###### Question 3.23.

Could a heart be isolated in its fan (assuming $\dim\Lambda_{\mathbb{R}}\geq
2$), i.e. can $\Sigma(\mathsf{H})$ have only $C(\mathsf{H})$ and
$C(\mathsf{H}[1])=-C(\mathsf{H})$ as maximal cones? Algebraically: can a heart
have no non-trivial torsion pairs?

This question is inspired by the convexity of the support of the heart fan in
all our examples.

###### Example 3.24 (coherent sheaves).

Let $X$ be a connected, smooth and projective variety,
$\mathsf{H}=\operatorname{\mathsf{coh}}(X)$ the category of coherent sheaves
on $X$ and $\Lambda=N(X)$ the numerical Grothendieck group of $X$ as in
Example 1.9. Since $\mathsf{H}$ is Noetherian, all torsion pairs in
$\mathsf{H}$ are of the form $(\mathsf{T},\mathsf{T}^{\perp})$ where
$\mathsf{T}\subseteq\mathsf{H}$ is an additive subcategory closed under
quotients and extensions. If $\mathsf{T}\neq 0$ then it must contain the
skyscraper sheaf $\mathcal{O}_{p}$ of a point $p\in X$, hence the effective
cone of the tilted heart $\mathsf{K}=\mathsf{T}^{\perp}[1]*\mathsf{T}$
contains the point class $[\mathcal{O}_{p}]\in E(\mathsf{K})$ (as $X$ is
connected, all points have the same class in $N(X)$.) Chose an orthonormal
basis for $\Lambda_{\mathbb{R}}=N(X)\otimes\mathbb{R}$ containing the point
class and denote its dual basis vector by
$v_{0}\in{\Lambda_{\mathbb{R}}^{*}}$. Then $C(\mathsf{K})$ is contained in the
half-space of ${\Lambda_{\mathbb{R}}^{*}}$ given by $v_{0}\geq 0$. With
$C(\mathsf{H}[1])=-C(\mathsf{H})$ being a ray by Example 1.9, this shows that
the support of the heart fan $\Sigma(X)$ is contained in a half-space.

A line bundle on $X$ induces an exact autoequivalence of
$\operatorname{\mathsf{coh}}(X)$ and hence, according to Remark 3.7, an
automorphism of the fan $\Sigma(X)$. In this way, the quotient group
$\mathrm{Pic}(X)/\mathrm{Pic}^{0}(X)$ acts faithfully on the fan. For example,
the action of $\mathrm{Pic}(\mathbb{P}^{1})$ on $\Sigma(\mathbb{P}^{1})$ fixes
the two rays $\pm C(\mathsf{H})$ and acts transitively on the remaining
maximal cones: twisting by $\mathcal{O}(1)$ maps $C(\mathsf{K}_{i})\mapsto
C(\mathsf{K}_{i+1})$ for all $i\in\mathbb{Z}$, using the notation of Example
3.19.

##  4\. $g$-fans

In the remaining sections, we compare geometric structures attached to
algebraic categories to our heart fans. Let $A$ be a finite-dimensional
algebra over an algebraically closed field ${\mathbf{k}}$. We write
$\Sigma(A)$ for the heart fan of the abelian category of finite-dimensional
$A$-modules $\mathsf{mod}(A)$ over $\Lambda\coloneqq K(A)\cong\mathbb{Z}^{n}$.
In order to define the $g$-fan of $A$ we require some silting theory.

###  4.1. Projective-simple duality

Without loss of generality, we may assume that $A$ is basic, i.e. $A\cong
P(1)\oplus\cdots\oplus P(n)$, where the $P(i)$ are a complete set of pairwise
non-isomorphic indecomposable projective $A$-modules. We consider two Hom-
finite triangulated categories associated to $A$ together with their
Grothendieck groups:

$\displaystyle\mathsf{D}^{p}$
$\displaystyle\coloneqq\mathsf{K}^{b}(\mathsf{proj}(A))$ the perfect derived
category of $A$, $\displaystyle\mathsf{D}^{b}$
$\displaystyle\coloneqq\mathsf{D}^{b}(\mathsf{mod}(A))$ the bounded derived
category of $A$, $\displaystyle K(A)$ $\displaystyle\coloneqq
K(\mathsf{D}^{b})=K(\mathsf{mod}(A))$ the Grothendieck group of $A$,
$\displaystyle K^{\text{split}}(A)$ $\displaystyle\coloneqq
K(\mathsf{D}^{p})=K(\mathsf{proj}(A))$ $\displaystyle\text{the split
Grothendieck group of $A$}.$

The classes $[P(1)],\ldots,[P(n)]$ form a basis of $K^{\text{split}}(A)$ and
the classes of simple modules $[S(1)],\ldots,[S(n)]$ form a basis of $K(A)$.
These bases are dual under the non-degenerate pairing

$K^{\text{split}}(A)\times
K(A)\to\mathbb{Z},\quad(P,M)=\dim{\mathrm{Hom}}_{A}(P_{i},M).$

Having set $\Lambda\coloneqq K(A)$, we identify
$\Lambda^{*}=K^{\text{split}}(A)$.

###  4.2. Two-term (pre)silting objects

Consider the full subcategory of two-term complexes of projective $A$-modules,
$\mathsf{C}\coloneqq\mathsf{K}^{[-1,0]}(\mathsf{proj}(A))=\mathsf{add}(A)*\mathsf{add}(A)[1]\subset\mathsf{D}^{p}$.
It is an extriangulated category with $K(\mathsf{C})=K(\mathsf{D}^{p})$. A
complex $P_{\bullet}\in\mathsf{C}$ is _presilting_ if
${\mathrm{Hom}}_{\mathsf{D}^{p}}(P_{\bullet},P_{\bullet}[{>}0])=0$. If
additionally $\mathsf{thick}_{\mathsf{D}^{p}}(P_{\bullet})=\mathsf{D}^{p}$, or
equivalently $|P_{\bullet}|=n=\mathrm{rk}\,K^{\text{split}}(A)$, then
$P_{\bullet}$ is called _silting_. Sometimes these complexes are called ‘two-
term (pre)silting’ and $\mathsf{C}$ is called the ‘two-term category’. We drop
the adjective but we stress that we are working in $\mathsf{C}$ throughout, so
that our complexes have non-zero terms in homological degrees $1$ and $0$
only.

###  4.3. $g$-vectors, $g$-cones and the $g$-fan

The _$g$ -vector_ of $P_{\bullet}\in\mathsf{C}$ is its class $[P_{\bullet}]\in
K(\mathsf{C})$; writing $P_{\bullet}=(P_{1}\to P_{0})$ and decomposing
$P_{0}=\bigoplus_{i=1}^{n}P(i)^{a_{i}}$ and
$P_{1}=\bigoplus_{i=1}^{n}P(i)^{b_{i}}$, this is the classical
$g^{P_{\bullet}}=(a_{1}-b_{1},\ldots,a_{n}-b_{n})\in\mathbb{Z}^{n}\cong
K(\mathsf{C})$, identified using the basis of indecomposable projective
modules.

For $P_{\bullet}\in\mathsf{C}$ presilting with indecomposable summands
$Q^{1}_{\bullet},\ldots,Q^{t}_{\bullet}$, the $g$-vectors
$[Q^{1}_{\bullet}],\ldots,[Q^{t}_{\bullet}]$ are linearly independent by [14]
and generate the _$g$ -cone_

$C(P_{\bullet})=\\{a_{1}[Q^{1}_{\bullet}]+\cdots+a_{t}[Q^{t}_{\bullet}]\mid
a_{i}\geq
0\\}\subset{\Lambda_{\mathbb{R}}^{*}}=K(\mathsf{C})\otimes\mathbb{R}.$

They form a basis if $t=n$, i.e. if $P_{\bullet}$ is silting.

The _$g$ -fan_ of $A$ is $\Sigma_{A}^{g}\coloneqq\\{C(P_{\bullet})\mid
P_{\bullet}\in\mathsf{C}\text{ is presilting}\\}$. We show that it coincides
with the subfan of the heart $\Sigma(A)$ consisting of full cones, i.e. those
coming from algebraic hearts, and their faces:

$\Sigma(A)^{\mathrm{full}}\coloneqq\\{\kappa\in\Sigma(A)\mid\exists\sigma\in\Sigma(A)\text{
full},\kappa\preceq\sigma\\}=\\{C(\mathsf{K}/\SS)\in\Sigma(A)\mid\mathsf{K}\text{
algebraic}\\}.$

By Proposition 1.10, a full cone is simplicial; the proposition applies in
view of Lemma 3.11. Recall that $\Sigma(A)$ was defined as a dual face fan but
as $\Sigma(A)^{\mathrm{full}}$ is generated by dual cones of polyhedral cones
$E(\mathsf{K})$, all faces of $C(\mathsf{K})$ are dual faces.

######  Proposition 4.1.

The $g$-fan of the finite-dimensional algebra $A$ is
$\Sigma_{A}^{g}=\Sigma(A)^{\mathrm{full}}$.

###### Remark 4.2.

According to [5, Thm. 4.7], the $g$-fan $\Sigma_{A}^{g}$ of a finite-
dimensional algebra $A$ is complete if and only if it is finite, which happens
if and only if there are finitely many silting complexes in $\mathsf{C}$. By
[22, Prop. 3.1], this implies that all hearts $\mathsf{K}$ with
$\mathsf{H}=\mathsf{mod}(A)\leq\mathsf{K}\leq\mathsf{H}[1]$ are algebraic and
$\Sigma_{A}^{g}=\Sigma(A)$; cf. Corollary 3.9.

In Theorem 3.8, we showed that the heart fan of any length category is
complete. Thus we can think of the heart fan of $\mathsf{mod}(A)$ as a natural
completion of the $g$-fan. Moreover, heart fans can be seen as generalisations
of $g$-fans to contexts without silting theory, like tube categories.

To prove the proposition, we need to describe the connection of silting theory
to Serre subcategories via simple-minded collections and $c$-vectors.

###  4.4. Koenig-Yang correspondences and Serre subcategories

In [21], Steffen Koenig and Dong Yang obtain correspondences between algebraic
hearts $\mathsf{K}$ in $\mathsf{D}^{b}$ and silting subcategories of
$\mathsf{D}^{p}$ which restrict to two-term versions by [12]. Recall that
$\\{X_{1},\ldots,X_{r}\\}$ is a _simple-minded collection_ in a
${\mathbf{k}}$-linear triangulated category if (a)
${\mathrm{Hom}}^{<0}(X_{i},X_{j})=0$ for all $i,j$ and (b)
${\mathrm{Hom}}(X_{i},X_{i})={\mathbf{k}}$, ${\mathrm{Hom}}(X_{i},X_{j})=0$
for $i\neq j$ and (c) the collection generates the triangulated category; see
[21, §3.2].

######  Proposition 4.3 ([21, Thm. 6.1] & [12, Cor. 4.3]).

Let $A$ be a finite-dimensional algebra and $\mathsf{H}=\mathsf{mod}(A)$.
There are bijections between the following sets:

  1. (1)

silting objects in $\mathsf{C}$, up to additive equivalence;

  2. (2)

algebraic hearts $\mathsf{K}$ with
$\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$;

  3. (3)

simple-minded collections in $\mathsf{H}[1]*\mathsf{H}$.

The map from (2) to (3) sends $\mathsf{K}\mapsto\\{\text{simple objects of
}\mathsf{K}\\}$; the inverse map from (3) to (2) sends a simple-minded
collection $\\{X_{1},\ldots,X_{n}\\}\mapsto{\langle
X_{1},\ldots,X_{n}\rangle}$, its extension closure in $\mathsf{D}^{b}$. The
objects in (1) and (3) obey a _generalised projective-simple duality_ : let
$P_{\bullet}=\bigoplus_{i=1}^{n}Q^{i}_{\bullet}$ be a silting object and
$\\{X_{1},\ldots,X_{n}\\}$ the corresponding simple-minded collection, then

(2)
${\mathrm{Hom}}(Q^{i}_{\bullet},X_{i})={\mathbf{k}}\qquad\text{and}\qquad{\mathrm{Hom}}(Q^{i}_{\bullet},X_{j})=0\text{
for }i\neq j.$

Moreover, $\mathsf{K}\simeq\mathsf{mod}({\mathrm{End}}(P_{\bullet}))$. In
particular, by Proposition 4.3 and [17, Prop. 5.3], there are bijections

$\\{\text{direct summands of }P_{\bullet}\\}\stackrel{{\scriptstyle
1-1}}{{\longleftrightarrow}}\\{\text{subsets of
}\\{X_{1},\ldots,X_{n}\\}\\}\stackrel{{\scriptstyle
1-1}}{{\longleftrightarrow}}\\{\text{Serre subcategories of }\mathsf{K}\\}.$

###  4.5. c-vectors

Following [4, p. 5035], given a simple-minded collection
$\\{X_{1},\ldots,X_{n}\\}$, the _$c$ -vector_ of $X_{i}$ is its class
$[X_{i}]\in K(A)$. Expressed in the basis of $K(A)$ of simple modules, this
gives the classical $c^{X_{i}}\in\mathbb{Z}^{n}$. Moreover, since
$\\{X_{1},\ldots,X_{n}\\}$ are a complete list of non-isomorphic simple
objects for an algebraic heart $\mathsf{K}$ in $\mathsf{D}^{b}$, the
$c$-vectors $\\{[X_{1}],\ldots,[X_{n}]\\}$ form a basis of
$\Lambda_{\mathbb{R}}$. By generalised simple-projective duality
$([Q^{i}_{\bullet}],[X_{j}])=\delta_{ij}$, and the $c$-vector basis is dual to
the $g$-vector basis of $\Lambda^{*}$ coming from the silting object
$P_{\bullet}=\bigoplus_{i=1}^{n}Q^{i}_{\bullet}$ corresponding to
$\\{X_{1},\ldots,X_{n}\\}$.

We now assemble the ingredients from silting theory to prove Proposition 4.1.

###### Proof of Proposition 4.1.

Let $\mathsf{K}$ be an algebraic heart with
$\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$, write
$P_{\bullet}=\bigoplus_{i=1}^{n}Q^{i}_{\bullet}$ for the corresponding two-
term silting complex in $\mathsf{C}$, and $\\{X_{1},\ldots,X_{n}\\}$ for the
corresponding two-term simple-minded collection in $\mathsf{H}[1]*\mathsf{H}$
under the bijections in Proposition 4.3. Then $C(\mathsf{K})=C(P_{\bullet})$
follows from the definitions and the duality
$([Q^{i}_{\bullet}],[X_{j}])=\delta_{ij}$.

Now, suppose $\SS\subseteq\mathsf{K}$ is a Serre subcategory. By [17, Prop.
5.3], we can re-order the two-term simple-minded collection
$\\{X_{1},\ldots,X_{n}\\}$ so that $\SS={\langle X_{1},\ldots,X_{t}\rangle}$
for some $1\leq t\leq n$. We order the summands of the corresponding two-term
silting complex $P_{\bullet}$ accordingly and write
$Q_{\bullet}=\bigoplus_{i=t+1}^{n}Q^{i}_{\bullet}$. It follows immediately
from the generalised projective-simple duality (2) that
$C(\mathsf{K}/\SS)=C(Q_{\bullet})$ using $C(\mathsf{K}/\SS)=C(\mathsf{K})\cap
E(\SS)^{\perp}$ from Lemma 2.6. ∎

##  5\. King semistability and wall-and-chamber structures

The heart fan $\Sigma(\mathsf{H})$ of an algebraic abelian category
$\mathsf{H}$ can be described in terms of Alastair King’s notion of
semistability [20], and we relate the heart fan to the wall-and-chamber
structure of [11]. Throughout, $\Lambda=K(\mathsf{H})$ and
$\mathsf{D}=\mathsf{D}^{b}(\mathsf{H})$.

###### Definition 5.1.

Let $\mathsf{H}$ be an abelian category and
$v\in{\Lambda_{\mathbb{R}}^{*}}={\mathrm{Hom}}(K(\mathsf{H}),\mathbb{R})$.

  1. (1)

$h\in\mathsf{H}$ is _$v$ -semistable_ if $v(h)=0$ and $v(h^{\prime})\leq 0$
for all subobjects $h^{\prime}\subseteq h$.

  2. (2)

$h\in\mathsf{H}$ is _$v$ -stable_ if it is $v$-semistable and if
$h^{\prime}\hookrightarrow h$, $v(h^{\prime})=0$ implies $h^{\prime}=0$ or
$h^{\prime}=h$.

  3. (3)

$\mathsf{H}^{\mathsf{ss}}(v)$ denotes the full subcategory of $v$-semistable
objects in $\mathsf{H}$.

###### Remark 5.2.

This is [20, Def. 1.1] although for compatibility with [9] we have reversed
the sign convention which in [20] was: $h$ is $v$-semistable $\iff$ $v(h)=0$
and $v(h^{\prime})\geq 0$ for all $h^{\prime}\subseteq h$.

The subcategory $\mathsf{H}^{\mathsf{ss}}(v)$ is a wide subcategory of
$\mathsf{H}$, i.e. closed under extensions, kernels and cokernels. In
particular, it is an abelian category.

The next proposition says that in the heart fan of an _algebraic_ abelian
category $\mathsf{H}$, among all face pairs describing the same cone there is
a unique one closest to the reference heart $\mathsf{H}$. Recall from
Definition 3.3 that a face pair $(\mathsf{K},\SS)$ consists of a heart
$\mathsf{K}$ with $\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$ and a face
subcategory $\SS\in\mathrm{Serre}_{\Lambda\\!}(\mathsf{K})$. By Theorem 3.8(1)
the heart fan is complete, i.e.
$\lvert\Sigma(\mathsf{H})\rvert={\Lambda_{\mathbb{R}}^{*}}$.

######  Proposition 5.3.

Let $\mathsf{H}$ be an algebraic abelian category and $\Lambda=K(\mathsf{H})$
its Grothendieck group.

For each $v\in{\Lambda_{\mathbb{R}}^{*}}$ there is a unique face pair
$(\mathsf{K}^{v},\SS^{v})$ in $\mathsf{D}=\mathsf{D}^{b}(\mathsf{H})$ such
that

  1. (1)

$C(\mathsf{K}^{v}\\!/\SS^{v})$ is the minimal dual face containing $v$,

  2. (2)

$\mathsf{K}^{v}$ is minimal amongst hearts
$\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$ with $v\in C(\mathsf{K})$ and

  3. (3)

$\SS^{v}=\mathsf{H}^{\mathsf{ss}}(v)\subseteq\mathsf{H}\cap\mathsf{K}^{v}$ is
the subcategory of $v$-semistable objects in $\mathsf{H}$.

Moreover, if $w\in C^{\circ}(\mathsf{K}^{v}\\!/\SS^{v})$ then
$(\mathsf{K}^{w},\SS^{w})=(\mathsf{K}^{v},\SS^{v})$, and there is a
distinguished choice of face pair for each cone in $\Sigma(\mathsf{H})$.

We call face pairs of the form $(\mathsf{K}^{v},\SS^{v})$ _distinguished_.
Choosing the distinguished pair determines a section of the map
$(\mathsf{K},\SS)\mapsto C(\mathsf{K}/\SS)$ from face pairs to cones in the
heart fan. While $\SS^{v}\subseteq\mathsf{K}^{v}$ is a face subcategory in the
sense of Definition 2.2, here the ambient abelian category $\mathsf{K}^{v}$
also depends on $v$.

###### Proof.

Let $\mathsf{H}=\mathsf{T}_{\\!v}*\mathsf{F}_{\\!v}$ be the torsion pair of
Lemma 3.10, i.e. $\mathsf{T}_{\\!v}\coloneqq\\{h\in\mathsf{H}\mid
v(h^{\prime\prime})\geq 0~{}\forall h\twoheadrightarrow h^{\prime\prime}\\}$
and $\mathsf{F}_{\\!v}\coloneqq\\{h\in\mathsf{H}\mid v(h^{\prime})<0~{}\forall
0\neq h^{\prime}\hookrightarrow h\\}.$ Consider the tilted heart
$\mathsf{K}^{v}\coloneqq\mathsf{F}_{\\!v}[1]*\mathsf{T}_{\\!v}$ and the face
subcategory $\SS^{v}\coloneqq\\{t\in\mathsf{T}_{\\!v}\mid
v(t)=0\\}\subseteq\mathsf{K}^{v}$. Then $C(\mathsf{K}^{v}\\!/\SS^{v})$ is the
minimal dual face of $C(\mathsf{K}^{v})$ containing $v$.

If $v\in C(\mathsf{K})$ with $\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$ then
$\mathsf{K}=\mathsf{F}[1]*\mathsf{T}$ for a torsion pair
$\mathsf{H}=\mathsf{T}*\mathsf{F}$ where $v|_{\mathsf{T}}\geq 0$. As
$\mathsf{T}$ is closed under quotientes, we get
$\mathsf{T}\subseteq\mathsf{T}_{\\!v}$, hence
$\mathsf{F}_{\\!v}\subseteq\mathsf{F}$, i.e. $\mathsf{K}^{v}\leq\mathsf{K}$.
Thus $\mathsf{K}^{v}$ is minimal amongst hearts
$\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$ with $v\in C(\mathsf{K})$.
Moreover, $v$ indeed determines $\mathsf{K}^{v}$ uniquely.

Now we show that the subcategory
$\SS^{v}\subseteq\mathsf{T}_{\\!v}\subseteq\mathsf{H}$ is the subcategory of
$v$-semistable objects. Suppose $h\in\SS^{v}$. If $h^{\prime}\hookrightarrow
h$ is a subobject then $0\leq
v(h/h^{\prime})=v(h)-v(h^{\prime})=-v(h^{\prime})$ by $h\in\mathsf{T}_{\\!v}$,
and hence $h\in\mathsf{H}^{\mathsf{ss}}(v)$. Conversely if
$h\in\mathsf{H}^{\mathsf{ss}}(v)$ then it has a short exact decomposition
sequence $0\to t\to h\to f\to 0$ in $\mathsf{H}$ with $t\in\mathsf{T}_{\\!v}$
and $f\in\mathsf{F}_{\\!v}$. Since $h$ is semistable, we have $v(t)\leq 0$ for
its subobject $t$. On the other hand, $v(t)\geq 0$ for all objects of
$\mathsf{T}_{\\!v}$. Therefore $v(t)=0$ and $v(f)=v(k)-v(t)=0$ too, so that
$f=0$ by definition of $\mathsf{F}_{\\!v}$. Hence $h=t\in\SS^{v}$.

Finally, if $C(\mathsf{K}^{v}\\!/\SS^{v})$ is the minimal dual face containing
$w$ then $\mathsf{K}^{w}=\mathsf{K}^{v}$ by minimality. Thus $\SS^{w}=\SS^{v}$
too, as each is the face subcategory corresponding to the dual face
$C(\mathsf{K}^{v}\\!/\SS^{v})$. ∎

###### Remark 5.4.

If the minimal dual face containing both $v$ and $w$ is $C(\mathsf{K}/\SS)$
for some $\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$ then the subcategories of
semistable objects $\mathsf{H}^{\mathsf{ss}}(v)=\mathsf{H}^{\mathsf{ss}}(w)$
are the same. The converse does not hold in general; for example
$\mathsf{H}^{\mathsf{ss}}(v)=0$ for any $v$ in the interior of a full cone in
$\Sigma(\mathsf{H})$.

###### Example 5.5 (distinguished face pair in Kronecker quiver heart fan).

Consider the heart fan of the module category over the Kronecker quiver,
$\mathsf{H}=\mathsf{mod}({\mathbf{k}}(\leavevmode\hbox to28pt{\vbox
to9.7pt{\pgfpicture\makeatletter\hbox{\hskip 13.99997pt\lower-7.47481pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.99997pt}{-5.2222pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
5.5pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\bullet}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
5.5pt\hfil&\hfil\hskip 17.49998pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\bullet}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
5.5pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-0.56943pt}\pgfsys@lineto{5.40005pt}{-0.56943pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-0.56943pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-5.79997pt}{-4.87497pt}\pgfsys@lineto{5.40005pt}{-4.87497pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.60004pt}{-4.87497pt}\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}}))$. This fan contains
a limiting ray denoted $C(\mathsf{K}_{\infty})$ in Example 3.18. This ray is
the dual cone of the infinitely many hearts of $\mathsf{D}^{b}(\mathsf{H})$
listed in Example 1.8, among them the geometric heart
$\operatorname{\mathsf{coh}}(\mathbb{P}^{1})$.

If $0\neq v\in C(\mathsf{K}_{\infty})$ then the distinguished face pair
$(\mathsf{K}^{v},\SS^{v})$ of Proposition 5.3 is
$\operatorname{\mathsf{coh}}(\mathbb{P}^{1})$ with its torsion subcategory:
note that the mixed and reversed geometric hearts are positive tilts of
$\operatorname{\mathsf{coh}}(\mathbb{P}^{1})$, so this heart is indeed minimal
amongst $\mathsf{K}$ with $\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$ and
$v\in C(\mathsf{K})$.

######  Lemma 5.6.

An object of $\mathsf{H}^{\mathsf{ss}}(v)$ is stable if and only it is simple
as an object of $\mathsf{K}^{v}$.

###### Proof.

Suppose $h\in\mathsf{H}^{\mathsf{ss}}(v)$. If $0\to h^{\prime}\to h\to
h/h^{\prime}\to 0$ is short exact in $\mathsf{K}^{v}$ then, since
$\mathsf{H}^{\mathsf{ss}}(v)=\SS^{v}$ is a Serre subcategory of
$\mathsf{K}^{v}$, both $h^{\prime}$ and $h/h^{\prime}$ are in
$\mathsf{H}^{\mathsf{ss}}(v)$ so the sequence is also short exact in
$\mathsf{H}$. It follows that if $h$ is stable it is simple in
$\mathsf{K}^{v}$.

Conversely, if $h^{\prime\prime}$ is a subobject of $h$ in $\mathsf{H}$ with
$v(h^{\prime\prime})=0$ then there is a short exact sequence $0\to
h^{\prime\prime}\to h\to h/h^{\prime\prime}\to 0$ in $\mathsf{H}$ with all
terms in $\mathsf{H}^{\mathsf{ss}}(v)$, which is therefore also short exact in
$\mathsf{K}^{v}$. Thus if $h$ is simple in $\mathsf{K}^{v}$ then it is stable.
∎

###  5.1. Wall-and-chamber structures

Defined in [11] for finite-dimensional algebras, we explain how to reconstruct
walls and chambers from the heart fan $\Sigma(\mathsf{H})$ of an algebraic
abelian category $\mathsf{H}$. The _stability space_ of an object
$M\in\mathsf{H}$ is [11, Def. 3.2]

$\mathcal{D}(M)\coloneqq\\{v\in{\Lambda_{\mathbb{R}}^{*}}\mid
M\in\mathsf{H}^{\mathsf{ss}}(v)\\}.$

This is a _wall_ if
$\operatorname{{\mathrm{codim}}}_{\Lambda_{\mathbb{R}}^{*}}\mathcal{D}(M)=1$.
We write $\mathcal{D}\coloneqq\bigcup_{M\neq 0}\mathcal{D}(M)$ for the union
of stability spaces of non-zero objects. A _chamber_ is a connected component
of the complement
${\Lambda_{\mathbb{R}}^{*}}\backslash\smash{\overline{\mathcal{D}}}$. The
_wall-and-chamber_ structure is the set of stability spaces $\mathcal{D}(M)$
for indecomposable modules $M$ together with the set of chambers.

######  Proposition 5.7.

The stability space $\mathcal{D}(M)$ of $M\in\mathsf{H}$ is a rational,
polyhedral cone. Moreover,

$\Delta(M)\coloneqq\\{C(\mathsf{K}/\SS)\mid(\mathsf{K},\SS)\text{
distinguished face pair with }M\in\SS\\}$

is a set of cones in ${\Lambda_{\mathbb{R}}^{*}}$ forming a dual face subfan
of $\Sigma(\mathsf{H})$ with support $\lvert\Delta(M)\rvert=\mathcal{D}(M)$.

The chambers are the open cones $C^{\circ}(\mathsf{K})$ for the algebraic
hearts $\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$. In particular each chamber
is the interior of a simplicial cone.

###### Remark 5.8.

This provides an alternative proof of [11, Prop. 3.15] and [5, Thm. 13.7] for
$\mathsf{H}=\mathsf{mod}(A)$ for a finite-dimensional algebra $A$ which
combined show that the chambers are in natural bijection with $\tau$-tilting
pairs, i.e. with algebraic hearts in the interval between $\mathsf{H}$ and
$\mathsf{H}[1]$. Note that $\Delta(M)\subseteq[M]^{\perp}$ but need not
contain all dual faces in $\Sigma(A)$ in this hyperplane; it only contains
those in heart cones $C(\mathsf{K})$ where $M\in\mathsf{K}$.

###### Proof.

_Stability spaces are rational, polyhedral cones:_ The set
$S\coloneqq\\{[M^{\prime}]\mid M^{\prime}\subseteq M\\}\subset K(\mathsf{H})$
of classes of subobjects of $M$ is finite because $\mathsf{H}$ is a length
category, and therefore it generates a rational, polyhedral cone
$E(S)\subset\Lambda_{\mathbb{R}}$. Hence its dual cone
$C(S)\subset{\Lambda_{\mathbb{R}}^{*}}$ is rational, polyhedral as well (this
is Gordon’s lemma in convex geometry), as is
$\mathcal{D}(M)=-C(S)\cap[M]^{\perp}$.

_$\Delta(M)$ is a dual face subfan:_ Let $(\mathsf{K},\SS)$ be a distinguished
face pair with $M\in\SS$. Then $M$ is $v$-semistable for any $v\in
C^{\circ}(\mathsf{K}/\SS)$ by Proposition 5.3; as semistability is a closed
condition, this holds for any $v\in C(\mathsf{K}/\SS)$. Hence every dual face
of $C(\mathsf{K}/\SS)$ in $\Sigma(\mathsf{H})$ is also in $\Delta(M)$. Because
$\Delta(M)\subseteq\Sigma(\mathsf{H})$ is a subset anyway, it is a dual face
subfan.

_Stability space as support:_ The support
$\lvert\Delta(M)\rvert\subseteq{\Lambda_{\mathbb{R}}^{*}}$ is the union of all
cones in $\Delta(M)$. We have $\mathcal{D}(M)\supseteq\lvert\Delta(M)\rvert$
by the previous step. Conversely, suppose $v\in\mathcal{D}(M)$. Then $M$ is
$v$-semistable and, as $\Sigma(\mathsf{H})$ is complete, there is a minimal
dual face $C(\mathsf{K}/\SS)$ containing $v$ where $(\mathsf{K},\SS)$ is a
distinguished face pair with $M\in\SS=\mathsf{H}^{\mathsf{ss}}(v)$ by
Proposition 5.3, and hence $\mathcal{D}(M)\subseteq\lvert\Delta(M)\rvert$.

_Chambers are interiors of maximal cones:_ Suppose $\mathsf{K}$ is an
algebraic heart with $\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$. Then
$\operatorname{{\mathrm{codim}}}_{{\Lambda_{\mathbb{R}}^{*}}}C(\mathsf{K})=0$,
so the relative interior
$C^{\circ}(\mathsf{K})\subseteq{\Lambda_{\mathbb{R}}^{*}}\backslash\mathcal{D}$
is an open subset. Each proper face of $C(\mathsf{K})$ has the form
$C(\mathsf{K}/\SS)$ for $\SS\neq 0$ and is therefore contained in
$\mathcal{D}$. Therefore $C^{\circ}(\mathsf{K})$ is a chamber.

Conversely, suppose $v\in{\Lambda_{\mathbb{R}}^{*}}$ is in a chamber
$\mathcal{C}$. As $\Sigma(\mathsf{H})$ is complete, $v$ is contained in a
minimal dual face $C(\mathsf{K}/\SS)$ where $(\mathsf{K},\SS)$ is a
distinguished face pair with $\SS=\mathsf{H}^{\mathsf{ss}}(v)=0$. It suffices
to show that $C^{\circ}(\mathsf{K})$ is open in ${\Lambda_{\mathbb{R}}^{*}}$
for then $\mathsf{K}$ is algebraic by Proposition 1.10, and
$\mathcal{C}=C^{\circ}(\mathsf{K})$ by the first part.

Fix a norm on ${\Lambda_{\mathbb{R}}^{*}}$. There is $\varepsilon>0$ with
$B_{\varepsilon}(v)\subset\mathcal{C}$, hence
$\SS^{w}=\mathsf{H}^{\mathsf{ss}}(w)=0$ for all $w\in B_{\varepsilon}(v)$. We
claim $\mathsf{T}_{\\!w}=\mathsf{T}_{\\!v}$ for all $w\in B_{\varepsilon}(v)$.
Suppose for a contradiction that $\mathsf{T}_{\\!w}\neq\mathsf{T}_{\\!v}$ for
some $w\in B_{\varepsilon}(v)$. As $\mathsf{H}$ is algebraic, we may assume
$t\in\mathsf{T}_{\\!v}\backslash\mathsf{T}_{\\!w}$, and that every proper
quotient of $t$ is in $\mathsf{T}_{\\!w}$, i.e. $v(t)\geq 0>w(t)$ and
$v(t^{\prime\prime}),w(t^{\prime\prime})\geq 0$ for every proper quotient
$t^{\prime\prime}$ of $t$. Thus there is $u\in B_{\varepsilon}(v)$ on the line
segment between $v$ and $w$ with $u(t)=0$ and $u(t^{\prime\prime})\geq 0$ for
all quotients $t^{\prime\prime}$ of $t$. Then
$u(t^{\prime})=u(t)-u(t/t^{\prime})=-u(t/t^{\prime})\leq 0$ for all subobjects
$t^{\prime}$ of $t$, hence $0\neq t\in\mathsf{H}^{\mathsf{ss}}(u)$,
contradicting $u\in B_{\varepsilon}(v)\subset\mathcal{C}$. We conclude that
$\mathsf{T}_{\\!w}=\mathsf{T}_{\\!v}$ for all $w\in B_{\varepsilon}(v)$ after
all. Therefore $\mathsf{K}^{w}=\mathsf{K}$ for all $w\in B_{\varepsilon}(v)$,
i.e. $B_{\varepsilon}(v)\subset C^{\circ}(\mathsf{K})$. Thus
$C^{\circ}(\mathsf{K})$ is open and we are done.∎

###### Remark 5.9.

Define the _stability fan_ of $\mathsf{H}$ to be the dual face subfan

$\Sigma^{\mathsf{ss}}(\mathsf{H})\coloneqq\\{C(\mathsf{K}/\SS)\in\Sigma(\mathsf{H})\mid(\mathsf{K},\SS)\text{
distinguished face pair with }\SS\neq 0\\}$

of the heart fan. We give it this name because its support
$\lvert\Sigma^{\mathsf{ss}}(\mathsf{H})\rvert=\bigcup_{M\neq 0}\mathcal{D}(M)$
is the union of stability spaces of non-zero modules. Note that this is a
union of rational polyhedral cones, even though
$\Sigma^{\mathsf{ss}}(\mathsf{H})$ may contain cones which are neither
rational nor polyhedral.

##  6\. Scattering diagrams

Let $A$ be a finite-dimensional algebra over $\mathbb{C}$. Since we are only
interested in invariants of the category $\mathsf{H}=\mathsf{mod}(A)$ of
finite-dimensional modules we may assume without loss of generality that
$A=\mathbb{C}Q/I$ where $(Q,I)$ is a quiver with relations. We explain how to
construct the Hall algebra scattering diagram defined in [9, Thm. 6.5], more
precisely a representative of its equivalence class, from the heart fan
$\Sigma(A)$. As usual we take $\Lambda=K(\mathsf{mod}(A))$, and define
$\Lambda^{+}=\Lambda\cap E(\mathsf{H})$ to be the submonoid of classes of
representations. Below, codimension one faces of cones will be important;
these are called _facets_.

Roughly, a scattering diagram $\mathfrak{D}=(\mathfrak{S},\varphi)$ consists
of a fan $\mathfrak{S}$ together with a choice of element
$\varphi(\mathfrak{d})$ in a pro-nilpotent Lie algebra for each facet
$\mathfrak{d}$ in the fan. In the particular case of the Hall algebra
scattering diagram, the pro-nilpotent Lie algebra is the completion with
respect to the natural grading by total dimension of the Hall algebra $H(A)$,
or rather of the Lie algebra given by its commutator bracket.

###  6.1. The Hall algebra

We review the construction of the Hall algebra and its completion. Essentially
this is a summary of [9, §4 and §5], to which we refer the reader for further
details.

The objects of $\mathsf{H}=\mathsf{mod}(A)$ are parametrised by an Artin stack
$\mathcal{M}$, locally of finite type over $\mathbb{C}$ and with affine
diagonal. The objects of $\mathcal{M}$ over a scheme $S$ are the algebra
homomorphisms $A\to{\mathrm{End}}_{S}(\mathcal{E})$ for locally free
$\mathcal{O}_{S}$-modules $\mathcal{E}$ of finite rank, i.e. representations
of $(Q,I)$ in vector bundles on $S$. This stack decomposes as a disjoint union

(3) $\mathcal{M}=\bigsqcup_{\lambda\in\Lambda^{+}}\mathcal{M}_{\lambda}$

of open and closed substacks parametrising the representations of each fixed
dimension vector. Each $\mathcal{M}_{\lambda}$ is a stack quotient of an
affine variety by a linear algebraic group. There is also a stack
$\mathcal{M}^{\text{ses}}$ parametrising short exact sequences $0\to M_{0}\to
M_{1}\to M_{2}\to 0$ in $\mathsf{mod}(A)$ and morphisms

${\mathcal{M}\times\mathcal{M}}$${\mathcal{M}^{\text{ses}}}$${\mathcal{M}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{0}\times\pi_{2}}$

sending a short exact sequence to its three objects.

###### Definition 6.1.

The _Hall algebra_ $H(A)$ is the $\mathbb{C}(t)$-algebra whose underlying
vector space has basis the isomorphism classes $[X\to\mathcal{M}]$ of
algebraic stacks $X$ over $\mathcal{M}$ of finite type over $\mathbb{C}$ and
with affine stabilisers, subject to the following relations:

  1. (a)

$[X\to\mathcal{M}]=[Y\to\mathcal{M}]+[X\backslash Y\to\mathcal{M}]$ for every
closed substack $Y\subset X$;

  2. (b)

$[Y_{1}\to\mathcal{M}]=[Y_{2}\to\mathcal{M}]$ whenever $Y_{1}\to X$ and
$Y_{2}\to X$ are locally trivial fibrations in the Zariski topology with
isomorphic fibres;

  3. (c)

$[X\times Y\to\mathcal{M}]=P_{t}(Y_{\text{an}})\cdot[X\to\mathcal{M}]$ when
$Y$ is a smooth projective variety over $\mathbb{C}$. Here
$P_{t}(Y_{\text{an}})$ is the Poincaré polynomial of the corresponding compact
complex manifold.

The algebra structure on $H(A)$ is given by the convolution product

$[X\to\mathcal{M}]*[Y\to\mathcal{M}]=[Z\to\mathcal{M}]$

where $Z$ is the fibre product defined by the Cartesian diagram

${Z}$${\mathcal{M}^{\text{ses}}}$${X\times
Y}$${\mathcal{M}\times\mathcal{M}}$$\scriptstyle{\pi_{0}\times\pi_{2}}$

and structure morphism
$Z\to\mathcal{M}^{\text{ses}}\xrightarrow{\pi_{1}}\mathcal{M}$.

The decomposition (3) induces a grading
$H(A)=\bigoplus_{\lambda\in\Lambda^{+}}H(A)_{\lambda}$ where $H(A)_{\lambda}$
is defined in the same way as the Hall algebra, but working over the substack
$\mathcal{M}_{\lambda}$. Let $d(\lambda)$ denote the dimension, as a complex
vector space, of any representation with dimension vector
$\lambda\in\Lambda^{+}$. For each $k\in\mathbb{N}$ the subspace
$\bigoplus_{d(\lambda)>k}H(A)_{\lambda}$ is an ideal. The quotient

$\mathfrak{g}_{\leq k}\coloneqq
H(A)\Big{/}\bigoplus_{\mathclap{d(\lambda)>k}}H(A)_{\lambda}$

is a nilpotent Lie subalgebra under the commutator bracket. The Lie
correspondence yields a corresponding unipotent algebraic Lie group $G_{\leq
k}$ whose exponential map $\exp\colon\mathfrak{g}_{\leq k}\to G_{\leq k}$ is a
bijection. Under this identification the group operation on $G_{\leq k}$ is
given by the Baker–Campbell–Hausdorff formula. The completion
$\widehat{\mathfrak{g}}_{\text{Hall}}\coloneqq\lim\mathfrak{g}_{\leq k}$ is a
pro-nilpotent Lie algebra, with corresponding pro-unipotent algebraic Lie
group $\smash{\widehat{G}}_{\text{Hall}}\coloneqq\lim G_{\leq k}$.

We construct a map
${\Lambda_{\mathbb{R}}^{*}}\backslash\\{0\\}\to\widehat{\mathfrak{g}}_{\text{Hall}}$,
$v\mapsto 1^{\mathsf{ss}}(v)$ using the subcategory
$\mathsf{H}^{\mathsf{ss}}(v)$ of $v$-semistable objects. Let
$\mathcal{M}^{\mathsf{ss}}(v)\subset\mathcal{M}$ be the open substack
parametrising $v$-semistable modules. Indeed,

(4)
$\mathcal{M}^{\mathsf{ss}}(v)\subset\bigoplus_{\mathclap{v(\lambda)=0}}\mathcal{M}_{\lambda}$

because if $M\in\mathsf{mod}(A)$ is $v$-semistable then $v([M])=0$. This
substack is not of finite type, so does not define an element of the Hall
algebra $H(A)$. However, the intersection
$\mathcal{M}^{\mathsf{ss}}(v)\cap\mathcal{M}_{\leq k}$ is of finite type with
affine stabilisers, for each $k\in\mathbb{N}$. Thus there are compatible
elements

$1_{\leq
k}^{\mathsf{ss}}(v)\coloneqq[\mathcal{M}^{\mathsf{ss}}(v)\cap\mathcal{M}_{\leq
k}\hookrightarrow\mathcal{M}_{\leq k}]\in\mathfrak{g}_{\leq k}$

which define an element
$1^{\mathsf{ss}}(v)\in\widehat{\mathfrak{g}}_{\text{Hall}}$. This illustrates
why we need to consider the completion, and not just the Hall algebra itself.

###  6.2. Scattering diagrams

We review the definition of scattering diagram as given in [9, §2]. For any
cone $\sigma\in{\Lambda_{\mathbb{R}}^{*}}$ we define a Lie subalgebra

$\mathfrak{g}_{\leq
k}(\sigma)\coloneqq\bigoplus_{0\neq\lambda\in\Lambda^{+}\cap\sigma^{\perp}}(\mathfrak{g}_{\leq
k})_{\lambda}$

where $(\mathfrak{g}_{\leq k})_{\lambda}$ denotes the image in the quotient of
the graded piece $H(A)_{\lambda}$.

###### Definition 6.2.

A _$\mathfrak{g}_{\leq k}$ -complex_ $\mathfrak{D}=(\mathfrak{S},\varphi)$
consists of a finite rational polyhedral fan $\mathfrak{S}$ in
${\Lambda_{\mathbb{R}}^{*}}$ together with a choice of element
$\varphi(\mathfrak{d})\in\mathfrak{g}_{\leq k}(\mathfrak{d})$ for each facet
$\mathfrak{d}\in\mathfrak{S}$. The _support_ $\mathrm{supp}(\mathfrak{D})$ is
the support $\lvert\mathfrak{S}\rvert$ of the underlying fan.

There are notions of consistency and equivalence for $\mathfrak{g}_{\leq
k}$-complexes defined as follows. A smooth path
$\gamma\colon[0,1]\to{\Lambda_{\mathbb{R}}^{*}}$ is _$\mathfrak{D}$ -generic_
if its endpoints do not lie in $\mathrm{supp}(\mathfrak{D})$, it does not meet
any cone of codimension two or higher, and it meets every facet transversely.
For any such path there is a finite sequence of points
$0<t_{1}<\cdots<t_{k}<1$ for which $\gamma(t_{i})$ lies in a facet
$\mathfrak{d}_{i}$. Therefore we can define the product

$\Phi_{\mathfrak{D}}(\gamma)\coloneqq\exp\varphi(\mathfrak{d}_{k})^{\varepsilon_{k}}\cdots\exp\varphi(\mathfrak{d}_{1})^{\varepsilon_{1}}\in
G_{\leq k}$

where $\varepsilon_{i}\in\\{\pm 1\\}$ depends on the orientation of the
crossing. Then

  * •

a $\mathfrak{g}_{\leq k}$-complex is _consistent_ if
$\Phi_{\mathfrak{D}}(\gamma)=1$ for any $\mathfrak{D}$-generic loop $\gamma$
and

  * •

two $\mathfrak{g}_{\leq k}$-complexes $\mathfrak{D}_{1}$ and
$\mathfrak{D}_{2}$ are _equivalent_ if
$\Phi_{\mathfrak{D}_{1}}(\gamma)=\Phi_{\mathfrak{D}_{2}}(\gamma)$ for any path
$\gamma$ which is both $\mathfrak{D}_{1}$-generic and
$\mathfrak{D}_{2}$-generic.

###### Definition 6.3.

A _$\widehat{\mathfrak{g}}$ -complex_ or _scattering diagram_ $\mathfrak{D}$
is a sequence $(\mathfrak{D}_{k})$ of $\mathfrak{g}_{\leq k}$-complexes for
$k\geq 1$ such that $\mathfrak{D}_{i}=(\mathfrak{S}_{i},\varphi_{i})$ and
$(\pi^{ji})_{*}\mathfrak{D}_{j}\coloneqq(\mathfrak{S}_{j},\pi^{ji}\circ\varphi_{j})$
are equivalent for all $i<j$ where $\pi^{ji}\colon\mathfrak{g}_{\leq
j}\to\mathfrak{g}_{\leq i}$ is the canonical quotient homomorphism. The
_support_ of the scattering diagram $\mathfrak{D}$ is
$\mathrm{supp}(\mathfrak{D})\coloneqq\bigcup_{i\in\mathbb{N}}\mathrm{supp}(\mathfrak{D}_{i})$.

A scattering diagram is consistent if each $\mathfrak{D}_{k}$ is consistent,
and two scattering diagrams are equivalent if their defining
$\mathfrak{g}_{\leq k}$-complexes are equivalent for all $k\geq 1$.

###  6.3. Hall algebra scattering diagram

The Hall algebra scattering diagram $\mathfrak{D}_{\text{Hall}}$ is
constructed from semistability data. Theorem 6.5 of [9] characterises it
uniquely, up to equivalence, by the two properties

  1. (1)

its support is the union $\bigcup_{0\neq M}\mathcal{D}(M)$ of stability spaces
of non-zero modules;

  2. (2)

the wall-crossing automorphism at a general point $v$ of the support is
$1^{\mathsf{ss}}(v)$.

Intuitively, $\mathfrak{D}_{\text{Hall}}$ consists of the stability fan
$\Sigma^{\mathsf{ss}}(A)=\\{C(\mathsf{K}/\SS)\mid\mathsf{K}\neq 0\\}$ together
with the assignments $C(\mathsf{K}/\SS)\mapsto 1^{\mathsf{ss}}(v)$ for (any)
$v$ in the relative interior of $C(\mathsf{K}/\SS)$. Naively, one could try to
make this precise by constructing the latter as the limit over the subfans

$\Sigma^{\mathsf{ss}}_{\leq
k}(A)\coloneqq\\{C(\mathsf{K}/\SS)\mid(\mathsf{K},\SS)\text{ distinguished,
}\mathsf{K}\neq 0,\ d(M)\leq k\text{ for all stable }M\in\mathsf{K}\\}.$

However, technically this fails because the above fan need not be finite, or
rational, or polyhedral. The issue is that it is too fine a decomposition;
instead we need to consider an alternative refinement of its support, which
takes account only of the semistability of modules of dimension less than $k$
rather than of all modules. What makes this possible is the fact — a
consequence of Proposition 5.7 — that the support
$\lvert\Sigma^{\mathsf{ss}}_{\leq k}(A)\rvert=\bigcup_{0<d(M)\leq
k}\mathfrak{D}(M)$ is a union of finitely many rational, polyhedral cones.

The alternative refinement is constructed, following [9, Ex. 2.5], using the
cones

$\sigma(P_{-},P_{0},P_{+})\coloneqq\\{v\in{\Lambda_{\mathbb{R}}^{*}}\mid
v(M)=0\ \forall\,M\in P_{0}\ \text{and}\ \pm v(M)\geq 0\ \forall M\in
P_{\pm}\\}$

where $P_{-}\cup P_{0}\cup P_{+}=\\{M\mid d(M)\leq k\\}$ is a partition into
unions of Grothendieck group classes. By construction each such cone is
rational and polyhedral, and the set of semistable modules $M$ with $d(M)\leq
k$ is constant on its relative interior.

###### Example 6.4.

The set of cones $\sigma(P_{-},P_{0},P_{+})$ for which there exists a non-zero
semistable $M$ with $d(M)\leq k$ forms a fan. We denote it by
$\mathfrak{S}_{\leq k}$. By [9, Lemma 6.2], or alternatively by Proposition
5.7, its support is $\bigcup_{0<d(M)\leq k}\mathfrak{D}(M)$.

For each facet $\mathfrak{d}$ in $\mathfrak{S}_{\leq k}$, set $\varphi_{\leq
k}(\mathfrak{d})\coloneqq 1_{\leq k}^{\mathsf{ss}}(v)$ for (any) $v$ in the
relative interior of $\mathfrak{d}$. It follows from (4) that $1_{\leq
k}^{\mathsf{ss}}(v)\in\mathfrak{g}_{\leq k}(\mathfrak{d})$. Hence the pair
$\mathfrak{D}_{\leq k}\coloneqq(\mathfrak{S}_{\leq k},\varphi_{\leq k})$ is a
$\mathfrak{g}_{\leq k}$-complex. The sequence $(\mathfrak{D}_{\leq k})$ of
$\mathfrak{g}_{\leq k}$-complexes defines a consistent scattering diagram, the
_Hall algebra scattering diagram_ $\mathfrak{D}_{\text{Hall}}$; see [9, §6].

##  Appendix A Homological algebra

A good source emphasising the similarity of abelian and triangulated
categories is [6, Ch. I]. Categories are supposed to be essentially small,
i.e. having a set of objects up to isomorphism. Subcategories are supposed to
be full and strict, i.e. closed under isomorphisms. The $n$-fold shift (or:
translation or suspension) functor of triangulated categories is denoted by
$[n]$.

###  A.1. Abelian categories.

Let $\mathsf{H}$ be an abelian category; it is _length_ if each object of
$\mathsf{H}$ admits a finite composition series (whose factors are simple
objects). Equivalently, $\mathsf{H}$ is Noetherian (all ascending chains of
subobjects stabilise) and Artinian (all descending chains stabilise). If in
addition, $\mathsf{H}$ has finitely many isomorphism classes of simple
objects, it is _algebraic_.

Let $\mathsf{U}\subseteq\mathsf{H}$ be a full subcategory; it is an abelian
subcategory if it is closed under kernels and cokernels. It is a _wide
subcategory_ if it is closed under kernels, cokernels and extensions. It is a
_Serre subcategory_ if it is closed under subobjects, quotients and
extensions; then the quotient category $\mathsf{H}/\mathsf{U}$ is abelian.
$\mathrm{Serre}(\mathsf{H})$ denotes the set of Serre subcategories. In
Definition 2.2, we will define $\mathrm{Serre}_{\Lambda\\!}(\mathsf{H})$ as
the subset of face subcategories.

###  A.2. Extension closure.

Let $\mathsf{C}$ be an abelian or triangulated category, so we can speak of
extensions in $\mathsf{C}$. If $M$ is a collection of objects of $\mathsf{C}$
then ${\langle M\rangle}$ denotes the _extension closure_ of $M$, i.e. the
smallest full subcategory of $\mathsf{C}$ that (a) contains $M$ and (b) is
closed under extensions. If $M_{1},M_{2}$ are two subcategories of
$\mathsf{C}$, their _ordered extension closure_ $M_{1}*M_{2}$ is the full
subcategory of all extensions of objects of $M_{2}$ by objects of $M_{1}$. If,
for example, $\mathsf{C}$ is triangulated then $M_{1}*M_{2}$ is the full
subcategory of objects $c\in\mathsf{C}$ sitting in exact triangles $m_{1}\to
c\to m_{2}\to m_{1}[1]$ with $m_{1}\in M_{1},m_{2}\in M_{2}$.

For $\mathsf{H}\subset\mathsf{C}$ a heart in a triangulated category and
$M\subseteq\mathsf{H}$, the two notions of ${\langle M\rangle}$ agree.

###  A.3. Torsion pairs.

Two full subcategories $(\mathsf{T},\mathsf{F})$ of an abelian category
$\mathsf{H}$ form a _torsion pair_ if (a)
${\mathrm{Hom}}(\mathsf{T},\mathsf{F})=0$ and (b)
$\mathsf{H}=\mathsf{T}*\mathsf{F}$. In this case, $\mathsf{T}$ is called the
_torsion class_ and $\mathsf{F}$ the _torsion-free class_ of the pair. Torsion
classes $\mathsf{T}$ are closed under quotients and extensions; torsion-free
classes $\mathsf{F}$ are closed under subobjects and extensions. For a torsion
pair $(\mathsf{T},\mathsf{F})$, we have
$\mathsf{F}=\mathsf{T}^{\perp}\coloneqq\\{h\in\mathsf{H}\mid{\mathrm{Hom}}(\mathsf{T},h)=0\\}$
and $\mathsf{T}={}^{\perp}\mathsf{F}$; e.g.
$\mathsf{F}\subseteq\mathsf{T}^{\perp}$ by (a) and
$\mathsf{F}\supseteq\mathsf{T}^{\perp}$ by (b).

If $\mathsf{H}$ is Noetherian and $\mathsf{T}\subseteq\mathsf{H}$ is an
additive subcategory closed under quotients and extensions then $\mathsf{T}$
defines a torsion pair $(\mathsf{T},\mathsf{T}^{\perp})$. Dually, if
$\mathsf{H}$ is Artinian and $\mathsf{F}\subseteq\mathsf{H}$ is closed under
subobjects and extensions then $({}^{\perp}\mathsf{F},\mathsf{F})$ is a
torsion pair.

###  A.4. T-structures.

Two subcategories $(\mathsf{T},\mathsf{F})$ of a triangulated category
$\mathsf{D}$ form a _t-structure_ if (a)
${\mathrm{Hom}}(\mathsf{T},\mathsf{F})=0$ and (b)
$\mathsf{C}=\mathsf{T}*\mathsf{F}$ and (c) $\mathsf{T}[1]\subseteq\mathsf{T}$.
The t-structure is _bounded_ if moreover (d)
$\bigcup_{n\in\mathbb{Z}}\mathsf{T}[n]=\mathsf{D}$. Any t-structure
$(\mathsf{T},\mathsf{F})$ on $\mathsf{D}$, bounded or not, induces an abelian
subcategory of $\mathsf{D}$ as
$\mathsf{H}\coloneqq\mathsf{T}\cap\mathsf{F}[1]$ called the _heart_. If the
t-structure is bounded then $\mathsf{T}$ and $\mathsf{F}$ can be reconstructed
from the heart $\mathsf{H}$ as
$\mathsf{T}={\langle\mathsf{H}[\mathit{\geq}0]\rangle}$ and
$\mathsf{F}={\langle\mathsf{H}[\mathit{<}0]\rangle}$. In this article, heart
always means the heart of a bounded t-structure.

Setting
$(\mathsf{T},\mathsf{F})\leq(\mathsf{T}^{\prime},\mathsf{F}^{\prime})\iff\mathsf{F}\subseteq\mathsf{F}^{\prime}$
defines a partial order on the bounded t-structures of a triangulated category
$\mathsf{D}$. Equivalently, this gives a partial order on hearts in
$\mathsf{D}$ via
$\mathsf{H}\leq\mathsf{H}^{\prime}\iff{\langle\mathsf{H}[{\leq}0]\rangle}\subseteq{\langle\mathsf{H}^{\prime}[{\leq}0]\rangle}$.
The convention we follow here makes $\mathsf{H}\leq\mathsf{H}[1]$.

###  A.5. Tilting.

Let $\mathsf{D}$ be a triangulated category, $\mathsf{H}$ a heart in
$\mathsf{D}$ and $(\mathsf{T},\mathsf{F})$ a torsion pair in the abelian
category $\mathsf{H}$, so that $\mathsf{H}=\mathsf{T}*\mathsf{F}$. From this
setup we get two new hearts of $\mathsf{D}$ as

$\mathsf{F}[1]*\mathsf{T}\text{, the \emph{{positive tilt}} of
$\mathsf{H}$};\qquad\mathsf{F}*\mathsf{T}[-1]\text{, the \emph{{negative
tilt}} of $\mathsf{H}$}.$

The terminology is justified by
$\mathsf{H}[-1]\leq\mathsf{F}*\mathsf{T}[-1]\leq\mathsf{H}\leq\mathsf{F}[1]*\mathsf{T}\leq\mathsf{H}[1]$.
Positive/negative tilts are also called left/right. The following basic fact
is crucial for us, see [23, Lemma 1.1.2]: If $\mathsf{K}$ and $\mathsf{H}$ are
two hearts in a triangulated category $\mathsf{D}$ with
$\mathsf{H}\leq\mathsf{K}\leq\mathsf{H}[1]$, then there is a unique torsion
pair $(\mathsf{T},\mathsf{F})$ on $\mathsf{H}=\mathsf{T}*\mathsf{F}$ with
$\mathsf{K}=\mathsf{F}[1]*\mathsf{T}$; it is given by
$\mathsf{T}=\mathsf{H}\cap\mathsf{K}$ and
$\mathsf{F}=\mathsf{H}\cap\mathsf{K}[-1]$.

##  Appendix B Convex geometry

We collect notions from convex geometry in a self-contained manner as we are
not aware of a source covering our needs: [16, §1] is inspired by applications
in toric geometry, so it discusses fans but cones are assumed to be
polyhedral; [18] is motivated by optimisation and treats a wider class of
convex sets and cones (although still often assumed to be closed) but no fans.
We need to pay more attention than usual as our effective cones may be neither
polyhedral nor closed, even in important examples. The notion of ‘dual face
fan’ is novel.

Throughout, $V$ is a finite-dimensional $\mathbb{R}$-vector space with dual
vector space $V^{*}={\mathrm{Hom}}(V,\mathbb{R})$.

###  B.1. Cones.

A _cone_ in $V$ is a subset $\sigma\subseteq V$ closed under (a) sums and (b)
multiples by positive real numbers, i.e. $\sigma+\sigma\subseteq\sigma$ and
$\mathbb{R}_{>0}\cdot\sigma\subseteq\sigma$. $\mathrm{Cones}(V)$ denotes the
set of all cones in $V$. Intersections and sums of cones are again cones, as
are the topological closure and interior of a cone in $V$; we allow the empty
set as a cone. The _linear hull_ of $\sigma$ is
$\mathbb{R}\sigma=\sigma-\sigma=\sigma^{\perp\perp}$ and its dimension
$\dim(\sigma)\coloneqq\dim_{\mathbb{R}}(\mathbb{R}\sigma)$. A cone is _full_
if it is full-dimensional, i.e. $\dim(\sigma)=\dim_{\mathbb{R}}(V)$;
equivalently $\sigma$ has non-empty interior:
$\mathrm{int}(\sigma)\neq\varnothing$. The _relative interior_
${\sigma}^{\circ}=\mathrm{relint}(\sigma)$ of a cone $\sigma\subseteq V$ is
the interior of $\sigma$ as a subset of its linear hull; it is again a cone.

Cones are convex. A cone $\sigma$ is _strictly convex_ if it contains no non-
trivial linear subspace, i.e. $\sigma\cap(-\sigma)\subseteq\\{0\\}$. It is
_pointed_ if $\sigma\cap(-\sigma)=\\{0\\}$. The cone _generated_ by a subset
$S\subseteq V$ is the convex hull of the rays $\mathbb{R}_{\geq 0}\cdot v$ for
$v\in S$; we denote it $E(S)$. We write $\overline{\mkern-4.0muE}(S)$ for the
closure of $E(S)$ in $V$. A cone is _polyhedral_ if it is generated by a
finite subset and _simplicial_ (or: simple) if it is generated by a linearly
independent subset.

The _dual cone_ (also: polar cone) of a cone $\sigma\subseteq V$ is
$\sigma^{\vee}\coloneqq\\{w\in V^{*}\mid w(v)\geq 0~{}\forall v\in\sigma\\}$,
i.e. the set of functionals $w\colon V\to\mathbb{R}$ with $w|_{\sigma}\geq 0$;
this is a closed cone in $V^{*}$. Note $\sigma^{\vee\vee}=\overline{\sigma}$
in $V^{**}=V$, using the hyperplane separation theorem for the inclusion
$\sigma^{\vee\vee}\subseteq\overline{\sigma}$. Clearly
$\overline{\sigma}^{\vee}=\sigma^{\vee}$.

For a subset $S\subseteq V$, we denote $C(S)\coloneqq E(S)^{\vee}\subseteq
V^{*}$ and $C^{\circ}(S)\coloneqq\mathrm{relint}(C(S))$.

Given a lattice $\Lambda\subset V$, i.e. a free abelian group of maximal rank,
we write $V=\Lambda\otimes\mathbb{R}=\Lambda_{\mathbb{R}}$ and
$V^{*}=\Lambda^{*}\otimes\mathbb{R}={\mathrm{Hom}}(\Lambda,\mathbb{R})={\Lambda_{\mathbb{R}}^{*}}$
with dual lattice $\Lambda^{*}={\mathrm{Hom}}(\Lambda,\mathbb{Z})\subset
V^{*}$. A cone in $V$ is _rational_ if it is generated by lattice elements.
The same notion applies to cones in $V^{*}$ via $\Lambda^{*}$.

###  B.2. Faces.

A _face_ of a cone $\sigma$ is a non-empty subset $\tau\subseteq\sigma$ such
that for any $v,v^{\prime}\in\sigma$, we have: $v+v^{\prime}\in\tau\iff
v,v^{\prime}\in\tau$. In particular $\tau$ is itself a cone. We denote by
$\tau\preceq\sigma$ that $\tau$ is a face of $\sigma$ and by $\tau\prec\sigma$
that it is a proper face, i.e. $\tau\neq\sigma$. A _facet_ of a cone is a face
of codimension one.

For faces of a cone, inclusion and face relation coincide: if
$\tau,\tau^{\prime}\preceq\sigma$ then
$\tau\subseteq\tau^{\prime}\iff\tau\preceq\tau^{\prime}$.

An _exposed face_ of $\sigma$ is a subset of the form $\sigma\cap w^{\perp}$
for $w\in\sigma^{\vee}$. When $w\neq 0$ the exposed face is cut out by
intersecting with the _supporting hyperplane_ $w^{\perp}\subset V$; when $w=0$
the corresponding exposed face is $\sigma$ itself. Every exposed face is a
face and the two notions coincide if $\sigma$ is polyhedral. See Example 2.1
for a non-exposed face.

The following properties are elementary: If $\tau\prec\sigma$ is a proper face
then $\tau\cap{\sigma}^{\circ}=\varnothing$ and $\dim(\tau)<\dim(\sigma)$. In
particular, a maximal chain of proper faces has length at most $\dim(V)$. A
non-exposed face is a subset of an exposed face, hence there is a finite
decreasing sequence of faces of the original cone, with each face exposed in
the previous one, ending at the non-exposed face. Therefore faces are closed
subsets of the cone, and the relative interior of a cone is the complement of
the union of its proper exposed faces.

$\mathrm{Faces}(\sigma)$ denotes the set of all faces of a cone $\sigma$ and
$\mathrm{ExFaces}(\sigma)$ the subset of exposed faces. The sets
$\mathrm{ExFaces}(\sigma)\subseteq\mathrm{Faces}(\sigma)\subset\mathrm{Cones}(V)$
are partially ordered by the $\preceq$ relation. Recall that in order theory,
a lattice is a partially ordered set such that each pair of elements has an
infimum and a supremum. This use of the word ‘lattice’ is unrelated to a free
abelian group of finite rank. Downward closures in the poset of faces and
upward closures in the poset of exposed faces have simple internal
descriptions:

######  Lemma B.1.

Let $\sigma\subseteq V$ be a cone with $0\in\sigma$; let
$\tau^{\prime}\preceq\sigma$ be a face and $\tau\preceq\sigma$ an exposed
face.

  1. (1)

$\mathrm{ExFaces}(\sigma)$ and $\mathrm{Faces}(\sigma)$ are lattices.

  2. (2)

$\\{\kappa\in\mathrm{Faces}(\sigma)\mid\kappa\preceq\tau^{\prime}\\}=\mathrm{Faces}(\tau^{\prime})$.

  3. (3)

$\\{\kappa\in\mathrm{ExFaces}(\sigma)\mid\kappa\succeq\tau\\}\cong\mathrm{ExFaces}(\sigma/\tau)$.

###### Proof.

(1) The infimum of two faces is their intersection which is non-empty by
$0\in\sigma$. The intersection of exposed faces $\sigma\cap w_{1}^{\perp}$ and
$\sigma\cap w_{2}^{\perp}$ is the exposed face
$\sigma\cap(w_{1}+w_{2})^{\perp}$, using
$w_{1}(v)=w_{2}(v)=0\iff(w_{1}+w_{2})(v)=0$ for any $v\in\sigma$, because of
$w_{1}|_{\sigma}\geq 0$ and $w_{2}|_{\sigma}\geq 0$.

The supremum of two faces of $\sigma$ is the unique minimal face containing
both. This exists because $\sigma$ is a face of itself. Given two exposed
faces then the smallest face containing them may not be exposed but it is
contained in a unique minimal exposed face. In particular,
$\mathrm{ExFaces}(\sigma)$ is not necessarily a sublattice of
$\mathrm{Faces}(\sigma)$.

(2) This follows immediately from the definitions.

(3) $\sigma/\tau\in\mathrm{Cones}(V/\mathbb{R}\tau)$ is the image of $\sigma$
under the projection $V\to V/\mathbb{R}\tau$. The bijection arises from the
straightforward formula $(\sigma/\tau)\cap w^{\perp}=(\sigma\cap
w^{\perp})/\tau$ for $w\in\sigma^{\vee}\cap\tau^{\perp}$. The map
$\\{\kappa\in\mathrm{ExFaces}(\sigma)\mid\kappa\succeq\tau\\}\to\mathrm{ExFaces}(\sigma/\tau)$,
$\kappa=\sigma\cap w^{\perp}\mapsto\kappa/\tau$ is surjective by the formula
and injective by $\tau\subseteq\kappa$. ∎

Formula (2) of the lemma can fail for exposed faces: see Example 2.1 where the
non-exposed face is an exposed face of an exposed face.

Likewise, (3) may fail for non-exposed faces: if $\sigma=C(\mathsf{H})$ is the
cone in Example 2.1 and $\tau$ is the non-exposed face given by the ray
through $A$ then $\sigma/\tau$ is a closed half-plane, and
$\\{\kappa\in\mathrm{Faces}(\sigma)\mid\kappa\succeq\tau\\}\not\cong\mathrm{Faces}(\sigma/\tau)$
because the respective cardinalities are $2$ and $3$.

###  B.3. Dual faces.

For a subset $\tau\subseteq\sigma$, we define a subset of the dual cone by

$\tau{{}^{\vartriangle}}\coloneqq\sigma^{\vee}\cap\tau^{\perp}=\\{w\in
V^{*}\colon w|_{\sigma}\geq 0,w|_{\tau}=0\\};$

this is called the _dual face_ (also: conjugated face) and is in fact even an
exposed face. We denote the set of dual faces by
$\mathrm{ExFaces}(\sigma){}^{\mathrm{op}}$; identifying the domain of the
order-reversing inclusion
$\mathrm{ExFaces}(\sigma)\hookrightarrow\mathrm{ExFaces}(\sigma^{\vee})$ with
its image. This set can be equivalently defined as the set of duals of:
exposed faces of $\sigma$; faces of $\sigma$; subsets of $\sigma$. We write
$\mathrm{ExFaces}(\sigma){}^{\mathrm{op}}$ because this gives a one-to-one
parametrisation of dual faces. See Proposition B.3 for these facts. As
$\mathrm{ExFaces}(\sigma)$ is a lattice by Lemma B.1, so is the poset
$\mathrm{ExFaces}(\sigma){}^{\mathrm{op}}$ of dual faces.

Faces and exposed faces are defined for arbitrary cones whereas dual faces
require the cone to be given as the dual of another cone. We always have the
following inclusions which are equalities if $\sigma$ is polyhedral
($\sigma^{\vee}$ polyhedral is not enough):

$\mathrm{ExFaces}(\sigma){}^{\mathrm{op}}\subseteq\mathrm{ExFaces}(\sigma^{\vee})\subseteq\mathrm{Faces}(\sigma^{\vee}).$

###### Example B.2.

Let $\sigma=\\{(x,y)\mid x,y>0\text{ or }x=y=0\\}$ be the open quadrant in the
plane together with the origin. Its dual is the closed quadrant
$\sigma^{\vee}=(\mathbb{R}_{\geq 0})^{2}$ which has four faces (all exposed):
the origin, the whole cone and two rays. However, only $\\{0\\}$ and
$\sigma^{\vee}$ are dual faces.

######  Proposition B.3.

Let $\sigma\subseteq V$ be a cone with $0\in\sigma$ and let
$\tau\subseteq\sigma$ be a non-empty subset.

  1. (1)

$\tau{{}^{\vartriangle}}\subseteq\sigma^{\vee}$ is an exposed face, and
$\tau{{}^{\vartriangle}}=\sigma^{\vee}\cap v^{\perp}$ for any
$v\in\mathrm{relint}(E(\tau))$.

  2. (2)

If $\nu\preceq\sigma$ is the minimal exposed face containing $\tau$ then
$\tau{{}^{\vartriangle}}=\nu{{}^{\vartriangle}}$.

  3. (3)

If $\tau=\sigma\cap w^{\perp}\preceq\sigma$ is an exposed face, where
$w\in\sigma^{\vee}$, then
$\tau^{\vartriangle\vartriangle}=\overline{\sigma}\cap
w^{\perp}\subseteq\overline{\sigma}=\sigma^{\vee\vee}$.

  4. (4)

If $w\in\sigma^{\vee}$ then $(\sigma\cap
w^{\perp}){{}^{\vartriangle}}\preceq\sigma^{\vee}$ is the minimal exposed face
containing $w$.

  5. (5)

$\mathrm{ExFaces}(\sigma)\to\mathrm{ExFaces}(\sigma^{\vee}){}^{\mathrm{op}}$,
$\tau\mapsto\tau{{}^{\vartriangle}}$ is a monotone and injective map.

  6. (6)

The map from (5) is bijective if $\sigma$ is closed.

###### Proof.

See [19, §2] for some statements and proofs in a similar but slightly less
general context; we give a self-contained proof for the convenience of the
reader.

  1. (1)

Without loss of generality, we assume that $\tau$ is closed in view of
$\overline{\tau}^{\perp}=\tau^{\perp}$. Since $V$ is a separable metric space,
so is its subset $\tau$, i.e. there is a dense sequence $(v_{i})$ in $\tau$.
Then
$\tau^{\perp}=\overline{\\{v_{0},v_{1},\ldots\\}}{}^{\perp}=\\{v_{0},v_{1},\ldots\\}^{\perp}=\bigcap_{i\geq
0}v_{i}^{\perp}=v_{0}^{\perp}\cap\cdots\cap v_{n}^{\perp}$ for some $n$, using
$\dim(V)<\infty$ in the last equality. We get
$\sigma^{\vee}\cap\tau^{\perp}=\sigma^{\vee}\cap v_{0}^{\perp}\cap\cdots\cap
v_{n}^{\perp}=\sigma^{\vee}\cap v_{\tau}^{\perp}$ with $v_{\tau}\coloneqq
v_{0}+\cdots+v_{n}$ because of $w(v_{\tau})=0\iff w(v_{0})=\cdots=w(v_{n})=0$
for any $w\in\sigma^{\vee}$.

Denote by $\tau^{\prime}$ the common relative interior of $E(\tau)\subseteq
E(\overline{\tau})$. If $w\in\sigma^{\vee}$ and $v\in\tau^{\prime}$ then
$w(v)=0\iff w|_{\tau^{\prime}}=0\iff w|_{\tau}=0$. We may assume
$v_{\tau}\in\tau^{\prime}$ by choosing $v_{i}\in\tau^{\prime}$ above. Varying
$\varepsilon>0$ and $v^{\prime}\in\tau^{\prime}$, the set of
$v\coloneqq\varepsilon v_{\tau}+v^{\prime}$ covers $\tau^{\prime}$. Now
$\sigma^{\vee}\cap
v^{\perp}=\sigma^{\vee}\cap\tau^{\perp}=\tau{{}^{\vartriangle}}$.

  2. (2)

From $\tau\subseteq\nu$ we get
$\tau{{}^{\vartriangle}}\supseteq\nu{{}^{\vartriangle}}$. For the other
inclusion, let $w\in\tau{{}^{\vartriangle}}$, i.e. $w\in V^{*}$ with
$w|_{\sigma}\geq 0$ and $w|_{\tau}=0$. Then the exposed face $\sigma\cap
w^{\perp}$ cut out by $w$ contains $\tau$. As $\nu$ is the minimal exposed
face over $\tau$, we have $\nu\subseteq\sigma\cap w^{\perp}$. Taking dual
faces again, we get $\nu{{}^{\vartriangle}}\supseteq(\sigma\cap
w^{\perp}){{}^{\vartriangle}}\ni w$.

  3. (3)

Let first
$v\in\tau^{\vartriangle\vartriangle}=\overline{\sigma}\cap(\sigma^{\vee}\cap\tau^{\perp})^{\perp}\subseteq
V$. From $w\in\tau{{}^{\vartriangle}}=\sigma^{\vee}\cap\tau^{\perp}$, we get
$w(v)=0$, i.e. $v\in\overline{\sigma}\cap w^{\perp}$. For the reverse
inclusion, let $v\in\overline{\sigma}\cap w^{\perp}$ and
$f\in\sigma^{\vee}\cap\tau^{\perp}$, i.e. $f|_{\sigma}\geq 0$ and
$0=f|_{\tau}=f|_{\sigma\cap w^{\perp}}$. Because $f$ is linear, hence
continuous, this implies $f|_{\overline{\sigma}\cap w^{\perp}}=0$. Therefore
$f(v)=0$ and so $v\in\tau^{\vartriangle\vartriangle}$.

  4. (4)

Let $\kappa=\sigma^{\vee}\cap v^{\perp}$ for some $v\in\overline{\sigma}$ be
an exposed face containing $w$. We show $(\sigma\cap
w^{\perp}){{}^{\vartriangle}}\subseteq\kappa$. Let $f\in(\sigma\cap
w^{\perp}){{}^{\vartriangle}}$, i.e. $f\in V^{*}$ with $f|_{\sigma}\geq 0$ and
$f|_{\sigma\cap w^{\perp}}=0$. We have $w(v)=0$ from $w\in\kappa$, hence
$v\in\overline{\sigma}\cap w^{\perp}$. Again by continuity of $f$, we find
$f(v)=0$, thus $f\in\kappa$, as claimed.

  5. (5)

The map $\tau\mapsto\tau{{}^{\vartriangle}}$ is clearly inclusion-reversing.
Among (exposed) faces, inclusion equals the face relation, so
$\mathrm{ExFaces}(\sigma)\to\mathrm{ExFaces}(\sigma^{\vee}){}^{\mathrm{op}}$
is monotone. To check injectivity, we iterate face duality and consider the
composition $\mathrm{ExFaces}(\sigma)\to\mathrm{ExFaces}(\overline{\sigma})$,
$\sigma\cap w^{\perp}\mapsto\overline{\sigma}\cap w^{\perp}$, using part (3)
above. The composition is injective, hence so is its first component.

  6. (6)

If $\sigma$ is closed then the composition of the previous step,
$\mathrm{ExFaces}(\sigma)\to\mathrm{ExFaces}(\sigma)$,
$\tau\mapsto\tau^{\vartriangle\vartriangle}$ is the identity, and so is the
composition
$\mathrm{ExFaces}(\sigma^{\vee})\to\mathrm{ExFaces}(\sigma)\to\mathrm{ExFaces}(\sigma^{\vee})$.
∎

The dual face $\tau{{}^{\vartriangle}}$ of a subset $\tau\subseteq\sigma$ is
itself a dual cone in a linear subspace, in view of
$\tau{{}^{\vartriangle}}=\sigma\cap\tau^{\perp}=(\sigma/\tau)^{\vee}$. Here
$\sigma/\tau\subseteq V/\mathbb{R}\tau$ and we consider its dual cone
$(\sigma/\tau)^{\vee}\subseteq(V/\mathbb{R}\tau)^{*}=\tau^{\perp}$ as a cone
in $V^{*}$. The diagram

(5)
${\mathrm{ExFaces}(\sigma/\tau){}^{\mathrm{op}}}$${\mathrm{ExFaces}(\sigma){}^{\mathrm{op}}}$${\kappa/\tau}$${\kappa}$${\mathrm{ExFaces}((\sigma/\tau)^{\vee})}$${\mathrm{ExFaces}(\sigma^{\vee})}$${(\kappa/\tau){{}^{\vartriangle}}}$${\kappa{{}^{\vartriangle}}}$

commutes; in particular dual faces of dual faces of $\sigma^{\vee}$ are
themselves dual faces of $\sigma^{\vee}$.

######  Lemma B.4.

If $V=\Lambda_{\mathbb{R}}$ for a lattice $\Lambda$ and $\tau$ is a rational
cone inside a cone $\sigma\subseteq\Lambda_{\mathbb{R}}$ then
$\sigma^{\vee}\cap\tau^{\perp}=\sigma^{\vee}\cap\lambda^{\perp}$ for some
$\lambda\in\sigma\cap\Lambda$.

###### Proof.

$\tau$ is rational in $V=\Lambda_{\mathbb{R}}$, i.e. generated by a subset of
$\Lambda$. Then the subset $\tau\cap\Lambda_{\mathbb{Q}}$ is dense in $\tau$.
Therefore we can choose the sequence of the proof of Proposition B.3(1) to be
of the form $v_{i}=\lambda_{i}\in\Lambda$. Following the previous argument,
$\sigma^{\vee}\cap\tau^{\perp}=\sigma^{\vee}\cap(\lambda_{0}+\cdots+\lambda_{n})^{\perp}$,
as claimed. ∎

###  B.4. Fans.

A _fan_ in $V$ is a subset $\Sigma\subset\mathrm{Cones}(V)$ such that (a)
$\Sigma$ is closed under taking faces and (b) any intersection of two cones in
$\Sigma$ is a face of each.

The _support_ of a fan $\Sigma$ is the union
$\lvert\Sigma\rvert\coloneqq\bigcup_{\sigma\in\Sigma}\sigma$ of all cones.
$\Sigma$ is _complete_ if $\lvert\Sigma\rvert=V$. A fan is called _polyhedral_
or _simplicial_ or _rational_ if all cones in the fan have this property. It
is called _finite_ if it is a finite collection of cones. Given a fan
$\Sigma$, we denote by $\Sigma^{\mathrm{full}}$ the subfan consisting of all
simplicial, full cones in $\Sigma$ together with their faces.

The next result shows that a fan $\Sigma$ is determined by its subset of
_maximal cones_ , i.e. the $\sigma\in\Sigma$ not contained in a larger cone in
$\Sigma$.

######  Lemma B.5.

Let $\Delta$ be a set of cones in $V$. Then the following are equivalent:

  1. (i)

The set $\Sigma$ of all faces of cones in $\Delta$ is a fan.

  2. (ii)

Any two cones in $\Delta$ intersect in a common face.

###### Proof.

If $\Sigma$ is a fan any two cones in $\Delta$ intersect in a common face. To
prove the converse we use the following two properties: (IF) the intersection
of two faces of a cone is a face and (FF) the faces of a face of a cone are
precisely the faces of the cone contained in that face.

By (FF), $\Sigma$ is closed under taking faces. Suppose $\sigma^{\prime}$ and
$\tau^{\prime}$ are faces respectively of $\sigma,\tau\in\Delta$. By
assumption $\sigma\cap\tau$ is a common face of $\sigma$ and of $\tau$. Thus
by (IF) $\sigma^{\prime}\cap\tau=\sigma^{\prime}\cap(\sigma\cap\tau)$ is a
face of $\sigma$. Since it is contained in $\sigma^{\prime}$ and
$\sigma\cap\tau$ it is a face of each by (FF). Applying (FF) again shows that
it is also a face of $\tau$.

Now consider
$\sigma^{\prime}\cap\tau^{\prime}=(\sigma^{\prime}\cap\tau)\cap\tau^{\prime}$.
By the above $\sigma^{\prime}\cap\tau$ is a face of $\tau$, and therefore by
(IF) so is $\sigma^{\prime}\cap\tau^{\prime}$. Thus by (FF)
$\sigma^{\prime}\cap\tau^{\prime}$ is a face of $\tau^{\prime}$ and of
$\sigma^{\prime}\cap\tau$. By the first part $\sigma^{\prime}\cap\tau$ is a
face of $\sigma^{\prime}$ so by (FF) $\sigma^{\prime}\cap\tau^{\prime}$ is a
face of $\sigma^{\prime}$ too. ∎

The analogue for exposed faces is false because property (FF) may fail: an
exposed face of an exposed face of a cone need not be exposed in the cone.

###  B.5. Dual face fans.

A _dual face fan_ is a set $\Sigma$ of dual faces of dual cones such that (a)
$\Sigma$ is closed under taking dual faces and (b) any intersection of two
cones in $\Sigma$ is a dual face of each.

In Section 3 we generate a dual face fan from a subset
$\mathcal{E}\subset\mathrm{Cones}(V)$ such that the set of all dual faces
$\Sigma\coloneqq\\{\tau{{}^{\vartriangle}}\mid\tau\preceq\sigma\in\mathcal{E}\\}\subset\mathrm{Cones}(V^{*})$
satisfies (a) and (b); we do not know if every dual face fan arises in this
way. Dual face fans are determined by their maximal cones:

######  Lemma B.6.

Let $\Delta$ be a set of dual faces of dual cones in $V^{*}$. Then the
following are equivalent:

  1. (i)

The set $\Sigma$ of all dual faces of cones in $\Delta$ is a dual face fan.

  2. (ii)

Any two cones in $\Delta$ intersect in a common dual face.

###### Proof.

The proof is identical to that of Lemma B.5, but using instead the facts that
(IF) the intersection of two dual faces of a dual cone is a dual face, because
dual faces form a lattice (the dual lattice of exposed faces of the original
cone), and (FF) the dual faces of a dual face $(\sigma/\tau)^{\vee}$ of
$\sigma^{\vee}$ are the dual faces of $\sigma^{\vee}$ contained in
$(\sigma/\tau)^{\vee}$, by Lemma B.1(3) and diagram (5). ∎

######  Corollary B.7.

Let $\Sigma$ be a dual face fan. Then the set $\Sigma^{\prime}$ of all faces
of cones in $\Sigma$ is a fan.

###### Proof.

Let $\Delta$ be the set of maximal cones in $\Sigma$. Clearly, the set of all
faces of cones in $\Delta$ is the set of all faces of cones in $\Sigma$. By
Lemma B.6 the intersection of any two cones in $\Delta$ is a common dual face,
in particular a common face. Therefore by Lemma B.5 the set of all faces of
cones in $\Delta$ is a fan. ∎

Clearly, $\Sigma\subseteq\Sigma^{\prime}$ with the same maximal cones and
hence the same support. If each maximal cone in $\Sigma$ is the dual of a
polyhedral cone then $\Sigma=\Sigma^{\prime}$ because each face of a cone in
$\Sigma$ is a dual face by Proposition B.3(6). This occurs, for example, with
the fans defining toric varities.

###### Remark B.8.

The converse is false: there are sets $\Delta$ of dual cones whose faces form
a fan, but whose dual faces do not form a dual face fan. For example, consider
$\Delta=\\{\sigma^{\vee},\tau^{\vee}\\}$ where $\sigma=\\{x<0,y>0\\}$ and
$\tau=\\{x\geq 0,y\geq 0\\}$ are cones in $\mathbb{R}^{2}$. Here
$\sigma^{\vee}\cap\tau^{\vee}=\\{x=0,y\geq 0\\}$ is a face of $\sigma^{\vee}$,
but not a dual face.

## References

  * [1]
  * [2] Klaus Altmann: _Torische Geometrie_ , Lecture HU Berlin, 1998.
  * [3] Toshitaka Aoki, Akihiro Higashitani, Osamu Iyama, Ryoichi Kase, Yuya Mizuno: _Fans and polytopes in tilting theory II: g-fans of rank 2_ , arXiv:2301.01498.
  * [4] Sota Asai: _Semibricks_ , Int. Math. Res. Not. IMRN 16, 4993–5054 (2020), also arXiv:1610.05860.
  * [5] Sota Asai: _The wall-chamber structures of the real Grothendieck groups_ , Adv. Math. 381, 107615 (2021), also arXiv:1905.02180.
  * [6] Apostolos Beligiannis, Idun Reiten: _Homological and Homotopical Aspects of Torsion Theories_ , Memoirs of the AMS 188 (2007).
  * [7] Grzegorz Bobiński, Christof Geiß, Andrzej Skowroński: _Classification of discrete derived categories_ , Cent. Eur. J. Math. 2, 19–49 (2004).
  * [8] Alexey Bondal, Michel van den Bergh: _Generators and representability of functors in commutative and noncommutative geometry_ , Moscow Math. J. 3, 1–36 (2003).
  * [9] Tom Bridgeland: _Scattering diagrams, Hall algebras and stability conditions_ , Algebr. Geom. 4, 523–561 (2017), also arXiv:1603.00416.
  * [10] Tom Bridgeland, Helge Ruddat: _Clusters, twistors and stability conditions_ , to appear.
  * [11] Thomas Brüstle, David Smith, Hipolito Treffinger: _Wall and Chamber Structure for finite-dimensional Algebras_ , Adv. Math. 354, 106746 (2019), also arXiv:1805.01880.
  * [12] Thomas Brüstle, Dong Yang: _Ordered exchange graphs_ , Advances in representation theory of algebras (ICRA XV Bielefeld), 135–193, EMS Ser. Congr. Rep. (2013), also arXiv:1302.6045.
  * [13] Bo Chen: _Dimension vectors in regular components over wild Kronecker quivers_ , Bull. Sci. Math. 137, 730–745 (2013), also arXiv:1209.0612.
  * [14] Laurent Demonet, Osamu Iyama, Gustavo Jasso: _$\tau$ -tilting finite algebras, bricks and $g$-vectors_, Int. Math. Res. Not. 3, 852–892 (2019), also arXiv:1503.00285.
  * [15] George Dimitrov, Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich: _Dynamical systems and categories_ , Contemp. Math. 621, 133–170 (2014), also arXiv:1307.8418.
  * [16] Günter Ewald: _Combinatorial Convexity and Algebraic Geometry_ , GTM 168, Springer (2011).
  * [17] Werner Geigle, Helmut Lenzing: _Perpendicular categories with applications to representations of sheaves_ , J. Algebra 144, 273–343 (1991).
  * [18] Jean-Baptiste Hiriart Urruty, Claude Lemaréchal: _Fundamentals of convex analysis_ , Springer (2001).
  * [19] Daniel Jaume, Rubén Puente: _Conjugacy for closed convex sets_ , Beitr. Algebra Geom. 46, 131–149 (2005).
  * [20] Alastair King: _Moduli of representations of finite dimensional algebras_ , Q. J. Math. 45, 515–530 (1994).
  * [21] Steffen Koenig, Dong Yang: _Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras_, Doc. Math. 19, 403–438 (2014), also arXiv:1203.5657.
  * [22] David Pauksztello, Manuel Saorín, Alexandra Zvonareva: _Contractibility of the stability manifold for silting-discrete algebras_ , Forum Math. 30, 1255–1263 (2018), also arXiv:1705.10604.
  * [23] Alexander Polishchuk: _Abelian varieties, Theta functions and the Fourier Transform_ , Cambridge University Press (2003).
  * [24] Alexander Polishchuk: _Constant families of t-structures on derived categories of coherent sheaves_ , Mosc. Math. J. 7(1), 109-134, 167, (2007), also arXiv:0606013.
  * [25] Shunya Saito: _The spectrum of Grothendieck monoid: classifying Serre subcategories and reconstruction theorem_ , arXiv:2206.15271.
  * [26] Gonzalo Tabuada: _Recent developments on noncommutative motives_ , Contemp. Math. 707, 143–173 (2018), also arXiv:1611.05439.

Contact: |<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
---|---|---
|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>