#  $\tau$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of
posets and concealed algebras

Erlend D. Børve  Institut Fourier, Université Grenoble Alpes, 100 rue des
mathématiques, 38610 Gières, France<EMAIL_ADDRESS>,  Jacob Fjeld
Grevstad  Department of Mathematical Sciences, NTNU, NO-7491 Trondheim,
Norway<EMAIL_ADDRESS>and  Endre S. Rundsveen∗ Department of
Mathematical Sciences, NTNU, NO-7491 Trondheim, Norway.  ∗Corresponding author
<EMAIL_ADDRESS>
###### Abstract.

We prove that any $\tau$-tilting finite incidence algebra of a finite poset is
representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a
finite simply connected poset is tame. As the converse of these assertions are
known to hold, we obtain characterizations of $\tau$-tilting finite incidence
algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both
results are proved using the theory of concealed algebras. The former will be
deduced from the fact that tame concealed algebras are $\tau$-tilting
infinite, and to prove the latter, we show that wild concealed algebras are
not $\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite
poset is wild if and only if it is not $\mathbf{g}$-tame, and prove a result
showing that there are relatively few possible counterexamples. In the
appendix, we determine the representation type of a $\tau$-tilting reduction
of a concealed algebra of hyperbolic type.

###### Key words and phrases:

partially ordered set, incidence algebra, tilting theory, concealed algebra,
$\tau$-tilting theory, $\mathbf{g}$-vector fan, $\mathbf{g}$-tame algebra,
$\tau$-tilting reduction

###### 2020 Mathematics Subject Classification:

16G20, 16G60, 16G70

E.D.B. acknowledges support from the French ANR grant CHARMS
(ANR-19-CE40-0017-02).

J.F.G. acknowledges support from The Research Council of Norway (RCN 301375).

###### Contents

  1. 0 Introduction
  2. 1 Generalities on incidence algebras of posets
  3. 2 $\tau$-tilting finiteness of incidence algebras
    1. 2.1 $\tau$-tilting theory
    2. 2.2 Tilted and concealed algebras
    3. 2.3 $\tau$-tilting finite incidence algebras are representation-finite
  4. 3 $\mathbf{g}$-tameness of incidence algebras and concealed algebras
    1. 3.1 Reduction from wild to hyperbolic type
    2. 3.2 Wild concealed algebras are not $\mathbf{g}$-tame
    3. 3.3 $\mathbf{g}$-tameness of incidence algebras
  5. A $\tau$-tilting reduction of concealed algebras

##  0\. Introduction

Historical notes and motivation

_Representation type._ Representation theory of finite-dimensional algebras is
at its core a tool to better understand algebras. Even though it can be traced
back further, its current state was formed by Gabriel [Gab72, Gab73, Gab80]
using his language of quivers. The quiver-theoretic manner of approaching
algebras provides the means of translating questions of algebraic structures
into questions of linear algebra. However, the main selling point was arguably
Gabriel’s classification of finite-dimensional hereditary algebras with
finitely many indecomposable modules in terms of Dynkin diagrams.

The question of how many indecomposable modules an algebra admits, has
historically been a driving factor in the study of algebras. The well known
Brauer–Thrall conjectures (see e.g. [Jan57, Rin80]) are directly tied to this
question. With regards to this question, algebras fall into one out of three
main categories of increasing complexity: finite, tame and wild type. Drozd
famously showed that any algebra is either tame or wild, but never both
[Dro80]. Hence, there is a dichotomy of algebras by representation type.

Around the same time as Gabriel proved his aforementioned result, the _Kyiv-
school_ studied and classified poset-representations of finite type [NRi72,
Kle72, NRi75]. This result was highly influential and valuable in
classifications of more general algebras, as a lot of results were proven by
reducing to these representations. The main example of interest for us is
Loupias’ classification of representation-finite incidence algebras of posets
[Lou75a, Lou75b].

_( $\tau$-)tilting_ Inspired by the methods used in Gabriel’s classification,
a new object was introduced, namely a tilting module [GP72, BGP73, APR79,
BB80, HR82], giving a method of constructing a new algebra from another while
carrying over known structure. This paved the way for a staggering amount of
progress, and is still an important tool for modern representation theorists.
It is indeed fundamental to the arguments in this paper.

In an effort to close gaps in the theory of tilting when the underlying
algebra is not hereditary, Adachi, Iyama and Reiten [AIR14] generalized it to
that of $\tau$-tilting theory. Their apporach took inspiration from the theory
of cluster algebras [FZ02a, FZ02b, BFZ05, FZ07], as well as cluster tilting
[BMR07]. By having a bijection with functorially finite torsion classes and
providing a module-theoretic interpretation of $2$-term silting objects,
$\tau$-tilting was soon accepted as an interesting and worthwhile direction of
study.

_$\mathbf{g}$ -representation type: $\tau$-tilting finiteness and
$\mathbf{g}$-tameness_ In a similar fashion as indecomposables, the amount of
$\tau$-rigid modules over an algebra has been closely studied. In fact, for
hereditary algebras this amount is directly related to the algebras
representation type. We say that an algebra is $\tau$-tilting finite (or
$\mathbf{g}$-finite) if it only admits a finite amount of $\tau$-rigid
indecomposable modules [DIJ19]. One easily observes that an infinite acyclic
component in the AR-quiver provides infinitely many $\tau$-rigid modules,
whence hereditary algebras of infinite representation type are $\tau$-tilting
infinite, since they admit two such components (one postprojective and one
preinjective).

Further, we say that an algebra is $\mathbf{g}$-tame if the
$\mathbf{g}$-vector fan is dense in the real Grothendieck group [AY23]. It has
been shown that tame representation type is sufficient for
$\mathbf{g}$-tameness [PYK23], and for hereditary algebras its in fact also
necessary [Hil06]. We do however have examples of $\mathbf{g}$-tame algebras
of wild representation type (for instance wild local algebras and certain
Jacobian algebras [Yur20]). At the time of writing, the authors are not
familiar with a definition of $\mathbf{g}$-wildness of an algebra, apart from
simply negating the notion of $\mathbf{g}$-tameness.

Having, for the hereditary case, two equivalent notions to finite and tame
representation type, it is natural to ask how big of a defect we might get for
less well-behaved algebras. We look at the slightly more rowdy class of
incidence algebras in this article. Both the classification of minimally
representation-infinite incidence algebras by Loupias [Lou75a, Lou75b] and the
criterion of tameness for incidence algebras by Leszczyński [Les03] will be
prominent in the following.

Original results

_$\tau$ -tilting finiteness._ Following the motivation outlined above we first
set forth on investigating when incidence algebras are $\tau$-tilting finite.
The strategy for our investigation relies on three observations. The first
being that representation-infinite concealed $\mathbf{k}$-algebras are
$\tau$-tilting infinite, see Lemma 2.6. The second being that the twelve
classes of minimal representation infinite incidence algebras provided by
Loupias (see Table 2.11.I on page 2.11.I) are tame concealed, see Corollary
2.12. The third and final being that the reduction techniques of Loupias
preserves $\tau$-tilting finiteness, see Lemma 2.14.

With this strategy we prove our first main result.

######  Theorem 2.15.

Let $\mathbf{k}$ be a field and let $(P,\leq)$ be a finite poset. The
incidence algebra $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is $\tau$-tilting
finite if and only if it is representation-finite.

_$\mathbf{g}$ -tameness._ Inspired by the situation for hereditary algebras
and the fact that $\tau$-tilting finite incidence algebras are representation
finite, we wondered whether $\mathbf{g}$-tame incidence algebras were tame as
well. When the underlying poset is simply connected, we answer this in the
affirmative. In the multiply connected case, we do not have a conclusive
answer, but we conjecture that the same will be true and provide some
evidence.

Mirroring the strategy for our first result we develop a means of reduction
here as well. We begin by slightly expanding a result of Happel and Unger
[HU89] telling us that for a wild concealed algebra we can find a suitable
factor algebra which is concealed of _hyperbolic type_ , see Definition 3.5.

######  Theorem 3.7.

Let $\mathbf{k}$ be an algebraically closed field. If $B$ is a concealed
$\mathbf{k}$-algebra of wild type, then there exists an idempotent $e\in B$
such that $B/(e)$ is concealed of hyperbolic type.

This is significant for us since $\mathbf{g}$-tameness is inherited by factor
algebras, [PYK23, Proposition 3.11]. Following this we show that for
hyperbolic hereditary algebras $H$ we can carry information from the
$\mathbf{g}$-vector fan over to their concealed algebras, Lemma 3.16. This,
together with Asai’s description of $\mathbf{g}$-vector fans through the wall
and chamber structure of $K_{0}(\operatorname{proj}\Lambda)$ [BST19, Asa21],
allows us to prove the following novel result.

######  Theorem 3.18.

Let $\mathbf{k}$ be an algebraically closed field and let $B$ be a wild
concealed $\mathbf{k}$-algebra. Then $B$ is not $\mathbf{g}$-tame.

This result is noteworthy in itself, but for us it is mainly in connection
with Leszczyński’s classification of wild incidence algebras [Les03, Theorem
1.4] where it really shines. Specifically, it leads to the following corollary
of Theorem 3.18.

######  Corollary 3.20.

Suppose that the field $\mathbf{k}$ is algebraically closed. Let $(P,\leq)$ be
a finite simply connected poset. Then the incidence $\mathbf{k}$-algebra
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is $\mathbf{g}$-tame if and only if
it is tame.

Corollary 3.20 is only concerned with simply connected posets. In 3.21, we
postulate that the result holds for any incidence algebras of a finite poset.
To justfy that our conjecture is a reasonable one, we include Proposition
3.25, where we argue that there are relatively few counterexamples, if any at
all.

_$\tau$ -tilting reduction._ At the end of the article we include an
application of our results. That is, in Appendix A, we use Theorem 3.7 to
prove results on the $\tau$-tilting reduction [Jas15] of concealed algebras.
Our main result in this appendix will be the following.

######  Theorem A.5.

Let $Q$ be a hyperbolic quiver, and let
$\Lambda=\operatorname{End}_{\mathbf{k}Q}(T)$ be a concealed algebra of type
$Q$. For $(M,\,R)\in\operatorname{\tau\mathchar 45\relax
rigid{\,}pair}\left(\Lambda\right)$ the following assertions hold:

  1. (2)

If $|M|+|R|\geq 1$, then the $\tau$-tilting reduction with respect to
$(M,\,R)$ is tame (or representation-finite).

  2. (3)

If $|M|+|R|\geq 2$, then the $\tau$-tilting reduction with respect to
$(M,\,R)$ is representation-finite.

Notation and conventions.  A field $\mathbf{k}$ is fixed throughout. All
modules over a finite-dimensional $\mathbf{k}$-algebra will be left modules,
as opposed to right modules. For a finite-dimensional $\mathbf{k}$-algebra
$\Lambda$, let $\operatorname{Mod}(\Lambda)$ denote the category of left
$\Lambda$-modules, and let $\operatorname{mod}(\Lambda)$ denote the
subcategory of finite-dimensional $\Lambda$-modules. Arrows in a quiver are
composed from right to left. A $\mathbf{k}$-linear category is an enriched
category over $\operatorname{Mod}(\mathbf{k})$. It need not admit direct sums.

### Acknowledgements.

We are grateful to Aslak Buan, Eric Hanson, Kaveh Mousavand, Steffen
Oppermann, Pierre-Guy Plamondon and Håvard Terland for insightful discussions.
E.D.B acknowledges support from the French ANR grant CHARMS (ANR-19-CE400017).
J.F.G. acknowledges support from The Research Council of Norway (RCN 301375).
Some of our diagrams were made with the help of https://q.uiver.app/.

##  1\. Generalities on incidence algebras of posets

A partially ordered set, or poset for short, is a tuple $(P,\leq)$, where $P$
is a set and $\leq$ is a binary relation on $P$ which is reflexive, transitive
and anti-symmetric. When the binary relation $\leq$ is clear from context, or
if the poset in question is arbitrary, we may omit $\leq$ from the notation,
so a reference to the underlying set $P$ is meant as reference to the entire
structure. An interval in a poset $P$ is a subposet of the form

$[a,b]\coloneqq\\{x\,\,|\,\,a\leq x\leq b\\}\subseteq P,$

where $a,b\in P$. We denote the set of non-empty intervals in $P$ by
$\operatorname{int}(P)$. A poset is locally finite if every interval therein
is finite.

######  Definition 1.1 ([Rot64, §3]).

Let $(P,\leq)$ be a locally finite poset. Denote the set of functions from
$\operatorname{int}(P)$ to $\mathbf{k}$ by
$\mathbf{k}^{\operatorname{int}(P)}$, and endow this set with a
$\mathbf{k}$-vector space structure using pointwise operations. The incidence
algebra of $P$ over $\mathbf{k}$ has $\mathbf{k}^{\operatorname{int}(P)}$ as
the underlying $\mathbf{k}$-vector space, and multiplication

${\mathbf{k}^{\operatorname{int}(P)}\otimes_{\mathbf{k}}\mathbf{k}^{\operatorname{int}(P)}}$${\mathbf{k}^{\operatorname{int}(P)}}$$\scriptstyle{\ast}$

given by $(f\ast g)([a,b])\coloneqq\sum\limits_{t\in[a,b]}f([t,b])g([a,t])$.
We denote the incidence $\mathbf{k}$-algebra of $P$ by
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$.

It is easy to see that the incidence algebra of $P$ is finite-dimensional
precisely when $P$ is finite. Indeed, it is equivalent to the assertion that
$\operatorname{int}(P)$ is a finite set. In this case, one can present the
$\mathbf{k}$-algebra $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ as the
quotient of the free $\mathbf{k}$-algebra generated by $\operatorname{int}(P)$
modulo the relations $\\{[c,d][a,b]-\delta_{b,c}[a,d]\\}$, where
$\delta_{\mathchar 45\relax,\mathchar 45\relax}$ is the Kronecker delta
symbol. In the following, all posets we consider will be finite.

We consistently refer to oriented multigraphs as quivers. Paths in a quiver
are composed from right to left. The Hasse quiver of a poset $P$ is the quiver
where the vertices are the elements in $P$, and we have an arrow $x\rightarrow
y$ if $x\leq y$ and no element lies strictly between $x$ and $y$. We will
denote this quiver by $Q_{P}$. Letting $I_{P}$ denote the ideal of the path
$\mathbf{k}$-algebra $\mathbf{k}Q_{P}$ generated by the relations
$\\{p_{1}-p_{2}\\}_{p_{1},p_{2}}$, where we index over all pairs of paths with
the same start and endpoints, one easily proves that
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is isomorphic to the bound path
$\mathbf{k}$-algebra $\mathbf{k}Q_{P}/I_{P}$. As a consequence, a left module
over the incidence algebra $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ can be
regarded as a covariant representation of the bound quiver $(Q_{P},I_{P})$
[Lad08, Lemma 2.7]. Finite-dimensional representations of $(Q_{P},I_{P})$
correspond exactly to finite-dimensional
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$-modules.

######  Remark 1.2.

We will refrain from referring to
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$-modules as “representations of
$P$.” For many authors, a representation of a poset $P$ in a
$\mathbf{k}$-vector space $V$ amounts to an order-preserving map from $P$ into
the poset of subspaces of $V$ [Rot64, §8] [Dro74, §1] [Naz74]. The term
$P$-space has also been used for this notion of representation. Equivalently,
they can be regarded as socle-projective modules of the incidence
$\mathbf{k}$-algebra of $P^{\star}$, where $P^{\star}$ is obtained from $P$ by
adding a maximal element. The maps in such representations of the bound quiver
$(Q_{P},I_{P})$ are all injective, which we, in contrast, will not require.

######  Example 1.3.

Let $n\geq 1$. A poset is of type $\mathbb{A}_{n}$ if it has $n$ elements and
the underlying unoriented graph of its Hasse quiver takes the form

${\bullet}$${\bullet}$${\cdots}$${\bullet.}$ ($\mathbb{A}_{n}$)

The incidence $\mathbf{k}$-algebra of such a poset is isomorphic to the path
algebra of the correspondingly oriented $\mathbb{A}_{n}$-quiver. Examples
include the totally ordered sets, whose Hasser quivers are given by

${1}$${2}$${\cdots}$${n}$ ($\overrightarrow{\mathbb{A}}_{n}$)

Their incidence algebras are isomorphic to path algebras of linearly ordered
$\mathbb{A}_{n}$-quivers. ∎

Given a connected bound quiver $(Q,I)$, let $N(Q,I)$ be the classifying space
of the category defined as the quotient of the free category of $Q$, modulo
the ideal identifying two paths $p_{1}$ and $p_{2}$ if there exists a minimal
relation in $I$ of the form $\sum_{p\in L}\lambda_{p}p$, where $p_{1},p_{2}\in
L$. If the bound quiver $(Q_{P},I_{P})$ is the bound Hasse quiver of a poset
$P$, then $N(Q_{P},I_{P})$ is simply the classifying space of $P$. A connected
bound quiver $(Q,I)$ is connected if $N(Q,I)$ is connected as a topological
space. We define the fundamental groupoid $\Pi_{1}(Q,I)$ of $(Q,I)$ as the
fundamental groupoid of $N(Q,I)$. A connected component of $Q$ can be defined
as a component of $\Pi_{1}(Q,I)$. One says that $(Q,I)$ is simply connected
(resp. multiply connected) if is connected and the fundamental groupoid
$\Pi_{1}(Q,I)$ is equivalent to the trivial groupoid (resp. not equivalent to
the trivial groupoid). We say that a finite poset $P$ is connected (resp.
simply connected, resp. multiply connected) if the bound quiver
$(Q_{P},I_{P})$ is connected (resp. simply connected, resp. multiply
connected). Posets admitting a unique maximal element, or a unique minimal
element, are simply connected.

##  2\. $\tau$-tilting finiteness of incidence algebras

###  2.1. $\tau$-tilting theory

$\tau$-tilting theory was developed by Adachi–Iyama–Reiten [AIR14]. We recall
the elementary definitions.

Let $\Lambda$ be a finite-dimensional $\mathbf{k}$-algebra. Recall that a
$\Lambda$-module is basic if no indecomposable direct summand is repeated in
its decomposition into indecomposable direct summands. A pair $(M,P)$ of
$\Lambda$-modules is basic if both $M$ and $P$ are basic as $\Lambda$-modules.

######  Definition 2.1.

Let $\Lambda$ be a finite-dimensional algebra over a field $\mathbf{k}$.

  1. (1)

A finitely generated left $\Lambda$-module $M$ is said to be rigid if
$\operatorname{Ext}_{\Lambda}^{1}({M},\,{M})=0$.

  2. (2)

A finitely generated left $\Lambda$-module $M$ is said to be $\tau$-rigid if
$\operatorname{Hom}_{\Lambda}({M},\,{\tau M})=0$, where $\tau$ is the
Auslander–Reiten translation of $\Lambda$. The set of isomorphism classes of
basic $\tau$-rigid $\Lambda$-modules is denoted by
$\operatorname{\tau\mathchar 45\relax rigid}\left(\Lambda\right)$.

  3. (3)

A pair $(M,\,P)$ is support $\tau$-rigid if $M$ is a $\tau$-rigid module and
$P$ is a finitely generated projective left $\Lambda$-module such that
$\operatorname{Hom}_{\Lambda}({P},\,{M})=0$. The set of isomorphism classes of
basic $\tau$-rigid pairs in $\operatorname{mod}(\Lambda)$ is denoted by
$\operatorname{\tau\mathchar 45\relax rigid{\,}pair}\left(\Lambda\right)$.

  4. (4)

A support $\tau$-rigid pair $(M,\,P)$ is support $\tau$-tilting if
$|M|+|P|=|\Lambda|$, where $|X|$ is the number of non-isomorphic
indecomposable direct summands of a left $\Lambda$-module $X$. The set of
isomorphism classes of basic support $\tau$-tilting pairs in
$\operatorname{mod}(\Lambda)$ is denoted by $\operatorname{s\tau\mathchar
45\relax tilt{\,}pair}\left(\Lambda\right)$.

  5. (5)

A left $\Lambda$-module $M$ is support $\tau$-tilting if there exists a
projective left $\Lambda$-module $P$ such that $(M,\,P)$ is a support
$\tau$-tilting pair. The set of isomorphism classes of basic support
$\tau$-tilting $\Lambda$-modules is denoted by $\mathrm{s\tau\mathchar
45\relax tilt}\left(\Lambda\right)$.

  6. (6)

A support $\tau$-rigid pair $(N,\,Q)$ is a direct summand of the support
$\tau$-rigid pair $(M,\,P)$ if $N$ is a direct summand of $M$ and $Q$ is a
direct summand of $P$.

######  Definition 2.2 ([DIJ19, Definition 1.1]).

We say that a finite-dimensional algebra $\Lambda$ is $\tau$-tilting finite if
the set $\mathrm{s\tau\mathchar 45\relax tilt}\left(\Lambda\right)$ is finite
(or equivalently that $\operatorname{\tau\mathchar 45\relax
rigid}\left(\Lambda\right)$ is finite). If $\Lambda$ is not $\tau$-tilting
finite, it will be said to be $\tau$-tilting infinite.

###  2.2. Tilted and concealed algebras

For a finite-dimensional $\mathbf{k}$-algebra $\Lambda$ and a finite-
dimensional $\Lambda$-module $M$ let $\operatorname{add}(M)$ denote the
smallest additive subcategory of $\operatorname{mod}(\Lambda)$ containing $M$.

######  Definition 2.3.

Let $\Lambda$ be a finite-dimensional $\mathbf{k}$-algebra. We say that
$T\in\operatorname{mod}(\Lambda)$ is a tilting $\Lambda$-module if the
following hold:

  1. (1)

The $\Lambda$-module $T$ is rigid.

  2. (2)

There exists a short-exact sequence in $\operatorname{mod}(\Lambda)$

${0}$${P^{T}_{1}}$${P^{T}_{0}}$${T}$${0}$

where the terms $P^{T}_{1}$ and $P^{T}_{0}$ are finite-dimensional projective
$\Lambda$-modules.

  3. (3)

There exists a short-exact sequence in $\operatorname{mod}(\Lambda)$

${0}$${\Lambda}$${{T}_{0}}$${T_{1}}$${0}$

where the terms $T_{0}$ and $T_{1}$ are in $\operatorname{add}(T)$.

Let $Q$ be an acyclic quiver. Recall that if the path $\mathbf{k}$-algebra
$\mathbf{k}Q$ is representation-infinite, the set of indecomposable
$\mathbf{k}Q$-modules can be partitioned into three distinct classes:

  * •

the postprojective $\mathbf{k}Q$-modules, namely the $\mathbf{k}Q$-modules $M$
for which there exists an indecomposable finite-dimensional projective
$\mathbf{k}Q$-module $P$ and a non-negative integer $i$ such that
$M\simeq\tau^{-i}P$.

  * •

the preinjective $\mathbf{k}Q$-modules, namely the $\mathbf{k}Q$-modules $M$
for which there exists an indecomposable finite-dimensional injective
$\mathbf{k}Q$-module $I$ and a non-negative integer $i$ such that
$M\simeq\tau^{i}I$.

  * •

the regular $\mathbf{k}Q$-modules, that are neither postprojective nor
preinjective.

######  Definition 2.4.

Let $\Lambda$ be a finite-dimensional $\mathbf{k}$-algebra.

  1. (1)

We say that $\Lambda$ has a postprojective component if the Auslander–Reiten
quiver of $\operatorname{mod}(\Lambda)$ has an acyclic connected component in
which every indecomposable module is isomorphic to a module of the form
$\tau^{-i}P$ for some indecomposable finite-dimensional projective
$\Lambda$-module $P$ and some non-negative integer $i$.

  2. (2)

We say that $\Lambda$ has a preinjective component if the Auslander–Reiten
quiver of $\operatorname{mod}(\Lambda)$ has an acyclic connected component in
which every indecomposable module is isomorphic to a module of the form
$\tau^{i}I$ for some indecomposable finite-dimensional injective
$\Lambda$-module $I$ and some non-negative integer $i$.

Path $\mathbf{k}$-algebras of quivers admit postprojective and preinjective
components. For each connected component of $Q$ there is a postprojective
component in the AR-quiver of $\mathbf{k}Q$, and likewise for the
preinjectives. Such a connected component $Q^{\prime}$ of $Q$ yields an
infinite postprojective (resp. preinjective) component if and only if
$Q^{\prime}$ is a representation-infinite quiver.

######  Definition 2.5.

Let $Q$ be an acyclic quiver.

  1. (1)

A tilted $\mathbf{k}$-algebra of type $Q$ is isomorphic to
$\operatorname{End}_{\mathbf{k}Q}(T)$ for some tilting $\mathbf{k}Q$-module
$T$.

  2. (2)

If $T$ is a postprojective tilting $\mathbf{k}Q$-module, we say that the
tilted algebra (and any $\mathbf{k}Q$-algebra to which it is isomorphic) is a
concealed $\mathbf{k}$-algebra of type $Q$.

  3. (3)

Let $\Lambda=\operatorname{End}_{\mathbf{k}Q}(T)$ be a concealed
$\mathbf{k}$-algebra of type $Q$. We say that $\Lambda$ is tame concealed
(resp. wild concealed) if $Q$ is a tame and representation-infinite (resp.
wild) quiver.

Let $Q$ be a representation-infinite acyclic quiver, and let $\Lambda$ be a
concealed $\mathbf{k}$-algebra of type $Q$. Then $\Lambda$ admits one
postprojetive component (resp. preinjective component) for every connected
component of the $\Lambda$ (i.e. if $\Lambda$ is a product of $m$ connected
$\mathbf{k}$-algebras, there are $m$ components), and at least one which
contains (countably) infinitely many indecomposable $\Lambda$-modules.

The following lemma is well known and immediately proved (c.f. [Mou23, Remark
2.9]).

######  Lemma 2.6.

Let $\Lambda$ be a tame concealed or a wild concealed $\mathbf{k}$-algebra.
Then $\Lambda$ is $\tau$-tilting infinite.

###### Proof.

Tame concealed and wild concealed $\mathbf{k}$-algebras admit at least one
infinite postprojective (and at least one infinite preinjective) component. It
follows easily from acyclicity that the indecomposable modules constituting
postprojective or preinjective components are all $\tau$-rigid. Thus, the
postprojective (or preinjective) component of $\Lambda$ provides a (countably)
infinite set of $\tau$-rigid $\Lambda$-modules. ∎

For a full subcategory $\mathcal{X}$ of $\operatorname{mod}(\Lambda)$, let
$\operatorname{Fac}\mathcal{X}$ denote the smallest full subcategory of
$\operatorname{mod}(\Lambda)$ containing $\mathcal{X}$ which is closed under
factor modules, and let $\mathcal{X}^{\perp}$ (resp.
${{}^{\perp}\mathcal{X}}$) denote the full subcategory of
$\operatorname{mod}(\Lambda)$ spanned by the $\Lambda$-modules $Y$ for which
$\operatorname{Hom}_{\Lambda}(X,Y)=0$ (resp.
$\operatorname{Hom}_{\Lambda}(Y,X)=0$) for all $X\in\mathcal{X}$. When
$\mathcal{X}$ is of the form $\operatorname{add}(T)$ for some $\Lambda$-module
$T$, we simplify the notation to $\operatorname{Fac}{T}$ and $T^{\perp}$,
respectively. Recall that a torsion class (resp. torsion-free class) of
$\operatorname{mod}(\Lambda)$ is a full subcategory of
$\operatorname{mod}(\Lambda)$ which is closed under factor modules (resp.
submodules) and extensions. A torsion pair in $\operatorname{mod}(\Lambda)$ is
a pair $(\mathcal{T},\,\mathcal{F})$ of full subcategories of
$\operatorname{mod}(\Lambda)$ such that $\mathcal{F}=\mathcal{T}^{\perp}$ and
$\mathcal{T}={{}^{\perp}\mathcal{F}}$. In a torsion pair
$(\mathcal{T},\,\mathcal{F})$, we have that $\mathcal{T}$ is a torsion class
and that $\mathcal{F}$ is a torsion-free class [Dic66]. A torsion pair
$(\mathcal{X},\,\mathcal{Y})$ in $\operatorname{mod}(\Lambda)$ is split if all
indecomposable finite-dimensional $\Lambda$-modules are either in
$\mathcal{X}$ or in $\mathcal{Y}$.

Tilting theory lets us compute the module categories of tilted
$\mathbf{k}$-algebras. This is thanks to the Brenner–Butler Theorem, namely
the following.

######  Theorem 2.7 (Brenner–Butler equivalences, [BB80, Bon81, HR82]).

Let $Q$ be an acyclic quiver, and let $T$ be a tilting $\mathbf{k}Q$-module.
Let $\Lambda=\operatorname{End}_{\mathbf{k}Q}(T)$ denote the tilted algebra
obtained from $T$. There is a torsion pair
$(\mathcal{T}_{T},\,\mathcal{F}_{T})\coloneqq(\operatorname{Fac}{T},\,T^{\perp})$
in $\operatorname{mod}{\mathbf{k}Q}$, and a split torsion pair
$(\mathcal{X}_{T},\,\mathcal{Y}_{T})$ in $\operatorname{mod}(\Lambda)$, such
that the following functors are mutually inverse additive equivalences:

$\mathcal{T}_{T}$$\mathcal{Y}_{T}$$\operatorname{Hom}_{\mathbf{k}Q}(T,\,-)$
$T\otimes_{\Lambda}-$
and$\mathcal{F}_{T}$$\mathcal{X}_{T}$.$\operatorname{Ext}_{\mathbf{k}Q}^{1}(T,\,-)$
$\operatorname{Tor}_{1}^{\Lambda}(T,\,-)$

An immediate corollary of Theorem 2.7 is the following: if $Q$ is a Dynkin
quiver, then any tilted $\mathbf{k}$-algebra of type $Q$ is representation-
finite.

###  2.3. $\tau$-tilting finite incidence algebras are representation-finite

Recall that a finite-dimensional $\mathbf{k}$-algebra $\Lambda$ is called
representation-finite if there are only finitely many isomorphism classes of
indecomposable left $\Lambda$-modules.

######  Remark 2.8.

For incidence algebras of posets, the choice of field does not matter. Indeed,
the incidence $\mathbf{k}$-algebra $\mathcal{I}_{\mathbf{k}}\left({P}\right)$
of a poset $P$ is representation-finite for some field $\mathbf{k}$ precisely
when it is representation-finite for all fields $\mathbf{k}$. This is shown by
setting up a correspondence between posets with representation-finite
incidence $\mathbf{k}$-algebras and hammocks [RV87, Corollary 4], the latter
being independent of the field $\mathbf{k}$.

Loupias devises two reduction procedures of incidence algebras that preserve
representation-finiteness [Lou75a, Propositions 1.2 and 1.3]. Given a poset
$P$, a subposet of $P$ is also called a subtraction of $P$. A poset $L$ is a
contraction of $P$ if there exists a surjective order-preserving map
$P\rightarrow L$ with connected fibers. An elementary contraction $c\colon
P\rightarrow R$ is a contraction for which there exists exactly one element
$x\in R$ such that $|c^{-1}(x)|=2$ and $|c^{-1}(r)|=1$ for all $r\in
R\setminus\\{x\\}$. Informally, an elementary contraction identifies two
adjacent elements in $P$. Thus, there are as many elementary contractions as
there are arrows in the Hasse quiver of $P$. It can be shown that all
contractions are composites of elementary contractions.

We say that a poset $P$ can be reduced to $L$ if it is the case that $L$ can
be obtained from $P$ by subtraction and contraction in a finite number of
steps. We may assume that each step either removes a single vertex or
contracts an arrow in the Hasse quiver.

If $L$ is a subposet of $P$, one sets up a $\mathbf{k}$-algebra isomorphism

$\mathcal{I}_{\mathbf{k}}\left({L}\right)\simeq
e\mathcal{I}_{\mathbf{k}}\left({P}\right)e,$

where $e$ is the idempotent $\sum_{i\in L}e_{i}$ and the terms $e_{i}$ in the
sum are primitive idempotents corresponding to the elements in $L$.
Consequently, we have fully faithful functor

$\operatorname{mod}(\mathcal{I}_{\mathbf{k}}\left({L}\right))\simeq\operatorname{mod}(e\mathcal{I}_{\mathbf{k}}\left({P}\right)e)\ext@arrow
<EMAIL_ADDRESS>e}({e\Lambda},\,{-})}\operatorname{mod}(\mathcal{I}_{\mathbf{k}}\left({P}\right)).$
(2.a)

If $R$ is a contraction of $P$, we have, by definition, an epimorphism of
posets $P\twoheadrightarrow R$. Since the module category
$\operatorname{mod}(\mathcal{I}_{\mathbf{k}}\left({P}\right))$ is equivalent
to the functor category $\operatorname{mod}(\mathbf{k})^{P}$, and similarly
$\operatorname{mod}(\mathcal{I}_{\mathbf{k}}\left({R}\right))\cong\operatorname{mod}(\mathbf{k})^{R}$,
one can induce a functor

$\operatorname{mod}(\mathcal{I}_{\mathbf{k}}\left({R}\right))\lhook\mathrel{\mkern-3.0mu}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow\operatorname{mod}(\mathcal{I}_{\mathbf{k}}\left({P}\right))$
(2.b)

directly from the map $P\twoheadrightarrow R$. Since this is an epimorphism
with connected fibers, it follows that the functor (2.b) is fully faithful
[Lou75b, Proposition 8]. The existence of the fully faithful functors
displayed in (2.a) and (2.b) is all one needs in order to prove the following
lemma.

######  Lemma 2.9 ([Lou75a, Propositions 1.2 and 1.3]).

Let $P$ be a poset such that the incidence $\mathbf{k}$-algebra
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is representation-finite.

  1. (1)

Any subposet of $P$ has a representation-finite incidence
$\mathbf{k}$-algebra.

  2. (2)

Any contraction of $P$ has a representation-finite incidence
$\mathbf{k}$-algebra.

######  Example 2.10.

Let $P$ be a poset of _type $\widetilde{\mathbb{A}}_{n}$_, where $n\geq 3$,
namely a poset with Hasse quiver as shown below, where the undirected arrows
may go in either direction:

$m_{1}$$a_{1}$$a_{l}$$m_{2}$$a_{l+1}$$a_{n-1}$ ($\widetilde{\mathbb{A}}_{n}$)

As a matter of definition, posets of type $\widetilde{\mathbb{A}}_{n}$ have
circular Hasse quivers. Note that there are at least two minimal elements,
denoted here by $m_{1}$ and $m_{2}$. Such posets can be contracted onto a
poset of type $\widetilde{\mathbb{A}}_{3}$, namely a poset of the following
form:

$m_{1}$$\\{a_{1},\,\ldots,\,a_{l}\\}$$m_{2}$$\\{a_{l+1},\,\ldots,\,a_{n-1}\\}$
($\widetilde{\mathbb{A}}_{3}$)

Similarly, a poset of _type $\widetilde{\mathbb{D}}_{n}$_, whose underlying
undirected graph is of the form

$a_{1}$$a_{2}$$d_{1}$$d_{n-3}$$b_{1}$$b_{2}$ ($\widetilde{\mathbb{D}}_{n}$)

where $n\geq 4$, can be contracted onto a poset of type
$\widetilde{\mathbb{D}}_{4}$, namely a poset for which the underlying
unoriented graph of the Hasse quiver admits the following shape:

$a_{1}$$a_{2}$$\\{d_{1},\,\ldots,\,d_{n-3}\\}$$b_{1}$$b_{2}$
($\widetilde{\mathbb{D}}_{4}$)

∎

We say that an incidence $\mathbf{k}$-algebra of a poset is minimally
representation-infinite if it is representation-infinite and all of its proper
subposets and non-trivial contractions have representation-finite incidence
$\mathbf{k}$-algebras. Now that the reduction techniques in Lemma 2.9 are
established, it is clear that all representation-infinite incidence
$\mathbf{k}$-algebras can be reduced to a minimal one, and that it suffices to
classify the minimally representation-infinite incidence $\mathbf{k}$-algebras
in order to distinguish representation-finiteness from representation-
infiniteness.

######  Theorem 2.11 ([Lou75a, Theorems 1.4 and 1.5]).

A finite poset $P$ has a representation-finite incidence $\mathbf{k}$-algebra
if and only if it cannot be reduced to a poset whose Hasse quiver is taken
from the list in Table 2.11.I, nor from the list of the opposites of these. In
other words, any minimally representation-infinite incidence
$\mathbf{k}$-algebra falls into one of these twelve types.

$\widetilde{\mathbb{A}}_{3}$ |  | $\widetilde{\mathbb{D}}_{4}$ |
---|---|---|---
$\widetilde{\mathbb{E}}_{6}$ |  | $\widetilde{\mathbb{E}}_{7}$ |
$\widetilde{\mathbb{E}}_{8}$ |  | $\mathbb{R}_{1}$ |
$\mathbb{R}_{2}$ |  | $\mathbb{R}_{3}$ |
$\mathbb{R}_{4}$ |  | $\mathbb{R}_{5}$ |
$\mathbb{R}_{6}$ |  | $\mathbb{R}_{7}$ |
Table 2.11.I. Hasse quivers of posets having minimally representation-infinite
incidence $\mathbf{k}$-algebras. Undirected edges indicate that the arrows may
go in either direction.

######  Corollary 2.12.

A finite poset $P$ has a representation-infinite incidence
$\mathbf{k}$-algebra if and only if it can be reduced to a poset $L$ for which
the incidence algebra $\mathcal{I}_{\mathbf{k}}\left({L}\right)$ is tame
concealed.

###### Proof.

Suppose that $P$ has a representation-infinite incidence $\mathbf{k}$-algebra.
Then by Theorem 2.11 $P$ can be reduced to one of the posets listed in Table
2.11.I. It can be shown that the incidence $\mathbf{k}$-algebras of the listed
posets are all tame concealed. Indeed, all of them (and their opposites)
appear on Happel–Vossieck’s list of tame concealed algebras [HV83], where we
find that the incidence algebra of type $\mathbb{R}_{i}$, where $1\leq i\leq
7$, is tame concealed of type $\widetilde{\mathbb{E}}$. This proves necessity.

Suppose that a finite number of subtractions and contractions of $P$ yields a
poset $L$ such that $\mathcal{I}_{\mathbf{k}}\left({L}\right)$ is tame
concealed. Since tame concealed algebras are representation-infinite, it
follows from Lemma 2.9 that $P$ has a representation-infinite incidence
$\mathbf{k}$-algebra. This proves the converse. ∎

######  Example 2.13.

To familiarize the reader with the notions recalled above, we will now apply
Theorem 2.11 to rediscover when the posets of the following form

$\prod\limits_{i=1}^{\ell}\overrightarrow{\mathbb{A}}_{n_{i}}$

have representation-finite incidence $\mathbf{k}$-algebras, as originally
shown by Leszczyński [Les94, Proposition 2.1 and Theorem 2.4]. We remind the
reader that the poset $\overrightarrow{\mathbb{A}}_{n_{i}}$ is totally ordered
and has $n_{i}$ elements.

We consider posets of the form
$\overrightarrow{\mathbb{A}}_{n}\times\overrightarrow{\mathbb{A}}_{m}$ first,
where $n\leq m$. If $n\geq 3$, then
$\overrightarrow{\mathbb{A}}_{n}\times\overrightarrow{\mathbb{A}}_{m}$ has a
subposet of type $\widetilde{\mathbb{D}}_{4}$, and consequently
$\overrightarrow{\mathbb{A}}_{n}\times\overrightarrow{\mathbb{A}}_{m}$ is of
infinite representation type. The poset
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{5}$; also
has a representation-infinite incidence $\mathbf{k}$-algebra, since the
subposet displayed by the solid nodes and arrows below

$\bullet$$\star$$\star$$\bullet$$\circ$$\circ$$\bullet$$\star$$\star$$\bullet$

can be contracted onto the poset

$\star$$\bullet$$\bullet$$\bullet$$\bullet$

which is of type $\widetilde{\mathbb{D}}_{4}$. A poset of the form
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{m}$ can be
reduced to
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{5}$ as long
as $m\geq 5$, whence all of these posets have representation-infinite
incidence $\mathbf{k}$-algebras. One checks that
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{4}$ is
iterated tilted of type $\mathbb{E}_{8}$, so it has a representation-finite
incidence $\mathbf{k}$-algebra. If $n=2$ and $m\leq 4$, one uses Theorem 2.11
to argue that
$\overrightarrow{\mathbb{A}}_{n}\times\overrightarrow{\mathbb{A}}_{m}$ has a
representation-finite incidence $\mathbf{k}$-algebra. The remaining cases are
those where $n=1$, which yields the class of finite totally ordered posets,
and all of these certainly have a representation-finite incidence
$\mathbf{k}$-algebra.

The poset
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}$
has a representation-infinite incidence $\mathbf{k}$-algebra; indeed, the
subposet obtained by removing the maximal and minimal elements can be
contracted onto a poset of type $\widetilde{\mathbb{A}}_{3}$, as shown below.

$\circ$$\bullet$$\bullet$$\star$$\bullet$$\star$$\star$$\circ$$\bullet$$\bullet$$\star$$\bullet$

In fact, by [CR19, Lemma 3.5] (see also [Len99, Example 18.6.2]) we have that
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}$
has a tame incidence $\mathbf{k}$-algebra. More generally, if we take any
tensor product of three or more totally ordered sets, all with cardinality
greater than or equal to 2, the result will always have a representation-
infinite incidence $\mathbf{k}$-algebra. Indeed, they can all be reduced to
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}$.
∎

We now turn to the question of $\tau$-tilting finiteness of incidence
algebras. _A priori_ , the $\tau$-tilting finiteness of an incidence algebra
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ depends both on the field
$\mathbf{k}$ and the poset $P$. However, when Theorem 2.15 has been proven, we
will have established that it only depends on $P$; by Remark 2.8, the
representation-finiteness of incidence $\mathbf{k}$-algebras does not depend
on the field.

The proof of Theorem 2.15 will rely on a variation of Lemma 2.9. The statement
in Lemma 2.14( ‣ 2.14) is slightly more general than the cited reference
[Pla19, Corollary 2.2], but the proof will remain identical, _mutatis
mutandis_. The key ingredient in the proof is the fact that a
$\mathbf{k}$-algebra is $\tau$-tilting finite precisely when it has finitely
many isomorphism classes of bricks [DIJ19, Theorem 1.4]. Recall that an object
$B$ in a $\mathbf{k}$-linear category $\mathcal{U}$ is a brick if its
endomorphism algebra is a division algebra over $\mathbf{k}$.

######  Lemma 2.14.

Let $\mathbf{k}$ be a field and let $\Lambda$ be a $\tau$-tilting finite
$\mathbf{k}$-algebra.

  1. (0)

Let $\mathcal{U}\hookrightarrow\operatorname{mod}(\Lambda)$ be a fully
faithful functor of $\mathbf{k}$-linear categories. Then $\mathcal{U}$
contains finitely many bricks. In particular, if $\mathcal{U}$ is equivalent
to a module category $\operatorname{mod}(\Lambda^{\prime})$, then
$\Lambda^{\prime}$ is $\tau$-tilting finite.

Let $P$ be a finite poset such that the incidence $\mathbf{k}$-algebra
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is $\tau$-tilting finite.

  1. (1)

Any subposet of $P$ has a $\tau$-tilting finite incidence
$\mathbf{k}$-algebra.

  2. (2)

Any contraction of $P$ has a $\tau$-tilting finite incidence
$\mathbf{k}$-algebra.

###### Proof.

As noted above, the assertion in ( ‣ 2.14) is a slight generalization of work
of Plamondon [Pla19, Corollary 2.2], and is proved in an identical manner. The
other assertions then follow from the existence of the fully faithful functors
in (2.a) and (2.b). ∎

######  Theorem 2.15.

Let $P$ be a finite poset. Then $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is
$\tau$-tilting finite if and only if it is representation-finite.

###### Proof.

All representation-finite algebras are $\tau$-tilting finite. We prove that
the converse assertion holds for incidence $\mathbf{k}$-algebras of finite
posets.

Suppose that $P$ has a representation-infinite incidence $\mathbf{k}$-algebra.
By Corollary 2.12, one can reduce $P$ to a poset $L$ such that the incidence
$\mathbf{k}$-algebra $\mathcal{I}_{\mathbf{k}}\left({L}\right)$ is tame
concealed. Since tame concealed algebras are $\tau$-tilting infinite, as we
remarked in Lemma 2.6, it follows directly from Lemma 2.14( ‣ 2.14) that $P$
has a $\tau$-tilting infinite incidence $\mathbf{k}$-algebra. ∎

There are other approaches to proving Theorem 2.15. For instance, one can
appeal to the fact that a simply connected $\mathbf{k}$-algebra is
$\tau$-tilting finite if and only if it is representation-finite [Wan22,
Theorem 1.1]. Since all minimally representation-infinite incidence
$\mathbf{k}$-algebras, except $\widetilde{\mathbb{A}}_{3}$, are simply
connected, we immediately deduce the $\tau$-tilting infiniteness for eleven of
the twelve cases listed in Table 2.11.I. The exception, namely the poset
$\widetilde{\mathbb{A}}_{3}$, has a tame and hereditary incidence
$\mathbf{k}$-algebra, and such $\mathbf{k}$-algebras are $\tau$-tilting
infinite (c.f. Lemma 2.6), whereby we conclude the alternative proof.

The key insight leading to Theorem 2.15 was Corollary 2.12, where we showed
that minimally representation-infinite incidence $\mathbf{k}$-algebras are
concealed. Since any proper quotient of a tame concealed $\mathbf{k}$-algebra
is representation-finite [Bon17, Proposition 6], we deduce that these
incidence $\mathbf{k}$-algebras are _minimally $\tau$-tilting infinite
$\mathbf{k}$-algebras_, which is to state that they are $\tau$-tilting
infinite $\mathbf{k}$-algebras for which all proper quotients are
$\tau$-tilting finite [MP23].

######  Example 2.16.

Mitamoto and Wang have previously classified posets of the form
$\overrightarrow{\mathbb{A}}_{n}\times\overrightarrow{\mathbb{A}}_{m}$ that
have $\tau$-tilting finite incidence $\mathbf{k}$-algebras [MW21]. This is
achieved by appealing to the fact that path $\mathbf{k}$-algebras of the
$\mathbb{A}_{n}$-quivers are simply connected. In 2.13 it was stated that
$\overrightarrow{\mathbb{A}}_{n}\times\overrightarrow{\mathbb{A}}_{m}$ is
representation-finite if and only if

$1\in\\{n,m\\}\quad\text{or}\quad\\{n,m\\}\in\big{\\{}\\{2,2\\},\\{2,3\\},\\{2,4\\}\big{\\}}.$
(2.c)

By Theorem 2.15, it follows that the condition stated in (2.c) is necessary
and sufficient for
$\overrightarrow{\mathbb{A}}_{n}\times\overrightarrow{\mathbb{A}}_{m}$ to have
a $\tau$-tilting finite incidence $\mathbf{k}$-algebra. ∎

######  Remark 2.17.

A related concept to $\tau$-tilting finiteness is that of silting discreteness
[Aih13, Definition 3.6]. A silting discrete $\mathbf{k}$-algebra is
$\tau$-tilting finite, but the converse is not true in general. There are
incidence $\mathbf{k}$-algebras that are $\tau$-tilting finite, but not
silting discrete. For example, consider the incidence $\mathbf{k}$-algebra of
the poset with the Hasse quiver displayed below.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

We can observe that the incidence $\mathbf{k}$-algebra has a representation-
finite incidence $\mathbf{k}$-algebra, hence also $\tau$-tilting finite.
However, we may also observe that it is tilted of type
$\widetilde{\mathbb{D}}_{4}$. Using that silting discreteness only depends on
the derived category and that a path $\mathbf{k}$-algebra of a quiver of type
$\widetilde{\mathbb{D}}_{4}$ has a $\tau$-tilting infinite incidence
$\mathbf{k}$-algebra, we conclude that the incidence $\mathbf{k}$-algebra is
not silting discrete.

##  3\. $\mathbf{g}$-tameness of incidence algebras and concealed algebras

For the remainder of this text, we assume the field $\mathbf{k}$ to be
algebraically closed. The main result in this section will be the following.

######  Theorem 3.18.

Let $B$ be a wild concealed $\mathbf{k}$-algebra. Then $B$ is not
$\mathbf{g}$-tame.

We then deduce Corollary 3.20, asserting that the incidence
$\mathbf{k}$-algebra of a finite simply connected poset is $\mathbf{g}$-tame
if and only if it is tame. We first recall the notion of
$\mathbf{g}$-tameness, so that we may develop the proof of Theorem 3.18 in
Sections 3.13 and 3.2 below.

Before defining the $\mathbf{g}$-vector fan of a finite-dimensional
$\mathbf{k}$-algebra, we need to recall the combinatorial notion of polyhedral
fans [Zie12]. Fixing a positive integer $n$, consider an $n$-dimensional
vector space $V$. Given a set of vectors
$S=\\{v_{1},\dots,v_{\ell}\\}\subseteq V$, its positive span
$C_{S}\subseteq\mathbb{R}^{n}$ is called a polyhedral cone in $V$, or simply a
cone. Such a cone is said to be simplicial if it is of the form $C_{S}$, where
$S$ is a set of linearly independent vectors in $V$. The dimension of a
simplicial cone is the dimension of the subspace spanned by it. A subset
$T\subseteq S$ determines a face $C_{T}$ of the cone $C_{S}$. A polyhedral fan
in $V$ is given by a family of polyhedral cones in $V$ denoted
$\mathcal{S}=\\{{C}_{X}\\}_{X}$, enjoying the following two properties:

  1. (1)

The family $\mathcal{S}$ is closed under the formation of faces.

  2. (2)

The intersection of two elements $C_{1}$ and $C_{2}$ in $\mathcal{S}$ is in
$\mathcal{S}$, and it is a face to both $C_{1}$ and $C_{2}$.

A polyhedral fan $\mathcal{S}$ is simplicial if all of its cones are
simplicial, and essential if it contains the cone $\\{0\\}$. We say that
$\mathcal{S}$ is finite if the number of elements in $\mathcal{S}$ is finite,
and complete if every vector in $V$ lies inside some cone $C\in\mathcal{S}$.

Let $\operatorname{proj}(\Lambda)$ denote the full subcategory of
$\operatorname{mod}(\Lambda)$ consisting of finite-dimensional projective
$\Lambda$-modules. Define the _Grothendieck group_ of
$\operatorname{proj}(\Lambda)$ as the free abelian group spanned by the
isomorphism classes of objects in $\operatorname{proj}(\Lambda)$. It will be
denoted by $\mathrm{K}_{0}(\operatorname{proj}\Lambda)$. One shows that
$\mathrm{K}_{0}(\operatorname{proj}\Lambda)$ is a free abelian group of rank
$n$, where $n$ is the number of isomorphism classes of indecomposable
projective $\Lambda$-modules. Using the indecomposable finite-dimensional
projective $\Lambda$-modules as a basis for
$\mathrm{K}_{0}(\operatorname{proj}\Lambda)$, we can identify the group
$\mathrm{K}_{0}(\operatorname{proj}\Lambda)$ with $\mathbb{Z}^{n}$.

There is an injection

${\mathbf{g}^{(-)}\colon\mathrm{ind}(\operatorname{\tau\mathchar 45\relax
rigid{\,}pair}\left(\Lambda\right))}$${\mathrm{K}_{0}(\operatorname{proj}\Lambda)}$
(3.a)

from the set of indecomposable $\tau$-rigid pairs in
$\operatorname{mod}(\Lambda)$ into the Grothendieck group of
$\operatorname{proj}(\Lambda)$ [AIR14, Theorem 5.5]. This injection can be
explicitly defined as follows: let $(M,Q)$ be an indecomposable $\tau$-rigid
pair and let $P_{1}\mathrel{\smash{\meno}}\mathrel{\mkern-3.0mu}\rightarrow
P_{0}$ be a minimal projective presentation of $M$. Set
$\mathbf{g}^{(M,Q)}\coloneqq[P_{0}]-[P_{1}]-[Q]\in\mathrm{K}_{0}(\operatorname{proj}\Lambda)$.
We define the real Grothendieck group of $\Lambda$ by

$\mathrm{K}_{0}(\operatorname{proj}\Lambda)_{\mathbb{R}}\coloneqq\mathrm{K}_{0}(\operatorname{proj}\Lambda)\otimes_{\mathbb{Z}}\mathbb{R}.$

As an $\mathbb{R}$-vector space, it is isomorphic to $\mathbb{R}^{n}$, where
$n$ is the rank of the algebra. For an indecomposable $\tau$-rigid pair
$(M,Q)$, we call the element $\mathbf{g}^{(M,Q)}\otimes
1\in\mathrm{K}_{0}(\operatorname{proj}\Lambda)_{\mathbb{R}}$ its
$\mathbf{g}$-vector. Since the map in (3.a) injects, we have that
$\mathbf{g}$-vectors determine indecomposable $\tau$-rigid pairs uniquely. A
$\tau$-rigid $\Lambda$-module can thus be identified with the ray of the
vector $\mathbf{g}^{(M,Q)}$ in
$\mathrm{K}_{0}(\operatorname{proj}\Lambda)_{\mathbb{R}}$. Equivalently, we
can regard it as a point on the unit $(n-1)$-sphere $\mathbb{S}^{n-1}$.

For a (possibly decomposable) $\tau$-rigid pair $(M,Q)$ in
$\operatorname{mod}(\Lambda)$, let $C_{(M,Q)}$ denote the cone in
$\mathrm{K}_{0}(\operatorname{proj}\Lambda)_{\mathbb{R}}$ positively spanned
by the $\mathbf{g}$-vectors of the indecomposable direct summands of $(M,Q)$.
The $\mathbf{g}$-vector fan of $\Lambda$ is defined as a the polyhedral fan
consisting of the cones $C_{(M,Q)}$, where $(M,Q)$ is a $\tau$-rigid pair
$(M,Q)$ in $\operatorname{mod}(\Lambda)$. We denote the $\mathbf{g}$-vector
fan of $\Lambda$ by $\mathcal{F}_{\mathbf{g}}(\Lambda)$. It is a simplicial
and essential fan.

######  Theorem 3.1 ([Asa21, Theorem 4.7]).

Let $\Lambda$ be a finite-dimensional $\mathbf{k}$-algebra. The
$\mathbf{g}$-vector fan $\mathcal{F}_{\mathbf{g}}(\Lambda)$ is finite
precisely when it is complete. Equivalently, the $\mathbf{k}$-algebra
$\Lambda$ is $\tau$-tilting finite if and only if the $\mathbf{g}$-vector fan
$\mathcal{F}_{\mathbf{g}}(\Lambda)$ is complete.

Having classified all $\tau$-tilting finite incidence $\mathbf{k}$-algebras of
finite posets in Theorem 2.15, we deduce from Theorem 3.1 that we can
characterize the incidence $\mathbf{k}$-algebras of finite posets admitting a
complete $\mathbf{g}$-vector fan. We now move on to the case where the
$\mathbf{g}$-vector fan may be incomplete.

######  Definition 3.2.

[AY23] We say that a finite-dimensional $\mathbf{k}$-algebra $\Lambda$ is
$\mathbf{g}$-tame if its $\mathbf{g}$-vector fan is dense in $\mathbb{R}^{n}$,
i.e. for any $x\in\mathbb{R}^{n}$ and for all $\varepsilon>0$, the
$\varepsilon$-ball centered at $x$ intersects a cone in
$\mathcal{F}_{\mathbf{g}}(\Lambda)$.

By Theorem 3.1, all $\tau$-tilting finite $\mathbf{k}$-algebras are
$\mathbf{g}$-tame. Furthermore, any tame $\mathbf{k}$-algebra is
$\mathbf{g}$-tame [PYK23]. However, the converses of these two assertions are
false; there are examples of both tame $\tau$-tilting finite
$\mathbf{k}$-algebras (for example a tame local $\mathbf{k}$-algebra) and wild
$\mathbf{k}$-algebras that are $\mathbf{g}$-tame (for example a wild local
$\mathbf{k}$-algebra). For some nice classes of $\mathbf{k}$-algebras, the
notions of tameness and $\mathbf{g}$-tameness coincide.

######  Proposition 3.3 ([Hil06, Theorem 5.1(4)], [Yur23, Theorem 1.2]).

Let $H$ be a hereditary finite-dimensional $\mathbf{k}$-algebra. Then $H$ is
$\mathbf{g}$-tame if and only if it is tame.

Our contribution to the theory of $\mathbf{g}$-tameness in Theorem 3.18 will
build on known results on hereditary $\mathbf{k}$-algebras. Indeed, we will
generalize Proposition 3.3 to the context of concealed $\mathbf{k}$-algebras.

###  3.1. Reduction from wild to hyperbolic type

In order to prove Theorem 3.18, we devise a reduction technique from concealed
$\mathbf{k}$-algebras to smaller concealed $\mathbf{k}$-algebras. Most of the
current section is due to Happel–Unger [HU89], but there are some edge cases
that we need to handle more carefully. Specifically, concealed
$\mathbf{k}$-algebras of type
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}$ will
need special attention.

The following lemma will be paramount. Recall that a subposet $P^{\prime}$ of
a poset $P$ is convex in $P$ if the following condition holds: if $x,y\in
P^{\prime}$ then $z\subseteq P^{\prime}$ provided that $x\leq z\leq y$.

######  Lemma 3.4.

  1.   2. (1)

Let $\Lambda$ be a $\mathbf{g}$-tame $\mathbf{k}$-algebra and let
$e\in\Lambda$ be an idempotent. Then the factor algebra $\Lambda/(e)$ is
$\mathbf{g}$-tame [PYK23, Proposition 3.11].

  3. (2)

Let $P$ be a finite poset, and let $L$ be a convex subposet of $P$. If $P$ has
a $\mathbf{g}$-tame incidence $\mathbf{k}$-algebra, so does $L$.

###### Proof.

We deduce assertion (2) from assertion (1). Let $e=\sum_{x\in L}e_{x}$ be the
sum of the primitive idempotents corresponding to the elements in $L$. Since
$L$ is convex, it follows that $\mathcal{I}_{\mathbf{k}}\left({L}\right)$ is
isomorphic to $\mathcal{I}_{\mathbf{k}}\left({P}\right)/(1-e)$. Since $1-e$ is
an idempotent in the $\mathbf{g}$-tame algebra
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$, we have by (1) that the algebra
$\mathcal{I}_{\mathbf{k}}\left({P}\right)/(1-e)$ is $\mathbf{g}$-tame. We
conclude that $\mathcal{I}_{\mathbf{k}}\left({L}\right)$ is $\mathbf{g}$-tame.
∎

######  Definition 3.5.

A finite-dimensional $\mathbf{k}$-algebra $\Lambda$ is called hyperbolic if
the following conditions hold.

  1. (1)

The $\mathbf{k}$-algebra $\Lambda$ is wild.

  2. (2)

For all primitive idempotents $e\in\Lambda$, the factor algebra $\Lambda/(e)$
is tame.

An acyclic quiver $Q$ is hyperbolic if the path algebra $\mathbf{k}Q$ is
hyperbolic, or equivalently if $Q$ is wild and any proper subquiver of $Q$ is
a union of tame quivers. If $Q$ is a hyperbolic quiver, a concealed
$\mathbf{k}$-algebra of type $Q$ is called _hyperbolically concealed of type
$Q$_.111Our terminology is that of Kac [Kac83] rather than that of
Happel–Unger [HU89] and Kerner [Ker88], who refer to hyperbolic quivers as
minimally wild.

######  Remark 3.6.

It does not follow directly from the definition that hyperbolically concealed
$\mathbf{k}$-algebras are hyperbolic, but this is the case [Ker88, Proposition
4.1]. Kerner provides an exhaustive list of hyperbolic quivers [Ker88, Remark
following Theorem 4.1], which enables a classification of hyperbolically
concealed $\mathbf{k}$-algebras by listing all concealed $\mathbf{k}$-algebras
that arise from these quivers. Such a classification is given by Unger
[Ung90].

In this section we prove the following result.

######  Theorem 3.7.

Let $Q$ be a wild quiver, and let $B$ be a concealed $\mathbf{k}$-algebra of
type $Q$. Then there exists an idempotent $e\in B$ such that $B/(e)$ is
concealed of hyperbolic type.

######  Definition 3.8.

Let $Q$ be a quiver.

  1. (1)

A vertex $v$ in a quiver $Q$ is called a tip if it has precisely one
neighboring vertex, to which it is connected by exactly one edge.222Clearly, a
tip must either be a sink or a source of the quiver.

  2. (2)

The quiver $Q$ is twice edge-connected if every vertex of $Q$ is adjacent to
at least two edges, or equivalently if $Q$ does not have a tip.

The following results of Happel–Unger will be instrumental in our proof of
Theorem 3.7.

######  Lemma 3.9.

[HU89] Let $Q$ be a connected acyclic wild quiver with more than two vertices,
let $T$ be a postprojective tilting $\mathbf{k}Q$-module and let
$B=\operatorname{End}_{\mathbf{k}Q}(T)$ be the corresponding concealed
$\mathbf{k}$-algebra of type $Q$.

  1. (1)

If $Q$ is not twice edge-connected and $v$ is a tip of $Q$, then the subquiver
$Q^{\prime}=Q\setminus\\{v\\}$ is connected and representation-infinite [HU89,
§1.1].

  2. (2)

If $Q$ is twice edge-connected, there exists a vertex $v$ in $Q$ such that the
subquiver $Q^{\prime}=Q\setminus\\{v\\}$ is connected and representation-
infinite [HU89, §1.2].

  3. (3)

Let $v$ be as in (1) or (2), depending on whether $Q$ is twice edge-connected.
There exists an indecomposable direct summand $T_{1}$ of $T$ belonging to the
$\tau$-orbit of the projective $\mathbf{k}Q$-module $P(v)$ [HU89, §1.1 and
§1.2].

######  Lemma 3.10.

[HU89] Let $Q$ be a quiver, let $v$ be a vertex such that $Q^{\prime}\coloneqq
Q\setminus\\{v\\}$ is connected representation-infinite, and let $P(v)$ denote
the indecomposable projective $\mathbf{k}Q$-module corresponding to $v$. Let
$T$ be a postprojective tilting ${\mathbf{k}Q}$-module and let $T_{1}$ be an
indecomposable direct summand of $T$ such that $T_{1}=\tau^{-r}P(v)$ for some
$r$. Let $B=\operatorname{End}_{\mathbf{k}Q}(T)$ and let $e$ be the idempotent
such that $eB=\operatorname{Hom}(T,T_{1})$. Then $B/(e)$ is concealed of type
$Q^{\prime}$.

###### Proof.

This is proven in [HU89]. Note that they assume that $v$ is a tip whenever $Q$
is not twice edge-connected, but that this assumption is only used to
guarantee the existence of a a summand equal to $\tau^{-r}P(v)$, and is not
necessary for the remaining argument. ∎

We will also need some technical lemmata.

######  Lemma 3.11.

Let $Q$ be a wild tree. Suppose that each subquiver of the form
$Q\setminus\\{v\\}$, where $v$ is a tip, is a tame quiver. Then $Q$ is a
hyperbolic quiver.

###### Proof.

If $Q$ were not hyperbolic, there would exist an internal vertex $x$ such that
$Q\setminus\\{x\\}$ is wild. The quiver $Q\setminus\\{x\\}$ has more than one
connected component, and at least one of which is wild. Let $C_{w}$ be a wild
connected component of $Q\setminus\\{x\\}$, let $C$ be a different connected
component and let $v$ be a tip in $C$. Then $Q\setminus\\{v\\}$ contains
$C_{w}$ as a subquiver, making $Q\setminus\\{v\\}$ wild, contradicting our
assumption on the tips of $Q$. Having reached a contradiction under the
assumption of non-hyperbolicity, we conclude that $Q$ must be hyperbolic. ∎

The previous two lemmata are already enough for us to reduce concealed
algebras of tree-type. We now turn our attention to understanding
postprojective modules in algebras of type
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}$. These
observations seem to be a part of the folklore, but we find it beneficial to
state and prove them explicitly.

######  Lemma 3.12.

Let $H$ be the path $\mathbf{k}$-algebra of a quiver $Q$ of type
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}_{n}$
for $n>3$, and $a\in Q_{0}$ the unique tip of the quiver. Consider an
indecomposable $H$-module $M$ in the $\tau$-orbit of $P(a)$, then for $r\geq
1$ $\operatorname{Ext}_{H}^{1}({\tau^{-r}M},\,{M})=0$ if only if $r=2$.

###### Proof.

Through the Auslander–Reiten-formula and $\tau^{-}$-shift we see that

$\operatorname{Ext}_{H}^{1}({\tau^{-r}M},\,{M})\cong
D\operatorname{Hom}_{H}({P(a)},\,{\tau^{-r+1}P(a)}),$

so we may assume that $M=P(a)$, and we are concerned with when
$\operatorname{Hom}_{H}({P(a)},\,{\tau^{-r+1}P(a)})=0$. By iterated APR-
tilting (see [APR79]), we know that $H$ is concealed of type
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}_{p,q}$
with $p\leq q$ such that the clockwise and anti-clockwise arrows are
collected. For the purposes of considering Hom-spaces in the postprojective
component, we may therefore assume that $Q$ is of the form indicated in (3.b).
Let us consider three cases: $p>2$, $p=2$, and $p=1$. In the first case, we
have $Q$ of the form

$\bullet$$a$$\bullet$$b$$\bullet$$\alpha$$\bullet$$\beta$$\bullet$$\hat{\beta}$$\bullet$$\hat{\alpha}$
(3.b)

At the beginning of the postprojective component in the AR-quiver of $H$, we
obtain the below subquiver, where we denote projectives by their corresponding
node in $Q$

$a$$b$$\alpha$$\beta$${\hat{\alpha}}$${\hat{\beta}}$$\tau^{-}a$$\tau^{-}b$$\tau^{-}\alpha$$\tau^{-}{\hat{\alpha}}$$\tau^{-2}a$$\tau^{-2}b$$\tau^{-3}a$

Applying $\dim\operatorname{Hom}_{H}({P(a)},\,{-})$ on the projectives and
then using the exactness of almost split sequences, we can manually compute
$\dim\operatorname{Hom}_{H}({P(a)},\,{-})$ for the nodes:

$1$$1$$1$$1$$1$$1$$0$$1$$1$$1$$1$$2$$1$

which shows us that $\operatorname{Hom}_{H}({P(a)},\,{\tau^{-r}P(a)})$ is
nonzero for $r=2,3$. Similar computation for $p=2$ and $p=1$ give the same
conclusion:

$p=2\colon$$1$$1$$1$$2$$1$$1$$0$$1$$2$$1$$1$$3$$2$and$p=1\colon$$1$$1$$2$$3$$1$$1$$0$$2$$3$$2$$2$$5$$3$

By the dual of [Ung86, Lemma 1], this means there exists $\tau^{-}$-stable
monomorphisms from $P(a)$ to $\tau^{-2}P(a)$ and $\tau^{-3}P(a)$. Now, if
$r>1$ is an integer, then it can be written as $2x+3y$ for $x,y\geq 0$. Then
the composite map

${{P(a)}}$${{\tau^{-2}P(a)}}$${\cdots}$${{\tau^{-2x}P(a)}}$${{\tau^{-(2x+3)}P(a)}}$${\cdots}$${{\tau^{-r}P(a)}}$

is a composition of monomorphisms, hence nonzero. Thus, the only value of
$r\geq 1$ such that $\operatorname{Hom}_{H}({P(a)},\,{\tau^{-r+1}P(a)})=0$ is
$r=2$. ∎

######  Lemma 3.13.

Let $H$ be the path $\mathbf{k}$-algebra of a quiver $Q$ of type
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}_{n}$
for $n>3$, and $a\in Q_{0}$ the unique tip of the quiver with unique neighbour
$b\in Q_{0}$. Let $M$ be any indecomposable $H$-module in the $\tau$-orbit of
$P(a)$, and $N$ any indecomposable $H$-module in the $\tau$-orbit of $P(b)$,
then $N\oplus M\oplus\tau^{-2}M$ is not rigid.

Moreover, for any postprojective tilting module $T\in\operatorname{mod}{H}$
and vertex $c\neq b$ of
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}_{n}$,
there is a summand $T^{\prime}$ of $T$ in the $\tau$-orbit of $P(c)$.

###### Proof.

Like before, we may assume that $P(a)$ is a simple $H$-module. By shifting
with $\tau^{-}$ if necessary, we may assume that $M$ and $N$ are not
projective. Then we have an almost split sequence

${0}$${\tau M}$${X}$${M}$${0}$

and an integer $r$ such that $\tau^{r}X=N$. If $r\geq 2$, then by Lemma 3.12
we have a morphism $\tau^{r+1}M\rightarrow\tau M$, and since every morphism
between postprojective $H$-modules is a composition of irreducible maps, this
must factor through $\tau^{r+1}M\rightarrow\tau^{r}X$. Hence, there is a
nonzero map $N\rightarrow\tau M$, meaning
$\operatorname{Ext}_{H}^{1}({M,\,N})\neq 0$. A similar argument applies for
$r\leq-3$. We have a map from $M$ to $\tau^{r+1}M$ which must factor through
$\tau^{r+1}X\rightarrow\tau^{r+1}M$, hence there is a nonzero map
$M\rightarrow\tau N$. If $r$ is $1$ or $-2$ we can immediately see that we
have non-zero maps $N\rightarrow\tau M$ and $M\rightarrow\tau N$ respectively.
Leaving the only possibilities as $r=0$ or $r=-1$. Doing similar analysis
replacing $M$ with $\tau^{-2}M$ leaves the only possibilities to be $r=-2$ or
$r-3$. Since these restrictions do not overlap, there can be no such $r$.
Hence $N\oplus M\oplus\tau^{-2}M$ is not rigid.

To prove the second part, we first wish to establish that $X\oplus\tau^{-r}X$
is never rigid when $X$ is a postprojective indecomposable $H$-module which is
outside of the $\tau$-orbit of $P(a)$ and $r\geq 1$. Let $X$ and $Y$ be
postprojective indecomposable $H$-modules outside of the $\tau$-orbit of
$P(a)$, and let $f\colon X\rightarrow Y$ be an irreducible morphism. We wish
to show that $f$ is injective. If $Y$ is projective, then this is immediate,
so let $Y$ be non-projective. Then we have an almost split sequence

${0}$${\tau Y}$${X\oplus E}$${Y}$${0,}$

where $E$ is not zero and has a summand in the $\tau$-orbit of some $P(c)$ for
some $c\neq a$. Then $f$ is a pushout of $\tau Y\rightarrow E$, and by
induction this is injective, hence $f$ is also injective.

Consequently, if $X$ is a postprojective indecomposable $H$-module not in the
$\tau$-orbit of $P(a)$, we can find a sequence of irreducible monomorphisms
from $X$ to $\tau^{-r}X$, and so $\operatorname{Hom}(X,\tau^{-r}X)\neq 0$,
when $r\geq 0$. Hence $X\oplus\tau^{-r}X$ is not rigid for $r\geq 1$.

Let now $T$ be a basic postprojective tilting $H$-module, and consider which
$\tau$-orbits the direct summands of $T$ occupy. By the preceding discussion,
$T$ can have at most one summand in the $\tau$-orbit of $P(c)$ for any $c\neq
a$. By Lemma 3.12 there can be at most two summands in the $\tau$-orbit of
$P(a)$. In the case where there are two summands, then by the first part of
this lemma there can be no summand in the $\tau$-orbit of $P(b)$.

Since the number of summands $T$ has is equal to the number of vertices in
$Q$, we see that $T$ must have exactly one summand in each $\tau$-orbit that
is not the $\tau$-orbit of $P(a)$ or $P(b)$, and that there are either one or
two summands in the $\tau$-orbit of $P(a)$. In particular, we have a summand
in the $\tau$-orbit of each $P(c)$ for $c\neq b$. ∎

###### Proof of Theorem 3.7.

One may assume $Q$ to be connected, since it is simply a matter of treating
each connected component otherwise. We proceed by induction on the number of
vertices $n$ in $Q$. If $n$ equals 1, then $Q$ cannot be wild so the statement
holds vacuously. If $n=2$, then $Q$ must be hyperbolic, and so $B$ is already
hyperbolically concealed. This establishes the base case.

To prove the inductive step, suppose that our claim holds for all quivers
having fewer than $\ell$ vertices, for some $\ell>2$. Suppose that $Q$ has
$\ell$ vertices. If $Q$ is hyperbolic, we are done. So assume that $Q$ is not
hyperbolic. Then we wish to show that there exists a proper wild subquiver
$Q^{\prime}$ and an idempotent $e$ such that $B/(e)$ is concealed of type
$Q^{\prime}$. This will be achieved by applying the previous lemmata.

If $Q$ is a tree, then Lemma 3.11 tells us that we can find a tip that yields
a wild subquiver $Q^{\prime}$ when removed. Then, by Lemma 3.9 and Lemma 3.10,
there is an idempotent $e$ such that $B/(e)$ is concealed of type
$Q^{\prime}$.

If $Q$ is twice edge-connected and not hyperbolic, then there is a vertex $v$
such that $Q^{\prime}\coloneqq Q\setminus\\{v\\}$ is wild. It is not hard to
see that we may pick $v$ so that $Q^{\prime}$ is connected. Again, by Lemma
3.9 and Lemma 3.10, there is an idempotent $e$ such that $B/(e)$ is concealed
of type $Q^{\prime}$.

Lastly, if $Q$ is not a tree and not twice edge-connected, then removing a tip
yields a subquiver $Q^{\prime}$ which is not a tree. In this case,
$Q^{\prime}$ is either wild or of type $\widetilde{\mathbb{A}}$. If
$Q^{\prime}$ is wild we may argue as before. If $Q^{\prime}$ is of type
$\widetilde{\mathbb{A}}$, then $Q$ is of type
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}$. If
$Q$ has fewer than 10 vertices, then $Q$ is hyperbolic and we are done. If $Q$
has 10 or more, then removing a vertex sufficiently far from the vertex of
order 3 (named $b$ in (3.b)) yields a subquiver $Q^{\prime}$ that has
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{E}}$}}_{7}$ as
a subquiver, so it is wild. By Lemma 3.13 and Lemma 3.10, there is an
idempotent $e$ such that $B/(e)$ is concealed of type $Q^{\prime}$.

This covers all the cases, and hence by induction we can always find an
idempotent $e$ such that $B/(e)$ is hyperbolically concealed. ∎

In Appendix A, we apply Theorem 3.7 to prove results on $\tau$-tilting
reduction of concealed $\mathbf{k}$-algebras.

###  3.2. Wild concealed algebras are not $\mathbf{g}$-tame

Let $\Lambda$ be a finite-dimensional $\mathbf{k}$-algebra. The
$\mathbb{Z}$-bilinear pairing

${\mathrm{K}_{0}(\operatorname{proj}{\Lambda})\times\mathrm{K}_{0}(\operatorname{mod}{\Lambda})}$${\mathbb{Z}}$${{([P],[M])}}$${\mathrm{dim}(\operatorname{Hom}_{\Lambda}(P,M))}$$\scriptstyle{\langle-,-\rangle}$${\in}$${\in}$

induces an $\mathbb{R}$-bilinear pairing

${\mathrm{K}_{0}(\operatorname{proj}{\Lambda})_{\mathbb{R}}\times\mathrm{K}_{0}(\operatorname{mod}{\Lambda})_{\mathbb{R}}}$${\mathbb{R}.}$$\scriptstyle{\langle-,-\rangle}$

Given an element
$\theta\in\mathrm{K}_{0}(\operatorname{proj}{\Lambda})_{\mathbb{R}}$ and
$N\in\operatorname{mod}{\Lambda}$, we write $\theta(N)$ for
$\langle\theta,[N]\rangle$.

Henceforth, we fix a hereditary $\mathbf{k}$-algebra $H=\mathbf{k}Q$ and a
tilting $H$-module $T$. The associated tilted $\mathbf{k}$-algebra of type $Q$
is then given by $\operatorname{End}_{H}(T)$, which we denote by $B$. We get a
linear isomorphism between the Grothendieck groups of $H$ and $B$. In this
section, we aim to use this, in the case that $B$ is concealed, to show that
$B$ is $\mathbf{g}$-tame if $H$ is ($\mathbf{g}$-)tame.

######  Lemma 3.14.

[Rin84, §4.1(7)] Let $H$ be a hereditary $\mathbf{k}$-algebra, let $T$ be a
tilting $H$-module and let $B=\operatorname{End}_{H}(T)$ be the associated
tilted $\mathbf{k}$-algebra. Then there are isomorphisms

${K_{0}(\operatorname{proj}H)}$${K_{0}(\operatorname{proj}B)}$${\theta}$${\theta_{B}}$$\scriptstyle{(-)_{B}}$${\in}$${\in}$

and

${K_{0}(\operatorname{mod}H)}$${K_{0}(\operatorname{mod}B)}$${d}$${\widehat{d}}$$\scriptstyle{(\widehat{-})}$${\in}$${\in}$

with the following properties:

  * •

$\theta(d)=\theta_{B}(\widehat{d}$),

  * •

If $M$ is in $\operatorname{Fac}T$, then
$\widehat{[M]}=[\operatorname{Hom}(T,M)]$ and $\mathbf{g}^{M}_{B}$ is the
$\mathbf{g}$-vector of $\operatorname{Hom}(T,M)$.

  * •

If $N$ is in $T^{\perp}$, then $\widehat{[N]}=-[\operatorname{Ext}(T,N)]$.

###### Proof.

By [Rin84, §4.1(7)] there is an isomorphism
$K_{0}(\operatorname{mod}H)\rightarrow K_{0}(\operatorname{mod}B)$ given by

$[M]\mapstochar\rightarrow[\operatorname{Hom}(T,M)]-[\operatorname{Ext}(T,M)].$

We take this to be our $(\widehat{-})$. Note that
$\widehat{[M]}=[\operatorname{Hom}(T,M)]$ and
$\widehat{[N]}=-[\operatorname{Ext}(T,N)]$ for $M\in\operatorname{Fac}T$ and
$N\in T^{\perp}$. Now, since $H$ and $B$ have finite global dimension, the
inclusions $K_{0}(\operatorname{proj}H)\hookrightarrow
K_{0}(\operatorname{mod}H)$ and $K_{0}(\operatorname{proj}B)\hookrightarrow
K_{0}(\operatorname{mod}B)$ are isomorphisms. Hence we can define $(-)_{B}$ as
the unique map making the square below commute.

${K_{0}(\operatorname{proj}H)}$${K_{0}(\operatorname{proj}B)}$${K_{0}(\operatorname{mod}H)}$${K_{0}(\operatorname{mod}B)}$$\scriptstyle{(-)_{B}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{(\widehat{-})}$

If $M\in\operatorname{Fac}T$ then both $M$ and $\operatorname{Hom}(T,M)$ have
projective dimension at most 1. Then the vertical isomorphisms identify $[M]$
and $[\operatorname{Hom}(T,M)]$ with their respective $\mathbf{g}$-vectors,
hence $(-)_{B}$ maps the $\mathbf{g}$-vector of $M$ to the $\mathbf{g}$-vector
of $\operatorname{Hom}(T,M)$. Lastly, note that when $T_{1}$ and $T_{2}$ are
in $\operatorname{add}T$, we have

$\displaystyle\mathbf{g}^{T_{1}}\left(T_{2}\right)=\operatorname{hom}\left(T_{1},T_{2}\right)=\operatorname{hom}\left(\operatorname{Hom}\left(T,T_{1}\right),\operatorname{Hom}\left(T,T_{2}\right)\right)=\mathbf{g}^{T_{1}}_{B}\left(\widehat{T_{2}}\right).$

Since the summands of $T$ form a basis for $K_{0}(\operatorname{mod}H)$, the
same holds for all $\theta$ and $d$ in $K_{0}(\operatorname{proj}H)$ and
$K_{0}(\operatorname{mod}H)$ respectively. ∎

Thinking of the $\mathbf{g}$-vector fans of $H$ and $B$ inside
$K_{0}(\operatorname{proj}H)$ and $K_{0}(\operatorname{proj}B)$, we would like
to relate them using the isomorphism $(-)_{B}$. To do this we must first
understand the $\mathbf{g}$-vector fan of $H$ a little better. Because $H$ is
hereditary, the isomorphism $K_{0}(\operatorname{proj}H)\rightarrow
K_{0}(\operatorname{mod}H)$ is particularly nice. It associates each
$H$-module with its $\mathbf{g}$-vector. This correspondence allows us to use
classical tools that study the dimension vectors of quivers.

Let us now recall the Tits forms of hereditary $\mathbf{k}$-algebras. Suppose
that $Q$ has $n$ vertices. The _Tits form_ of $Q$ is the $\mathbb{Z}$-linear
quadratic form defined by

${\mathbb{Z}^{n}}$${\mathbb{Z}}$${x}$${\sum_{i\in Q_{0}}x_{i}^{2}-\sum_{i\in
Q_{0}}d_{i,j}x_{i}x_{j},}$$\scriptstyle{q_{H}}$${\in}$${\in}$

where $d_{i,j}$ is the number of arrows in $Q$ from the vertex $i$ to the
vertex $j$. If $x$ is the dimension vector of an indecomposable $H$-module
$M$, one can express $q_{H}(x)$ by

$q_{H}(x)=\dim_{\mathbf{k}}\operatorname{Hom}_{H}(M,M)-\dim_{\mathbf{k}}\operatorname{Ext}^{1}_{H}(M,M)$

We write $\operatorname{hom}$ for
$\dim_{\mathbf{k}}\operatorname{Hom}_{H}(M,M)$ and $\operatorname{ext}$ for
$\dim_{\mathbf{k}}\operatorname{Ext}^{1}_{H}$, so that the expression above
simplifies to

$q_{H}(x)=\operatorname{hom}(M,M)-\operatorname{ext}(M,M).$ (3.c)

We often think of $q_{H}$ as a function from $K_{0}(\operatorname{mod}H)$ to
$\mathbb{Z}$. With this formulation, we have that
$q_{H}([M])=\mathbf{g}^{M}(M)$, and hence $q_{H}([M])$ is positive whenever
$M$ is rigid. Using the isomorphism between $K_{0}(\operatorname{proj}H)$ and
$K_{0}(\operatorname{mod}H)$, we may think of $q_{H}$ as a $\mathbb{Z}$-valued
function on $K_{0}(\operatorname{proj}H)$.

Suppose now that $H$ is hyperbolic. The hyperbolicity of $H$ famously enables
the following characterization of the dimensions vectors in
$\operatorname{mod}({H})$ in terms of the Tits form: for any integer vector
with $q_{H}(x)\leq 1$, we have that $x$ or $-x$ is the dimension vector of an
indecomposable $H$-module [Kac83, page 78]. In particular, if $x$ is a
positive vector with integer coefficients such that $q_{H}(x)\leq 1$, then $x$
is the dimension vector of an indecomposable $H$-module. Since $q_{H}(x)=1$
for all dimension vectors $x$ of an indecomposable $H$-module which is either
postprojective or preinjective, we have that $x$ is the dimension vector of a
regular $H$-module whenever $q_{H}(x)\leq 0$.

The Tits form $q_{H}$ extends to a quadratic $\mathbb{R}$-linear form

${q_{H,\mathbb{R}}\colon\mathbb{R}^{n}}$${\mathbb{R}.}$

We define the negative cone of the Tits form to be the set of positive vectors
in the negative locus of the Tits form, namely the set

$C^{<0}_{H}\coloneqq\\{x\in\mathbb{R}^{n}\,\,|\,\,q_{H,\mathbb{R}}(x)<0,x\text{
positive}\\}\subseteq\mathbb{R}^{n}.$

It is now easy to see that, when $H$ is hyperbolic, any rational vector in
$C^{<0}_{H}$ is a scalar multiple of an indecomposable regular $H$-module.
Using the isomorphism $K_{0}(\operatorname{proj}H)_{\mathbb{R}}\cong
K_{0}(\operatorname{mod}H)_{\mathbb{R}}\cong\mathbb{R}^{n}$, where $[P]$ is
sent to its dimension vector $\underline{\dim}(P)$, we will think of
$C^{<0}_{H}$ as a subset of $K_{0}(\operatorname{proj}H)_{\mathbb{R}}$, and
relate it to the $\mathbf{g}$-vector fan.

######  Remark 3.15.

It is well known that a hereditary $\mathbf{k}$-algebra $H$ is wild iff
$C^{<0}_{H}$ is nonempty. See for example [Rin84, page 7], [DDPW08, Theorem
1.11] or [Kac80, Lemma 1.2].

######  Lemma 3.16.

Let $H$ be a hyperbolic hereditary $\mathbf{k}$-algebra, let $N$ be an
indecomposable postprojective $H$-module and let $\theta\in C^{<0}_{H}$ be a
vector in the negative cone. Then $\theta(N)<0$.

###### Proof.

Let us first assume that $\theta$ is a rational vector. Then by the preceding
discussion, we deduce that $\theta$ is a multiple of the $\mathbf{g}$-vector
of a regular $H$-module $M$, and so we may assume that
$\theta=\mathbf{g}^{M}$. Then
$\theta(N)=\operatorname{hom}(M,N)-\operatorname{ext}(N,M)=-\operatorname{ext}(N,M)\leq
0$. Now, since $N$ is postprojective, it must be rigid, whence
$\mathbf{g}^{N}(N)>0$. Since the negative cone is an open subset of
$\mathbb{R}^{n}$, we may pick a rational number $\varepsilon>0$ such that
$\theta+\varepsilon\mathbf{g}^{N}$ is still in $C^{<0}_{H}$. By the same
argument as before, we have $(\theta+\varepsilon\mathbf{g}^{N})(N)\leq 0$, but
this implies that

$\displaystyle\theta(N)\leq-\varepsilon\mathbf{g}^{N}(N)<0.$

We conclude that $\theta(N)<0$. ∎

The last tool we will need is the concept of a wall. First utilized in the
context of $\tau$-tilting by Brüstle–Smith–Treffinger [BST19] and Asai
[Asa21].

######  Definition 3.17.

[BST19] For a $\mathbf{k}$-algebra $\Lambda$ and a nonzero $\Lambda$-module
$M$ we define

$\displaystyle\Theta_{M}\coloneqq\\{\theta\in
K_{0}(\operatorname{proj}\Lambda)_{\mathbb{R}}\mid\theta(M)=0,\theta(X)\geq
0\text{ for }X\in\operatorname{Fac}M\\}.$

We also write $\mathrm{Walls}$ for the union of all walls in
$K_{0}(\operatorname{proj}\Lambda)$, and $\overline{\mathrm{Walls}}$ for its
closure in $K_{0}(\operatorname{proj}\Lambda)_{\mathbb{R}}$.

An important fact about walls is that the complement of
$\overline{\mathrm{Walls}}$ is exactly given by the union of the interiors of
maximal cones in the $\mathbf{g}$-vector fan [Asa21, Thm 3.17]. This means
that $\overline{\mathrm{Walls}}^{\mathrm{o}}$ is exactly the exterior of the
$\mathbf{g}$-vector fan. In particular, if $\Lambda=H$ is hereditary, then
$C^{<0}_{H}$ is contained in $\overline{\mathrm{Walls}}$. Further, if we only
consider rational vectors, then $C^{<0}_{H}\cap
K_{0}(\operatorname{proj}H)_{\mathbb{Q}}\subseteq\mathrm{Walls}$.

######  Theorem 3.18.

Let $Q$ be a wild acyclic quiver, let $H=\mathbf{k}Q$ be its path
$\mathbf{k}$-algebra, and let $B$ be a concealed $\mathbf{k}$-algebra of type
$Q$ with $B=\operatorname{End}_{H}(T)$ for a postprojective tilting $H$-module
$T$. Then $B$ is not $\mathbf{g}$-tame.

###### Proof.

Theorem 3.7 provides an idempotent $e\in B$ such that $B/(e)$ is a concealed
$\mathbf{k}$-algebra of hyperbolic type. By Lemma 3.4(2), the
$\mathbf{k}$-algebra $B$ will not be $\mathbf{g}$-tame if $B/(e)$ is not
$\mathbf{g}$-tame. We may therefore make the additional assumption that $H$ is
hyperbolic, since it suffices to consider this special case to prove the
general statement.

Suppose therefore that $H$ is hyperbolic. Now consider a rational vector
$\theta\in C^{<0}_{H}$, and let $M$ be an indecomposable $H$-module such that
$\theta\in\Theta_{M}$. By Lemma 3.16, we have that $\theta(N)<0$ for every
postprojective $H$-module $N$, so it must be the case that $M$ is in
$\operatorname{Fac}T$. We wish to show that $\theta_{B}$ is contained in
$\Theta_{\operatorname{Hom}(T,M)}$.

Since $M\in\operatorname{Fac}T$ we have
$\theta_{B}(\operatorname{Hom}(T,M))=\theta(M)=0$. Any $B$-module in
$\operatorname{Fac}\operatorname{Hom}(T,M)$ will be of the form
$\operatorname{Hom}(T,M^{\prime})\oplus\operatorname{Ext}(T,N)$ with
$M^{\prime}$ in $\operatorname{Fac}M$ and $N$ in $T^{\perp}$. By Lemma 3.14,

$\displaystyle\theta_{B}(\operatorname{Hom}(T,M^{\prime})\oplus\operatorname{Ext}(T,N))=\theta_{B}(\widehat{M^{\prime}})+\theta_{B}(-\widehat{N})=\theta(M^{\prime})-\theta(N)\geq
0.$

Thus, we have that $\theta_{B}$ is in $\Theta_{\operatorname{Hom}(T,M)}$.

This means that the image of $C^{<0}_{H}\cap
K_{0}(\operatorname{proj}H)_{\mathbb{Q}}$ under $(-)_{B}$ does not intersect
the interior of any maximal cone. Since the subset of rationals vectors is
dense in $C^{<0}_{H}$, it does not intersect any maximal cone either. Hence,
the image of $C^{<0}_{H}$ is an open set in the exterior of the
$\mathbf{g}$-vector fan of $B$. In conclusion, we have that $B$ is not
$\mathbf{g}$-tame. ∎

###  3.3. $\mathbf{g}$-tameness of incidence algebras

We move on to classify $\mathbf{g}$-tame incidence $\mathbf{k}$-algebras of
finite simply connected posets. We will use Leszczyński’s characterization of
tameness for incidence $\mathbf{k}$-algebras, which we restate in Theorem 3.19
below. This result makes reference to the notion of universal Galois covering
of a $\mathbf{k}$-algebra, in the sense of Martinez-Villa–de la Peña
[MVdlPn83]. Since the universal Galois covering of a simply connected poset is
itself [MVdlPn83, Theorem 4.2], one can prove Corollary 3.20 below without
relying on a detailed account of covering theory.

######  Theorem 3.19 ([Les03, Theorem 1.4]).

Let $P$ be a finite poset. Then the incidence $\mathbf{k}$-algebra
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is tame if and only if its
universal Galois covering does not contain a convex subalgebra $L$ which is a
concealed $\mathbf{k}$-algebra of one of the types appearing in Table 3.19.I.

$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{A}}$}}_{p,q}$ |  | $\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{D}}$}}_{n}$ |
---|---|---|---
|  where the number of vertices is $p+q+1$, and among the arrows in the unoriented cycle, exactly $p$ arrows have clockwise orientation and exactly $q$ have counter-clockwise orientation. |  |  where $4\leq n\leq 8$, and the number of vertices is $n+2$.
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{E}}$}}_{6}$ |  | $\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{E}}$}}_{7}$ |
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{E}}$}}_{8}$ |  | $\mathbb{T}_{5}$ |
Table 3.19.I. Quivers appearing in Leszczyński’s classification of wild
posets. Undirected edges indicate that the arrows may go in either direction.

Since the universal Galois covering of a finite simply connected poset is just
the poset itself, we deduce that a simply connected poset has a
$\mathbf{g}$-tame indcidence $\mathbf{k}$-algebra if and only if it does not
contain a convex subposet whose incidence $\mathbf{k}$-algebra is a concealed
$\mathbf{k}$-algebra of one of the types in Table 3.19.I.

######  Corollary 3.20.

Let $P$ be a finite simply connected poset. Then the incidence
$\mathbf{k}$-algebra $\mathcal{I}_{k}\left({P}\right)$ is $\mathbf{g}$-tame if
and only if it is tame.

###### Proof.

Plamondon and Yurikusa have shown that any tame finite-dimensional
$\mathbf{k}$-algebra is $\mathbf{g}$-tame [PYK23]. We show that the converse
holds for incidence $\mathbf{k}$-algebras of finite simply connected posets.

Suppose that the incidence $\mathbf{k}$-algebra
$\mathcal{I}_{k}\left({P}\right)$ is wild. Since the universal Galois covering
of $P$ is just $P$, it follows from Theorem 3.19 that $P$ must contain a
concealed $\mathbf{k}$-algebra of wild type as a convex subposet. By Theorem
3.18, this concealed $\mathbf{k}$-algebra cannot be $\mathbf{g}$-tame.
Applying Lemma 3.4(2), we conclude that $\mathcal{I}_{k}\left({P}\right)$
cannot be $\mathbf{g}$-tame. ∎

In the case that $P$ is multiply connected, we might not be able to reduce $P$
to a concealed $\mathbf{k}$-algebra of wild type, as achieved above. However
we still formulate the following conjecture:

######  Conjecture 3.21.

Let $P$ be an arbitrary finite poset such that the incidence
$\mathbf{k}$-algebra $\mathcal{I}_{k}\left({P}\right)$ is $\mathbf{g}$-tame.
Then $\mathcal{I}_{k}\left({P}\right)$ is tame.

One could conceivably aim to prove this conjecture by examining multiply
connected posets with hyperbolic incidence $\mathbf{k}$-algebras. Here we give
such an example, taken from the literature [Les03, Sim10].

######  Example 3.22.

Let $P$ be the multiply connected poset displayed below.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

The $\mathbf{k}$-algebra $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is not a
concealed $\mathbf{k}$-algebra. To see this, observe that the
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$-module with dimension vector

$1$$1$$0$$0$$0$$1$$1$

has projective dimension 2 and injective dimension 2 and hence by [HRS96,
Lemma 2.1] the $\mathbf{k}$-algebra is not quasi-tilted, and in particular not
concealed. Therefore, we may not apply Theorem 3.18 to decide whether this
poset has a $\mathbf{g}$-tame incidence $\mathbf{k}$-algebra. The universal
Galois cover of $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ contains a
hereditary $\mathbf{k}$-algebra of type
$\widetilde{\raisebox{0.0pt}[0.85pt][0.1pt]{$\widetilde{\mathbb{D}}$}}_{7}$ as
a convex subcategory [Les03, §1], so using the characterization of wild posets
in Theorem 3.19, we observe that $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is
a wild $\mathbf{k}$-algebra. Moreover, for any primitive idempotent
$e\in\mathcal{I}_{\mathbf{k}}\left({P}\right)$, we have that
$\mathcal{I}_{\mathbf{k}}\left({P}\right)/(e)$ not wild, whence
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is hyperbolic. ∎

Another approach might be to examine the relationship between the
$\mathbf{g}$-vector fan of a $\mathbf{k}$-algebra and the $\mathbf{g}$-vector
fan of (finite quotients) of its universal Galois cover. We urge those
interested to explore this further.

We finish this section by arguing that the class of possible counterexamples
to 3.21 is rather constrained. Before our argument culminates in Proposition
3.25, we develop the necessary preparation.

Given an integer $\ell\geq 2$, let $C_{\ell}$ denote the following poset of
type $\widetilde{\mathbb{A}}_{2\ell-1}$:

${p^{+}_{1}}$${p^{+}_{2}}$${\cdots}$${p^{+}_{\ell-1}}$${p^{+}_{\ell}}$${p^{\div}_{1}}$${p^{\div}_{2}}$${\cdots}$${p^{\div}_{\ell-1}}$${p^{\div}_{\ell}}$
($C_{\ell}$)

Adding a maximal element $\omega$ and a minimal element $0$ yields a poset we
denote by $C_{\ell}^{\lozenge}$, as displayed below.

${\omega}$${p^{+}_{1}}$${p^{+}_{2}}$${\cdots}$${p^{+}_{\ell-1}}$${p^{+}_{\ell}}$${p^{\div}_{1}}$${p^{\div}_{2}}$${\cdots}$${p^{\div}_{\ell-1}}$${p^{\div}_{\ell}}$${0}$
($C_{\ell}^{\lozenge}$)

The existence of a unique maximal element (or indeed a unique minimal element)
ensures that $C_{\ell}^{\lozenge}$ is simply connected. Note that the poset
$C_{2}^{\lozenge}$ is a subposet of the poset with the following Hasse quiver:

${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$
($\mathbf{8}$)

This poset is often called the “figure 8”, and we will denote it here by
$\mathbf{8}$.

One can use posets of the form $C_{\ell}^{\lozenge}$ to characterize the
posets whose incidence $\mathbf{k}$-algebras are of global dimension 2 or
greater. Recall that the _global dimension_ of a finite-dimensional
$\mathbf{k}$-algebra $\Lambda$ is defined as the supremum of the projective
dimension of all finitely generated $\Lambda$-modules. The incidence
$\mathbf{k}$-algebra of a finite poset is always of finite global dimension
[Mit65, IX.10.3], but the global dimension may depend on the characteristic of
$\mathbf{k}$ [IZ90, Propostion 2.3]. One can nevertheless characterize the
incidence $\mathbf{k}$-algebras of global dimension greater than 2, in a
manner that is independent of the characteristic.

######  Theorem 3.23 ([IZ90, Theorem 3.3]).

Let $P$ be a finite poset. Then $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ has
global dimension at most 2 precisely when the following two criteria both
hold.

  1. (IZ1)

no subposet of $P$ is isomorphic to $C_{\ell}^{\lozenge}$ for any $\ell\geq
3$,

  2. (IZ2)

any subposet of $P$ which is isomorphic to $C_{2}^{\lozenge}$ is contained in
a subposet of $P$ which is isomorphic to $\mathbf{8}$.

If $P$ is a poset and $P^{\prime}\subseteq P$ is a subposet, one defines the
convex hull $P^{\prime}$ in $P$ as the smallest convex subposet of $P$
containing $P^{\prime}$. It is denoted $\overline{P^{\prime}}$.

######  Lemma 3.24.

Let $P$ be a finite poset.

  1. (1)

Let $P^{\prime}$ be a subposet of $P$. If $P^{\prime}$ has a unique maximal or
a unique minimal element, so does its convex hull $\overline{P^{\prime}}$ in
$P$. In particular, the poset $\overline{P^{\prime}}$ is simply connected.

  2. (2)

Let $P^{\prime}$ be a poset which is a subposet or a contraction of $P$. If
$P^{\prime}$ is wild, so is $P$.

  3. (3)

For $\ell\geq 2$, incidence $\mathbf{k}$-algebra of the poset
$C_{\ell}^{\lozenge}$ is representation-infinite, and wild precisely when
$\ell\geq 5$.

  4. (4)

Given an integer $\ell\geq 5$ and a poset embedding
${C_{\ell}^{\lozenge}}$${P,}$ then the convex hull of the image of
$C_{\ell}^{\lozenge}$ in $P$ is simply connected and wild. In particular, by
Corollary 3.20, no convex hull of this form may have a $\mathbf{g}$-tame
incidence $\mathbf{k}$-algebra.

###### Proof.

Since (1) immediately follows from the definitions, and (2) is known to hold
[CR19, Lemma 2.1] [Lou75a, Proposition 1.3], we first prove (3). Since
$C_{\ell}^{\lozenge}$ contains a subposet $C_{\ell}$ of type
$\widetilde{\mathbb{A}}$, we deduce from Lemma 2.9 that $C_{\ell}^{\lozenge}$
has a representation-infinite incidence $\mathbf{k}$-algebra for all $\ell\geq
2$. If $\ell\geq 5$, the subposet consisiting of the elements in
$\\{\omega,p^{+}_{1},\dots,p^{+}_{5}\\}$ forms a convex subposet of type
$\mathbb{T}_{5}$, whence it follows from Lemma 3.4(2) and Theorem 3.19 that
$\mathcal{I}_{\mathbf{k}}\left({C_{\ell}^{\lozenge}}\right)$ is wild. On the
other hand, we have that $C_{4}^{\lozenge}$ has a tame incidence
$\mathbf{k}$-algebra. Suppose to the contrary that the $C_{4}^{\lozenge}$ has
a wild incidence $\mathbf{k}$-algebra. By simple-connectedness and Theorem
3.19, there should exists a convex subposet of $C_{4}^{\lozenge}$ whose
incidence $\mathbf{k}$-algebra is wild concealed. It is readily checked that
no proper convex subposet of $C_{4}^{\lozenge}$ has a wild incidence
$\mathbf{k}$-algebra, whence
$\mathcal{I}_{\mathbf{k}}\left({C_{4}^{\lozenge}}\right)$ must be a wild
concealed $\mathbf{k}$-algebra itself. However, since Theorem 3.23 proves that
the global dimension of
$\mathcal{I}_{\mathbf{k}}\left({C_{4}^{\lozenge}}\right)$ is larger than 2, it
cannot be a concealed $\mathbf{k}$-algebra [HRS96, Lemma 2.1], a
contradiction, so $\mathcal{I}_{\mathbf{k}}\left({C_{4}^{\lozenge}}\right)$ is
indeed tame. It is easy to see that $C_{2}^{\lozenge}$ and $C_{3}^{\lozenge}$
are contractions of $C_{4}^{\lozenge}$, whereby it now follows from (2) that
$\mathcal{I}_{\mathbf{k}}\left({C_{2}^{\lozenge}}\right)$ and
$\mathcal{I}_{\mathbf{k}}\left({C_{3}^{\lozenge}}\right)$ are tame as
well.333We note in passing that the poset $C_{3}^{\lozenge}$ is exactly the
poset
$\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}\times\overrightarrow{\mathbb{A}}_{2}$,
see 2.13.

We conclude the proof by addressing (4). Since $C_{\ell}^{\lozenge}$ is wild
for $\ell\geq 5$, the assertion in (4) now follows from (1), (2) and (3)
above. ∎

We are now in a position to argue that most wild posets do not have
$\mathbf{g}$-tame incidence $\mathbf{k}$-algebras. In conclusion, we contend
that our 3.21 is justified. In an informal sense, Proposition 3.25 shows that
at most only a handful wild posets may have a $\mathbf{g}$-tame incidence
$\mathbf{k}$-algebra, since we will be to exclude all posets admitting a
subposet which is isomorphic to $C_{\ell}^{\lozenge}$ for some $\ell\geq 5$.

######  Proposition 3.25.

Let $P$ be a finite multiply connected poset such that
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is both wild and $\mathbf{g}$-tame.
Then $P$ must have one of the following three properties (if it exists at
all):

  1. (1)

the global dimension of $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is exactly
2,

  2. (2)

the global dimension of $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is greater
than 2 and $P$ contains a subposet of the form $C_{3}^{\lozenge}$ or
$C_{4}^{\lozenge}$, so that the condition (IZ1) in Theorem 3.23 is not met.

  3. (3)

the global dimension of $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is greater
than 2 and the condition (IZ2) in Theorem 3.23 is not met, i.e. there is a
subposet of $P$ which is isomorphic to $C_{2}^{\lozenge}$ which cannot be
extended to a subposet of $P$ which is isomorphic to $\mathbf{8}$.

###### Proof.

If the global dimension of $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is less
than 2, then $\mathcal{I}_{\mathbf{k}}\left({P}\right)$ is hereditary, so it
follows directly from Proposition 3.3 that
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ cannot be $\mathbf{g}$-tame if it
is wild, which would contradict our assumptions on $P$. It now suffices to
show that $P$ does not have a $\mathbf{g}$-tame incidence $\mathbf{k}$-algebra
whenever $P$ contains a subposet $P^{\prime}$ of the form
$C_{\ell}^{\lozenge}$ for some $\ell\geq 5$. In this case, we have that
$\mathcal{I}_{\mathbf{k}}\left({\overline{P^{\prime}}}\right)$ is not
$\mathbf{g}$-tame by Lemma 3.24(4), whence
$\mathcal{I}_{\mathbf{k}}\left({P}\right)$ cannot be $\mathbf{g}$-tame, as a
result of Lemma 3.4(2). ∎

##  Appendix A $\tau$-tilting reduction of concealed algebras

This appended section builds on Lemma 3.13. We will use Theorem 3.7 to prove
results on $\tau$-tilting reducions of concealed $\mathbf{k}$-algebras. For
simplicity, we keep the assumption that the field $\mathbf{k}$ is
algebraically closed, as imposed in Section 3.

######  Definition A.1.

Let $\Lambda$ be a finite-dimensional $\mathbf{k}$-algebra and let $(M,\,R)$
be a $\tau$-rigid pair in $\operatorname{mod}({\Lambda})$. The Bongartz
completion of $(M,\,R)$ is the unique support $\tau$-tilting pair
$(M^{+},\,R)$ satisfying $\operatorname{Fac}{M^{+}}={{}^{\perp}}(\tau M)\cap
R^{\perp}$. The Bongartz complement of $(M,\,R)$ is the $\tau$-rigid
$\Lambda$-module $M^{+}/M$.

The Bongartz completion of a $\tau$-rigid pair always exists [AIR14, Theorem
2.10]. The notion of Bongartz completion has its origin in the study of
tilting modules over hereditary $\mathbf{k}$-algebras. In this setting, the
Bongartz completion of a partial tilting module $U$ can be defined in terms of
a universal extension of the $\mathbf{k}$-algebra $\Lambda$ in
$\operatorname{add}(U)$ [Bon81, Lemma 2.1]. In the more general setting of
$\tau$-tilting theory, such a universal extension does not exist, but the
Bongartz completion can be recovered from a right exact sequence. Although
this fact is well-known, we give a proof in Lemma A.2 with reference to the
relevant literature.

Recall that an additive subcategory $\mathcal{X}$ of
$\operatorname{mod}(\Lambda)$ is said to be contravariantly finite (resp
covariantly finite) if for every $\Lambda$-module $Y$, there exists a
$\Lambda$-homomorphism $X\xrightarrow{f}Y$ (resp. $Y\xrightarrow{f}X$) such
that the induced map $\operatorname{Hom}_{\Lambda}(X,f)$ (resp.
$\operatorname{Hom}_{\Lambda}(f,X)$) is surjective. The $\Lambda$-homomorphism
$f$ is called a right $\mathcal{X}$-approximation (resp. left
$\mathcal{X}$-approximation) of $Y$. A functorially finite subcategory of
$\operatorname{mod}(\Lambda)$ is by definition a full subcategory that is both
contravariantly and covariantly finite.

######  Lemma A.2.

Let $\Lambda$ be a finite-dimensional $\mathbf{k}$-algebra, $(M,\,R)$ a
$\tau$-rigid pair in $\operatorname{mod}(\Lambda)$ and $M^{+}/M$ the Bongartz
complement of $(M,\,R)$. Then, there exists a right exact sequence

${\Lambda}$${L}$${M^{\prime}}$${0,}$$\scriptstyle{f}$

where $f$ is a minimal left $\operatorname{add}(M^{+})$-approximation, the
object $L$ is in $\operatorname{add}(M^{+})$ and $M^{\prime}$ is in
$\operatorname{add}(M)$.

###### Proof.

Since the Bongartz completion $M^{+}$ is a support $\tau$-tilting
$\Lambda$-module, we have that
$\mathcal{T}=\operatorname{Fac}{M^{+}}={{}^{\perp}}(\tau M)\cap R^{\perp}$ is
a functorially finite torsion class of $\operatorname{mod}(\Lambda)$ [AIR14,
Theorem 2.7]. By the covariant finiteness of $\mathcal{T}$, we can construct
the following exact sequence

${\Lambda}$${T_{0}}$${T_{1}}$${0,}$$\scriptstyle{f}$

where $f$ is a minimal left $\mathcal{T}$-approximation. By [MŠ17, Lemma 3.7],
we know that $\operatorname{add}(T_{0}\oplus T_{1})$ is the class of
$\mathrm{Ext}$-projectives in $\mathcal{T}$, i.e.
$\operatorname{add}(T_{0}\oplus
T_{1})=\\{X\in\mathcal{T}\,\,|\,\,\operatorname{Ext}_{\Lambda}^{1}(X,\mathcal{T})=0\\}$.
Also, any $\Lambda$-module $X\in\operatorname{add}(T_{0}\oplus T_{1})$ is a
direct summand of $T_{0}$ if and only if it is split projective in
$\mathcal{T}$. Now, since all summands of $M^{+}/M$ are split projective by
[BM21, Lemma 4.13], we are done. ∎

Let $(M,R)$ be a $\tau$-rigid pair in $\operatorname{mod}(\Lambda)$. The
_$\tau$ -tilting reduction_ of $\Lambda$ with respect to $(M,R)$ is defined as
the $\mathbf{k}$-algebra
$C_{(M,R)}\coloneqq\operatorname{End}_{\Lambda}({M^{+}}/[M])$, where
$(M^{+},R)$ is the Bongartz completion of $(M,R)$ [Jas15, DIR+23]. Let
$e\in\Lambda$ be a primitive idempotent and $\Lambda e$ the corresponding
indecomposable projective $\Lambda$-module. For the $\tau$-rigid pair
$(\Lambda e,0)$, one can consider the $\tau$-tilting reduction $C_{(\Lambda
e,0)}$, which is obviously isomorphic to the factor $\mathbf{k}$-algebra
$\Lambda/(e)$.

It is known that the $\tau$-tilting reduction of a hereditary
$\mathbf{k}$-algebra is hereditary [GL91, §4.16(3)]. We include a result
letting us determine the representation-type.

######  Lemma A.3.

Let $Q$ be a quiver and let $(M,\,R)$ be a $\tau$-rigid pair in
$\operatorname{mod}(\mathbf{k}Q)$ in which $M$ is postprojective.

  1. (1)

Suppose that $Q$ is hyperbolic. Then the $\tau$-tilting reduction of
$\mathbf{k}Q$ with respect to $(M,\,R)$ is tame.

  2. (2)

Suppose that $Q$ is tame. Then the $\tau$-tilting reduction of $\mathbf{k}Q$
with respect to $(M,\,R)$ is tame.

###### Proof.

We only prove (1), since (2) is proved by employing precisely the same
techniques.

Suppose first that $R=0$. Let $M^{+}$ be the Bongartz completion of $M$. Since
we are working over a hereditary $\mathbf{k}$-algebra, we can consider the
universal extension of $\Lambda$ in $\operatorname{add}(M)$

${0}$${\Lambda}$${M^{\prime}}$${M^{\prime\prime}}$${0,}$

and assert that
$\operatorname{add}(M^{+})=\operatorname{add}({M^{\prime}\oplus
M^{\prime\prime}})$ by Lemma A.2. It follows from the extension-closure of the
postprojective component that $M^{+}$ is a postprojective ($\tau$-)tilting
$\mathbf{k}Q$-module. The $\tau$-tilting reduction of $\mathbf{k}Q$ with
respect to $(M,0)$ is given by
$C_{M}=\operatorname{End}_{\mathbf{k}Q}({M^{+}})/(e_{M})$, where $e_{M}$ is
the idempotent endomorphism corresponding to $M$. We have that $C_{M}$ is tame
for the reason that $\operatorname{End}_{\mathbf{k}Q}(M^{+})$ is a hyperbolic
(concealed) $\mathbf{k}$-algebra and $C_{M}$ is obtained through factoring out
a non-zero sum of primitive idempotents.

In the case where $R\neq 0$, the $\tau$-tilting reduction of $\mathbf{k}Q$
with respect to $(M,\,R)$ is given by the $\tau$-tilting reduction of
$\mathbf{k}Q/(e_{R})$ with respect to $(M,\,0)$. The hyperbolicity of
$\mathbf{k}Q$ makes $\mathbf{k}Q/(e_{R})$ tame hereditary
$\mathbf{k}$-algebra. Repeating the argument in the previous paragraph shows
that the $\tau$-tilting reduction of $\mathbf{k}Q/(e_{R})$ with respect to
$(M,\,0)$ is a factor of a representation-finite or tame concealed
$\mathbf{k}$-algebra. In particular, it is tame, and are done. ∎

We note that the assumption of postprojectivity in Lemma A.3 is necessary for
the conclusions to hold. A counterexample for (1) would be the following
quiver

$\bullet$$1$$\bullet$$2$$\bullet$$3$$\bullet$$4$

with the $\tau$-rigid pair $(M,\,0)$, where $M$ is the indecomposable
representation with dimension vector $(1,0,1,0)$. For (2), consider the quiver
obtained by removing the vertex $4$, and the indecomposable representation
with dimension vector $(1,0,1)$.

######  Lemma A.4.

Let $B$ be a concealed $\mathbf{k}$-algebra of type $Q$.

  1. (1)

If $Q$ is euclidean and $e$ is a non-zero primitive idempotent, then $B/(e)$
is representation-finite.

  2. (2)

If $Q$ is hyperbolic and $e=e_{1}+e_{2}$ is a sum of distinct primitive
idempotents in $B$, then $B/(e)$ is representation-finite.

###### Proof.

The assertion in (1) is shown by Happel–Vossieck [HV83, Theorem 2]. We deduce
the assertion in (2) from those in (1) above and Lemma 3.10. We first consider
$B/(e_{1})$. Pick a postprojective tilting $\mathbf{k}Q$-module $T$ such that
$B=\operatorname{End}_{\mathbf{k}Q}(T)$. Then $e_{1}$ corresponds to a summand
$T_{1}$ of $T$, and let $v$ be the vertex in $Q$ such that $T_{1}$ is in the
$\tau$-orbit of $P(v)$. If $Q\setminus\\{v\\}$ is connected representation-
infinite, then by Lemma 3.10 $B/(e_{1})$ is tame concealed, and hence by (1)
we get that $B/(e)\cong\frac{B/(e_{1})}{(e_{2})}$ is representation finite.

If $Q^{\prime}\coloneqq Q\setminus\\{v\\}$ is representation-finite, then
following the construction in [HU89], we can construct a tilting
$\mathbf{k}Q^{\prime}$-complex $\tilde{N}^{\bullet}$, such that
$\operatorname{End}_{\mathcal{D}^{\mathrm{b}}(\mathbf{k}Q^{\prime})}(\tilde{N}^{\bullet})\cong
B/(e_{1})$. Now, the $\mathbf{k}$-algebra $B/(e_{1})$ is derived equivalent to
a representation-finite hereditary $\mathbf{k}$-algebra, hence representation-
finite, and therefore so is $B/(e)$.

By examining the list of hyperbolic quivers [Ker88, Remark following Theorem
4.1], we see that whenever $Q\setminus\\{v\\}$ is representation-infinite it
is also connected, so this covers all the cases, hence $B/(e)$ is
representation-finite. ∎

We are now ready to prove the main result of this appended section.

######  Theorem A.5.

Let $Q$ be a hyperbolic quiver, and let
$\Lambda=\operatorname{End}_{\mathbf{k}Q}(T)$ be a concealed
$\mathbf{k}$-algebra of type $Q$. For $(M,\,R)\in\operatorname{\tau\mathchar
45\relax rigid{\,}pair}\left(\Lambda\right)$ the following assertions hold:

  1. (1)

Let $\mathcal{Y}_{T}$ be the torsion class in $\operatorname{mod}(\Lambda)$ as
defined in Theorem 2.7. Then the Bongartz complement $M^{+}/M$ of $(M,\,R)$ is
in $\mathcal{Y}_{T}$.

  2. (2)

If $|M|+|R|\geq 1$, then the $\tau$-tilting reduction with respect to
$(M,\,R)$ is tame.

  3. (3)

If $|M|+|R|\geq 2$, then the $\tau$-tilting reduction with respect to
$(M,\,R)$ is representation-finite.

###### Proof.

We first address our claim in (1). By Lemma A.2, we have a right exact
sequence of the form

${\Lambda}$${L_{\mathcal{X}}\oplus L_{\mathcal{Y}}}$${N}$${0,}$

where $N\in\operatorname{add}(M)$ and $L_{\mathcal{X}}\oplus L_{\mathcal{Y}}$
is a not necessarily basic object in $\operatorname{add}(M^{+})$ with
$L_{\mathcal{X}}\in\mathcal{X}_{T}$ and $L_{\mathcal{Y}}\in\mathcal{Y}_{T}$.
Applying $\operatorname{Tor}^{\Lambda}_{1}({T},\,{-})$ gives a right exact
sequence in $\mathbf{k}Q$

${0}$${\operatorname{Tor}^{\Lambda}_{1}({T},\,{L_{\mathcal{X}}})}$${\operatorname{Tor}^{\Lambda}_{1}({T},\,{N})}$${0,}$

having used that $\operatorname{Tor}^{\Lambda}_{1}({T},\,{\Lambda})$ and
$\operatorname{Tor}^{\Lambda}_{1}({T},\,{L_{\mathcal{Y}}})$ vanish, since both
$\Lambda$ and $L_{\mathcal{Y}}$ are in $\mathcal{Y}_{T}$. Hence, we have that
$\operatorname{Tor}^{\Lambda}_{1}({T},\,{L_{\mathcal{X}}})\simeq\operatorname{Tor}^{\Lambda}_{1}({T},\,{N})$.
From the Brenner–Butler Theorem (Theorem 2.7), we deduce that
${L_{\mathcal{X}}}\simeq{N}$. Thus, we have that
$L_{\mathcal{X}}\in\operatorname{add}(M)$. We can therefore take
$L_{\mathcal{X}}$ to be $0$ if we insist that the Bongartz completion be
basic. This proves that $M^{+}/M\in\mathcal{Y}_{T}$, as claimed.

We move on to prove (2) and (3). Write $M=M_{\mathcal{X}}\oplus
M_{\mathcal{Y}}$ where $M_{\mathcal{X}}\in\mathcal{X}_{T}$ and
$M_{\mathcal{Y}}\in\mathcal{Y}_{T}$. Using (1) and the fact that the torsion
pair $(\mathcal{X}_{T},\,\mathcal{Y}_{T})$ splits, one calculates that

$\displaystyle C_{(M,R)}=\operatorname{End}_{\Lambda}({M^{+}})/[M]$
$\displaystyle\simeq\operatorname{End}_{\Lambda}({M^{+}/M})/[M_{\mathcal{Y}}].$

Transferring endomorphisms along the equivalence
$T\otimes_{\Lambda}-\colon\mathcal{Y}_{T}\rightarrow\mathcal{T}_{T}$ provides
an isomorphism

$\operatorname{End}_{\Lambda}({M^{+}/M})/[M_{\mathcal{Y}}]\simeq\operatorname{End}_{\mathbf{k}Q}\left({T\otimes_{\Lambda}\frac{M^{+}}{M}}\right)/[T\otimes_{\Lambda}M_{\mathcal{Y}}].$

The $\mathbf{k}Q$-module $T\otimes_{\Lambda}\frac{M^{+}}{M}\in\mathcal{T}_{T}$
is partial tilting. As a result of Theorem 3.7 and Lemma A.4(2), the
endomorphism $\mathbf{k}$-algebra
$\operatorname{End}_{\mathbf{k}Q}\left({T\otimes_{\Lambda}\frac{M^{+}}{M}}\right)$
is either tame concealed or representation-finite. This establishes (2), since
quotients of tame $\mathbf{k}$-algebras are tame. Our claim in (3) is deduced
from Lemma A.4(2). ∎

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