$, respectively [BHM+,
Theorem 1.9]. It would be interesting to define functors to the abelian
category of finite dimensional graded vector spaces over $\mathbb{Q}$ taking a
matroid $M$ to $\operatorname{IH}(M)$ and
$\operatorname{IH}(M)_{\varnothing}$, and then to strengthen Conjecture 8.20
along the same lines as Conjecture 8.19 by conjecturing that the linear maps
induced by a morphism in $\mathcal{M}$ are surjective. Unfortunately, we do
not currently know how to define these functors on morphisms that are not
isomorphisms (other than by defining all such maps to be zero, which would be
neither valuative nor surjective). This is why we work instead with the
functors $\operatorname{KL}$ and $\Sigma$ to the category of bigraded vector
spaces.

###  8.5 The Grothendieck group of the valuative category

We observed in Remark 4.10 that the Grothendieck group of the valuative
category $\mathcal{V}(E)$ is a priori isomorphic to a quotient of the
valuative group $V(E)$. The subtlety here is that localizing subcategories are
by definition closed under passing to direct summands, which means that the
Grothendieck classes of objects of $\mathcal{I}(E)$ could in theory include
elements of the group $M(E)\cong
K(\operatorname{Ch}_{b}(\mathcal{M}_{\operatorname{id}}^{+}(E)))$ that do not
lie in the subgroup $I(E)$. In this section, we prove that the Grothendieck
group of $\mathcal{V}(E)$ is in fact isomorphic to $V(E)$; we thank Matt
Larson for outlining this argument.

Consider an increasing $r$-tuple $\textbf{k}=(k_{1},\ldots,k_{r})$ of natural
numbers and an increasing $r$-tuple $\textbf{S}=(S_{1},\ldots,S_{r})$ of
subsets of $E$. The functor
$\Psi_{E,\textbf{k},\textbf{S}}:\mathcal{M}_{\operatorname{id}}(E)\to\operatorname{Vec}_{\mathbb{Q}}$
was defined in Example 4.3 and proved to be valuative in Lemma 8.3. In the
proof of Proposition 8.2, we fixed k and considered the direct sum of these
functors over all possible S. In this section, we do the opposite: we fix S,
and define
$\Psi_{E,\textbf{S}}:\mathcal{M}_{\operatorname{id}}(E)\to\operatorname{Vec}_{\mathbb{Q}}$
by putting

$\Psi_{E,\textbf{S}}(M):=\bigoplus_{\textbf{k}}\Psi_{E,\textbf{k},\textbf{S}}(M).$

Concretely, $\Psi_{E,\textbf{S}}(M)$ is equal to $\mathbb{Q}$ if $S_{j}$ is a
flat (of any rank) for all $j$, and zero otherwise.

Now fix a subset $I\subset S_{1}$ of cardinality $k$. Let $\textbf{S}\setminus
I:=(S_{1}\setminus I,\ldots,S_{r}\setminus I)$, which we regard as an
increasing chain of subsets of $E\setminus I$. We will be interested in the
functor

$\Psi_{E\setminus I,\textbf{S}\setminus
I}:\mathcal{M}_{\operatorname{id}}(E\setminus
I)\to\operatorname{Vec}_{\mathbb{Q}}.$

We will also consider the functor
$[\tau]_{k}:\mathcal{M}_{\operatorname{id}}(I)\to\operatorname{Vec}_{\mathbb{Q}}$,
which takes the Boolean matroid on $I$ to $\mathbb{Q}$, and all other matroids
on $I$ to zero. Convolving them, we obtain a new functor

$\Psi_{I\leq\textbf{S}}:=[\tau_{k}]*\Psi_{E\setminus I,\textbf{S}\setminus
I}:\mathcal{M}_{\operatorname{id}}(E)\to\operatorname{Vec}_{\mathbb{Q}}.$

On the level of objects, $\Psi_{I\leq\textbf{S}}(M)$ is equal to $\mathbb{Q}$
if $I$ is independent in $M$ and $S_{j}\setminus I$ is a flat of the
contraction $M_{I}$ for all $j$, or equivalently if $I$ is independent in $M$
and $S_{j}$ is a flat of $M$ for all $j$. Otherwise,
$\Psi_{I\leq\textbf{S}}(M)=0$.

######  Lemma 8.22.

The functor $\Psi_{I\leq\textbf{S}}$ is valuative.

###### Proof.

Valuativity of the first convolution factor follows from Proposition 4.11 and
Lemma 8.1, while valuativity of the second convolution factor follows from
Lemma 8.3. The result then follows from Corollary 7.7. ∎

The stellahedral fan [EHL, Definition 3.2] $\Sigma_{E}$ is a complete
unimodular fan in $\mathbb{R}_{E}$ with cones $\sigma_{I\leq\textbf{S}}$
indexed by chains

$\emptyset\subseteq I\subseteq S_{1}\subsetneq S_{1}\subsetneq\cdots\subsetneq
S_{r}\subsetneq E.$

Note that we are allowed to have $r=0$, in which case $I$ is allowed to
coincide with $E$. Explicitly, we have

$\sigma_{I\leq\textbf{S}}:=\mathbb{R}_{\geq 0}\\{v_{e}\mid e\in
I\\}+\mathbb{R}_{\geq 0}\\{-v_{E\setminus S_{j}}\mid 1\leq j\leq r\\}.$

For any matroid $M$ on $E$, the augmented Bergman fan [BHM+22, Definition 2.4]
$\Sigma_{M}$ is defined to be the subfan of $\Sigma_{E}$ consisting of those
$\sigma_{I\leq\textbf{S}}$ for which $I$ is independent and $S_{j}$ is a flat
for all $j$. Note that the stellahedral fan is equal to the augmented Bergman
fan of the Boolean matroid.

Let $\mathbb{Z}^{\Sigma_{E}}$ be the vector space with basis
$\\{w_{\sigma}\mid\sigma\in\Sigma_{E}\\}$. The following theorem follows
immediately from [EHL, Theorems 1.5 and 5.6].

######  Theorem 8.23.

The homomorphism $M(E)\to\mathbb{Z}^{\Sigma_{E}}$ taking $M$ to
$\sum_{\sigma\in\Sigma_{M}}w_{\sigma}$ has kernel equal to $I(E)$, and
therefore descends to an embedding of $\operatorname{Val}(E)$ into
$\mathbb{Z}^{\Sigma_{E}}$.

We are now ready to prove our proposition.

######  Proposition 8.24.

The Grothendieck group of the valuative category $\mathcal{V}(E)$ is
isomorphic to the valuative group $\operatorname{Val}(E)$.

###### Proof.

We have already noted that there is a canonical surjection
$\operatorname{Val}(E)\to K(\mathcal{V}(E))$ taking the class of a matroid $M$
to the Grothendieck class of the object $M$ of $\mathcal{V}(E)$. We will now
prove that this is an isomorphism by constructing its inverse.

Let $\operatorname{Vec}_{\mathbb{Q}}^{\Sigma_{E}}$ be the category of finite
dimensional $\mathbb{Q}$-vector spaces equipped with a direct sum
decomposition indexed by cones $\sigma\in\Sigma_{E}$, with morphisms given by
linear maps that are compatible with these decompositions. Consider the
functor
$\Psi:\mathcal{M}_{\operatorname{id}}(E)\to\operatorname{Vec}_{\mathbb{Q}}^{\Sigma_{E}}$
obtained as the direct sum of the functors $\Psi_{I\leq\textbf{S}}$. Lemma
8.22 immediately implies that $\Psi$ is valuative, thus $\Psi$ descends to the
valuative category $\mathcal{V}(E)$, and therefore the induced homomorphism

$M(E)\cong K(\mathcal{M}_{\operatorname{id}}(E))\to
K(\operatorname{Vec}_{\mathbb{Q}}^{\Sigma_{E}})\cong\mathbb{Z}^{\Sigma_{E}}$

descends to $K(\mathcal{V}(E))$. This is precisely the homomorphism from
Theorem 8.23, whose image is isomorphic to $\operatorname{Val}(E)$. The
induced homomorphism from $K(\mathcal{V}(E))$ to $\operatorname{Val}(E)$ takes
the Grothendieck class of $M$ to the class of $M$ in $\operatorname{Val}(E)$,
and is therefore inverse to the homomorphism in the first paragraph. ∎

##  9 Characters

In this section, we explain how to use valuative categorical matroid
invariants to obtain relations among isomorphism classes of graded
representations of the symmetry group of a matroid decomposition. Everything
that follows works equally well for decompositions of polyhedra, but since our
examples are all matroid invariants, we will work entirely in that setting.

###  9.1 The oriented case

Let $\mathcal{N}$ be a decomposition of a matroid $M$ on the ground set $E$,
and let $\Omega$ be an orientation of $\mathcal{N}$. Let $\Gamma$ be a finite
group acting via permutations of $E$ that preserve $\mathcal{N}$ and $\Omega$.
In other words, for every $N\in\mathcal{N}$ and $\gamma\in\Gamma$,
$\gamma(N)\in\mathcal{N}$, and the map $P(N)\to P(\gamma(N))$ on base
polytopes induced by $\gamma$ is orientation preserving. This implies that
$\gamma$ acts on the complex $C_{\bullet}^{\Omega}(\mathcal{N})$. Fix an
abelian group, and let $\mathcal{A}$ be the category of finite dimensional
vector spaces over $\mathbb{Q}$ that are graded in that group.101010In
practice, we will always take our gradings in $\mathbb{Z}$ or
$\mathbb{Z}^{2}$. Suppose that $\Phi:\mathcal{M}(E)\to\mathcal{A}$ is a
categorical valuative invariant. Then
$\Phi(C_{\bullet}^{\Omega}(\mathcal{N}))$ is exact, and we therefore have an
equation

$\displaystyle
0=\chi^{\Gamma}\\!\Big{(}\Phi(C_{\bullet}^{\Omega}(\mathcal{N}))\Big{)}$
$\displaystyle:=$
$\displaystyle\sum_{k}(-1)^{k}\,\Phi(C_{k}^{\Omega}(\mathcal{N}))$
$\displaystyle=$
$\displaystyle\sum_{k}(-1)^{k}\bigoplus_{N\in\mathcal{N}_{k}}\Phi(N)$
$\displaystyle=$
$\displaystyle\sum_{k}(-1)^{k}\\!\bigoplus_{N\in\mathcal{N}_{k}/\Gamma}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(N)$

of virtual graded representations of $\Gamma$, where for the last sum one
takes a single representative of each $\Gamma$ orbit in $\mathcal{N}_{k}$.

Examples from previous sections of valuative categorical invariants include
$\operatorname{OS}$ (Theorem 5.5), $\underline{\operatorname{CH}}$ (Corollary
8.6), $\operatorname{CH}$ (Corollary 8.10), all of which take values in finite
dimensional graded vector spaces. In these cases, Equation (9.1) gives a
relationship between Orlik–Solomon algebras, Chow rings, and augmented Chow
rings of all of the matroids in $\mathcal{N}$, regarded as isomorphism classes
of representations of $\Gamma$. We also have the examples $\operatorname{KL}$
(Corollary 8.14) and $\Sigma$ (Corollary 8.14), both of which take values in
bigraded vector spaces. The virtual graded $\Gamma$-representations

$\sum_{j}(-1)^{j}\operatorname{KL}^{i,j}(M)\qquad\text{and}\qquad\sum_{j}(-1)^{j}\,\Sigma^{i,j}(M)$

are equal to the coefficients of $t^{i}$ in the $\Gamma$-equivariant
Kazhdan–Lusztig polynomial $P_{M}^{\Gamma}(t)$ and the $\Gamma$-equivariant
$Z$-polynomial $Z_{M}^{\Gamma}(t)$, respectively [PXY18, Theorem 6.1]. Thus
Equation (9.1) also allows us to relate these equivariant matroid invariants
for $M$ to those of the various $N\in\mathcal{N}$.

There is, however, a serious limitation to the usefulness of Equation (9.1).
When given a decomposition $\mathcal{N}$ with an action of $\Gamma$, it is
usually impossible to choose an orientation $\Omega$ that is preserved by
$\Gamma$. For instance, consider the decomposition in Examples 1.1 and 3.6. We
have $E=\\{1,2,3,4\\}$, and the group $\mathfrak{S}_{2}$ acts by swapping 1
with 3 and 2 with 4. This action preserves $\mathcal{N}$, swapping $N$ with
$N^{\prime}$ and taking $N^{\prime\prime}$ to itself. However, the action of
$\mathfrak{S}_{2}$ on $P(N^{\prime\prime})$ is orientation reversing, so there
is no way to choose $\Omega$. If there were, then Equation (9.1) applied to
the trivial categorical invariant $\tau$ would tell us that
$\tau(M)\oplus\tau(N^{\prime\prime})$ (two copies of the trivial
representation of $\mathfrak{S}_{2}$) is isomorphic to
$\tau(N)\oplus\tau(N^{\prime})$ (the regular representation of
$\mathfrak{S}_{2}$), which is of course false.

What we would like to do is replace $\tau(N^{\prime\prime})$ with the sign
representation of $\mathfrak{S}_{2}$, to reflect the fact that the action on
$P(N^{\prime\prime})$ is orientation reversing (see Example 9.5). The rest of
this section is devoted to developing the machinery needed to make this
precise, in the form of Corollary 9.3, and then applying it to certain
families of examples.

###  9.2 The determinant category

If $M$ is a matroid on the ground set $E$, consider the vector space

$A(M):=\operatorname{Span}\\{x-y\mid x,y\in P(M)\\}\cap\mathbb{Q}^{E}.$

If $P(M^{\prime})\subset P(M)$, we define the 1-dimensional vector space

$L(M,M^{\prime}):=\wedge^{d(M)-d(M^{\prime})}\left(A(M)/A(M^{\prime})\right).$

Given $M$, $M^{\prime}$, and $M^{\prime\prime}$ with
$P(M^{\prime\prime})\subset P(M^{\prime})\subset P(M)$, wedge product induces
a canonical isomorphism

$L(M,M^{\prime})\otimes L(M^{\prime},M^{\prime\prime})\to
L(M,M^{\prime\prime}).$ (24)

We now define a new category $\mathcal{M}_{\operatorname{id}}^{\wedge}(E)$ by
taking the objects to be formal direct sums of matroids on $E$, and taking

$\operatorname{Hom}_{\mathcal{M}_{\operatorname{id}}^{\wedge}(E)}(M,M^{\prime}):=L(M,M^{\prime}).$

Composition is given by Equation (24), and morphisms between formal direct
sums are matrices whose entries consist of morphisms between the individual
matroids.

######  Remark 9.1.

The category $\mathcal{M}_{\operatorname{id}}^{\wedge}(E)$ should be regarded
as a variant of $\mathcal{M}_{\operatorname{id}}^{+}(E)$. In both categories,
the space of homomorphisms from $M$ to $M^{\prime}$ is a 1-dimensional
$\mathbb{Q}$-vector space. The difference is that, in
$\mathcal{M}_{\operatorname{id}}^{+}(E)$, this vector space is canonically
identified with $\mathbb{Q}$, whereas the same is not true in
$\mathcal{M}_{\operatorname{id}}^{\wedge}(E)$.

Given a decomposition $\mathcal{N}$ of a matroid $M$ on the ground set $E$, we
defined a large collection of complexes $C_{\bullet}^{\Omega}(M)$ in
$\mathcal{M}_{\operatorname{id}}^{+}(E)$, depending on an $\Omega$ of
$\mathcal{N}$. In contrast, we will define a single complex
$C_{\bullet}^{\wedge}(\mathcal{N})$ in
$\mathcal{M}_{\operatorname{id}}^{\wedge}(E)$ that does not depend on any
choices. The objects will be the same; that is, we put

$C_{k}^{\wedge}(\mathcal{N}):=\bigoplus_{N\in\mathcal{N}_{k}}N.$

For each maximal face $N\in\mathcal{N}_{d(M)}$, $L(M,N)$ is canonically
isomorphic to $\mathbb{Q}$, and we define $C_{d(M)+1}^{\wedge}(\mathcal{N})\to
C_{d(M)}^{\wedge}(\mathcal{N})$ to be the diagonal map. Given $k\leq d(M)$,
$N\in\mathcal{N}_{k}$, and $N^{\prime}\in\mathcal{N}_{k-1}$ with
$P(N^{\prime})\subset P(N)$, we define the $(N,N^{\prime})$ component of the
differential $C_{k}^{\wedge}(\mathcal{N})\to C_{k-1}^{\wedge}(\mathcal{N})$ to
be the class of the outward unit normal vector to $P(N^{\prime})$ inside of
$P(N)$ in

$A(N)/A(N^{\prime})=Q(N,N^{\prime})=\operatorname{Hom}_{\mathcal{M}_{\operatorname{id}}^{\wedge}(E)}(N,N^{\prime}).$

It is straightforward to check that the differential squares to zero.

###  9.3 The determinant functor

In this section, we define an additive functor $\operatorname{Det}$ from
$\mathcal{M}_{\operatorname{id}}^{\wedge}(E)$ to the category
$\operatorname{Vec}_{\mathbb{Q}}$. For any matroid $M$ on $E$, we put

$\operatorname{Det}(M):=\wedge^{d(M)}A(M)^{*}.$

If $P(M^{\prime})\subset P(M)$ and

$\sigma\in\wedge^{d(M)-d(M^{\prime})}(A(M)/A(M^{\prime}))=L(M,M^{\prime})=\operatorname{Hom}_{\mathcal{M}_{\operatorname{id}}^{\wedge}(E)}(M,M^{\prime}),$

then wedge product with $\sigma$ gives an isomorphism from
$\wedge^{d(M^{\prime})}A(M^{\prime})$ to $\wedge^{d(M)}A(M)$. Dualizing, we
obtain our map

$\operatorname{Det}(\sigma):\operatorname{Det}(M)\to\operatorname{Det}(M^{\prime}).$

Let $\Phi:\mathcal{M}_{\operatorname{id}}(E)\to\mathcal{A}$ be a categorical
matroid invariant valued in a $\mathbb{Q}$-linear category $\mathcal{A}$.
There is a monoidal action of $\operatorname{Vec}_{\mathbb{Q}}$ on
$\mathcal{A}$, which we denote with $\otimes$. We define a functor
$\Phi^{\wedge}:\mathcal{M}_{\operatorname{id}}^{\wedge}(E)\to\mathcal{A}$ by
putting $\Phi^{\wedge}:=\Phi\otimes\operatorname{Det}$. More precisely, we
define $\Phi^{\wedge}(M):=\Phi(M)\otimes\operatorname{Det}(M)$ for all $M$,
and given an element

$\sigma\in
L(M,M^{\prime})=\operatorname{Hom}_{\mathcal{M}_{\operatorname{id}}^{\wedge}(E)}(M,M^{\prime}),$

we put

$\Phi^{\wedge}(\sigma):=\Phi(\iota_{M,M^{\prime}})\otimes\operatorname{Det}(\sigma):\Phi^{\wedge}(M)\to\Phi^{\wedge}(M^{\prime}).$

######  Proposition 9.2.

If $\Phi:\mathcal{M}_{\operatorname{id}}(E)\to\mathcal{A}$ is valuative and
$\mathcal{N}$ is a decomposition of a matroid $M$ on the ground set $E$, then
$\Phi^{\wedge}(C_{\bullet}^{\wedge}(\mathcal{N}))$ is contractible.

###### Proof.

Let $\Omega$ be an orientation of $\mathcal{N}$. This induces an isomorphism
$\operatorname{Det}(M)\cong\mathbb{Q}$, along with isomorphisms
$\operatorname{Det}(N)\cong\mathbb{Q}$ for each $N\in\mathcal{N}$. These
isomorphisms fit together to form an isomorphism of complexes
$\Phi^{\wedge}(C_{\bullet}^{\wedge}(\mathcal{N}))\cong\Phi(C_{\bullet}^{\Omega}(\mathcal{N}))$,
and the latter is contractible by definition of valuativity. ∎

Proposition 9.2 has a corollary that allows us to generalize Equation (9.1) to
situations where it is not possible to choose an orientation $\Omega$ in a way
that is fixed by symmetries. Fix an abelian group, and let $\mathcal{A}$ be
the category of finite dimensional vector spaces over $\mathbb{Q}$ that are
graded by that group.

######  Corollary 9.3.

Let $\mathcal{N}$ be a decomposition of a matroid $M$ on the ground set $E$.
Let $\Gamma$ be a finite group acting on $E$ preserving $\mathcal{N}$. Let
$\Phi:\mathcal{M}\to\mathcal{A}$ be a valuative categorical invariant. Then we
have an equation

$\displaystyle
0=\chi^{\Gamma}\\!\Big{(}\Phi(C_{\bullet}^{\wedge}(\mathcal{N}))\Big{)}$
$\displaystyle:=$
$\displaystyle\sum_{k}(-1)^{k}\,\Phi(C_{k}^{\wedge}(\mathcal{N}))$
$\displaystyle=$
$\displaystyle\sum_{k}(-1)^{k}\bigoplus_{N\in\mathcal{N}_{k}}\Phi(N)\otimes\operatorname{Det}(N)$
$\displaystyle=$
$\displaystyle\sum_{k}(-1)^{k}\\!\bigoplus_{N\in\mathcal{N}_{k}/\Gamma}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(N)\otimes\operatorname{Det}(N)$

of virtual graded representations of $\Gamma$.

######  Remark 9.4.

If there exists an orientation $\Omega$ of $\mathcal{N}$ that is fixed by the
action of $\Gamma$, then for all $N\in\mathcal{N}$, $\operatorname{Det}(N)$ is
isomorphic to the 1-dimensional trivial representation of the stabilizer of
$N$. In particular, Equation (9.1) is a special case of Corollary 9.3.

######  Example 9.5.

Let $\mathcal{N}$ be the decomposition of $M$ in Examples 1.1 and 3.6, and let
$\Gamma=\mathfrak{S}_{2}$ act by swapping 1 with 3 and 2 with 4. Applying
Corollary 9.3 to the trivial categorical invariant $\tau$, we obtain the
equation

$0=\operatorname{Det}(M)-\operatorname{Det}(N)-\operatorname{Det}(N^{\prime})+\operatorname{Det}(N^{\prime\prime})$

of virtual representations of $\mathfrak{S}_{2}$. Since $\mathfrak{S}_{2}$
acts on $P(M)$ in an orientation preserving way, $\operatorname{Det}(M)$ is
the trivial representation. Since $\mathfrak{S}_{2}$ acts on
$P(N^{\prime\prime})$ in an orientation reversing way,
$\operatorname{Det}(N^{\prime\prime})$ is the sign representation. Finally,
since $\mathfrak{S}_{2}$ swaps $N$ with $N^{\prime}$,
$\operatorname{Det}(N)\oplus\operatorname{Det}(N^{\prime})$ is the regular
representation.

###  9.4 Equivariant relaxation

Let $M$ be a matroid of rank $k$ on the ground set $E$, equipped with an
action of the group $\Gamma$. Let $F$ be a stressed flat of rank $r$, and let
$\mathcal{F}:=\\{\gamma F\mid\gamma\in\Gamma\\}$. Let $\tilde{M}$ be the
matroid obtained by relaxing $M$ with respect to every $G\in\mathcal{F}$, as
described in Section 3.3. Let $\Gamma_{\\!F}\subset\Gamma$ denote the
stabilizer of $F$. Note that $\Gamma_{\\!F}$ acts on the sets $F$ and
$E\setminus F$, and therefore on the matroids $\Pi_{r,k,E,F}$ and
$\Lambda_{r,k,E,F}$ from Section 3.3.

Fix an abelian group, and let $\mathcal{A}$ be the category of finite
dimensional vector spaces over $\mathbb{Q}$ that are graded by that group. Let
$\Phi:\mathcal{M}\to\mathcal{A}$ be a valuative categorical invariant.

######  Proposition 9.6.

We have the following equality of virtual graded $\Gamma$-representations:

$\Phi(\tilde{M})=\Phi(M)+\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(\Lambda_{r,k,E,F})-\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(\Pi_{r,k,E,F}).$

###### Proof.

If $M=\Pi_{r,k,E,F}$, then $\tilde{M}=\Lambda_{r,k,E,F}$ and the statement is
trivial. Assume now that this is not the case. Let $\mathcal{N}$ be the
decomposition of $\tilde{M}$ described in Theorem 3.10. By Corollary 9.3, we
have

$\displaystyle(-1)^{d(\tilde{M})}\Phi(\tilde{M})\otimes\operatorname{Det}(\tilde{M})$
$\displaystyle=$ $\displaystyle(-1)^{d(M)}\Phi(M)\otimes\operatorname{Det}(M)$
$\displaystyle+\;\;(-1)^{d(\Lambda_{r,k,E,F})}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(\Lambda_{r,k,E,F})\otimes\operatorname{Det}(\Lambda_{r,k,E,F})$
$\displaystyle+\;\;(-1)^{d(\Pi_{r,k,E,F})}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(\Pi_{r,k,E,F})\otimes\operatorname{Det}(\Pi_{r,k,E,F}).$

To simplify this, we first note that

$d(\tilde{M})=d(M)=d(\Lambda_{r,k,E,F})=d(\Pi_{r,k,E,F})+1.$

Next, we observe that

$A(\tilde{M})=A(M)=A(\Lambda_{r,k,E,F})=A(\Pi_{r,k,E,F})\oplus\mathbb{Q}\cdot
x_{F},$

which implies that $\operatorname{Det}(\tilde{M})=\operatorname{Det}(M)$ as
representations of $\Gamma$. Since $\Gamma_{\\!F}$ fixes $x_{F}$, it also
implies that
$\operatorname{Det}(\Lambda_{r,k,E,F})=\operatorname{Det}(\Pi_{r,k,E,F})=\operatorname{Res}^{\Gamma}_{\Gamma_{\\!F}}\operatorname{Det}(M)$
as representations of $\Gamma_{\\!F}$. Thus

$\displaystyle(-1)^{d(M)}\Phi(\tilde{M})\otimes\operatorname{Det}(M)$
$\displaystyle=$ $\displaystyle(-1)^{d(M)}\Phi(M)\otimes\operatorname{Det}(M)$
$\displaystyle+\;\;(-1)^{d(M)}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Big{(}\Phi(\Lambda_{r,k,E,F})\otimes\operatorname{Res}^{\Gamma}_{\Gamma_{\\!F}}\operatorname{Det}(M)\Big{)}$
$\displaystyle-\;\;(-1)^{d(M)}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Big{(}\Phi(\Pi_{r,k,E,F})\otimes\operatorname{Res}^{\Gamma}_{\Gamma_{\\!F}}\operatorname{Det}(M)\Big{)}$
$\displaystyle=$ $\displaystyle(-1)^{d(M)}\Phi(M)\otimes\operatorname{Det}(M)$
$\displaystyle+\;\;(-1)^{d(M)}\Big{(}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(\Lambda_{r,k,E,F})\Big{)}\otimes\operatorname{Det}(M)$
$\displaystyle-\;\;(-1)^{d(M)}\Big{(}\operatorname{Ind}_{\Gamma_{\\!F}}^{\Gamma}\Phi(\Pi_{r,k,E,F})\Big{)}\otimes\operatorname{Det}(M).$

Dividing both sides by $(-1)^{d(M)}\operatorname{Det}(M)$, which is an
invertible element of the ring of virtual graded representations of $\Gamma$,
yields the statement of the proposition. ∎

###  9.5 Relaxing a stressed hyperplane

We conclude by applying Proposition 9.6 to the functors $\operatorname{OS}$
and $\operatorname{KL}$ in the special case where the stressed flat $F=H$ is a
hyperplane, meaning that $r=k-1$. In this case, we have

$\Pi_{k-1,k,E,H}=U_{1,E\setminus H}\sqcup U_{k-1,H}.$

The matroid $\Lambda_{k-1,k,E,H}$ obtained by relaxing $\Pi_{k-1,k,E,H}$ with
respect to the stressed hyperplane $H$ has as its bases those subsets
$B\subset E$ of cardinality $k$ with $|B\cap H|\geq k-1$. It has no loops, and
its simplification is the uniform matroid of rank $k$ on the ground set
$\bar{H}:=H\sqcup\\{*\\}$ obtained from $E$ by identifying all of the elements
of $E\setminus H$.

The group $\operatorname{Aut}(E\setminus H)\times\operatorname{Aut}(H)$ acts
on the matroids $\Pi_{k-1,k,E,H}$ and $\Lambda_{k-1,k,E,H}$, and therefore on
their Orlik–Solomon algebras. Since all of the elements of $E\setminus H$ are
parallel in both $\Pi_{k-1,k,E,H}$ and $\Lambda_{k-1,k,E,H}$, the actions on
the Orlik–Solomon algebra factor through the projection to
$\operatorname{Aut}(H)$.

For the rest of this section, we will write $h=|H|$ and identify $H$ with
$\\{1,\ldots,h\\}$. If $\lambda$ is a partition of $h$, we write $V_{\lambda}$
to denote the corresponding Specht module for
$\mathfrak{S}_{h}=\operatorname{Aut}(H)$. Recall that $V_{[h]}$ is the trivial
representation.

######  Lemma 9.7.

We have

$\operatorname{OS}(\Lambda_{k-1,k,E,H})-\operatorname{OS}(\Pi_{k-1,k,E,H})=\left(\wedge^{k-1}\,V_{[h-1,1]}\right)\otimes\Big{(}\mathbb{Q}(1-k)\oplus\mathbb{Q}(-k)\Big{)}$

as graded virtual representations of $\mathfrak{S}_{h}$.

###### Proof.

The matroids $\Lambda_{k-1,k,E,H}$ and $\Pi_{k-1,k,E,H}$ have the same
independent sets of cardinality less than $k$, so their Orlik–Solomon algebras
differ only in degrees $k-1$ and $k$. Let us consider the degree $k$ part
first.

The Orlik–Solomon algebra of a direct sum is isomorphic to the tensor product
of the Orlik–Solomon algebras, and the degree $k-1$ part of the Orlik–Solomon
algebra of $U_{k-1,H}$ is isomorphic to $\wedge^{k-2}V_{[h-1,1]}$, thus

$\displaystyle\operatorname{OS}^{k}(\Pi_{k-1,k,E,H})$
$\displaystyle\cong\operatorname{OS}^{1}(U_{1,E\setminus
H})\otimes\operatorname{OS}^{k-1}(U_{k-1,H})$ $\displaystyle\cong
V_{[h]}\otimes\wedge^{k-2}\,V_{[h-1,1]}$
$\displaystyle\cong\wedge^{k-2}\,V_{[h-1,1]}.$

The Orlik–Solomon algebra is unaffected by simplification, thus

$\displaystyle\operatorname{OS}^{k}(\Lambda_{k-1,k,E,H})$
$\displaystyle\cong\operatorname{Res}^{\mathfrak{S}_{h+1}}_{\mathfrak{S}_{h}}\operatorname{OS}^{k}(U_{k,\bar{H}})$
$\displaystyle\cong\operatorname{Res}^{\mathfrak{S}_{h+1}}_{\mathfrak{S}_{h}}\wedge^{k-1}\,V_{[h,1]}$
$\displaystyle\cong\wedge^{k-1}\operatorname{Res}^{\mathfrak{S}_{h+1}}_{\mathfrak{S}_{h}}V_{[h,1]}$
$\displaystyle\cong\wedge^{k-1}\left(V_{[h-1,1]}\oplus V_{[h]}\right)$
$\displaystyle\cong\wedge^{k-1}\,V_{[h-1,1]}\oplus\wedge^{k-2}\,V_{[h-1,1]}.$

This proves that the lemma is correct in degree $k$. The statement in degree
$k-1$ follows from the fact that, for any matroid $M$ on a nonempty ground
set, $\sum(-1)^{i}\operatorname{OS}^{i}(M)=0$ as a virtual representation of
the automorphism group of $M$. ∎

Let $\Gamma_{H}:=\Gamma\cap\operatorname{Aut}(E\setminus
H)\times\operatorname{Aut}(H)$ be the stabilizer of $H$, which maps via
projection to $\mathfrak{S}_{h}=\operatorname{Aut}(H)$. We write
$\operatorname{Res}_{\Gamma_{\\!H}}^{\mathfrak{S}_{h}}$ for the functor that
pulls back a representation along the homomorphism from $\Gamma_{\\!H}$ to
$\mathfrak{S}_{h}$. Proposition 9.6 and Lemma 9.7 have the following immediate
corollary.

######  Corollary 9.8.

Let $M$ be a matroid equipped with an action of the group $\Gamma$. Let $H$ be
a stressed hyperplane of $M$, and let $\tilde{M}$ be the matroid obtained from
$M$ by relaxing $\gamma H$ for all $\gamma\in\Gamma$. Then

$\operatorname{OS}(\tilde{M})=\operatorname{OS}(M)+\left(\operatorname{Ind}_{\Gamma_{\\!H}}^{\Gamma}\operatorname{Res}_{\Gamma_{\\!H}}^{\mathfrak{S}_{h}}\wedge^{k}\,V_{[h-1,1]}\right)\otimes\Big{(}\mathbb{Q}(1-k)\oplus\mathbb{Q}(-k)\Big{)}$

as virtual graded representations of $\Gamma$.

We now turn our attention to the functor $\operatorname{KL}$. Recall that, if
$M$ is a matroid equipped with an action of a group $\Gamma$, then the
coefficient of $t^{i}$ in the $\Gamma$-equivariant Kazhdan–Lusztig polynomial
$P_{M}^{\Gamma}(t)$ is equal to the virtual $\Gamma$-representation
$\sum_{j}(-1)^{j}\Sigma^{i,j}(M)$. Given a skew partition $\lambda/\mu$ of
$h$, we write $V_{\lambda/\mu}$ to denote the corresponding skew Specht module
for $\mathfrak{S}_{h}$.

######  Lemma 9.9.

If $k>1$, then we have

$P^{\mathfrak{S}_{h}}_{\Lambda_{k-1,k,E,H}}(t)-P^{\mathfrak{S}_{h}}_{\Pi_{k-1,k,E,H}}(t)=\sum_{0<i<k/2}V_{[h-2i+1,(k-2i+1)^{i}]/[k-2i,(k-2i-1)^{i-1}]}\,t^{i}.$

###### Proof.

Equivariant Kazhdan–Lusztig polynomials of matroids are multiplicative under
direct sums and the equivariant Kazhdan–Lusztig polynomial of a rank 1
loopless matroid is the trivial representation in degree zero, hence

$P^{\mathfrak{S}_{h}}_{\Pi_{k-1,k,E,H}}(t)=P^{\mathfrak{S}_{h}}_{U_{k-1,H}}(t)=\sum_{0\leq
i<(k-1)/2}V_{[h-2i,(k-2i)^{i}]/[(k-2i-2)^{i}]}\,t^{i},$

where the second equality is proved in [GXY, Theorem 3.7]. On the other hand,
equivariant Kazhdan–Lusztig polynomials of loopless matroids are unchanged by
simplification, hence

$P^{\mathfrak{S}_{h}}_{\Lambda_{k-1,k,E,H}}(t)=P^{\mathfrak{S}_{h}}_{U_{k,\bar{H}}}(t)=\sum_{0\leq
i<k/2}\operatorname{Res}_{\mathfrak{S}_{h}}^{\mathfrak{S}_{h+1}}V_{[h-2i+1,(k-2i+1)^{i}]/[(k-2i-1)^{i}]}\,t^{i},$

where the last equality again follows from [GXY, Theorem 3.7]. By [KNPV23,
Lemma 4.1] and [Kle05, Proposition 2.3.5, Lemma 2.3.12], we may rewrite this
as

$P^{\mathfrak{S}_{h}}_{\Lambda_{k-1,k,E,H}}(t)=\sum_{0\leq
i<(k-1)/2}V_{[h-2i,(k-2i+1)^{i}]/[(k-2i-1)^{i}]}\,t^{i}+\sum_{0<i<k/2}V_{[h-2i+1,(k-2i+1)^{i}]/[k-2i,(k-2i-1)^{i-1}]}\,t^{i}.$

The lemma is obtained by taking the difference between the our expressions for
$P^{\mathfrak{S}_{h}}_{\Pi_{k-1,k,E,H}}(t)$ and
$P^{\mathfrak{S}_{h}}_{\Lambda_{k-1,k,E,H}}(t)$. ∎

Proposition 9.6 and Lemma 9.9 have the following immediate corollary, which
was first proved in [KNPV23, Theorems 1.3 and 1.4].

######  Corollary 9.10.

Let $M$ be a matroid of rank $k>1$ equipped with an action of the group
$\Gamma$. Let $H$ be a stressed hyperplane of $M$, and let $\tilde{M}$ be the
matroid obtained from $M$ by relaxing $\gamma H$ for all $\gamma\in\Gamma$.
Then

$P_{\tilde{M}}^{\Gamma}(t)=P_{M}^{\Gamma}(t)+\bigoplus_{0<i<k/2}\operatorname{Ind}_{\Gamma_{\\!H}}^{\Gamma}\operatorname{Res}_{\Gamma_{\\!H}}^{\mathfrak{S}_{h}}V_{[h-2i+1,(k-2i+1)^{i}]/[k-2i,(k-2i-1)^{i-1}]}\,t^{i}.$

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# A Note On Gorenstein Projective Conjecture II

Xiaojin Zhang
School of Mathematics and Statistics, Nanjing University of Information
Science and Technology,
Nanjing 210044, P. R. China E-mail address<EMAIL_ADDRESS>
Abstract In this paper, we prove that Gorenstein projective conjecture is left
and right symmetric and the co-homology vanishing condition can not be reduced
in general. Moreover, Gorenstein projective conjecture is proved to be true
for CM-finite algebras.

Keywords Gorenstein projective, CM-finite algebras, Gorenstein projective
conjecture

AMS(2000) Subject Classification 16G10, 16E05.

1 Introduction

For the representation theory of Artinian algebras, the Auslander-Reiten
conjecture (ARC) which is related to generalized Nakayama conjecture (GNC) is
everything. It was proposed by Auslander and Reiten, which says that $M$ is
projective if ${\rm Ext}_{\Lambda}^{i}(M\bigoplus\Lambda,M\bigoplus\Lambda)=0$
for any $i\geq 1$(See [2,3]). Achievements for special cases have been got by
K. R. Fuller, B. Zimmermann-Huisgen, A. Mar$\dot{o}$ti and G. Wilson…(See
[10,15,16]). In general it is still open now. As a special case of Auslander-
Reiten conjecture, Luo and Huang proposed the following Gorenstein projective
conjecture (GPC) in 2008:

Let $\Lambda$ be an Artinian algebra and let $M$ be a Gorenstein projective
module. Then $M$ is projective if and only if ${\rm Ext}_{\Lambda}^{i}(M,M)=0$
for any $i\geq 1$(See [14]).

It is still unknown whether the Auslander-Reiten conjecture is left and right
symmetric. But as we stated Gorenstein projective conjecture is a special
case. So what about the left and right symmetric property of Gorenstein
projective conjecture? In this paper, we will give a positive answer to this
question.

By the definition of Gorenstein projective conjecture, for an algebra
$\Lambda$ the truth of Auslander-Reiten conjecture implies the truth of
Gorenstein projective conjecture. So we can get a large class of algebras
satisfying Gorenstein projective conjecture. It is interesting to ask: Is
there an algebra satisfying Gorenstein projective conjecture while for which
the Auslander-Reiten conjecture is unknown?

Recall that an algebra is called CM-finite (of finite Cohen-Macaulay type) if
there are only finitely many isomorphism classes of indecomposable finitely
generated Gorenstein projective modules. CM-finite algebras are studied by
several authors recently (see [5,6,7,12,13]). Although the Auslander-Reiten
conjecture for this class of algebras is unknown, we will give a positive
answer to the second question above.

The paper is organized as follows:

In Section 2, based on some facts of Gorenstein projective modules, we will
show the symmetric property of Gorenstein projective conjecture. Moreover, an
example is given to show that the condition ’${\rm Ext}_{\Lambda}^{i}(M,M)=0$
for any $i\geq 1$’ in Gorenstein projective conjecture can not be reduced to
’${\rm Ext}_{\Lambda}^{i}(M,M)=0$ for some positive integer $t$ and any $1\leq
i\leq t$ .’

In Section 3, we will prove that CM-finite algebras satisfy the Gorenstein
projective conjecture by showing the Gorenstein projective conjecture holds
for algebras with finite self-orthogonal indecomposable Gorenstein projective
modules (up to isomorphisms).

Throughout the paper, $\Lambda$ is an Artinian algebra and all modules are
finitely generated left $\Lambda$-modules.

2 Symmetric property of Gorenstein projective conjecture

In this section we will show the symmetric property of Gorenstein projective
conjecture. First, we need to recall some notions and lemmas. The following
definition is due to Auslander, Briger, Enochs and Jenda (see [1,8,9]).

Definition 2.1 A module $M$ is called Gorenstein projective if for any $i\geq
1$

(1) ${\rm Ext}_{\Lambda}^{i}(M,\Lambda)=0$

(2) ${\rm Ext}_{\Lambda}^{i}(\mathop{\rm Tr}\nolimits M,\Lambda)=0$

Where $\mathop{\rm Tr}\nolimits M$ denotes the Auslander transpose of $M$.

Let $\dots\rightarrow P_{2}(M)\rightarrow P_{1}(M)\rightarrow M\rightarrow 0$
be a minimal projective resolution of $M$. Denoted by $\Omega^{i}M$ the $i$-th
syzygy of $M$. Dually, one can define $\Omega^{-i}M$. We remark that for any
$i\geq 0$ $\Omega^{i}M$ is a Gorenstein projective if so is $M$. Let
$\mathscr{C}$ be the subcategory of $\mathop{\rm mod}\nolimits\Lambda$
consisting of modules $M$ such that ${\rm Ext}_{\Lambda}^{j}(M,\Lambda)=0$ for
any $j\geq 1$ and $\mathscr{D}$ a subcategory of $\mathop{\rm
mod}\nolimits\Lambda$ consisting of Gorenstein projective modules. We use
$\underline{\mathscr{C}}$ and $\underline{\mathscr{D}}$ to denote the stable
subcategory of $\mathscr{C}$ and $\mathscr{D}$ modulo projectives,
respectively. We recall the following proposition from [1].

Proposition 2.2 (1)
$\Omega:\underline{\mathscr{C}}\rightarrow\underline{\mathscr{C}}$ is a fully
faithful functor .

(2) $\Omega:\underline{\mathscr{D}}\rightarrow\underline{\mathscr{D}}$ is an
equivalence.

(3) $(-)^{*}={\rm Hom}(-,\Lambda):\mathscr{D}\rightarrow\mathscr{D}^{o}$ is a
duality, where $\mathscr{D}^{o}$ denotes the subcategory of Gorenstein
projective right $\Lambda$-modules.

Proof. (1) is a result of Auslander and Bridger. One can get (2) by (1) and
the remark above. (3) is well-known. $\hfill{\square}$

Recall that a module $M$ is called self-orthogonal if ${\rm
Ext}_{\Lambda}^{j}(M,M)=0$ for any $j\geq 1$. The following self-orthogonal
property is essential to the main result in this section.

Lemma 2.3 Let $\Lambda$ be an algebra. Then for any $M\in\mathscr{D}$ and
$i\geq 1$, $M$ is self-orthogonal if and only if $M^{*}$ is self-orthogonal.

Proof. Since $(-)^{*}$ is a duality between $\mathscr{D}$ and
$\mathscr{D}^{o}$, it is enough to show that ${\rm Ext}_{\Lambda}^{i}(M,M)=0$
implies ${\rm Ext}_{\Lambda}^{i}(M^{*},M^{*})=0$.

One can take the following minimal projective resolution of $M$:

$\dots\rightarrow P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0\ \ \ \ (1)$

Applying the functor ${\rm Hom}(-,M)$ to sequence $(1)$ above, since ${\rm
Ext}_{\Lambda}^{i}(M,M)=0$ we get the following exact sequence

$0\rightarrow{\rm Hom}(M,M)\rightarrow{\rm Hom}(P_{0},M)\rightarrow{\rm
Hom}(P_{1},M)\rightarrow\cdots\ \ \ (2)$

On the other hand, applying the functor $(-)^{*}$ to the sequence $(1)$, since
$M\in\mathscr{D}\subseteq\mathscr{C}$ one can show the following exact
sequence

$0\rightarrow
M^{*}\rightarrow{P_{0}}^{*}\rightarrow{P_{1}}^{*}\rightarrow\cdots\ \ \ (3)$

Then by using the functor ${\rm Hom}(M^{*},-)$ on the exact sequences (3), one
has the following exact sequence

$0\rightarrow{\rm Hom}(M^{*},M^{*})\rightarrow{\rm
Hom}(M^{*},{P_{0}}^{*})\rightarrow{\rm
Hom}(M^{*},{P_{1}}^{*})\rightarrow\cdots\ \ \ (4)$

Using Proposition 2.2(3), we get ${\rm
Ext}_{\Lambda}^{i}(M^{*},M^{*})\simeq{\rm Ext}_{\Lambda}^{i}(M,M)=0$ by
comparing sequences (2) with (4). $\hfill{\square}$

Although the symmetric property of Auslander-Reiten conjecture is still
unknown, we are able to show the symmetric properties of Gorenstein projective
conjecture.

Theorem 2.4 Let $\Lambda$ be an algebra and let $\Lambda^{o}$ be the opposite
ring of $\Lambda$. Then $\Lambda$ satisfies the Gorenstein projective
conjecture if and only if $\Lambda^{o}$ satisfies the Gorenstein projective
conjecture.

Proof. $\Rightarrow$ Assume that $N\in\mathscr{D}^{o}$ and ${\rm
Ext}_{\Lambda}^{i}(N,N)=0$ for any $i\geq 1$. By Proposition 2.2, there is a
$M\in\mathscr{D}$ such that $M^{*}\simeq N$. By Lemma 2.3 one gets that ${\rm
Ext}_{\Lambda}^{i}(M,M)=0$. Note that $\Lambda$ satisfies the Gorenstein
projective conjecture, we have $M$ is projective, and hence $N\simeq M^{*}$ is
projective. Conversely, one can formula the proof above. $\hfill{\square}$

Notice that Gorenstein projective conjecture is a special case of Auslander-
Reiten conjecture. It is natural to consider whether the assumption of
Gorenstein projective conjecture can be reduced. In particular, whether can
the condition ’${\rm Ext}_{\Lambda}^{i}(M,M)=0$ for any $i\geq 1$’ in GPC be
reduced to ’${\rm Ext}_{\Lambda}^{i}(M,M)=0$ for some positive integer $t$ and
any $1\leq i\leq t$ ? At the end of this section, we construct an example to
give a negative answer to the question.

Example 2.5 Let $n>t+1$ be a positive integer and let $\Lambda$ be the algebra
generated by the following quiver

$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{n}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\textstyle{n-1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{n-1}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{2}}$$\textstyle{n-2}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{5\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{4\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

modulo the ideal $\\{a_{n}a_{1}=0,a_{i}a_{i+1}=0|1\leq i\leq n-1\\}$. Denoted
by $S(j)$ the simple module according to the dot $j$. Then

(1) $\Lambda$ is a Nakayama self-injective algebra.

(2) $S(j)$ is Gorenstein projective such that ${\rm
Ext}_{\Lambda}^{i}(S(j),S(j))=0$ for $t\geq i\geq 1$ and $1\leq j\leq n$, but
it is not projective.

3 Gorenstein projective conjecture for CM-finite algebras

In this section we try to find a class of algebras which satisfy Gorenstein
projective conjecture and for which the Auslander-Reiten conjecture is
unknown. They are CM-finite algebras. We begin with the following definition
due to Beligiannis

Definition 3.1 An algebra is called CM-finite (of finite Cohen-Macaulay type )
if there are only finite number of indecomposable Gorenstein projective
modules (up to isomorphisms).

Remark 3.2 (1) Algebras of finite representation type or finite global
dimension are CM-finite.

(2) There does exist a CM-finite algebra $\Lambda$ such that $\Lambda$ is of
infinite type and the global dimension of $\Lambda$ is infinite [13].

(3) There does exist a CM-finite algebra which is not Gorenstein [5].

(4)An algebra with a trivial maximal n-orthogonal subcategory for some
positive integer $n$ is CM-finite [11].

Let $\mathscr{C}$, $\mathscr{D}$ and $\mathscr{D}^{o}$ be as in Section 2. The
following lemma partly from [1] plays an important role in the proof of the
main results.

Lemma 3.3 For any $M\in\mathscr{C}$ and $N\in\mathop{\rm
mod}\nolimits\Lambda$, then ${\rm Ext}_{\Lambda}^{1}(M,N)\simeq\underline{{\rm
Hom}}_{\Lambda}(\Omega^{1}M,N)$ and hence ${\rm
Ext}_{\Lambda}^{i}(M,N)\simeq\underline{{\rm Hom}}_{\Lambda}(\Omega^{i}M,N)$
for any $i\geq 1$.

Proof. The first assertion is a result of Auslander and Bridger. For the
second one, the case $i=1$ is clear. We only need to show the case $i\geq 2$.
Taking a minimal projective resolution of $M$, one gets ${\rm
Ext}_{\Lambda}^{i}(M,N)\simeq{\rm Ext}_{\Lambda}^{1}(\Omega^{i-1}M,N)$ for any
$i\geq 2$. Notice that $M\in\mathscr{C}$, by Proposition 2.2 one can show
$\Omega^{i-1}M\in\mathscr{C}$. Using the first assertion, we are done.
$\hfill{\square}$.

The following proposition gives a connection between the self-orthogonal
property of $M$ and that of $\Omega^{i}M$ for any $i\geq 0$.

Proposition 3.4  Let $M\in\mathscr{C}$ $(\mathscr{D})$. Then

(1) $\Omega^{i}M$ is self-orthogonal in $\mathscr{C}$ $(\mathscr{D})$ for any
$i\geq 0$ if $M$ is self-orthogonal.

(2) If $M\in\mathscr{D}$ is self-orthogonal, then ${\rm Tr}M$ is self-
orthogonal in $\mathscr{D}^{o}.$

Proof. (1) For the case $i=0$, there is nothing to prove. By Proposition 2.2,
we only need to prove the case $i=1$, that is, ${\rm Ext}_{\Lambda}^{j}(\Omega
M,\Omega M)=0$ for any $j\geq 1$. One gets ${\rm Ext}_{\Lambda}^{j}(\Omega
M,\Omega M)\simeq\underline{{\rm Hom}}_{\Lambda}(\Omega^{j+1}M,\Omega
M)\simeq\underline{{\rm Hom}}_{\Lambda}(\Omega^{j}M,M)$ by Proposition 2.2 and
Lemma 3.3. Using the second equation of Lemma 3.3, one can show
$\underline{{\rm Hom}}_{\Lambda}(\Omega^{j}M,M)\simeq{\rm
Ext}_{\Lambda}^{j}(M,M)=0$ since M is self-orthogonal.

(2) Taking a minimal projective resolution of $M$, it is not difficult to show
that ${\rm Tr}M\simeq(\Omega^{2}M)^{*}$ since $M\in\mathscr{D}$. By
Propositions 2.2 and 3.4(1), $\Omega^{2}M$ is also self-orthogonal in
$\mathscr{D}$. Then by Lemma 2.3 and Proposition 2.2 one gets the assertion.
$\hfill{\square}$

The following proposition is crucial to the main results.

Proposition 3.5 Let $\Lambda$ be an algebra with only finite (up to
isomorphism) self-orthogonal indecomposable modules in $\mathscr{D}$
($\mathscr{C}$) and let $M$ be a self-orthogonal indecomposable module in
$\mathscr{D}$ ($\mathscr{C}$). Then $M$ is projective.

Proof. Denoted by $\\{M_{1},M_{2},...,M_{t}\\}$ the complete set of non-
isomorphic self-orthogonal indecomposable modules in $\mathscr{D}$
($\mathscr{C}$). Then $M\simeq M_{i}$ for some $1\leq i\leq t$.

Suppose that M is not projective. Then by Proposition 2.2 and Proposition 3.4,
we have the following set of self-orthogonal indecomposable modules
$\mathcal{S}=\\{\Omega^{i}M|1\leq i\\}$ in $\mathscr{D}$ ($\mathscr{C}$).

We claim that there are two modules $\Omega^{i}M,\Omega^{j}M$ in $\mathcal{S}$
such that $\Omega^{i}M\simeq\Omega^{j}M$ for some $i<j$. Otherwise, one gets
infinite number of non-isomorphic self-orthogonal indecomposable modules in
$\mathscr{D}$ ($\mathscr{C}$), a contradiction. Again by Proposition 2.2, one
gets $M\simeq\Omega^{j-i}M$.

Considering the following exact sequence $0\rightarrow\Omega^{j-i}M\rightarrow
P\rightarrow\Omega^{j-i-1}M\rightarrow 0$, we will show ${\rm
Ext}_{\Lambda}^{1}(\Omega^{j-i-1}M,\Omega^{j-i}M)=0$. Since
$\Omega^{j-i-1}M\in\mathscr{D}$ ($\mathscr{C}$) and $M\simeq\Omega^{j-i}M$, we
get ${\rm
Ext}_{\Lambda}^{1}(\Omega^{j-i-1}M,\Omega^{j-i}M)\simeq\underline{{\rm
Hom}}_{\Lambda}(\Omega^{j-i}M,M)\simeq{\rm Ext}_{\Lambda}^{j-i}(M,M)=0$ by
Proposition 2.2 and Lemma 3.3, and hence $M$ is projective, a contradiction.
The assertion holds true.$\hfill\square$

Now we are in the position to show the main result of this section.

Theorem 3.6 Let $\Lambda$ be CM-finite. Then $\Lambda$ satisfies Gorenstein
projective conjecture.

Proof.  Since $\Lambda$ is CM-finite, then there are only finite (up to
isomorphisms) indecomposable modules in $\mathscr{D}$. One can show the result
by Proposition 3.5. $\hfill{\square}$

Although the Auslander-Reiten conjecture for CM-finite algebras is unknown
now, we have the following

Proposition 3.7 Let $\Lambda$ be a CM-finite algebra and let $M$ be a
$\Lambda$-module satisfying ${\rm Ext}_{\Lambda}^{i}(M,M\bigoplus\Lambda)=0$
for any $i\geq 1$. Then the following are equivalent.

(1) $M$ is projective.

(2) $M$ is Gorenstein projective.

Proof. $(1)\Rightarrow(2)$ is trivial. The converse follows from Theorem 3.6.
$\hfill{\square}$

We end this section with two open questions related to this paper.

Question 1 Does the Gorenstein projective conjecture hold for virtually
Gorenstein algebras (see [5])?

Question 2 Does the Auslander-Reiten conjecture hold for CM-finite algebras?

Acknowledgements Part of the paper was finished when the author stayed in
University of Bielefeld with the support of DAAD Fellowship. The author would
like to thank people in Bielefeld for their help. He also wants to thank Prof.
Xiaowu Chen, Prof. Zhaoyong Huang, Prof. Shengyong Pan and Prof. Changchang Xi
for useful suggestions. The research of the author is supported by NSFC(Grant
No.11101217) and NSF for Colleges and Universities in Jiangsu Province (Grant
No.11KJB110007).

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