11institutetext: The University of Sydney, Sydney NSW 2000, Australia
11email<EMAIL_ADDRESS>
#  Service-Based Wireless Energy Crowdsourcing

Amani Abusafia  Abdallah Lakhdari  Athman Bouguettaya

###### Abstract

We propose a novel service-based ecosystem to crowdsource wireless energy to
charge IoT devices. We leverage the service paradigm to abstract wireless
energy crowdsourcing from nearby IoT devices as energy services. The proposed
energy services ecosystem offers convenient, ubiquitous, and cost-effective
power access to charge IoT devices. We discuss the impact of a crowdsourced
wireless energy services ecosystem, the building components of the ecosystem,
the energy services composition framework, the challenges, and proposed
solutions.

###### Keywords:

Service ComputingEnergy ServicesWireless Energy Charging CrowdsourcingIoT
Wireless Power Transfer

##  1 Introduction

Internet of Things (IoT) is a paradigm that enables everyday objects (i.e.,
_things_) to connect to the internet and exchange data [1]. IoT devices
usually have capabilities, such as sensing, networking, and processing [1].
The number of connected IoT devices is expected to reach 125 billion in 2030
[2]. This potential pervasiveness of IoT provides opportunities to _abstract_
their capabilities using the _service paradigm_ as _IoT services_[3]. IoT
services are defined by their functional and non-functional attributes. The
functional attributes define the purpose of the service, such as sharing
internet access using WiFi. The non-functional attributes are the properties
that assess the Quality of Service (QoS), e.g., signal strength, reliability,
etc. For example, an IoT device owner may offer their WiFi as a hotspot (i.e.,
service provider) to other nearby IoT devices (i.e., service consumers). A
multitude of novel IoT services may be used to enable intelligent systems in
several domains, including smart cities, smart homes, and healthcare [3].
Examples of IoT services are WiFi hotspots [4], environmental sensing [5], and
energy services [6]. Of particular interest is the use of energy services.

Energy service, also known as _Energy-as-a-Service (ES)_ , refers to the
wireless power transfer among nearby IoT devices [6]. We consider a particular
set of IoT devices named wearables. Wearables refer to anything worn or hand-
held like smart shirts, smartwatches, and smartphones [7]. Wearables may
harvest energy from natural resources such as kinetic activity, solar power,
or body heat [8][9]. For instance, a smart shoe using a PowerWalk kinetic
energy harvester may produce 10-12 watts on-the-move power111bionic-power.com.
In this respect, wearing a PowerWalk harvester may generate energy to charge
up to four smartphones from an hour’s walk at a comfortable speed. Energy
services may be deployed through the newly developed “Over-the-Air” wireless
power transfer technologies [6][10]. Several companies focus on developing the
wireless charging technology of IoT devices over a distance, including
WiTricity222witricity.com, Energous333energous.com, Cota444ossia.com,
Powercastco555powercastco.com. For example, WiTricity started based on the
work of [11] where they succeeded in transferring 60 W of power wirelessly to
power a light bulb. Another example is Energous which developed a device that
can charge up to 3 Watts power within a 5-meter distance to multiple
receivers.

Figure 1: The components of IoT energy services ecosystem

Crowdsourcing is an efficient way to leverage IoT energy services to create a
self-sustained environment [6][3]. IoT users may collaborate to share their
spare energy to charge nearby IoT devices and extend their battery endurance
[12][13]. Crowdsourced IoT energy services present a convenient and adaptable
solution as devices do not need to be tethered to a power point or use
charging cords, and power banks [14]. In addition, crowdsourcing energy offers
an ubiquitous power access for IoT users as they may be charged anytime and
anywhere, even while moving [6][14]. Charging IoT devices wirelessly from a
central source usually requires a high-frequency magnetic field to transfer
the energy over a distance [15]. Studies have shown that a strong magnetic
field has a harmful impact on humans [16][15]. On the contracts, crowdsourcing
IoT energy services enables charging by aggregating energy from multiple
close-by devices. As the devices are near, transferring the energy will
require a low-frequency magnetic field. Hence, crowdsourcing energy services
offer an alternative solution to charge devices wirelessly without
compromising users’ health [17][18]. In this paper, we propose to leverage the
service paradigm to enable a self-sustained IoT energy services ecosystem by
utilizing three components, crowdsourcing, IoT, and wireless power transfer
technologies (see Fig.1).

This paper maps out a strategy to leverage the service paradigm to utilize IoT
energy services in smart cities. We envision a sustainable ecosystem that
allows on-the-go wireless energy crowdsourcing to recharge IoT devices in
smart cities. First, we highlight the benefits of the energy services
ecosystem. Then, we present a holistic service-based ecosystem to
conceptualize the idea and call for future validation. Next, we describe the
ecosystem in terms of the environment, service-oriented architecture, and
enabling technologies. We also present the envisioned framework and the
contemporary approaches for composing IoT-based energy services. Finally, we
discuss the uprising challenges to implementing the envisioned ecosystem and
highlight the potential future research directions that may address these
challenges.

##  2 Impact of Crowdsourcing Energy Services Ecosystem

Crowdsourcing wireless energy services provides numerous benefits to both the
environment and IoT users. Each of the benefits is discussed in depth below.

  * •

Enabling Sustainable IoT Ecosystem: The proliferation of IoT devices leads to
a significant increase in energy consumption [19][20]. IoT devices’ global
energy demand is predicted to reach 46TWh by 2025 [20]. This tremendous energy
demand accounts for 6% to 8% of the global carbon footprint generated by
information and communication technologies [21] [22]. Crowdsourcing energy
services reduce carbon footprint by reducing dependence on fossil fuels to
charge IoT devices. In addition, crowdsourcing energy services rely on
renewable or spare energy resources. The renewable energy may be harvested
from natural resources such as body heat, solar energy, or kinetic movement
[23].

Figure 2: Usage-time using wire vs wireless charging technologies

  * •

Extending Battery Endurance:IoT devices are constrained by the size of their
batteries [24][25]. The small size of the battery results in a limited energy
storage capacity. This limited capacity hinders the capabilities of IoT
devices. Increasing the battery capacity for IoT devices faces several
challenges,including, safety, weight, cost, and recycling [26]. Other charging
methods, such as carrying a power cord and searching for a power outlet, are
inconvenient for users. Hence, crowdsourcing energy services becomes an
attractive solution to improve battery endurance [27].

  * •

Unlimited Usage Time: Crowdsourcing energy services enable recharging IoT
devices without interrupting their usage time. In contrast, wired charging may
require IoT users to stop the usage of their devices in order for them to be
charged. Hence, wireless recharging of IoT devices may result in unlimited
usage time, i.e., constantly using them to provide or access services (See
Fig.2). Thus, crowdsourcing energy services prolong the IoT device’s usage
lifetime, especially when the external energy supply is unavailable.

  * •

Spatial Freedom: Crowdsourcing wireless energy services provide spatial
independence for IoT users. As previously mentioned, IoT devices are
frequently required to be tethered to power points or carry power banks to be
charged. The last obstacle for IoT devices to achieve their spatial freedom is
both charging wires and pads. The prospected environment is expected to enable
wireless power transfer delivery up to five meters [28].

  * •

Flexible Contracts:  Unlike traditional services, crowdsourced energy services
environment don’t have lock-in contracts [29]. The flexible contracts enable
consumers and providers to request or offer full or partial services according
to their preferences [29][30]. In addition, providers and consumers may extend
their stay-time to offer or receive an energy service [31].

  * •

Ubiquitous Power Access: As aforementioned, energy services offer spatial
freedom for IoT devices. Therefore, crowdsourcing energy services offer an
ubiquitous power access to charge IoT devices anytime and anywhere. This
ubiquity of wireless charging facilitates access to energy for IoT users.

  * •

Convenience Charging: The ubiquity of wireless charging offers a convenient
alternative to charge IoT devices [14]. In addition, as previously discussed,
the flexibility to participate in crowdsourcing energy services without any
long term contract offers convenience for IoT users. IoT users might offer or
request energy services without any lock-in contract. They can also move
around freely while providing or receiving energy [32]. This convenience
ensures that the ecosystem will expand widely.

  * •

Business Edge: As previously mentioned, energy services offer a convenient
solution to charge IoT devices. Thus, energy services may be used as a
complementary service to provide customers with the best quality of experience
when visiting a business [17]. For example, energy services, like WiFi, may be
used to charge customers’ wearables in a cafe. Such a service will increase
customers’ satisfaction with a business. Customer satisfaction is the main
factor in ensuring that customers keep coming back to businesses, i.e.,
maintaining or increasing foot traffic [33]. For example, a case study showed
that ”Sacred”, a cafe in London, had a noticeable increase in foot traffic
after installing wireless charging points666air-charge.com. The foot traffic
in a business has a direct impact on its revenue [34].

##  3 Crowdsourcing Energy Services Ecosystem

The energy services ecosystem setup consists of three major components: (1)
The context of the envisioned prospective environment, (2) The abstraction of
the ecosystem as an energy service-oriented architecture, and (3) The enabling
technologies to implement the foreseen ecosystem, In what follows, we discuss
each component in detail.

Figure 3: Crowdsourcing IoT energy services scenario

###  3.1 Prospective Environment:

The envisioned crowdsourced IoT energy environment is a dynamic environment
that consists of IoT users congregating and moving across microcells [6][35].
A microcell is any confined area in a smart city where people may gather
(e.g., coffee shops, restaurants, museums, libraries) (see Fig.3 (A)). In this
environment, IoT users are assumed to act as energy providers or consumers
(see Fig.3 (B)). Their IoT devices are assumed to be equipped with wireless
energy transmitters and receivers. Energy providers may use their wearables to
harvest energy [36]. The IoT users might share their spare or harvested energy
to fulfill the requirements of nearby IoT devices. Energy Providers advertise
services and consumers submit requests to the IoT coordinator. The super
provider manages the IoT coordinator. A super provider is typically a
microcell’s owner who manages the exchange of energy services between
providers and consumers. The IoT coordinator is assumed to be deployed one hop
away from the energy providers and consumers (e.g., router at the edge) to
minimize the communication overhead and latency while advertising energy
services and requests. The super provider limits a consumer’s requested energy
and uses a reward system to encourage providers to share energy. Rewards come
in the form of stored credits to providers. The collected credits may be used
later by the provider to increase the limit on the amount of requested energy
when they are in the consumer role. Providers may also be a consumer and vice-
versa. Providers receive rewards based on the amount of shared energy.

In the energy services environment, IoT devices may share energy using the
energy service model. Note that energy services exhibit _functional_ and _non-
functional_ (Quality of service (QoS)) properties [29]. In this respect, the
functional property of energy service is represented by the wireless service
delivery of energy to nearby IoT devices. The non-functional (QoS) properties
may include the amount of energy, location, duration of sharing, etc.

###  3.2 Energy Service Oriented Architecture:

A key aspect to unlocking the full potential of the energy services ecosystem
is to design an end-to-end Service Oriented Architecture (SOA) to share
crowdsourced energy services. We identify three key components of the SOA:
energy service provider, energy service consumer, and super provider. (see
Fig.4). An _energy provider_ refers to an IoT device that may share their
energy. An _energy consumer_ refers to an IoT device that requires energy. A
super provider is typically a microcell’s owner who manages the exchange of
energy services between providers and consumers.

Figure 4:  Energy services oriented architecture

###  3.3 Enabling Technologies:

The implementation of a crowdsourcing energy services ecosystem strongly
depends on the recent technology of wireless charging. Wireless charging
technologies are the transmitters and receivers that enable the wireless
transfer of energy [10]. For example, Xiaomi’s Mi Air charger transmits energy
wirelessly to nearby IoT devices777mi.com.

Energy harvesting technologies are another technology to enable a sustainable
crowdsourcing energy services ecosystem. Energy harvesters may enable a green
crowdsourcing ecosystem to charge IoT devices [37]. The IoT devices may
harvest energy from natural resources, including kinetic movement, solar
power, or body heat [8][9]. The harvested energy may be used to charge the
device itself or to be shared with nearby devices [38].

##  4 Crowdsourcing Energy Services Framework

The energy service framework is responsible for composing and managing the
received energy services and requests. An energy service is described by the
IoT owner’s spatio-temporal preferences, available energy, usage, and mobility
model (See Fig.5). Similarly, an energy request is described by the IoT
owner’s spatio-temporal preferences, amount of requested energy, usage, and
mobility model. We envision the composition framework to consist of four
components: (1) an incentive model. (2) a reliability and trust assessment,
(3) a spatio-temporal composability model, and (4) a service composition
approach. In what follows, we discuss each component in detail.

Figure 5: Crowdsourcing energy services framework

###  4.1 Incentive Model

Typically, in crowdsourcing environments, users resist sharing their resources
[39]. Similarly, IoT users may resist sharing their energy since energy is a
scarce resource [30][40]. Hence, an incentive model is needed to encourage
providers to share their energy. Designing an incentive model shall consider
the context of the environment and the behavior of the IoT users [41][30].

###  4.2 Reliability and Trust Assessment

The participation of IoT users in the energy crowdsourcing ecosystem is
influenced by reliable and trustworthy providers, and consumers [42]. To
guarantee high-quality services, both reliability and trust are required in
crowdsourced IoT systems. High-quality services will encourage and maintain
users’ participation in the energy services ecosystems. Therefore, when
managing and composing energy services, it is imperative to assess the
reliability and trustworthiness of IoT energy users.

Figure 6: Energy consumers may be charged from energy providers within the
power transfer range

###  4.3 Spatio-temporal Composability Model

Energy services and requests may have different times and locations [29][6].
Successfully delivering energy services requires providers and consumers to be
within the power transfer range. For example, if a provider’s location is out
of the power transfer range from a given consumer, the system shouldn’t match
them (See Fig.6). Thus, an effective filtering method is needed in order to
efficiently match services to requests. A composability model was proposed to
index services nearby a signal request based on time and space [29]. The same
model may be used to filter multiple services and multiple requests.

###  4.4 Services Composition Approach

A single energy request may not be fulfilled by one energy service due to the
limited resources of IoT devices [29]. In such cases, the system may utilize
the shareable nature of energy by composing multiple nearby energy services to
one request or vice versa [17]. Similarly, a single energy service may be used
to fulfill one or multiple energy requests [29]. Hence, we may have different
modes of composition (See Fig.7). Additionally, energy service providers and
consumers may have different Spatio-temporal preferences. Therefore, the
service composition approach should consider these preferences in matching and
composing energy services and requests.

Figure 7: Modes of energy services composition

##  5 Challenges in Crowdsourcing Energy Services

To exploit the potential of crowdsourced energy services for charging IoT
devices, multiple challenges must be tackled. These challenges can be
classified into three main categories enabling technologies, human-in-the-
Loop, and system deployment. The following subsections describe the challenges
encountered in each category and propose possible solutions.

###  5.1 Enabling Technologies:

The deployment of a crowdsourcing energy services ecosystem highly relies on
the current Wireless Power Transfer (WPT) technologies. Although WPT
technologies are advancing rapidly, there are technological constraints that
impede the ecosystem from being fully deployed [43]. Challenges posed by
technology include energy loss, miniature energy delivery, and restricted
power transmission range. Each of these constraints is discussed further
below.

####  5.1.1 Energy Loss

Using the recently developed wireless power transfer technologies, consumers
may spend more energy to receive the requested energy [44]. The consumed
energy may include the energy required for service discovery, energy delivery,
and connection establishment between the consumer and the provider [45]. The
huge energy loss impedes the energy services ecosystem from fulfilling its
purpose of charging nearby IoT devices. Therefore, minimizing energy loss is a
challenge yet to be addressed.

####  5.1.2 Miniature Delivered Energy

Even though there is rapid development in wireless power transfer
technologies, the amount of delivered energy is insufficient to what is
currently required [14]. The small amount of delivered energy may be justified
by the energy loss in delivering the energy service as aforementioned. Hence,
Delivering a considerable amount of energy wirelessly over a distance is still
a challenge that hinders the full deployment of the energy services ecosystem.

####  5.1.3 Limited Power Transfer Range

The near-field magnetic coupling is an efficient technology to transfer energy
wirelessly [14]. However, the magnetic coupling has a short power transfer
distance. The limited transfer distance restricts the spatial freedom we
envision as part of this ecosystem.

###  5.2 Human-in-the-loop

Crowdsourcing IoT energy services is a human-centered application as it
depends on their participation behavior. Modeling human behavior is
challenging as it requires modeling complex behavioral, psychological and
physiological characteristics of human nature [46]. In our environment, the
human factor has several influences on the ecosystem. First, the deployment of
the ecosystem depends on the IoT users’ willingness to participate in the
energy sharing process [30]. Second, the mobility of IoT users in this dynamic
environment could impede the delivery of energy services [32]. Third, energy
services are offered by IoT devices that are simultaneously in use by their
owners. Therefore, the consistent wireless energy delivery from one IoT device
to another depends on the device owners’ usage frequency. Finally, the spatial
and temporal preferences of IoT users differ, which affects the availability
of energy services. In what follows, we discuss each of the aforementioned
challenges.

####  5.2.1 Provisioning Resistance

Individuals are likely motivated to engage in the energy services ecosystem
since they can easily access nearby energy resources. However, since energy is
a scarce resource, they may be reluctant to share it [30]. Service resistance
is the reluctance to provide services due to limited resources [47]. In the
energy services context, service resistance refers to the unwillingness to
provide energy to IoT devices [30]. A provider’s resistance is influenced by
the provider’s available energy and the size of the requested energy. As a
result, one of the challenges is to predict providers’ resistance to offering
their energy service and devise a correct incentive to overcome it. Since
humans get impacted differently by different incentives, determining the
effective incentive for energy providers is a challenge that still needs to be
addressed.

####  5.2.2 Modelling Human Mobility

IoT devices change their locations regularly according to the mobility of
their owners. The mobility of the crowd has multiple impacts on the
crowdsourcing ecosystem. First, the mobility of the crowd across microcells
determines the availability of energy services and requests. Second, The IoT
users’ mobility affects the connectivity between IoT devices. Highly moving
IoT devices within a microcell may result in disconnecting the wireless energy
transfer between them. A disconnection in the delivery of the energy transfer
may result in its failure [42][32]. Therefore, the mobility of IoT users
should be modeled and analyzed to manage the energy services in the ecosystem
better. The mobility model of IoT users should be studied and investigated on
a micro level as indoor mobility and a macro level as mobility patterns across
microcells. The indoor mobility models should reflect the users’ movement
within a confined space [48]. Indoor mobility models will enable the ecosystem
to filter services and requests from a spatial perspective within the power
transfer range. Moreover, it will allow the ecosystem to predict users’
movement and their impact on disconnecting a service delivery. On the other
side, the outdoor mobility pattern of an IoT user may be used to proactively
plan for them when to request energy [35]. Furthermore, it enables us to
understand the crowd movement and predict the undersupplied areas [49]. Thus,
mobility models are needed to unlock the full potential of the ecosystem.

####  5.2.3 Modelling Usage Patterns

: The energy usage pattern of providers may impact the consistency of
provisioning energy services [31]. An energy provider with highly dynamic
usage of their device impacts the delivered energy service quality (QoS).
Providers may offer their spare energy while simultaneously using it for their
devices. If a provider is heavily using their device, they may end up
consuming their advertised service while sharing it. Thus, energy services may
fluctuate due to the providers’ usage. As a result, IoT devices’ energy usage
behavior must be modeled to ensure consistent provision of energy services
[48][49].

####  5.2.4 Determining Services Availability:

Matching energy services and requests in a crowdsourced environment is
challenging due to the uncertain availability of energy services and requests
[17]. Predicting the availability of energy services is challenging as it
relies on detecting the mobility patterns of IoT Users and their energy usage
patterns. Defining the mobility patterns mainly focuses on analyzing the
spatio-temporal attributes and potential regularities hidden in individual
movement trajectories [50]. However, the existing literature to determine
mobility patterns has low accuracy due to the flexible nature of human
movement [49]. Thus, the challenge of detecting human mobility patterns
hinders the prediction of service availability.

###  5.3 Deployment of Energy Services Ecosystem

The deployment of the energy crowdsourcing ecosystem raises several
challenges. In what follows, we discuss each challenge and its impact of on
the ecosystem.

####  5.3.1 Balancing Local Provisioning and Demand:

Naturally, energy users’ preferences to provide or receive energy vary in time
and space. The variety in energy users’ preferences may severely affect the
balance between the amount of required and provided energy within a confined
area. The imbalance between the energy demand and the available services may
result in oversupplied and undersupplied areas in a smart city. Under-supplied
areas will result in unfulfilled energy requests, which may discourage the
participation of IoT users in the ecosystem. Hence, ensuring the availability
of energy services to fulfill the requirement of consumers is essential to the
sustainability of the ecosystem. A redistribution of energy service across
over-supplied and under-supplied areas may achieve better-balanced energy
availability. Another possible direction is using an incentive model to
increase participation in provisioning energy services [30].

####  5.3.2 Loose Contracts

As previously mentioned, energy services offer flexibility as they do not
require lock-in contracts [31]. For instance, providers may offer full or
partial services according to their preferences [29]. However, this
flexibility may impact the commitment of IoT users. The uncertain commitment
of IoT users may add uncertainty to the energy transfer process and thereby
impact the participation of IoT users in the ecosystem.

####  5.3.3 Reliability

Considering reliability in the energy ecosystem encourages the participation
of energy users in crowdsourcing energy services [42][6]. The reliability of
an energy provider refers to the probability that an energy service will be
successfully delivered with the advertised Quality of Service (QoS)
attributes. Similarly, the reliability of an energy consumer refers to the
probability that an IoT energy service will be successfully received with the
same submitted requirements. The reliability of the energy provider is
impacted by their usage pattern and mobility. As aforementioned, a provider
that heavily uses their device might consume their advertised energy. Also, a
highly mobile provider will cause frequent disconnections in the energy
transfer between the devices—similarly, the reliability of an energy consumer
is impacted by their mobility patterns. Like providers, a highly mobile
consumer will cause frequent disconnections in the energy transfer between the
devices. Accordingly, it is vital to consider the reliability of energy users
while managing and composings energy services.

####  5.3.4 Trust

The definition of trust is ”the confidence, belief, and expectation regarding
the reliability, integrity, and other characteristics of an entity”[51]. To
guarantee high-quality services, trust assessment is required in crowdsourced
IoT systems [4]. High-quality services will encourage and maintain users’
participation in the energy services ecosystems. The trust assessment must
evaluate the trustworthiness of energy providers and consumers. The
trustworthiness of energy providers may represent the provider’s reputation,
their service reliability, and their security level [4]. Likewise, energy
consumers’ trustworthiness may represent their reputation and their security
level. Although trust assessment has been studied in other areas, it remains a
challenge in IoT ecosystems due to several reasons [52]. First, the dynamic
nature of the crowdsourced IoT environment makes it challenging to keep an
accurate record of the devices’ reputations. For instance, IoT devices are
usually moving, and their existence may be temporary. Second, IoT devices
typically lack a global identifier, which makes it difficult to maintain a
globally accessible profile for these devices. As a result, a novel trust
assessment framework is required to assess the reputation of IoT devices [52].

####  5.3.5 Security

The goal of IoT device security is to prevent and protect against IoT attacks
and service failures [53]. Attacks on IoT devices may compromise the privacy
and confidentiality of users, infrastructures, data, and IoT devices [54].
Furthermore, IoT attacks such as denial-of-service attacks may impede the
delivery of services to IoT Users. Securing IoT systems is a critical
challenges because users may not adopt many IoT systems if they are not secure
[55]. Existing security architectures and protocols are difficult to integrate
in IoT devices due to their limited computing power and storage size[56].

####  5.3.6 Privacy

The global digital data generated by IoT devices is expected to reach 180
Zettabytes by 2020 [57]. As a result, protecting IoT data and the privacy of
users who generate or consume data has become a major concern in research and
industry [58]. Privacy refers to “The claim of individuals, groups, or
institutions to determine for themselves when, how and to what extent
information about them is communicated to others” [59]. Maintaining the
privacy in IoT ecosystems is challenging as it conflict with the need to use
the IoT data to achieve their functions. The data utilization may invade the
privacy of IoT users [52]. For example, IoT Data are required to enhance the
energy crowdsourcing process by profiling the energy providers and consumers.
The collected IoT data may include the IoT users’ mobility behavior and their
energy consumption model. Collecting such personal information might violate
the privacy of the IoT users. Therefore, maintaining a balance between IoT
data protection and use is a continuing challenge.

##  6 Conclusion

We presented a novel service-based framework to crowdsource wireless energy
from neighboring IoT devices to charge low-in-battery IoT devices. First, we
highlighted the benefits of adopting a crowdsourced wireless energy services
ecosystem. We then envisioned and designed an architecture to implement the
proposed ecosystem. Lastly, we discussed the open challenges and recommended
research directions for possible solutions.

## Acknowledgment

This research was partly made possible by LE220100078 and LE180100158 grants
from the Australian Research Council. The statements made herein are solely
the responsibility of the authors.

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# Stratifying systems and $g$-vectors

Octavio Mendoza, Corina Sáenz and Hipolito Treffinger

###### Abstract.

In this paper we study the Cartan matrix associated to the Ext-projective
stratifying system induced by a basic and $\tau$-rigid object $M$ in
$\mathrm{mod}\,(A)$ by means of the $g$-vectors of the indecomposable direct
summands of $M$. In particular we show that the Cartan group of a stratifying
system associated to a $\tau$-rigid module can be calculated directly using
these vectors. Moreover we characterise the stratifying systems coming from
$\tau$-rigid modules that have a diagonal Cartan matrix.

2020 Mathematics Subject Classification. Primary 18G20, 18E40, 16D10.
Secondary 16E99.
Keywords: Stratifying systems; $g$-vectors; $\tau$-rigid modules; Cartan
matrices and groups; weakly triangular algebras.

##  1\. Introduction

The notion of an Ext-injective stratifying system was introduced in [8] as an
axiomatisation of the standard objects of the standardly stratified algebras
defined in [6]. It was quickly realised that Ext-injective stratifying systems
are equivalent to the notions of Ext-projective stratifying systems [14] and
what is now known simply as stratifying systems [13]. Since then, stratifying
systems were deeply studied in [7, 20, 12, 17, 18, 21, 10]. Throughout this
paper, $A$ will denote to a basic finite dimensional $k$-algebra over an
algebraically closed filed $k.$ We will work on the abelian category
$\mathrm{mod}(A)$ of all the finitely generated left $A$-modules. Given a
positive integer $t,$ we denote by $[1,t]$ the set of all the natural numbers
between $1$ and $t,$ and it is also considered the natural order $\leq$ on the
set $[1,t].$

The notion of Ext-projective stratifying system in $\mathrm{mod}(A)$ is
defined in [14]. We recall this notion as follows.

######  Definition 1.1.

[14, Definition 2.1] An Ext-projective stratifying system in
$\mathrm{mod}(A),$ of size $t,$ is a triple $(\Theta,\underline{Q},\leq),$
where $\Theta:=\\{\Theta(i)\\}_{i=1}^{t}$ is a family of non-zero $A$-modules,
$\underline{Q}:=\\{Q(i)\\}_{i=1}^{t}$ is a family of indecomposable
$A$-modules and $\leq$ is the natural order on $[1,t]$ satisfying the
following conditions:

  1. (a)

$\mathrm{Hom}_{A}(\Theta(j),\Theta(i))=0$ for $j>i;$

  2. (b)

for each $i\in[1,t]$, there is an exact sequence $0\to
U(i)\xrightarrow{\alpha_{i}}Q(i)\xrightarrow{\beta_{i}}\Theta(i)\to 0$ in
$\mathrm{mod}\,(A)$ such that $U(i)$ is filtered by the set
$\\{\Theta(j):j>i\\}$;

  3. (c)

$\mathrm{Ext}^{1}_{A}(Q,X)=0$ where $Q:=\bigoplus_{i=1}^{t}Q(i)$ and $X$ is an
$A$-module filtered by $\\{\Theta(i)\\}_{i=1}^{t}$.

Although it has been shown that stratifying systems are objects having very
nice properties, until very recently there was no systematic method to
construct stratifying systems in the module category of a given algebra $A$.
This problem has been addressed in [19] and [22] using tools from
$\tau$-tilting theory and higher homological algebra, respectively.

It was shown in [15] that it can be associated to every Ext-projective
stratifying system $(\Theta,\underline{Q},\leq)$ two invariants $C_{\Theta}$
and $G_{\Theta}$, known as the Cartan matrix and the Cartan group of $\Theta$,
respectively (see Section 3). In this paper, we study the Cartan matrix
$C_{\Delta_{M}}$ and the Cartan group $G_{\Delta_{M}}$ for a stratifying
system $\Delta_{M}$ associated to a (basic) $\tau$-rigid module
$M\in\mathrm{mod}(A).$ In particular, we show that the matrix $C_{\Delta_{M}}$
can be obtained using the $g$-matrix $G^{M}$ (formed by the $g$-vectors of the
indecomposable direct summands of $M$), the $M$-standard matrix $S^{M}$
(formed by the dimension vectors of the $M$-standard modules $\Delta_{M}$) and
the $M$-residual matrix $R^{M}$ (see Section 5). This result can be summarized
as follows, for the complete version, see Theorem 5.3 and Proposition 5.4.

######  Theorem 1.2.

Let $M\in\mathrm{mod}\,(A)$ be basic and $\tau$-rigid with a TF-admissible
decomposition $M=\bigoplus_{i=1}^{t}M_{i},$ and let
$(\\{\Delta_{M}\\}_{i=1}^{t},\underline{Q},\leq)$ be its Ext-projective
stratifying system. Then,

$C_{\Delta_{M}}=(G^{M})^{tr}\,S^{M}+R^{M}\quad\text{and}\quad|G_{\Delta_{M}}|=\prod_{i=1}^{t}[(G^{M})^{tr}\,S^{M}]_{i,i}$

Moreover, if $M$ is filtered by the family $\Delta_{M},$ then $R^{M}=0$ and
$Q(i)\simeq M_{i}$ $\forall\,i.$

Then we apply this theorem to characterise algebras $A$ and stratifying
systems in $\mathrm{mod}(A)$ coming from $\tau$-tilting modules having a
diagonal Cartan matrix, generalising [15, Theorem 5.9]. In what follows, we
summarize this result. For more details, see Theorem 5.8.

######  Theorem 1.3.

For a basic $\tau$-tilting module $M\in\mathrm{mod}\,(A)$ with a TF-admissible
decomposition $M=\bigoplus_{i=1}^{n}M_{i}$ such that
$M\in\mathcal{F}(\Delta_{M}),$ the following statements are equivalent.

  * $\mathrm{(a)}$

The matrix $C_{\Delta_{M}}$ is diagonal.

  * $\mathrm{(b)}$

$\mathrm{Hom}_{A}(M_{i},M_{j})=0$ for $i<j.$

  * $\mathrm{(c)}$

$\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))=0$ for $i<j.$

Moreover, if one of the above conditions holds true, then $M\simeq{}_{A}A$ and
$A$ is a weakly triangular algebra.

The paper is organised as follows. In Section 2 we establish the setting and
notation in which we work and we recall some basic representation theory.
Then, in Section 3 we recall some theory on stratifying systems and we prove
some preparatory results. Later, in Section 4 we give a brief overview of the
necessary results on $\tau$-tilting theory that we need in Section 5 where we
state and prove the main results of this paper. Finally, in Section 6 we give
some examples illustrating Theorem 5.3.

### Ackowledgements

Most of the mathematical content of this paper was performed while the third
named author visited the other two authors at the UNAM in México in February
2020. We hope that in the near future the sanitary conditions worldwide will
allow for more in person collaborations.

The authors thank the research project PAPIIT-Universidad Nacional Autónoma de
México IN100520. H. Treffinger was partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy Programme – EXC-2047/1 – 390685813. H. Treffinger is also
supported by the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No 893654. He was
also partially supported by the EPSRC through the Early Career Fellowship,
EP/P016294/1. H. Treffinger would like to thank the Isaac Newton Institute for
Mathematical Sciences, Cambridge, for support and hospitality during the
programme Cluster Algebras and Representation Theory where work on this paper
was undertaken. This work was supported by EPSRC grant no EP/K032208/1 and the
Simons Foundation.

##  2\. Setting and background

For $M\in\mathrm{mod}\,(A),$ we denote by $rk(M)$ the number of pairwise non-
isomorphic indecomposable direct summands of $M.$ Finally, we say that
$M\in\mathrm{mod}\,(A)$ is basic if there is a decomposition
$M=\bigoplus_{i=1}^{t}M_{i},$ with $M_{i}$ indecomposable for every $i$ and
$M_{i}\not\simeq M_{j}$ if $i\neq j.$

Given an algebra $A$, we denote by $M\twoheadrightarrow N$ an epimorphism from
$M$ to $N$ in $\mathrm{mod}\,(A);$ and for each $M\in\mathrm{mod}\,(A),$ we
consider the class

$\mathrm{Fac}(M):=\\{X\in\mathrm{mod}\,(A)\;:\;\exists\,M^{n}\twoheadrightarrow
X\text{ for some $n\in\mathbb{N}$}\\}.$

Dually, the arrow $\hookrightarrow$ stands for a monomorphism in
$\mathrm{mod}\,(A),$ and we have the class

$\mathrm{Sub}(M):=\\{X\in\mathrm{mod}\,(A)\;:\;\exists\,X\hookrightarrow
M^{n}\text{ for some $n\in\mathbb{N}$}\\}.$

For any subclass $\mathcal{X}\subseteq\mathrm{mod}\,(A),$ we have the right
$\mathrm{Hom}$-perpendicular complement of $\mathcal{X}$

$\mathcal{X}^{\perp}:=\\{M\in\mathrm{mod}\,(A)\;:\;\mathrm{Hom}_{A}(-,M)|_{\mathcal{X}})=0\\}.$

Dually, we have the left $\mathrm{Hom}$-perpendicular complement
${}^{\perp}\mathcal{X}$ of $\mathcal{X}.$ For a single element class
$\mathcal{X}=\\{M\\},$ we just write $M^{\perp}$ and ${}^{\perp}M.$ It is said
that $N\in\mathrm{mod}\,(A)$ admits an $\mathcal{X}$-filtration if there is a
chain of submodules $0=M_{0}\subseteq M_{1}\subseteq\ldots\subseteq
M_{n-1}\subseteq M_{n}=M$ satisfying that each quotient $M_{i}/M_{i-1}\simeq
X_{i}\in\mathcal{X}.$ The class of all the $N\in\mathrm{mod}\,(A)$ which
admits an $\mathcal{X}$-filtration will be denoted by
$\mathcal{F}(\mathcal{X}).$

Let $\mathcal{X}\subseteq\mathrm{mod}\,(A)$ and $M\in\mathrm{mod}\,(A).$ A
morphism $f:X\to M$ in $\mathrm{mod}\,(A)$ is called $\mathcal{X}$-precover
(or right $\mathcal{X}$-approximation) of $M$ if $X\in\mathcal{X}$ and
$\mathrm{Hom}_{A}(X^{\prime},f):\mathrm{Hom}_{A}(X^{\prime},X)\to\mathrm{Hom}_{A}(X^{\prime},M)$
is surjective, for any $X^{\prime}\in\mathcal{X}$. Moreover, if the equality
$fh=f$ holds true only for an automorphism $h:X\to X,$ it is said that the
$\mathcal{X}$-precover is an $\mathcal{X}$-cover. The class $\mathcal{X}$ is
precovering (or contravariantly finite) if any $N\in\mathrm{mod}\,(A)$ admits
an $\mathcal{X}$-precover. Dually, we have the notions of
$\mathcal{X}$-preenvelope (or left $\mathcal{X}$-approximation),
$\mathcal{X}$-envelope and preenveloping (or covariantly finite) class. If
$\mathcal{X}$ is precovering and preenveloping, it is said that $\mathcal{X}$
is functorially finite.

We recall that a pair $(\mathcal{X},\mathcal{Y})$ of full subcategories in
$\mathrm{mod}\,(A)$ is a torsion pair if $\mathcal{X}={}^{\perp}\mathcal{Y}$
and $\mathcal{Y}=\mathcal{X}^{\perp}.$ Given a torsion pair
$(\mathcal{X},\mathcal{Y})$ in $\mathrm{mod}\,(A),$ it is said that
$\mathcal{X}$ is a torsion class and $\mathcal{Y}$ is a torsion free class.
Moreover, in this case, it can be shown that $\mathcal{X}$ is closed under
quotients and extensions, while $\mathcal{Y}$ is closed under submodules and
extensions. It is also well known that, for every subclass
$\mathcal{X}\subseteq\mathrm{mod}\,(A)$ which is closed under quotients and
extensions, there exists a subclass $\mathcal{Y}\subseteq\mathrm{mod}\,(A)$
such that $(\mathcal{X},\mathcal{Y})$ is a torsion pair in
$\mathrm{mod}\,(A).$ In particular, if $\mathcal{X}=\mathrm{Fac}(M)$ for some
$M\in\mathrm{mod}\,(A),$ then $\mathcal{Y}=M^{\perp}.$

It is also well known that each torsion pair $(\mathcal{X},\mathcal{Y})$ in
$\mathrm{mod}\,(A)$ has associated two additive functors
$\mathrm{t},\mathrm{f}:\mathrm{mod}\,(A)\to\mathrm{mod}\,(A).$ The torsion
functor $\mathrm{t}$ is a subfunctor of the identity functor
$1_{\mathrm{mod}\,(A)},$ and the torsion free functor $\mathrm{f}$ is equal to
the quotient functor $1/\mathrm{t}.$ Moreover, for every
$M\in\mathrm{mod}\,(A),$ there exists the so-called canonical short exact
sequence

$0\to\mathrm{t}(M)\to M\to\mathrm{f}(M)\to 0,$

where $\mathrm{t}(M)\in\mathcal{X}$ and $\mathrm{f}(M)\in\mathcal{Y}$ and such
exact sequence is unique up to isomorphisms.

Let $\mathcal{X}$ be a full subcategory of $\mathrm{mod}\,(A).$ It is said
that $X\in\mathcal{X}$ is Ext-projective in $\mathcal{X}$ if
$\mathrm{Ext}^{1}_{A}(X,-)|_{\mathcal{X}}=0.$ Moreover, if $\mathcal{X}$ is
functorially finite and a torsion class, it is well known that there are only
finitely many (up to isomorphisms) indecomposable Ext-projective modules in
$\mathcal{X}$ and we denote by $\mathbb{P}(\mathcal{X})$ their direct sum.

##  3\. Stratifying systems

The concept of stratifying system for module categories was first introduced
in [8], and further studied in [13, 14]. The original definition is stated in
Definition 3.2, but it was shown in [13] that this definition is equivalent to
the following one.

######  Definition 3.1.

[13, Characterisation 1.6] A stratifying system of size $t,$ in
$\mathrm{mod}\,(A),$ consists of a pair $(\Theta,\leq),$ where
$\Theta:=\\{\Theta(i)\\}_{i=1}^{t}$ is a family of indecomposable objects in
$\mathrm{mod}\,(A)$ and $\leq$ is the natural order on the set
$[1,t]:=\\{1,2,\dots,t\\}$ satisfying the following conditions:

  1. $\mathrm{(a)}$

$\mathrm{Hom}_{A}(\Theta(j),\Theta(i))=0$ if $j>i$;

  2. $\mathrm{(b)}$

$\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j))=0$ if $j\geq i$.

Let $(\Theta,\leq)$ be a stratifying system in $\mathrm{mod}\,(A).$ The theory
of stratifiying systems, developed in [8, 13, 14], shows that
$\mathcal{F}(\Theta)$ is equivalent to the full subcategory
$\mathcal{F}(\Delta)$ of $\mathrm{mod}\,(B)$, where $B$ is a standardly
stratified algebra and $\mathcal{F}(\Delta)$ is the category of the
$\Delta$-filtered objects, of key importance in the theory of quasi-hereditary
and standardly stratified algebras. In order to determine the standardly
stratified algebra $B$, it is necessary to consider the so-called Ext-
projective and Ext-injective stratifying systems. The notion of Ext-projective
stratifying system was given in Definition 1.1.

######  Definition 3.2.

[8, Definition 1.1] An Ext-injective stratifying system in $\mathrm{mod}(A),$
of size $t,$ is a triple $(\Theta,\underline{Y},\leq),$ where
$\Theta:=\\{\Theta(i)\\}_{i=1}^{t}$ is a family of non-zero $A$-modules,
$\underline{Y}:=\\{Y(i)\\}_{i=1}^{t}$ is a family of indecomposable
$A$-modules and $\leq$ is the natural order on $[1,t]$ satisfying the
following conditions:

  1. $\mathrm{(a)}$

$\mathrm{Hom}_{A}(\Theta(j),\Theta(i))=0$ for $j>i;$

  2. $\mathrm{(b)}$

for each $i\in[1,t]$, there is an exact sequence
$0\to\Theta(i)\xrightarrow{\lambda_{i}}Y(i)\xrightarrow{\gamma_{i}}Z(i)\to 0$
in $\mathrm{mod}\,(A)$ such that $Z(i)\in\mathcal{F}(\\{\Theta(j):j<i\\});$

  3. $\mathrm{(c)}$

$\mathrm{Ext}^{1}_{A}(-,Y)|_{\mathcal{F}(\Theta)}=0,$ for
$Y:=\bigoplus_{i=1}^{t}Y(i).$

Given a stratifying system $(\Theta,\leq),$ it is well known [8, 13, 14] that
there exist a unique (up to isomorphism) Ext-projective stratifying system
$(\Theta,\underline{Q},\leq)$ and an Ext-injective stratifying system
$(\Theta,\underline{Y},\leq).$ Furthermore, for a given Ext-projective
stratifying system $(\Theta,\underline{Q},\leq),$ we have that the pair
$(\Theta,\leq)$ is a stratifying system. Moreover the same statement holds
true for Ext-injective stratifying systems.

Let $(\Theta,\\{Q(i)\\}_{i=1}^{t},\leq)$ be an Ext-projective stratifying
system in $\mathrm{mod}\,(A).$ By following [15, Section 4], we consider the
$\Theta$-Cartan matrix $C_{\Theta}\in\mathrm{Mat}_{t\times t}(\mathbb{Z}),$
naturally associated to this system, which is defined as follows: the
$(i,j)$-coordinate of $C_{\Theta}$ is
$[C_{\Theta}]_{i,j}:=\dim_{k}\mathrm{Hom}_{A}(Q(i),\Theta(j)).$

The main idea in [15] behind the introduction of the Cartan matrix
$C_{\Theta},$ for an Ext-projective stratifying system
$(\Theta,\\{Q(i)\\}_{i=1}^{t},\leq),$ is that we can use it to construct a
finite abelian group $G_{\Theta}$, known as the _Cartan Group_ of $\Theta$
which is defined as $G_{\Theta}:=\mathrm{CoKer}(C_{\Theta}),$ where the matrix
$C_{\Theta}$ is identified with the $\mathbb{Z}$-linear map
$\mathbb{Z}^{t}\to\mathbb{Z}^{t},\;X\mapsto C_{\Theta}\,X.$ It is shown in
[15], that $|G_{\Theta}|=1$ if and only if the standardly stratified algebra
$\mathrm{End}_{A}(Q)^{op}$ is quasi-hereditary. Then the size $|G_{\Theta}|$
of $G_{\Theta}$ give us a measure of how far is $\mathrm{End}_{A}(Q)^{op}$
from being quasi-hereditary.

######  Lemma 3.3.

[14, 15, 16] Let $(\Theta,\\{Q(i)\\}_{i=1}^{t},\leq)$ be an Ext-projective
stratifying system in $\mathrm{mod}\,(A).$ Then, $C_{\Theta}$ is an upper
triangular matrix and
$[C_{\Theta}]_{i,i}=\dim_{k}(\mathrm{End}_{A}(\Theta(i)))$
$\forall\,i\in[1,t].$ Moreover
$|G_{\Theta}|=\prod_{i=1}^{n}[C_{\Theta}]_{i,i}.$

###### Proof.

It follows from [16, Lemma 2.5 (b)] and [14, Lemma 2.6 (b)]. ∎

Let $(\Theta,\\{Q(i)\\}_{i=1}^{t},\leq)$ be an Ext-projective stratifying
system in $\mathrm{mod}\,(A).$ In what follows, for each $i\in[1,t],$ we
consider the exact sequence $\eta_{i}:\;0\rightarrow
U(i)\xrightarrow{\alpha_{i}}Q(i)\xrightarrow{\beta_{i}}\Theta(i)\rightarrow 0$
associated to each $\Theta(i),$ see Definition 1.1 (b), and the map

$\partial_{i,j}:\mathrm{Hom}_{A}(U(i),\Theta(j))\to\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j)),\;f\mapsto
f\cdot\eta_{i},$

where $f\cdot\eta_{i}$ is the push-out of $f$ and $\eta_{i}.$

######  Proposition 3.4.

For an Ext-projective stratifying system $(\Theta,\\{Q(i)\\}_{i=1}^{t},\leq)$
in $\mathrm{mod}\,(A),$ the following statements are equivalent.

  * $\mathrm{(a)}$

The matrix $C_{\Theta}$ is diagonal.

  * $\mathrm{(b)}$

$\mathrm{Hom}_{A}(Q(i),Q(j))=0$ for $i<j.$

  * $\mathrm{(c)}$

$\mathrm{Hom}_{A}(Q(i),\Theta(j))=0$ for $i<j.$

  * $\mathrm{(d)}$

For $i<j,$ we have that $\mathrm{Hom}_{A}(\Theta(i),\Theta(j))=0$ and

$\partial_{i,j}:\mathrm{Hom}_{A}(U(i),\Theta(j))\to\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j)),\;f\mapsto
f\cdot\eta_{i}$

is an isomorphism.

Moreover, if one of the above conditions holds true, then
$\mathrm{Hom}_{A}(Q(i),U(j))=0$ for $i<j.$

###### Proof.

(a) $\Leftrightarrow$ (c): It follows from Lemma 3.3.

(b) $\Rightarrow$ (c): Let $\mathrm{Hom}_{A}(Q(i),Q(j))=0$ for $i<j.$ We prove
now that $\mathrm{Hom}_{A}(Q(i),\Theta(j))=0.$ If $Q(j)=\Theta(j)$ there is
nothing to prove. Otherwise, take $f:Q(i)\to\Theta(j)$ and consider the exact
sequence

$\eta_{j}:\quad 0\rightarrow
U(j)\xrightarrow{\alpha_{j}}Q(j)\xrightarrow{\beta_{j}}\Theta(j)\rightarrow 0$

associated to $\Theta(j),$ where $U(j)$ is nonzero and
$U(j)\in\mathcal{F}(\\{\Theta(k):k>j\\}).$ Since $Q(i)$ is Ext-projective in
$\mathcal{F}(\Theta)$, there exist $f^{\prime}:Q(i)\to Q(j)$ such that
$f^{\prime}\beta_{j}=f$. By hypothesis, $f^{\prime}=0$. Hence $f=0$.

(c) $\Rightarrow$ (b): Let $\mathrm{Hom}_{A}(Q(i),\Theta(j))=0$ for $i<j.$ We
carry on the proof by reverse induction on $j.$ If $j=n$ then $\Theta(n)=Q(n)$
and thus (b) follows. Therefore, we can assume that $j<n.$ Let us prove that
$\mathrm{Hom}_{A}(Q(i),Q(j))=0.$ In order to do that, take a map $g:Q(i)\to
Q(j)$ and consider the exact sequence $\eta_{j}.$ Since $g\beta_{j}=0$, there
exist $h:Q(i)\to U(j)$ such that $\alpha_{j}h=g.$ Moreover, since
$U(j)\in\mathcal{F}(\\{\Theta(k):k>j\\})$, [14, Proposition 2.10] implies the
existence of an exact sequence

$0\rightarrow N\to Q_{0}(U(j))\xrightarrow{\epsilon_{U(j)}}U(j)\rightarrow 0$

in $\mathcal{F}(\Theta),$ where
$Q_{0}(U(j))\in\mathrm{add}\left(\bigoplus_{k>j}Q(k)\right)$. Once again, by
using that $Q(i)$ is Ext-projective in $\mathcal{F}(\Theta),$ there exist
$h^{\prime}:Q(i)\to Q_{0}(U(j))$ such that $\epsilon_{U(j)}h^{\prime}=h$. By
induction, $h^{\prime}=0$. Therefore
$g=\alpha_{j}\epsilon_{U(j)}h^{\prime}=0$.

(c) $\Leftrightarrow$ (d): Let $i<j.$ By applying the functor
$\mathrm{Hom}_{A}(-,\Theta(j))$ to the exact sequence $\eta_{i},$ we get the
following exact sequence

$0\to\mathrm{Hom}_{A}(\Theta(i),\Theta(j))\to\mathrm{Hom}_{A}(Q(i),\Theta(j))\to\mathrm{Hom}_{A}(U(i),\Theta(j))\to\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j))\to
0$

Thus, the equivalence between (c) and (d) follows from the above exact
sequence.

Finally, assume that (b) holds true. Then, by applying the functor
$\mathrm{Hom}_{A}(Q(i),-)$ to $\eta_{j}$ we get that
$\mathrm{Hom}_{A}(Q(i),U(j))=0$ for $i<j.$

∎

######  Corollary 3.5.

For an Ext-projective stratifying system $(\Theta,\\{Q(i)\\}_{i=1}^{t},\leq)$
in $\mathrm{mod}\,(A),$ such that the matrix $C_{\Theta}$ is diagonal, the
following statements are equivalent.

  * $\mathrm{(a)}$

$[U(i):\Theta(j)]=0$ for $i<j.$

  * $\mathrm{(b)}$

$\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j))=0$ $\forall\,i,j.$

  * $\mathrm{(c)}$

$Q(i)\simeq\Theta(i)$ as $A$-modules $\forall\,i.$

  * $\mathrm{(d)}$

$U(i)=0$ $\forall\,i.$

Moreover, if one of the above conditions holds true, then
$\mathcal{F}(\Theta)=\mathrm{add}(Q)$ and
$\mathrm{End}_{A}(Q)\simeq\times_{i=1}^{t}\,\mathrm{End}_{A}(Q(i))$ as
$k$-algebras.

###### Proof.

By Proposition 3.4, we get that $\mathrm{Hom}_{A}(\Theta(i),\Theta(j))=0$
$\forall\,i\neq j,$ and moreover
$\partial_{i,j}:\mathrm{Hom}_{A}(U(i),\Theta(j))\xrightarrow{\simeq}\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j))$
for $i<j.$

(a) $\Rightarrow$ (b): Let $i,j\in[1,t].$ By [14, Lemma 2.6 (b)], we know that
$\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j))=0$ for $i\geq j.$ Hence, we can
assume that $i<j.$ Since $U(i)\in\mathcal{F}(\Theta(l)\;:\;l>i),$ by (a) we
have that $\mathrm{Hom}_{A}(U(i),\Theta(j))=0.$ Therefore
$\mathrm{Ext}^{1}_{A}(\Theta(i),\Theta(j))=0$ for $i<j.$

(b) $\Rightarrow$ (c): Let $i\in[1,t].$ Then, by (b), we get that the exact
sequence $\eta_{i}:\quad 0\rightarrow
U(i)\xrightarrow{\alpha_{i}}Q(i)\xrightarrow{\beta_{i}}\Theta(i)\rightarrow 0$
splits. Thus $Q(i)\simeq\Theta(i)$ as $A$-modules since $Q(i)$ and $\Theta(i)$
are indecomposable (see [14, Corollary 2.13]).

(c) $\Rightarrow$ (d): From (c) and the exact sequence $\eta_{i},$ we have
that $\dim_{k}U(i)=0$ and hence $U(i)=0.$

(d) $\Rightarrow$ (a): It is trivial.

Finally, If one of the above conditions holds true, then we get that
$Q(i)\simeq\Theta(i)$ as $A$-modules $\forall\,i.$ Hence
$\mathrm{Hom}_{A}(Q(i),Q(j))=0$ for $i\neq j$ and thus
$\mathrm{End}_{A}(Q)\simeq\times_{i=1}^{t}\,\mathrm{End}_{A}(Q(i))$ as
$k$-algebras. ∎

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

Most of the results of this paper are proven using the tools provided by
$\tau$-tilting theory [1]. We now give a brief summary of the definitions and
results on $\tau$-tilting theory that will be needed later. For a broader
survey on this topic, the reader is encouraged to see [23].

######  Definition 4.1.

[1, Definition 0.1] Let $M\in\mathrm{mod}\,(A).$ If $\mathrm{Hom}_{A}(M,\tau
M)=0,$ it is said that $M$ is $\tau$-rigid. A $\tau$-rigid module $M$ is
$\tau$-tilting if $rk(M)=rk({}_{A}A).$ A pair $(M,P)$ is called $\tau$-rigid
if $P\in\mathrm{mod}\,(A)$ is projective, $M$ is $\tau$-rigid and
$\mathrm{Hom}_{A}(P,M)=0.$ Finally, the $\tau$-rigid pair $(M,P)$ is
$\tau$-tilting if $rk(M)+rk(P)=rk({}_{A}A).$

One of the main features of $\tau$-tilting theory is that all functorially
finite torsion classes in $\mathrm{mod}\,(A)$ can be described by using
$\tau$-tilting pairs, as stated in the following result that has been taken
from [1, Theorem 2.7] and [3, Theorem 5.10].

######  Theorem 4.2.

For any algebra $A,$ there is a well defined function

$\Phi:\mathrm{\tau\text{-}rig}(A)\to\mathrm{f\text{-}tors}(A),\;M\mapsto\mathrm{Fac}(M),$

from $\tau$-rigid basic pairs to functorially finite torsion classes in
$\mathrm{mod}\,(A).$ Moreover, $\Phi$ is a bijection if we restrict it to the
class $\mathrm{s\tau\text{-}tilt}(A)$ of $\tau$-tilting pairs, and in this
case $\Phi^{-1}(\mathcal{X})=\mathbb{P}(\mathcal{X}).$

In [19], it was shown how to produce a stratifying system from a basic
$\tau$-rigid module $M\in\mathrm{mod}\,(A).$ We start by recalling the notion
of torsion free admissible decomposition.

######  Definition 4.3.

[19, Definition 3.1] Let $M\in\mathrm{mod}\,(A)$ be basic and $\tau$-rigid. A
decomposition $M=\bigoplus_{i=1}^{t}M_{i}$ as the direct sum of
indecomposables $A$-modules is called torsion free admissible (TF-admissible,
for short) if $M_{i}\not\in\mathrm{Fac}\left(\bigoplus_{j>i}M_{j}\right),$ for
every $i\in[1,t].$

The existence of such TF-admissible decompositions is supported by the
following proposition proved in [19].

######  Proposition 4.4.

[19, Proposition 3.2] Every basic and $\tau$-rigid $A$-module
$M\in\mathrm{mod}\,(A)$ admits a TF-admissible decomposition.

The following result, proved in [19], tells us how to build a stratifying
system $(\Delta_{M},\leq)$ from a TF-admissible decomposition of
$M\in\mathrm{mod}\,(A).$ The stratifying system $(\Delta_{M},\leq)$ is known
as the $M$-standard stratifying system associated to this TF-admissible
decomposition.

######  Theorem 4.5.

[19, Theorem 3.4] Let $M\in\mathrm{mod}\,(A)$ be a basic and $\tau$-rigid with
a TF-admissible decomposition $M=\bigoplus_{i=1}^{t}M_{i},$ and let
$\mathrm{f}_{k}$ be the torsion free functor associated to the torsion pair
$\left(\mathrm{Fac}(\bigoplus_{j\geq k}M_{j}),(\bigoplus_{j\geq
k}M_{j})^{\perp}\right).$ Then, the family
$\Delta_{M}:=\\{\Delta_{M}(i):=\mathrm{f}_{i+1}(M_{i})\\}_{i=1}^{t}$ and the
natural order on $[1,t]$ form a stratifying system in $\mathrm{mod}\,(A)$ of
size $t$.

In this paper we study the relation between the homological properties of a
basic and $\tau$-rigid module $M\in\mathrm{mod}\,(A)$ and its induced
$M$-stratifying system $(\Delta_{M},\leq).$ In order to do that, we need to
introduce a vector with integer coordinates, known as the $g$-vector of $M$,
associated to each module $M$ in $\mathrm{mod}\;(A).$ For defining such
$g$-vector, we denote by $P(1),\ldots,P(n)$ the iso-classes of indecomposable
projective $A$-modules.

######  Definition 4.6.

[1, Section 5] Let $M\in\mathrm{mod}\,(A)$ and $P_{1}\to P_{0}\to M\to 0$ be
the minimal projective presentation of $M,$ where
$P_{0}=\bigoplus\limits_{i=1}^{n}P(i)^{a_{i}}$ and
$P_{1}=\bigoplus\limits_{i=1}^{n}P(i)^{a^{\prime}_{i}}.$ The $g$-vector of $M$
is set to be
$g^{M}:=(a_{1}-a^{\prime}_{1},a_{2}-a^{\prime}_{2},\dots,a_{n}-a^{\prime}_{n})\in\mathbb{Z}^{n}.$

Another useful vector associated with $M\in\mathrm{mod}\,(A)$ is the dimension
vector $\underline{\mathrm{dim}}(M)\in\mathbb{Z}^{n}$ whose $i$-th coordinate
is the composition factor of the simple $S(i)$ in $M,$ where $S(i)$ is the
simple top of $P(i).$ There is important homological information that arises
when $g$-vectors and dimension vectors interplay, as shown in [2]. For
$u,v\in\mathbb{Z}^{n},$ we denote by $\langle u,v\rangle$ the standard inner
product $\sum_{i=1}^{n}u_{i}v_{i}$ of these vectors.

######  Theorem 4.7.

[2, Theorem 1.4.(a)] Let $M$ and $N$ be in $\mathrm{mod}\,(A).$ Then

$\langle
g^{M},\underline{\mathrm{dim}}(N)\rangle=\dim_{k}(\emph{Hom}_{A}(M,N))-\dim_{k}(\emph{Hom}_{A}(N,\tau_{A}M)).$

##  5\. Main results

In this section we state and prove our main results. From now on, let
$n:=rk({}_{A}A)$ and fix some basic and $\tau$-rigid module $M$ in
$\mathrm{mod}\,(A)$ with some TF-admissible decomposition
$M=\bigoplus_{i=1}^{t}M_{i}.$ As we mentioned in the previous section, this
determines a stratifying system $(\Delta_{M},\leq)$ which, in turn, induces an
Ext-projective stratifying system $(\Delta_{M},\underline{Q},\leq)$. Moreover,
for each $i\in[1,t],$ we consider the following exact sequences:

(1) the canonical exact sequence

$\varepsilon_{i}:\;0\to
K(i)\xrightarrow{\iota_{i}}M_{i}\xrightarrow{\pi_{i}}\Delta_{M}(i)\rightarrow
0$

of $M_{i}$ with respect to the torsion pair
$\left(\mathrm{Fac}(\bigoplus_{l>i}M_{l}),(\bigoplus_{l>i}M_{l})^{\perp}\right);$

(2) the exact sequence

$\eta_{i}:\;0\to
U(i)\xrightarrow{\alpha_{i}}Q(i)\xrightarrow{\beta_{i}}\Delta_{M}(i)\to 0$

associated to each $\Delta_{M}(i),$ see Definition 1.1 (b). We also consider
the following integer matrices:

(3) the $G$-matrix $G^{M}\in\mathrm{Mat}_{n\times t}(\mathbb{Z})$ of $M$ whose
$i$-th column is $[G^{M}]^{i}:=(g^{M_{i}})^{tr};$

(4) the $M$-standard matrix $S^{M}\in\mathrm{Mat}_{n\times t}(\mathbb{Z})$
whose $i$-th column is
$[S^{M}]^{i}:=\underline{\mathrm{dim}}(\Delta_{M}(i))^{tr};$

(5) the $M$-residual matrix $R^{M}\in\mathrm{Mat}_{t\times t}(\mathbb{Z})$
whose $(i,t)$-coordinate is

$[R^{M}]_{i,j}:=\dim_{k}\mathrm{Hom}_{A}(U(i),\Delta_{M}(j))-\dim_{k}\mathrm{Hom}_{A}(K(i),\Delta_{M}(j))$

######  Remark 5.1.

We note that in (3) we are incurring in an abuse of notation, since the
$G$-matrices are only defined for $\tau$-tilting pairs, see [5, 9, 24].

An important lemma in the proof of our main theorem is the following.

######  Lemma 5.2.

For a basic and $\tau$-rigid module $M\in\mathrm{mod}\,(A)$ with a TF-
admissible decomposition $M=\bigoplus_{i=1}^{t}M_{i},$ and the Ext-projective
stratifying system $(\Delta_{M},\underline{Q},\leq)$ induced by
$(\Delta_{M},\leq),$ the following statements hold true.

  * $\mathrm{(a)}$

$\mathrm{Hom}_{A}(U(i),\Delta_{M}(j))=0$ and
$\mathrm{Hom}_{A}(K(i),\Delta_{M}(j))=0$ if $i\geq j.$

  * $\mathrm{(b)}$

We have the isomorphisms of $k$-vector spaces

$\mathrm{Hom}_{A}(M_{i},\Delta_{M}(i))\simeq\mathrm{Hom}_{A}(Q(i),\Delta_{M}(i))\simeq\mathrm{End}_{A}(\Delta_{M}(i))\quad\forall\,i\in[1,t].$

  * $\mathrm{(c)}$

For all $i,j\in[1,t],$ we have

$[C_{\Delta_{M}}]_{i,j}=\dim_{k}\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))+[R^{M}]_{i,j}$

  * $\mathrm{(d)}$

$\mathrm{Hom}_{A}(\Delta_{M}(j),\tau M_{i})=0$ and
$\mathrm{Ext}^{1}_{A}(M_{i},\Delta_{M}(j))=0$ $\forall\,i,j.$

###### Proof.

Fix some $i\in[1,t].$ Then, we have the exact sequence

$\eta_{i}:\;0\rightarrow
U(i)\xrightarrow{\alpha_{i}}Q(i)\xrightarrow{\beta_{i}}\Delta_{M}(i)\rightarrow
0,$

where $U(i)\in\mathcal{F}(\Delta_{M}(j):j>i),$ and the canonical exact
sequence

$\varepsilon_{i}:\;0\rightarrow
K(i)\xrightarrow{\iota_{i}}M_{i}\xrightarrow{\pi_{i}}\Delta_{M}(i)\rightarrow
0$

of $M_{i}$ with respect to the torsion pair
$(\mathcal{X}_{i},\mathcal{Y}_{i}):=\left(\mathrm{Fac}(\bigoplus_{l>i}M_{l}),(\bigoplus_{l>i}M_{l})^{\perp}\right).$
Moreover, by [19, Corallary 3.6(c)], we have that $\mathcal{F}(\Delta_{M})$ is
a subcategory of $\mathrm{Fac}(M).$

We know that $M_{i}$ is Ext-projective in $\mathrm{Fac}(M)$. Since
$\mathcal{F}(\Delta_{M})\subseteq\mathrm{Fac}(M)$, we have in particular that
$M_{i}$ is Ext-projective in $\mathcal{F}(\Delta_{M})$. Hence, we can complete
the following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{i}}$$\scriptstyle{q}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{i}}$$\scriptstyle{p}$$\textstyle{\Delta_{M}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{i}}$$\textstyle{Q(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{i}}$$\textstyle{\Delta_{M}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Note that the first square of the above diagram is the push-out of $q$ and
$\alpha_{i}.$ Hence, there exist an exact sequence
$0\xrightarrow{}K(i)\xrightarrow{}U(i)\bigoplus
M_{i}\xrightarrow{}Q(i)\xrightarrow{}0$ to which we apply the functor
$\mathrm{Hom}_{A}(-,\Delta_{M}(j))$ to obtain the following one

$(*):\;0\xrightarrow{}\mathrm{Hom}_{A}(Q(i),\Delta_{M}(j))\to\mathrm{Hom}_{A}(U(i)\oplus
M_{i},\Delta_{M}(j))\to\mathrm{Hom}_{A}(K(i),\Delta_{M}(j))\to 0.$

(a) Let $i\geq j.$ Since $U(i)\in\mathcal{F}(\Delta_{M}(r):r>i)$ and
$\mathrm{Hom}_{A}(\Delta_{M}(r),\Delta_{M}(j))=0$ $\forall\,r>i,$ we can
conclude that $\mathrm{Hom}_{A}(U(i),\Delta_{M}(j))=0.$ On the other hand, by
using that

$K(i)\in\mathrm{Fac}\left(\bigoplus_{l>i}M_{l}\right)\subseteq\mathrm{Fac}\left(\bigoplus_{l>j}M_{l}\right),$

it follows that $\mathrm{Hom}_{A}(K(i),\Delta_{M}(j))=0.$

(b) Take $i=j$ in $(*).$ Then, from (a),
$\mathrm{Hom}_{A}(K(i),\Delta_{M}(i))=0$ and
$\mathrm{Hom}_{A}(U(i),\Delta_{M}(i))=0.$ Hence, by $(*)$ we get that
$\mathrm{Hom}_{A}(Q(i),\Delta_{M}(i))\simeq\mathrm{Hom}_{A}(M_{i},\Delta_{M}(i)).$
Finally, the isomorphism
$\mathrm{Hom}_{A}(Q(i),\Delta_{M}(i))\simeq\mathrm{End}_{A}(\Delta_{M}(i))$
follows from Lemma 3.3.

(c) It follows by applying $\dim_{k}$ on the exact sequence $(*).$

(d) Since $\tau M\simeq\bigoplus_{r=1}^{t}\tau M_{r},$ it is enough to show
that $\mathrm{Hom}_{A}(\Delta_{M}(j),\tau(M))=0$ $\forall\,j\in[1,t].$ But,
the aforementioned statement holds true since
$\Delta_{M}(j)\in\mathrm{Fac}(M_{j})\subseteq\mathrm{Fac}(M)$ and $M$ is
$\tau$-rigid. Finally, we have
$\mathrm{Ext}^{1}_{A}(M_{i},\Delta_{M}(j))\simeq
D\overline{\mathrm{Hom}}_{A}(\Delta_{M}(j),\tau M_{i})=0.$ ∎

Now, we are ready to announce and prove our first main result.

######  Theorem 5.3.

For a basic and $\tau$-rigid module $M\in\mathrm{mod}\,(A)$ with a TF-
admissible decomposition $M=\bigoplus_{i=1}^{t}M_{i},$ and the Ext-projective
stratifying system $(\Delta_{M},\underline{Q},\leq)$ induced by
$(\Delta_{M},\leq),$ the following statements hold true.

  * $\mathrm{(a)}$

The matrix $(G^{M})^{tr}\,S^{M}$ is upper triangular and
$[(G^{M})^{tr}\,S^{M}]_{i,j}=\dim_{k}\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))$
$\forall\,i,j.$

  * $\mathrm{(b)}$

$R^{M}$ is an upper triangular matrix with zeros in its diagonal. Moreover, we
have that

$R^{M}=0$ $\Leftrightarrow$
$\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))\simeq\mathrm{Hom}_{A}(Q(i),\Delta_{M}(j))$
$\forall\,i<j.$

  * $\mathrm{(c)}$

$C_{\Delta_{M}}=(G^{M})^{tr}\,S^{M}+R^{M}.$

  * $\mathrm{(d)}$

$R^{M}=0$ if $M\in\mathcal{F}(\Delta_{M}).$

  * $\mathrm{(e)}$

$|G_{\Delta_{M}}|=\prod_{i=1}^{t}[(G^{M})^{tr}\,S^{M}]_{i,i}.$

###### Proof.

Let $i,j\in[1,t].$ Then, by Theorem 4.7 we get

$[(G^{M})^{tr}\,S^{M}]_{i,j}=\langle
g^{M_{i}},\underline{\mathrm{dim}}(\Delta_{M}(j))\rangle=\dim_{k}(\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j)))-\dim_{k}(\mathrm{Hom}_{A}(\Delta_{M}(j),\tau_{A}M_{i})).$

On the other hand, from Lemma 5.2 (d), we know that
$\mathrm{Hom}_{A}(\Delta_{M}(j),\tau_{A}M_{i})=0,$ proving that

$[(G^{M})^{tr}\,S^{M}]_{i,j}=\dim_{k}(\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))).$

Thus, from Lemma 5.2 (c), we also get the equality
$C_{\Delta_{M}}=(G^{M})^{tr}\,S^{M}+R^{M},$ proving (c). Moreover, by Lemma
5.2 (a) and the exact sequence $(*)$ in the proof of Lemma 5.2, we can obtain
(b). Also, since $C_{\Delta_{M}}$ is upper triangular (see Lemma 3.3), then by
(b) and (c) it follows that the matrix $(G^{M})^{tr}\,S^{M}$ is upper
triangular. Finally, (e) follows directly from (b), (c) and Lemma 3.3.

Assume now that $M\in\mathcal{F}(\Delta_{M}).$ Then, by the proof of [19,
Corollary 3.8], we get that the exact sequences $\eta_{i}$ and
$\varepsilon_{i}$ are isomorphic, for each $i\in[1,t].$ Therefore we obtain
(d). ∎

As we have seen in Theorem 5.3 (d), the condition
$M\in\mathcal{F}(\Delta_{M})$ is enough to get that $R^{M}=0.$ In the
following, we characterise when $M\in\mathcal{F}(\Delta_{M}).$

######  Proposition 5.4.

Let $M\in\mathrm{mod}\,(A)$ be a basic and $\tau$-rigid module with a TF-
admissible decomposition $M=\bigoplus_{i=1}^{t}M_{i},$ and let
$(\Delta_{M},\underline{Q},\leq)$ be the Ext-projective stratifying system
induced by $(\Delta_{M},\leq).$ Then, for any $i\in[1,t],$ the following
statements are equivalent.

  * (a)

$M_{i}\in\mathcal{F}(\Delta_{M}).$

  * (b)

The exact sequences $\eta_{i}$ and $\varepsilon_{i}$ are isomorphic.

  * (c)

$Q(i)\simeq M_{i}.$

  * (d)

$\mathrm{Ext}^{1}_{A}(Q(i),K(i))=0.$

  * (e)

$K(i)\simeq U(i).$

###### Proof.

(a) $\Rightarrow$ (b) It follows by the proof of [19, Corollary 3.8].

(b) $\Rightarrow$ (c) It is trivial.

(c) $\Rightarrow$ (d) It follows from
$\mathrm{Ext}^{1}_{A}(M,\mathrm{Fac}(M))=0$ since $K(i)\in\mathrm{Fac}(M).$

(d) $\Rightarrow$ (e) Since $\mathrm{Ext}^{1}_{A}(Q(i),K(i))=0,$ there is some
$q:Q(i)\to M_{i}$ with $\pi_{i}q=\beta_{i}.$ On the other hand by the proof of
Lemma 5.2, there is some $p:M_{i}\to Q(i)$ with $\beta_{i}p=\pi_{i}.$ Since
$\beta_{i}(pq)=\beta_{i}$ and $\beta_{i}:Q(i)\to\Delta_{M}(i)$ is right
minimal [14, Lemma 2.3], we get that $pq$ is an isomorphism, and thus
$q:Q(i)\xrightarrow{\thicksim}M_{i}$ since $Q(i)$ and $M_{i}$ are
indecomposable. Thus, we have that the exact sequences $\eta_{i}$ and
$\varepsilon_{i}$ are isomorphic. In particular, $K(i)\simeq U(i).$

(e) $\Rightarrow$ (a) By the exact sequence $\varepsilon_{i}$ and the fact
that $\mathcal{F}(\Delta_{M})$ is closed under extensions, we conclude that
$M_{i}\in\mathcal{F}(\Delta_{M}).$ ∎

######  Remark 5.5.

Recall from [11] that given $M\in\mathrm{mod}(A)$ and a vector
$v\in\mathbb{R}^{n},$ we say that $M$ is $v$-semistable if $\langle
v,\underline{\mathrm{dim}}M\rangle=0$ and $\langle
v,\underline{\mathrm{dim}}L\rangle\leq 0$ for all submodule $L$ of $M$. Using
this notion of stability, one can construct (for every finite dimesional
algebra $A$) a geometric invariant known as the wall-and-chamber structure of
$A$. It was shown in [4] that the wall-and-chamber structure of an algebra $A$
can be described by its $\tau$-tilting theory, via the $g$-vectors of the
indecomposable $\tau$-rigid objects.

Now, let $M\in\mathrm{mod}(A)$ be basic and $\tau$-rigid, with a TF-admissible
decomposition $M=\bigoplus_{i=1}^{t}M_{i},$ and let $\Delta_{M}$ be its
associated stratifying system. From the construction of $\Delta_{M}$ and the
results of [4], it follows that the $A$-module $\Delta_{M}(i)$ is
$g^{M_{j}}$-semistable, for all $1\leq i\leq t$ and $j>i$. It would be
interesting to see if the Cartan matrix $C_{\Delta_{M}}$ has a geometric
realisation in the wall-and-chamber structure of $A$.

Let $\Lambda$ be a finite dimensional $k$-algebra. We recall from [15] that
the Cartan group $G_{\Lambda}$ of $\Lambda$ is the Cokernel of the Cartan map
$C_{\Lambda}:K_{0}(\mathrm{proj}(\Lambda))\to
K_{0}(\Lambda),\;[P_{i}]\mapsto\underline{\mathrm{dim}}(P_{i}),$ where
$\\{P_{i}\\}_{i=1}^{n}$ is a representative set of pairwise non-isomorphic
indecomposable projective $\Lambda$-modules.

######  Corollary 5.6.

Let $M\in\mathrm{mod}\,(A)$ be basic and $\tau$-tilting and consider a TF-
admissible decomposition $M=\bigoplus_{i=1}^{t}M_{i}$ such that
$M\in\mathcal{F}(\Delta_{M}).$ Then, for the algebra
$B:=\mathrm{End}_{A}(M)^{op},$ we have that
$G_{B}\simeq\mathrm{CoKer}(S^{M}).$

###### Proof.

Let $(\Delta_{M},\underline{Q},\leq)$ be the Ext-projective stratifying system
induced by $(\Delta_{M},\leq).$ By Theorem 5.3 and Proposition 5.4, we get
that $C_{\Delta_{M}}=(G^{M})^{tr}\,S^{M}$ and $M_{i}\simeq Q(i)$
$\forall\,i\in[1,t].$ On the other hand, by [1, Theorem 5.1], we conclude that
the matrix $G^{M}\in\mathrm{Mat}_{n\times n}(\mathbb{Z})$ induces an
isomorphism $\mathbb{Z}^{n}\to\mathbb{Z}^{n},\;X\mapsto G^{M}\,X,$ of abelian
groups. Therefore, $\mathrm{CoKer}(C_{\Delta_{M}})\simeq\mathrm{CoKer}(S^{M})$
and then the result follows from [15, Theorem 4.2]. ∎

As we have already seen, the Cartan matrix $C_{\Theta}$ associated to a
stratifying system $\Theta$ is an upper-triangular matrix. We now will
characterise in Theorem 5.8 the stratifying systems coming from $\tau$-tilting
modules $M$ whose associated Cartan matrix $C_{\Theta}$ is diagonal. Before
proving our second main theorem, we need the following intermediate result.

######  Proposition 5.7.

For a basic and $\tau$-rigid module $M\in\mathrm{mod}\,(A),$ with a TF-
admissible decomposition $M=\bigoplus_{i=1}^{t}M_{i},$ the following
statements (a), (b) and (c) are equivalent.

  * $\mathrm{(a)}$

The matrix $(G^{M})^{tr}\,S^{M}$ is diagonal and
$\mathrm{Hom}_{A}(M_{i},K(j))=0$ for $i<j.$

  * $\mathrm{(b)}$

$\mathrm{Hom}_{A}(M_{i},M_{j})=0$ for $i<j.$

  * $\mathrm{(c)}$

For $i<j,$ we have that $\mathrm{Hom}_{A}(\Delta_{M}(i),\Delta_{M}(j))=0,$
$\mathrm{Hom}_{A}(M_{i},K(j))=0$ and the map

$\mathrm{Hom}_{A}(K(i),\Delta_{M}(j))\to\mathrm{Ext}^{1}_{A}(\Delta_{M}(i),\Delta_{M}(j)),\;f\mapsto
f\cdot\varepsilon_{i}$

is an isomorphism.

Moreover, for $B:=\mathrm{End}_{A}(M)^{op}$ and
$d_{i}:=\dim_{k}\mathrm{End}_{A}(\Delta_{M}(i)),$ we have that
$G_{B}\simeq\bigoplus_{i=1}^{t}\,\mathbb{Z}/d_{i}\mathbb{Z}$ if
$M\in\mathcal{F}(\Delta_{M})$ and $(G^{M})^{tr}\,S^{M}$ is diagonal.

###### Proof.

Let $i,j\in[1,t].$ Consider the canonical exact sequence

$\varepsilon_{j}:\;0\to K(j)\to M_{j}\to\Delta_{M}(j)\to 0$

of $M_{j}$ with respect to the torsion pair
$\left(\mathrm{Fac}(\bigoplus_{l>j}M_{l}),(\bigoplus_{l>j}M_{l})^{\perp}\right).$
Note that $\varepsilon_{j}$ lies in $\mathrm{Fac}(M);$ and since
$\mathrm{Ext}^{1}_{A}(M,\mathrm{Fac}(M))=0,$ from $\varepsilon_{j}$ we get the
exact sequence

$(*):\;0\to\mathrm{Hom}_{A}(M_{i},K(j))\to\mathrm{Hom}_{A}(M_{i},M_{j})\to\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))\to
0.$

(a) $\Leftrightarrow$ (b): It follows from (*) and Theorem 5.3 (a).

(a) $\Leftrightarrow$ (c): Let $i<j.$ Consider the exact sequence
$\varepsilon_{i}:\;0\to K(i)\to M_{i}\to\Delta_{M}(i)\to 0.$ Then, we get the
exact sequence

$0\to\mathrm{Hom}_{A}(\Delta_{M}(i),\Delta_{M}(j))\to\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))\to\mathrm{Hom}_{A}(K(i),\Delta_{M}(j))\to\mathrm{Ext}^{1}_{A}(\Delta_{M}(i),\Delta_{M}(j))\to
0$

Since $\mathrm{Ext}^{1}_{A}(M_{i},\Delta_{M}(j))=0$ (see Lemma 5.2 (d)). Thus,
the equivalence between (a) and (c) follows from the above exact sequence and
Theorem 5.3 (a).

Let $M\in\mathcal{F}(\Delta_{M})$ and $(G^{M})^{tr}\,S^{M}$ be a diagonal
matrix. By Theorem 5.3 and Proposition 5.4, we get that
$C_{\Delta_{M}}=(G^{M})^{tr}\,S^{M}$ and $M_{i}\simeq Q(i)$ $\forall\,i.$
Then, by [15, Theorem 4.2] and Lemma 5.2 (b), we conclude that
$G_{B}\simeq\bigoplus_{i=1}^{t}\,\mathbb{Z}/d_{i}\mathbb{Z}.$ ∎

######  Theorem 5.8.

For a basic $\tau$-tilting module $M\in\mathrm{mod}\,(A),$ with a TF-
admissible decomposition $M=\bigoplus_{i=1}^{n}M_{i}$ such that
$M\in\mathcal{F}(\Delta_{M}),$ the following statements are equivalent.

  * $\mathrm{(a)}$

The matrix $C_{\Delta_{M}}$ is diagonal.

  * $\mathrm{(b)}$

$\mathrm{Hom}_{A}(M_{i},M_{j})=0$ for $i<j.$

  * $\mathrm{(c)}$

$\mathrm{Hom}_{A}(M_{i},\Delta_{M}(j))=0$ for $i<j.$

  * $\mathrm{(d)}$

For $i<j,$ we have that $\mathrm{Hom}_{A}(\Delta_{M}(i),\Delta_{M}(j))=0$ and
the map

$\mathrm{Hom}_{A}(K(i),\Delta_{M}(j))\to\mathrm{Ext}^{1}_{A}(\Delta_{M}(i),\Delta_{M}(j)),\;f\mapsto
f\cdot\eta_{i}$

is an isomorphism.

Moreover, if one of the above conditions holds true, then $M\simeq{}_{A}A,$
$A$ is a weakly triangular algebra and $\mathrm{Hom}_{A}(M_{i},K(j))=0$ for
$i<j.$

###### Proof.

Let $(\Delta_{M},\underline{Q},\leq)$ be the Ext-projective stratifying system
induced by $(\Delta_{M},\leq).$ By Theorem 5.3 and Proposition 5.4, we get
that $C_{\Delta_{M}}=(G^{M})^{tr}\,S^{M},$ $U(i)\simeq K(i)$ and $M_{i}\simeq
Q(i)$ $\forall\,i.$ Thus, the fact that $\mathrm{Hom}_{A}(M_{i},K(j))=0$ for
$i<j$ (by assuming one of the above conditions) and the equivalences between
(a), (b), (c) and (d) follow from Proposition 3.4.

Now, assume that one of the above equivalent conditions holds true. Let us
proof, firstly, that $M\simeq{}_{A}A.$ Indeed, since the decomposition
$M=\bigoplus_{i=1}^{n}M_{i}$ is TF-admissible, we get from (b) that
$M_{i}\not\in\mathrm{Fac}(\bigoplus_{j\neq i}M_{j})$ $\forall\,i\in[1,n].$ On
the other hand, it is known that $\bigoplus_{j\neq i}M_{j}$ admits two torsion
classes, namely, $\mathrm{Fac}(\bigoplus_{j\neq i}M_{j})$ and
${}^{\perp}(\bigoplus_{j\neq i}\tau M_{j}).$ Moreover, since
$\mathrm{Fac}(\bigoplus_{j\neq i}M_{j})\subsetneq{}^{\perp}(\bigoplus_{j\neq
i}\tau M_{j})$ and $M_{i}\not\in\mathrm{Fac}(\bigoplus_{j\neq i}M_{j}),$ we
have that $M_{i}\in{}^{\perp}(\bigoplus_{j\neq i}\tau M_{j})$ and thus the
mutation of $M=\bigoplus_{i=1}^{t}M_{i}$ at $M_{i}$ is descendent, for all
$i\in[1,n].$ Let us show that $\mathrm{Fac}(M)=\mathrm{mod}(A).$ Suppose that
$\mathrm{Fac}(M)\neq\mathrm{mod}(A).$ Since $(\mathrm{Fac}(M),M^{\perp})$ is a
torsion pair in $\mathrm{mod}(A),$ it follows from [5, Theorem 3.1], that
there exists a mutation $M^{\prime}$ of $M$ such that
$\mathrm{Fac}(M)\subsetneq\mathrm{Fac}(M^{\prime})$ contradicting that any
mutation of $M$ is descendent. Therefore $\mathrm{Fac}(M)=\mathrm{mod}(A)$ and
then $M\simeq{}_{A}A.$ In particular, $A$ is a standardly stratified algebra
since $M_{i}\simeq Q(i)$ $\forall\,i;$ and by [15, Remark 4.5], we conclude
that $A$ is a weakly triangular algebra. ∎

##  6\. Examples

We finish the paper by calculating the Cartan matrix of stratifying systems
induced by $\tau$-rigid modules. But before we go to the examples, we recall
the notation of Lemma 5.2 of the two short exact sequences naturally
associated to $\Delta_{M}(i)$ for $i\in\\{1,\dots,t\\}$.

$\varepsilon_{i}:\;0\to
K(i)\xrightarrow{\iota_{i}}M_{i}\xrightarrow{\pi_{i}}\Delta_{M}(i)\rightarrow
0,$ $\eta_{i}:\;0\to
U(i)\xrightarrow{\alpha_{i}}Q(i)\xrightarrow{\beta_{i}}\Delta_{M}(i)\to 0.$

######  Example 6.1.

Let $A$ be the quotient path $k$-algebra given by the quiver

$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

and the third power of the ideal generated by all the arrows. The Auslander-
Reiten quiver of $A$ can be seen in Figure 1.

${\tiny\begin{matrix}2\end{matrix}}$${\tiny\begin{matrix}1\end{matrix}}$${\tiny\begin{matrix}3\end{matrix}}$${\tiny\begin{matrix}2\\\
3\end{matrix}}$${\tiny\begin{matrix}1\\\
2\end{matrix}}$${\tiny\begin{matrix}3\\\
1\end{matrix}}$${\tiny\begin{matrix}2\\\
3\end{matrix}}$${\tiny\begin{matrix}1\\\ 2\\\
3\end{matrix}}$${\tiny\begin{matrix}3\\\ 1\\\
2\end{matrix}}$${\tiny\begin{matrix}2\\\ 3\\\ 1\end{matrix}}$ Figure 1. The
Auslander-Reiten quiver of $A$

Note that every module is represented by its Loewy series and both copies of
${\tiny\begin{matrix}2\\\ 3\end{matrix}}$ should be identified. Thus, the
Auslander-Reiten quiver of $A$ has the shape of a cylinder.

Now take the $\tau$-rigid modules $A={\tiny\begin{matrix}1\\\ 2\\\
3\end{matrix}}\oplus{\tiny\begin{matrix}2\\\ 3\\\
1\end{matrix}}\oplus{\tiny\begin{matrix}3\\\ 1\\\ 2\end{matrix}}$ and
$M={\tiny\begin{matrix}1\\\ 2\\\ 3\end{matrix}}\oplus{\tiny\begin{matrix}1\\\
2\end{matrix}}\oplus{\tiny\begin{matrix}1\end{matrix}}.$ It can be shown that,
in both cases, the $\tau$-tilting modules are written in a TF-admissible
fashion. Hence, their corresponding induced stratifying system are
$\Delta_{A}=\left\\{{\tiny\begin{matrix}1\end{matrix}},{\tiny\begin{matrix}2\end{matrix}},{\tiny\begin{matrix}3\\\
1\\\ 2\end{matrix}}\right\\}$ and $\Delta_{M}=\left\\{{\tiny\begin{matrix}1\\\
2\\\
3\end{matrix}},{\tiny\begin{matrix}1\end{matrix}},{\tiny\begin{matrix}2\end{matrix}}\right\\}$.

$i$ | $\Delta_{A}(i)$ | $M_{i}$ | $K(i)$ | $Q(i)$ | $U(i)$
---|---|---|---|---|---
$1$ | ${\tiny\begin{matrix}1\end{matrix}}$ | ${\tiny\begin{matrix}1\\\ 2\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}2\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}2\end{matrix}}$
$2$ | ${\tiny\begin{matrix}2\end{matrix}}$ | ${\tiny\begin{matrix}2\\\ 3\\\ 1\end{matrix}}$ | ${\tiny\begin{matrix}3\\\ 1\end{matrix}}$ | ${\tiny\begin{matrix}2\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$
$3$ | ${\tiny\begin{matrix}3\\\ 1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}3\\\ 1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$ | ${\tiny\begin{matrix}3\\\ 1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$
Table 1. Modules associated to the stratifying system $\Delta_{A}$ $i$ | $\Delta_{M}(i)$ | $M_{i}$ | $K(i)$ | $Q(i)$ | $U(i)$
---|---|---|---|---|---
$1$ | ${\tiny\begin{matrix}1\\\ 2\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}1\\\ 2\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$ | ${\tiny\begin{matrix}1\\\ 2\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$
$2$ | ${\tiny\begin{matrix}1\end{matrix}}$ | ${\tiny\begin{matrix}1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}2\end{matrix}}$ | ${\tiny\begin{matrix}1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}1\end{matrix}}$
$3$ | ${\tiny\begin{matrix}2\end{matrix}}$ | ${\tiny\begin{matrix}2\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$ | ${\tiny\begin{matrix}2\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$
Table 2. Modules associated to the stratifying system $\Delta_{M}$

Using the modules in Table 1 and Table 2 and the Auslander-Reiten quiver of
$A,$ one can calculate easily the following matrices.

$C_{\Delta_{A}}=\left(\begin{matrix}1&0&1\\\ 0&1&1\\\
0&0&1\end{matrix}\right)\qquad G^{A}=\left(\begin{matrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{matrix}\right)\qquad S^{A}=\left(\begin{matrix}1&0&1\\\ 0&1&1\\\
0&0&1\end{matrix}\right)\qquad R^{A}=\left(\begin{matrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}\right).$

$C_{\Delta_{M}}=\left(\begin{matrix}1&1&0\\\ 0&1&0\\\
0&0&1\end{matrix}\right)\qquad G^{M}=\left(\begin{matrix}1&0&0\\\ 0&1&-1\\\
0&1&-1\end{matrix}\right)\qquad S^{M}=\left(\begin{matrix}1&1&0\\\ 1&0&1\\\
1&0&0\end{matrix}\right)\qquad R^{M}=\left(\begin{matrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}\right).$

In particular, it is straightforward to verify that
$C_{\Delta_{A}}=(G^{A})^{tr}S^{A}+R^{A}$ and
$C_{\Delta_{M}}=(G^{M})^{tr}S_{M}+R^{M}$, as shown in Theorem 5.3.

######  Example 6.2.

Let $A$ be the quotient path $k$-algebra given by the quiver

$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\beta_{2}}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{2}}$

and the ideal generated by the set of relations
$\\{\beta_{1}\alpha_{1}\beta_{1},$ $\beta_{1}\alpha_{1}-\alpha_{2}\beta_{2},$
$\alpha_{2}\beta_{2}\alpha_{2}\\}.$ Now take the $\tau$-rigid module
$A={\tiny\begin{matrix}\;1\;\\\ 3\;\;2\\\
\;1\;\end{matrix}}\oplus{\tiny\begin{matrix}2\\\ 1\\\
2\end{matrix}}\oplus{\tiny\begin{matrix}3\\\ 1\\\ 3\end{matrix}}.$ Hence, its
corresponding induced stratifying system are
$\Delta_{A}=\left\\{{\tiny\begin{matrix}1\end{matrix}},{\tiny\begin{matrix}2\\\
1\\\ 2\end{matrix}},{\tiny\begin{matrix}3\\\ 1\\\ 3\end{matrix}}\right\\}.$ A
quick calculation shows that the modules associated to the stratifying system
$\Delta_{A}$ are the following.

$i$ | $\Delta_{A}(i)$ | $M_{i}$ | $K(i)$ | $Q(i)$ | $U(i)$
---|---|---|---|---|---
$1$ | ${\tiny\begin{matrix}1\end{matrix}}$ | ${\tiny\begin{matrix}\;1\;\\\ 3\;\;2\\\ \;1\;\end{matrix}}$ | ${\tiny\begin{matrix}3\;\;2\\\ \;1\;\end{matrix}}$ | ${\tiny\begin{matrix}1\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$
$2$ | ${\tiny\begin{matrix}2\\\ 1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}2\\\ 1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$ | ${\tiny\begin{matrix}2\\\ 1\\\ 2\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$
$3$ | ${\tiny\begin{matrix}3\\\ 1\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}3\\\ 1\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$ | ${\tiny\begin{matrix}3\\\ 1\\\ 3\end{matrix}}$ | ${\tiny\begin{matrix}0\end{matrix}}$
Table 3. Modules associated to the stratifying system $\Delta_{A}$

From Table 3 we can calculate the matrices associated to $\Delta_{A}$, which
are below. In particular, it is straightforward to verify that
$C_{\Delta_{A}}=(G^{A})^{tr}S^{A}+R^{A},$ as shown in Theorem 5.3.

$C_{\Delta_{A}}=\left(\begin{matrix}1&0&0\\\ 0&2&0\\\
0&0&2\end{matrix}\right)\qquad G^{A}=\left(\begin{matrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{matrix}\right)\qquad S^{A}=\left(\begin{matrix}1&1&1\\\ 0&2&0\\\
0&0&2\end{matrix}\right)\qquad R^{A}=\left(\begin{matrix}0&-1&-1\\\ 0&0&0\\\
0&0&0\end{matrix}\right).$

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Octavio Mendoza:
Instituto de Matemáticas,
Universidad Nacional Autónoma de México,
Circuito Exterior, Ciudad Universitaria,
México D.F. 04510, MÉXICO.

<EMAIL_ADDRESS>
Corina Sáenz:
Departamento de Matemáticas, Facultad de Ciencias,
Universidad Nacional Autónoma de México,
Circuito Exterior, Ciudad Universitaria,
México D.F. 04510, MÉXICO.

<EMAIL_ADDRESS>
Hipolito Treffinger:
Université de Paris et Sorbonne Université.
CNRS, IMJ-PRG,
Bâtiment Sophie Germain,
5 rue Thomas Mann,
75205 Paris Cedex 13, FRANCE

<EMAIL_ADDRESS>