# On $d$-term silting objects and (co) torsion pairs in truncated derived
categories

Esha Gupta

###### Abstract.

For a finite-dimensional algebra $\Lambda$ over an algebraically closed field
$K$, it is known that the poset of $2$-term silting objects in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ is isomorphic to the poset
of functorially finite torsion classes in $\operatorname{mod}\Lambda$, and to
that of complete cotorsion classes in
$\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$. In this work, we
generalise this result to the case of $d$-term silting objects for arbitrary
$d\geq 2$ by introducing the notion of torsion classes for extriangulated
categories. In particular, we show that the poset of $d$-term silting objects
in $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ is isomorphic to the
poset of complete and hereditary cotorsion classes in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, and to that of
positive and functorially finite torsion classes in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$, an extension-closed
subcategory of $\mathcal{D}^{b}(\operatorname{mod}\Lambda)$. We further show
that the posets
$\operatorname{cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$
and $\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ are
lattices, and that the truncation functor $\tau_{\geq-d+2}$ gives an
isomorphism between the two.

###### Contents

  1. 1 Introduction
  2. 2 Notation and definitions
  3. 3 Bijection between torsion and cotorsion pairs
    1. 3.1 Truncation functors
    2. 3.2 From cotorsion pairs to torsion pairs
    3. 3.3 The bijections
  4. 4 Silting objects, cotorsion pairs, and torsion pairs
    1. 4.1 Definition of $\psi$
    2. 4.2 Definition of $\psi^{\prime}$
    3. 4.3 Proof of Theorem 4.1
    4. 4.4 Lattices
  5. 5 Examples

##  1\. Introduction

The theory of $\tau$-tilting was first introduced in [2] as a mutation
completion of tilting theory. In that seminal work, the authors provided
isomorphisms between the following posets for an arbitrary finite-dimensional
algebra $\Lambda$.

  1. (1)

Isomorphism classes of basic support $\tau$-tilting modules in
$\operatorname{mod}\Lambda$.

  2. (2)

Functorially finite torsion pairs in $\operatorname{mod}\Lambda$.

  3. (3)

Isomorphism classes of basic $2$-term silting objects in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.

For path algebras of acyclic quivers, it was proven in [18] that finitely
generated torsion classes in $\operatorname{mod}\Lambda$ are in bijection with
finitely generated wide subcategories of $\operatorname{mod}\Lambda$. This was
generalized in [23] to all finite-dimensional algebras, adding the following
class to the above list.

  1. (4)

Left finite wide subcategories of $\operatorname{mod}\Lambda$.

In [14], the author interpreted the class of thick subcategories as the
‘mirror’ of wide subcategories, adding the following class to the mix.

  1. (5)

Thick subcategories in $\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$
with enough injectives.

Finally, in [27], the authors gave a map from the poset of torsion classes in
$\operatorname{mod}\Lambda$ and the poset of cotorsion pairs in
$\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$, which restricts to an
isomorphism of the above posets with

  1. (6)

Complete cotorsion pairs in
$\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$.

Recently, in [13], the same map was shown to provide a bijection between all
torsion pairs in $\operatorname{mod}\Lambda$ and all cotorsion pairs in
$\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$. The proof of this
fact relies on the following equivalence of categories.

(1)
$\mathrm{H}^{0}:\frac{\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)}{\Sigma\operatorname{proj}\Lambda}\to\operatorname{mod}\Lambda$

The above isomorphisms can be summarized in the diagram below, where
$\operatorname{K}_{\Lambda}=\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$.

${{2\mbox{-}\operatorname{silt}\operatorname{K}_{\Lambda}}}$${{\operatorname{K}_{\Lambda}}}$${{c\mbox{-}\operatorname{cotors}\operatorname{K}_{\Lambda}}}$${{\operatorname{inj-
thick}\operatorname{K}_{\Lambda}}}$${{\operatorname{s\tau-
tilt}\Lambda}}$${{\operatorname{mod}\Lambda}}$${\operatorname{f-tors}\Lambda}$${\operatorname{l-wide}\Lambda}$[14][27][14][2][23][2][23]
Figure 1. Figure taken from [14]

Given the variety of families in bijection with $2$-term silting objects in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$, it is natural to consider
if the poset of $d$-term silting objects in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ enjoys similar properties
for arbitrary $d\geq 2$. It was proven in [19, Corollary 3.4] that this poset
is isomorphic to the poset of simple-minded collections in
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ lying in
$\mathcal{D}^{[-d+1,0]}(\operatorname{mod}\Lambda)$. Furthermore, they show
that if $\Lambda$ is hereditary of Dynkin type, silting objects with
cohomology concentrated in $[-d+1,0]$ are in bijection with $d$-simple-minded
systems in the $(-d)$-cluster category $\mathcal{C}_{-d}(\Lambda)$, which are
then proven to be counted by the positive Fuss-Catalan number $C^{+}_{d}(W)$,
where $W$ is the Weyl group of $\Lambda$ [19, Theorem 1.1, 1.2]. For an
acyclic quiver $Q$, it was shown in [20, Theorem 5.14] that the hearts of
$\mathcal{D}^{b}(\operatorname{mod}kQ)$ lying between the canonical heart
$\mathcal{H}_{Q}=\operatorname{mod}kQ$ and $\mathcal{H}_{Q}[d-1]$ are in
bijection with basic $d$-term silting complexes as well as $d$-cluster tilting
sets.

In [6]111This is a work in progress building up on the previous work [5] of
the same authors, and whose results were announced at the online talk
https://www.fd-seminar.xyz/talks/2023-05-04/, the authors study the subset of
$d$-term silting objects whose cohomology is concentrated in degrees $0$ and
$-d+1$ using $(d-1)$-torsion classes in a $(d-1)$-cluster tilting subcategory
$\mathcal{M}$ of $\operatorname{mod}\Lambda$. In particular, they show that
the set of functorially finite $(d-1)$-torsion classes in $\mathcal{M}$
injects into this set. Furthermore, for Dynkin types, $d$-term silting objects
were shown to be in bijection with several combinatorial objects in [30],
including $d$-eralized clusters and $d$-eralized noncrossing partitions. These
objects were then shown to be counted by the Fuss-Catalan numbers of type $W$,
where $W$ is the Coxeter group associated to $\Lambda$.

In this work, we start by generalising the notion of torsion pairs for abelian
categories [12] and triangulated categories [8, 19] to extriangulated
categories with an additional structure that plays the role of extension
spaces in all integer degrees satisfying some compatibility conditions. This
is axiomatised in the definition of a _bivariant $\delta$-functor_ in [15,
Definition 4.5]. Although the definition of a $\delta$-functor is more
general, for simplification, we restrict ourselves to the case where the
degree $0$ extensions are given by the $\mathrm{Hom}$-functors while the
degree $1$ extensions coincide with the extriangulated structure. We also
introduce the notion of positive torsion pairs for such categories. For
extriangulated categories equipped with negative first extensions, there
exists the notion of $s$-torsion classes as introduced in [1], and we provide
a comparison between these two notions in Theorem 3.20.

Recall that for an extriangulated category
$(\mathcal{C},\mathbb{E},\mathfrak{s})$, for any $\mathbb{E}$-extension
$\delta\in\mathbb{E}(C,A)$, there are canonical natural transformations
$A_{\delta}:\mathrm{Hom}(-,C)\to\mathbb{E}(-,A)$ and
$B_{\delta}:\mathrm{Hom}(A,-)\to\mathbb{E}(C,-)$ [25, Definition 3.1].

######  Definition

(Definition 3.14) Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an
extriangulated category equipped with a bivariant $\delta$-functor
$(G^{\bullet},\delta_{\\#},\delta^{\\#})$ such that
$G^{0}=\mathrm{Hom}_{\mathcal{C}}$, $G^{1}=\mathbb{E}$, and for any
$\mathbb{E}$-extension $\delta\in\mathbb{E}(C,A)$,
$\delta^{0}_{\\#}=A_{\delta}$ and $\delta^{\\#,0}=B_{\delta}$. Let
$\mathcal{T},\mathcal{F}$ be full subcategories of $\mathcal{C}$.

  1. (1)

The pair $(\mathcal{T},\mathcal{F})$ is called a _torsion pair_ if

    1. (a)

$\mathrm{Hom}(t,\mathcal{F})=0$ if and only if $t\in\mathcal{T}$;

    2. (b)

$\mathrm{Hom}(\mathcal{T},f)=0$ if and only if $f\in\mathcal{F}$.

  2. (2)

A torsion pair $(\mathcal{T},\mathcal{F})$ is called _positive_ if
$G^{k}(\mathcal{T},\mathcal{F})=0$ for all $k\leq-1$.

In order to relate this notion to $d$-term silting objects, the main step is
finding a ‘higher’ generalization of Equation 2. It is natural to replace
$\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$ by
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, and the key step is
replacing $\operatorname{mod}\Lambda$ by
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. These two categories are
then related by the truncation functor $\tau_{\geq-d+2}$ and we have the
following equivalence (Proposition 3.1).

(2)
$\tau_{\geq-d+2}:\frac{\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)}{\Sigma^{d-1}\operatorname{proj}\Lambda}\to\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$

Our main results are the following.

######  Theorem

(3.20) The map $\Phi$ defined as
$(\mathcal{X},\mathcal{Y})\mapsto(\tau_{\geq-d+2}\mathcal{Y},\leavevmode\nobreak\
(\tau_{\geq-d+2}\mathcal{Y})^{\perp})$ gives an isomorphism between the poset
of cotorsion pairs in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ and the poset of
torsion pairs in $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

In [21], the authors provided isomorphisms $\psi$ (resp. $\psi^{\prime}$)
between the poset of silting objects in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ and algebraic
$t$-structures in $\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ (resp. bounded
co-$t$-structures in $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$) (See
definitions 4.2 and 4.5).

######  Theorem

(4.1) The map $\Phi$ defined above along with the maps $\psi$ and
$\psi^{\prime}$ introduced in [21] restrict to the following commutative
triangle of poset isomorphisms.

${d\mbox{-}\operatorname{silt}\Lambda}$${\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)}$${\operatorname{(f,p)-tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)}$$\scriptstyle{\psi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\Phi}$

The above results can be summarized in the following commutative diagram of
poset morphisms, where c stands for complete, h for hereditary, f for
functorially finite, p for positive, cov for covariantly, and contr for
contravariantly finite.

${{\operatorname{(c,h)-cotors}\Lambda}}$${{\operatorname{cotors}\Lambda}}$${{\operatorname{(contr,h)-cotors}\Lambda}}$hereditarycomplete${d\mbox{-}\operatorname{silt}\Lambda}$positivecov.
finite${{\operatorname{(f,p)-tors}\Lambda}}$${{\operatorname{tors}\Lambda}}$${{\operatorname{stors}\Lambda}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Furthermore, it is known that the poset of torsion classes in
$\operatorname{mod}\Lambda$ is a lattice, and its lattice-theoretic properties
have been extensively studied ([11], [31], [10]). The corresponding result
also holds for torsion and cotorsion pairs in extriangulated categories.

######  Theorem

(4.9) The poset of torsion pairs in an extriangulated category $\mathcal{C}$
is a lattice.

######  Theorem

(4.10) The poset of cotorsion pairs in an extriangulated category
$\mathcal{C}$ is a lattice.

We also have the following result which implies that the poset of $d$-term
silting objects embeds into a smaller lattice.

######  Theorem

(4.11) The subposet of positive torsion pairs in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ is a sublattice of
$\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

######  Corollary

(4.12) The subposet of hereditary cotorsion pairs in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ is a sublattice of
$\operatorname{cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.

Since the class of $s$-torsion classes is contained in the class of positive
torsion classes, it would be interesting to know if it is closed under meets
and joins, or if it forms a lattice.

## Acknowledgements

I would like to express my most sincere gratitude towards Pierre-Guy Plamondon
for his guidance, availability, and constant support throughout the
development of this work. I am also thankful to Vincent Pilaud for pointing
out the non-semidistributivity of the lattice of silting objects (Example
5.1), to Hugh Thomas for referring to the study of $d$-term silting objects
for Dynkin cases [30], and to Yann Palu for his several interesting questions,
including the one about the relation of this work to higher torsion classes.
Finally, I would like to thank Monica Garcia for sharing the manuscript of her
Ph.D. thesis, which provided the idea for the proof of Theorem 3.20.

##  2\. Notation and definitions

Throughout this work, $\Lambda$ will denote a finite-dimensional algebra over
an algebraically closed field $K$. For a morphism $f$ in a triangulated
category, $C(f)$ will denote the cone of $f$. We denote by
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ the bounded derived category of
$\operatorname{mod}\Lambda$, which is equivalent to
$\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)$, the homotopy category of
right bounded complexes of finitely generated projectives over $\Lambda$ with
bounded cohomology. We denote the category of bounded complexes of finitely
generated projectives over $\Lambda$ by
$\mathrm{C}^{b}(\operatorname{proj}\Lambda)$, and the homotopy category of
bounded complexes of finitely generated projectives over $\Lambda$ by
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$. For $n,m\in\mathbb{Z}$, we
set

$\begin{split}\mathcal{D}^{\leq
n}(\operatorname{mod}\Lambda)&:=\\{X\in\mathcal{D}^{b}(\operatorname{mod}\Lambda)\mid\mathrm{H}^{>n}(X)=0\\}\\\
\mathcal{D}^{\geq
m}(\operatorname{mod}\Lambda)&:=\\{X\in\mathcal{D}^{b}(\operatorname{mod}\Lambda)\mid\mathrm{H}^{<m}(X)=0\\}\\\
\mathcal{D}^{[m,n]}(\operatorname{mod}\Lambda)&:=\mathcal{D}^{\leq
n}(\operatorname{mod}\Lambda)\cap\mathcal{D}^{\geq
m}(\operatorname{mod}\Lambda),\end{split}$

and call them the _truncated derived categories_ of $\Lambda$. Similarly, we
define

$\begin{split}\operatorname{K}^{\geq
m}(\operatorname{proj}\Lambda)&:=\\{X\in\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)\mid
X^{<m}=0\\}\\\ \operatorname{K}^{\leq
m}(\operatorname{proj}\Lambda)&:=\\{X\in\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)\mid
X^{>m}=0\\}\\\
\operatorname{K}^{[m,n]}(\operatorname{proj}\Lambda)&:=\operatorname{K}^{\geq
m}(\operatorname{proj}\Lambda)\cap K^{\leq
n}(\operatorname{proj}\Lambda),\end{split}$

and call them the _truncated homotopy categories_ of $\Lambda$. Being
extension closed subcategories of triangulated categories, both
$\mathcal{D}^{[m,n]}(\operatorname{mod}\Lambda)$ and
$\operatorname{K}^{[m,n]}(\operatorname{proj}\Lambda)$ inherit extriangulated
structures in the sense of [25].

We recall that for any $d\in\mathbb{Z}$, the natural $t$-structure [7]

$(\mathcal{D}^{\leq d}(\operatorname{mod}\Lambda),\mathcal{D}^{\geq
d+1}(\operatorname{mod}\Lambda))$

on $\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ induces truncation functors
$\tau^{\leq d}$ and $\tau^{\geq d+1}$ defined as follows.

Let $A^{\bullet}=(\cdots\to
A^{-2}\xrightarrow{\delta^{-2}}A^{-1}\xrightarrow{\delta^{-1}}A^{0}\xrightarrow{\delta^{0}}A^{1}\xrightarrow{\delta^{1}}A^{2}\to\cdots)\in\mathcal{D}^{b}(\operatorname{mod}\Lambda)$.
Then

$\begin{split}\tau^{\leq d}A^{\bullet}&:=(\cdots\to A^{d-2}\to
A^{d-1}\to\mathrm{Ker}(\delta^{d})\to 0\to 0\to\cdots)\\\ \tau^{\geq
d+1}A^{\bullet}&:=(\cdots\to 0\to 0\to A^{d}/\mathrm{Ker}(\delta^{d})\to
A^{d+1}\to A^{d+2}\to\cdots)\end{split}$

and we have a triangle $\tau^{\leq d}A^{\bullet}\to A^{\bullet}\to\tau^{\geq
d+1}A^{\bullet}$.

Similarly, the natural co-$t$-structure ([26, 9][21, § 3.4])

$(\operatorname{K}^{\geq d}(\operatorname{proj}\Lambda),\operatorname{K}^{\leq
d}(\operatorname{proj}\Lambda))$

on $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ induces the hard
truncations $t^{\leq d}$ and $t^{\geq d-1}$ defined below.

Let $P^{\bullet}=(\cdots\to
P^{-2}\xrightarrow{\delta^{-2}}P^{-1}\xrightarrow{\delta^{-1}}P^{0}\xrightarrow{\delta^{0}}P^{1}\xrightarrow{\delta^{1}}P^{2}\to\cdots)\in\mathrm{C}^{b}(\operatorname{proj}\Lambda)$.
Then

$\begin{split}t^{\leq d}P^{\bullet}&:=(\cdots\to P^{d-2}\to P^{d-1}\to
P^{d}\to 0\to 0\to\cdots)\in\mathrm{C}^{b}(\operatorname{proj}\Lambda)\\\
t^{\geq d-1}P^{\bullet}&:=(\cdots\to 0\to 0\to P^{d-1}\to P^{d}\to
P^{d+1}\to\cdots)\in\mathrm{C}^{b}(\operatorname{proj}\Lambda),\end{split}$

and we get a triangle $t^{\geq d}P^{\bullet}\to P^{\bullet}\to t^{\leq
d-1}P^{\bullet}$ in $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.

We now recall the definition of silting objects in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.

######  Definition 2.1

Let $M\in\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.

  1. (1)

$M$ is called a _silting object_ if

    1. (a)

$\mathrm{Hom}(M,\Sigma^{i}M)=0$ for all $i>0$;

    2. (b)

$\operatorname{thick}M=\operatorname{K}^{b}(\operatorname{proj}\Lambda)$,
where $\operatorname{thick}M$ is the smallest full triangulated subcategory of
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ containing $M$, and closed
under summands and isomorphism.

  2. (2)

A silting object is called $d$_-term_ if it is isomorphic to a complex of
projectives concentrated in degrees $\\{-d+1,\cdots,0\\}$.

We denote by $\operatorname{silt}\Lambda$ the set of isomorphism classes of
basic silting objects in $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.
This can be equipped with a poset structure $\leq$ as follows ([4, Theorem
2.11]).

$P\leq Q:\iff\mathrm{Hom}(Q,\Sigma^{i}P)=0\ \forall\ i>0$

We will denote the subposet of $\operatorname{silt}\Lambda$ of $d$-term
silting objects by $d$-$\operatorname{silt}\Lambda$.

##  3\. Bijection between torsion and cotorsion pairs

###  3.1. Truncation functors

The goal of this section is to generalize the relation between
$K^{[-1,0]}(\operatorname{proj}\Lambda)$ and $\operatorname{mod}\Lambda$ to
higher truncations of the homotopy category. Fix $d\geq 2$. Then we note that
the truncation functor
$\tau_{\geq-d+2}:\mathcal{D}^{b}(\operatorname{mod}\Lambda)\to\mathcal{D}^{b}(\operatorname{mod}\Lambda)$
defined in § 2 gives a restriction

$\tau_{\geq-d+2}:\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\to\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda).$

The following proposition recovers, for $d=2$, some known properties of the
cohomology functor
$\mathrm{H}^{0}:\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)\to\operatorname{mod}\Lambda$.

######  Proposition 3.1

The truncation functor

$\tau_{\geq-d+2}:\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\to\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$

induces an equivalence between the categories
$\frac{\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)}{\operatorname{add}\Lambda[d-1]}$
and $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

###### Proof.

We divide the proof into three steps. Step 1. The functor $\tau_{\geq-d+2}$ is
essentially surjective on $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$:
Let $M\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Then, using the
fact that the inclusion of
$\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)$ into
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ is an equivalence, $M$ is
isomorphic to a complex of projectives bounded below, say, $(\cdots\to
P^{-d+1}\to\cdots P^{-1}\to P^{0}\to 0)$. Then $P^{\prime}=(\cdots\to 0\to
P^{-d+1}\to\cdots\to P^{0}\to
0)\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ and the
following map gives a quasi-isomorphism between $M$ and
$\tau_{\geq-d+2}(P^{\prime})$.

${\cdots}$${P^{-d}}$${P^{-d+1}}$${P^{-d+2}}$${\cdots}$${P^{0}}$${0}$${\cdots}$${0}$${\frac{P^{-d+1}}{\mathrm{Ker}(\delta^{-d+1})}}$${P^{-d+2}}$${\cdots}$${P^{0}}$${0}$$\scriptstyle{\delta^{-d}}$$\scriptstyle{\pi}$$\scriptstyle{\delta^{-d+1}}$$\scriptstyle{id}$$\scriptstyle{id}$$\scriptstyle{\overline{\delta^{-d+1}}}$

Step 2. The functor $\tau_{\geq-d+2}$ is full: Let $X,Y\in
K^{[-d+1,0]}(\operatorname{proj}\Lambda)$ and
$f:\tau_{\geq-d+2}X\to\tau_{\geq-d+2}Y\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.
By definition,

$\tau_{\geq-d+2}X=0\to\frac{X^{-d+1}}{\mathrm{Ker}(\delta_{X}^{-d+1})}\xrightarrow{\overline{\delta_{X}^{-d+1}}}X^{-d+2}\to\cdots\to
X^{0}\to 0$

and

$\tau_{\geq-d+2}Y=0\to\frac{Y^{-d+1}}{\mathrm{Ker}(\delta_{Y}^{-d+1})}\xrightarrow{\overline{\delta_{Y}^{-d+1}}}Y^{-d+2}\to\cdots\to
Y^{0}\to 0.$

Let $\cdots\to P^{-d-1}\to P^{-d}\to X^{-d+1}\to 0$ and $\cdots\to Q^{-d-1}\to
Q^{-d}\to Y^{-d+1}\to 0$ be projective resolutions of
$\frac{X^{-d+1}}{\mathrm{Ker}(\delta_{X}^{-d+1})}$ and
$\frac{Y^{-d+1}}{\mathrm{Ker}(\delta_{Y}^{-d+1})}$ respectively. Set
$X^{\prime}=\cdots\to P^{-d}\to X^{-d+1}\to X^{-d+2}\to\cdots\to X^{0}\to 0$
and $Y^{\prime}=\cdots\to Q^{-d}\to Y^{-d+1}\to Y^{-d+2}\to\cdots\to Y^{0}\to
0$. Then there exist canonical quasi-isomorphisms
$i_{X}:X^{\prime}\to\tau_{\geq-d+2}X$ and
$i_{Y}:Y^{\prime}\to\tau_{\geq-d+2}Y$. Using the equivalence between
$\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)$ and
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ again, we get that
$g:=(i_{Y})^{-1}fi_{X}\in\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)$.
Let $g^{\prime}:X\to Y$ be the map shown below.

${0}$${X^{-d+1}}$${X^{-d+2}}$${\cdots}$${X^{0}}$${0}$${0}$${Y^{-d+1}}$${Y^{-d+2}}$${\cdots}$${Y^{0}}$${0}$$\scriptstyle{g^{-d+1}}$$\scriptstyle{g^{-d+2}}$$\scriptstyle{g^{0}}$

Then it can be checked that $\tau_{\geq-d+2}(g^{\prime})\circ i_{X}=i_{Y}\circ
g$. Thus $\tau_{\geq-d+2}(g^{\prime})=i_{Y}\circ g\circ(i_{X})^{-1}=f$.

Step 3. The kernel of $\tau_{\geq-d+2}$ is the set of maps factoring through
$\operatorname{add}\Lambda[d-1]$: Clearly, if a map factors through
$\operatorname{add}\Lambda[d-1]$, then it lies in
$\mathrm{Ker}(\tau_{\geq-d+2})$. Let $f:X\to
Y\in\mathrm{Ker}(\tau_{\geq-d+2})$. As before, we can define
$X^{\prime},Y^{\prime}\in\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)$
with canonical quasi-isomorphisms $i_{X}:X^{\prime}\to\tau_{\geq-d+2}X$ and
$i_{Y}:Y^{\prime}\to\tau_{\geq-d+2}Y$. Since a morphism of modules can be
lifted to a morphism of their projective resolutions, we can find maps
$g^{i}:P^{i}\to Q^{i}$ for $i\leq-d$ such that the following is a morphism $h$
of complexes.

${\cdots}$${P^{-d}}$${X^{-d+1}}$${X^{-d+2}}$${\cdots}$${X^{0}}$${0}$${\cdots}$${Q^{-d}}$${Y^{-d+1}}$${Y^{-d+2}}$${\cdots}$${Y^{0}}$${0}$$\scriptstyle{g^{-d}}$$\scriptstyle{f^{-d+1}}$$\scriptstyle{f^{-d+2}}$$\scriptstyle{f^{0}}$

Again it can be checked that $\tau_{\geq-d+2}(f)\circ i_{X}=i_{Y}\circ h$,
which implies that $h=0$. Thus, there exist maps $k^{i}:X^{i}\to Y^{i-1}$ such
that the homotopy relation holds.

${\cdots}$${P^{-d}}$${X^{-d+1}}$${X^{-d+2}}$${\cdots}$${X^{0}}$${0}$${\cdots}$${Q^{-d}}$${Y^{-d+1}}$${Y^{-d+2}}$${\cdots}$${Y^{0}}$${0}$$\scriptstyle{g^{-d}}$$\scriptstyle{f^{-d+1}}$$\scriptstyle{k^{-d+1}}$$\scriptstyle{f^{-d+2}}$$\scriptstyle{k^{-d+2}}$$\scriptstyle{f^{0}}$$\scriptstyle{\delta_{Y}^{-d}}$

Consider the following two maps.

${0}$${X^{-d+1}}$${X^{-d+2}}$${\cdots}$${X^{0}}$${0}$${0}$${X^{-d+1}}$${0}$${\cdots}$${0}$${0}$${0}$${Y^{-d+1}}$${Y^{-d+2}}$${\cdots}$${Y^{0}}$${0}$$\scriptstyle{id}$$\scriptstyle{\delta^{-d}_{Y}\circ
k^{-d+1}}$

The diagram below shows that the composition of the above two maps is equal to
$f$ in the homotopy category.

${0}$${X^{-d+1}}$${X^{-d+2}}$${\cdots}$${X^{0}}$${0}$${0}$${Y^{-d+1}}$${Y^{-d+2}}$${\cdots}$${Y^{0}}$${0}$$\scriptstyle{f^{-d+1}-\delta^{-d}_{Y}\circ
k^{-d+1}}$$\scriptstyle{f^{-d+2}}$$\scriptstyle{k^{-d+2}}$$\scriptstyle{f^{0}}$

Thus, $f$ factors through $\operatorname{add}\Lambda[d-1]$, and we get an
induced equivalence

$\tau_{\geq-d+2}:\frac{\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)}{\operatorname{add}\Lambda[d-1]}\to\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda).$

∎

Our next goal is to show that we can lift conflations in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ to conflations in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ across this
equivalence. We will need the following lemma about inflations in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.

######  Lemma 3.2

Let $f:X\to Y$ be a morphism in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ such that $f^{-d+1}$
is a split mono. Then $f$ is an inflation.

###### Proof.

We need to show that
$C(f)\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. By
definition, $C(f)$ is given as

$0\to X^{-d+1}\xrightarrow{\begin{bmatrix}0\\\
f^{-d+1}\end{bmatrix}}X^{-d+2}\oplus Y^{-d+1}\to\cdots X^{0}\oplus Y_{1}\to
Y^{0}.$

Since $f^{-d+1}$ is a split mono, $C(f)$ is isomorphic to

$0\to X^{-d+1}\xrightarrow{\begin{bmatrix}0\\\ 1_{X^{-d+1}}\\\
0\end{bmatrix}}X^{-d+2}\oplus X^{-d+1}\oplus Z\to\cdots X^{0}\oplus Y_{1}\to
Y^{0}$

for some $Z\in\operatorname{proj}\Lambda$. Thus $C(f)=$

${0}$${X^{-d+1}}$${X^{-d+1}}$${\cdots}$${0}$${0}$${\oplus}$${0}$${0}$${X^{-d+2}\oplus
Z}$${\cdots}$${X^{0}\oplus Y^{1}}$${Y^{0}}$$\scriptstyle{1_{X^{-d+1}}}$

Hence, it is quasi-isomorphic to

$(X^{-d+2}\oplus Z\to\cdots\to X^{0}\oplus Y^{1}\to
Y^{0})\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda).$

∎

######  Proposition 3.3

Let $\epsilon:X\xrightarrow{f}Y\xrightarrow{g}Z$ be a conflation in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Then there exists a
conflation
$\epsilon^{\prime}:X^{\prime}\xrightarrow{f^{\prime}}Y^{\prime}\xrightarrow{g^{\prime}}Z^{\prime}$
in $\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ such that
$\tau_{\geq-d+2}\epsilon^{\prime}\cong\epsilon$.

###### Proof.

Using the equivalence between
$\operatorname{K}^{b,-}(\operatorname{proj}\Lambda)$ and
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$, we can assume that $f$ is of the
following form:

${\cdots}$${P^{-d+1}}$${P^{-d+2}}$${\cdots}$${P^{0}}$${0}$${\cdots}$${Q^{-d+1}}$${Q^{-d+2}}$${\cdots}$${Q^{0}}$${0}$$\scriptstyle{f^{-d+1}}$$\scriptstyle{f^{-d+2}}$$\scriptstyle{f^{0}}$

This implies that $\epsilon\cong X\xrightarrow{f}Y\xrightarrow{g}C(f)$. Set

$X^{\prime}=t_{\geq-d+1}(X)=0\to P^{-d+1}\to P^{-d+2}\to\cdots\to P^{0}\to 0,$

and

$Y^{\prime\prime}=t_{\geq-d+1}(Y)=0\to Q^{-d+1}\to Q^{-d+2}\to\cdots\to
Q^{0}\to 0,$

and $f^{\prime\prime}=t_{\geq-d+1}(f)$. Set
$Y^{\prime}:=Y^{\prime\prime}\oplus P^{-d+1}[d-1]$. Let $i:X^{\prime}\to
P^{-d+1}[d-1]$ be the map that is identity in degree $-d+1$, and $0$ in the
others. Then the map $f^{\prime}:=\begin{bmatrix}f^{\prime\prime}\\\
i\end{bmatrix}:X^{\prime}\to Y^{\prime}$ is an inflation in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ by Lemma 3.2 because
it is a split mono in degree $-d+1$. We claim that
$\tau_{\geq-d+2}(X^{\prime}\xrightarrow{f^{\prime}}Y^{\prime}\xrightarrow{g^{\prime}}C(f^{\prime}))\cong\epsilon$.

Using the previous proposition, we know that
$\tau_{\geq-d+2}(f^{\prime})\cong\tau_{\geq-d+2}(f^{\prime\prime})\cong f$. By
definition,

$\displaystyle\tau_{\geq-d+2}C(f^{\prime})$
$\displaystyle=0\to\frac{P^{-d+2}\oplus Q^{-d+1}\oplus
P^{-d+1}}{\mathrm{Ker}(\delta_{C(f)}^{-d+1})\oplus P^{-d+1}}\to P^{-d+3}\oplus
Q^{-d+2}\to\cdots Q^{0}\to 0$ $\displaystyle\cong 0\to\frac{P^{-d+2}\oplus
Q^{-d+1}}{\mathrm{Ker}(\delta_{C(f)}^{-d+1})}\to P^{-d+3}\oplus
Q^{-d+2}\to\cdots Q^{0}\to 0,$

where $\delta_{C(f)}^{-d+1}$ denotes the $-d+1$ boundary map of $C(f)$. Since
$C(f)\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$, the following map
gives a quasi-isomorphism between $\tau_{\geq-d+2}C(f^{\prime})$ and $C(f)$.

${\cdots}$${P^{-d+1}\oplus Q^{-d}}$${P^{-d+2}\oplus
Q^{-d+1}}$${\cdots}$${Q^{0}}$${0}$${\cdots}$${0}$${\frac{P^{-d+2}\oplus
Q^{-d+1}}{\mathrm{Ker}(\delta_{C(f)}^{-d+1})}}$${\cdots}$${Q^{0}}$${0}$$\scriptstyle{\pi}$$\scriptstyle{id}$

Denoting the above map by $i_{C(f^{\prime})}$, it can be easily checked that
the following diagram commutes,

${X}$${Y}$${C(f)}$${\tau_{\geq-d+2}X^{\prime}}$${\tau_{\geq-d+2}Y^{\prime}}$${\tau_{\geq-d+2}C(f^{\prime})}$$\scriptstyle{f}$$\scriptstyle{i_{X^{\prime}}}$$\scriptstyle{g}$$\scriptstyle{i_{Y^{\prime}}}$$\scriptstyle{i_{C(f^{\prime})}}$$\scriptstyle{\tau_{\geq-d+2}f^{\prime}}$$\scriptstyle{\tau_{\geq-d+2}g^{\prime}}$

where $i_{X^{\prime}}$ and $i_{Y^{\prime}}$ are the canonical quasi-
isomorphisms defined in the previous proposition.

Setting $\epsilon^{\prime}=X^{\prime}\xrightarrow{f^{\prime}}Y^{\prime}\to
C(f^{\prime})$, we get that $\tau_{\geq-d+1}\epsilon^{\prime}\cong\epsilon$. ∎

The following corollary gives the dual of the above two propositions for the
homotopy category of injectives.

######  Corollary 3.4

  1. (1)

The functor $\tau_{\leq 0}$ induces an equivalence between the categories
$\frac{\operatorname{K}^{[-d+2,1]}(\operatorname{inj}\Lambda)}{\operatorname{inj}\Lambda[-1]}$
and $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

  2. (2)

Given a conflation $\epsilon:X\xrightarrow{f}Y\xrightarrow{g}Z$ in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$, there exists a conflation
$\epsilon^{\prime}:X^{\prime}\xrightarrow{f^{\prime}}Y^{\prime}\xrightarrow{g^{\prime}}Z^{\prime}$
in $\operatorname{K}^{[-d+2,1]}(\operatorname{inj}\Lambda)$ such that
$\tau_{\leq 0}\epsilon^{\prime}\cong\epsilon$.

Our next goal is to characterize the extensions between any two objects in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ in terms of the
morphisms between their images under the truncation functors. This will be
used in the next section to give a correspondence between cotorsion pairs in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ and torsion pairs in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

######  Theorem 3.5

[16, § I.4.6] Let $X,Y\in\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.
Then

$\mathrm{Hom}_{\operatorname{K}^{b}(\operatorname{proj}\Lambda)}(X,Y)\cong\mathrm{D}\mathrm{Hom}_{\mathcal{D}^{b}(\operatorname{mod}\Lambda)}(Y,\nu
X).$

In the following lemma, we view
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ as an extriangulated
category with $\mathbb{E}(X,Y)$ denoting the group of extensions from $X$ to
$Y$. We will use this notation again in § 3.2.

######  Lemma 3.6

Let $X,Y\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Then

$\mathbb{E}(X,Y)\cong\mathrm{D}\mathrm{Hom}_{\mathcal{D}^{b}(\operatorname{mod}\Lambda)}(\tau_{\geq-d+2}Y,\Sigma^{-1}\tau_{\leq-1}\nu
X).$

###### Proof.

Using Theorem 3.5, we get that

$\mathbb{E}(X,Y)\cong\mathrm{D}\mathrm{Hom}_{\operatorname{K}^{b}(\operatorname{proj}\Lambda)}(X,\Sigma
Y)\cong\mathrm{D}\mathrm{Hom}_{\mathcal{D}^{b}(\operatorname{mod}\Lambda)}(\Sigma
Y,\nu X).$

The triangle $\tau_{\leq-d}\Sigma Y\to\Sigma Y\to\tau_{\geq-d+1}\Sigma Y$
gives the exact sequence

$\mathbb{E}^{-1}(\tau_{\leq-d}\Sigma Y,\nu
X)\to\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\nu X)\to\mathrm{Hom}(\Sigma Y,\nu
X)\to\mathrm{Hom}(\tau_{\leq-d}\Sigma Y,\nu X)$

. By comparing the support of the cohomology of $\tau_{\leq-d}\Sigma Y$ and
$\nu X$, we see that $\mathbb{E}^{-1}(\tau_{\leq-d}\Sigma Y,\nu X)$and
$\mathrm{Hom}(\tau_{\leq-d}\Sigma Y,\nu X)$ vanish. Thus,

$\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\nu X)\cong\mathrm{Hom}(\Sigma Y,\nu
X).$

Now, the triangle $\tau_{\leq-1}\nu X\to\nu X\to\tau_{\geq 0}\nu X$ gives the
exact sequence

${\mathbb{E}^{-1}(\tau_{\geq-d+1}\Sigma Y,\tau_{\geq 0}\nu
X)}$${\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\tau_{\leq-1}\nu
X)}$${\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\nu
X)}$${\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\tau_{\geq 0}\nu X).}$

Again, we see that $\mathbb{E}^{-1}(\tau_{\geq-d+1}\Sigma Y,\tau_{\geq 0}\nu
X)$ and $\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\tau_{\geq 0}\nu X)$ vanish.
Thus,

$\begin{split}\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\nu
X)&\cong\mathrm{Hom}(\tau_{\geq-d+1}\Sigma Y,\tau_{\leq-1}\nu X)\\\
&\cong\mathrm{Hom}(\Sigma\tau_{\geq-d+2}Y,\tau_{\leq-1}\nu X)\\\
&\cong\mathrm{Hom}(\tau_{\geq-d+2}Y,\Sigma^{-1}\tau_{\leq-1}\nu
X).\end{split}$

∎

We need the following technical lemmas for the subsequent sections.

######  Lemma 3.7

Let $f:X\to Y\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Then

$\tau_{\geq-d+2}C(f)\cong\tau_{\geq-d+2}C(\tau_{\geq-d+2}f).$

###### Proof.

Suppose $f$ is of the following form.

${X^{-d+1}}$${\cdots}$${X^{-1}}$${X^{0}}$${Y^{-d+1}}$${\cdots}$${Y^{-1}}$${Y^{0}}$$\scriptstyle{\delta_{X}^{-d+1}}$$\scriptstyle{f^{-d+1}}$$\scriptstyle{\delta_{X}^{-1}}$$\scriptstyle{f^{-1}}$$\scriptstyle{f^{0}}$$\scriptstyle{\delta_{Y}^{-d+1}}$$\scriptstyle{\delta_{Y}^{-1}}$

Then, by definition, $\tau_{\geq-d+2}C(f)$ is

${\frac{X^{-d+2}\oplus
Y^{-d+1}}{\mathrm{Ker}(\begin{bmatrix}-\delta_{X}^{-d+2}&&0\\\
f^{-d+2}&&\delta_{Y}^{-d+1}\end{bmatrix})}}$${X^{-d+3}\oplus
Y^{-d+2}}$${\cdots}$${X^{0}\oplus Y^{-1}}$${Y^{0}.}$

On the other hand, $\tau_{\geq-d+2}C(\tau_{\geq-d+2}f)$ is given as

${\frac{X^{-d+2}\oplus\frac{Y^{-d+1}}{\mathrm{Ker}(\delta_{Y}^{-d+1})}}{\mathrm{Ker}(\begin{bmatrix}-\delta_{X}^{-d+2}&&0\\\
f^{-d+2}&&\overline{\delta_{Y}^{-d+1}}\end{bmatrix})}}$${X^{-d+3}\oplus
Y^{-d+2}}$${\cdots}$${X^{0}\oplus Y^{-1}}$${Y^{0}.}$

The following map gives an isomorphism between the two.

${\frac{X^{-d+2}\oplus
Y^{-d+1}}{\mathrm{Ker}(\begin{bmatrix}-\delta_{X}^{-d+2}&&0\\\
f^{-d+2}&&\delta_{Y}^{-d+1}\end{bmatrix})}}$${X^{-d+3}\oplus
Y^{-d+2}}$${\cdots}$${X^{0}\oplus
Y^{-1}}$${Y^{0}}$${\frac{X^{-d+2}\oplus\frac{Y^{-d+1}}{\mathrm{Ker}(\delta_{Y}^{-d+1})}}{\mathrm{Ker}(\begin{bmatrix}-\delta_{X}^{-d+2}&&0\\\
f^{-d+2}&&\overline{\delta_{Y}^{-d+1}}\end{bmatrix})}}$${X^{-d+3}\oplus
Y^{-d+2}}$${\cdots}$${X^{0}\oplus
Y^{-1}}$${Y^{0}}$$\scriptstyle{\overline{(x,y)}\mapsto\overline{(x,\overline{y})}}$$\scriptstyle{id}$$\scriptstyle{id}$$\scriptstyle{id}$

∎

Recall that a full subcategory $\mathcal{X}$ of a category $\mathcal{C}$ is
called _contravariantly finite_ (resp. _covariantly finite_) if for all
$C\in\mathcal{C}$, there exists $f_{C}:X_{C}\to C$ (resp. $g_{C}:C\to X_{C}$)
with $X_{C}\in\mathcal{X}$ such that for all $X\in\mathcal{X}$,
$\mathrm{Hom}(X,X_{C})\xrightarrow{\mathrm{Hom}(X,f_{C})}\mathrm{Hom}(X,C)$
(resp.
$\mathrm{Hom}(X_{C},X)\xrightarrow{\mathrm{Hom}(g_{C},X)}\mathrm{Hom}(C,X)$)
is surjective.

######  Lemma 3.8

Let
$\mathcal{Y}\subseteq\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$
be an additive subcategory containing $\operatorname{add}\Lambda[d-1]$. Then
$\mathcal{Y}$ is covariantly (resp. contravariantly) finite in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ if and only if
$\tau_{\geq-d+2}\mathcal{Y}$ is covariantly (resp. contravariantly) finite in
$\mathcal{D}^{[-d+2,0]}(\operatorname{proj}\Lambda)$.

###### Proof.

We will only prove the case for covariant finiteness. The case for
contravariant finiteness will be dual. $(\implies)$ Follows from the fact that
the functor $\tau_{\geq-d+2}$ is full and essentially surjective.
$(\impliedby)$ Let
$Z\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ and
$f^{\prime}:\tau_{\geq-d+2}Z\to Y^{\prime}$ be a left
$\tau_{\geq-d+2}\mathcal{Y}$-approximation of $\tau_{\geq-d+2}Z$. Since
$\tau_{\geq-d+2}$ is full and essentially surjective, there exists $f:Z\to
Y\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ such that
$\tau_{\geq-d+2}f=f^{\prime}$. Let $g:Z\to P$ be a left
$\operatorname{add}\Lambda[d-1]$-approximation of $Z$, which exists since
$\operatorname{add}\Lambda[d-1]$ is functorially finite. We claim that
$\begin{bmatrix}f\\\ g\end{bmatrix}:Z\to Y\oplus P$ is a left
$\mathcal{Y}$-approximation of $Z$.

Let $f_{1}:Z\to Y_{1}$ be a morphism with $Y_{1}\in\mathcal{Y}$. Since
$f^{\prime}$ is an approximation, there exists $g_{1}:Y\to Y_{1}$ such that
$f^{\prime}\tau_{\geq-d+2}g_{1}=\tau_{\geq-d+2}f_{1}$. Therefore, the map
$g_{1}f-f_{1}$ factors through some
$P^{\prime}\in\operatorname{add}\Lambda[d-1]$ as shown below.

${Z}$${Y_{1}}$${P^{\prime}}$$\scriptstyle{g_{1}f-f_{1}}$$\scriptstyle{h}$$\scriptstyle{k}$

Since $g:Z\to P$ is a left $\operatorname{add}\Lambda[d-1]$-approximation of
$Z$, there exists some $l:P\to P^{\prime}$ such that $lg=h$. This gives the
following commutative triangle and we are done.

${Z}$${Y\oplus P}$${Y_{1}}$$\scriptstyle{\begin{bmatrix}f\\\
g\end{bmatrix}}$$\scriptstyle{f_{1}}$$\scriptstyle{\begin{bmatrix}g_{1}\
-kl\end{bmatrix}}$

∎

######  Lemma 3.9

For any morphism $f:A\to
B\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, there exists an
inflation $f^{\prime}:A\to
B^{\prime}\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ such
that $\tau_{\geq-d+2}f\cong\tau_{\geq-d+2}f^{\prime}$.

###### Proof.

Let $f^{\prime}:A\to A^{-d+1}[d-1]\oplus B$ be the following morphism.

${A^{-d+1}}$${A^{-d+2}}$${\cdots}$${A^{-1}}$${A^{0}}$${A^{-d+1}\oplus
B^{-d+1}}$${B^{-d+2}}$${\cdots}$${B^{-1}}$${B^{0}}$$\scriptstyle{\begin{bmatrix}id\\\
f^{-d+1}\end{bmatrix}}$$\scriptstyle{\delta^{-d+1}_{A}}$$\scriptstyle{f^{-d+1}}$$\scriptstyle{f^{-1}}$$\scriptstyle{f^{0}}$$\scriptstyle{[0\
\ \delta_{B}^{-d+1}]}$

Then, clearly, $\tau_{\geq-d+2}f\cong\tau_{\geq-d+2}f^{\prime}$. Since
$f^{\prime-d+1}$ is a split mono, using Lemma 3.2, we get that $f^{\prime}$ is
an inflation in $\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. ∎

###  3.2. From cotorsion pairs to torsion pairs

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category
equipped with a bivariant $\delta$-functor
$(G^{\bullet},\delta_{\\#},\delta^{\\#})$ such that
$G^{0}=\mathrm{Hom}_{\mathcal{C}}$, $G^{1}=\mathbb{E}$, and for any
$\mathbb{E}$-extension $\delta\in\mathbb{E}(C,A)$,
$\delta^{0}_{\\#}=A_{\delta}$ and $\delta^{\\#,0}=B_{\delta}$. In particular,
$(\mathcal{C},\mathbb{E},\mathfrak{s},G^{-1})$ is an extriangulated category
with a negative first extension in the sense of [1, Definition 2.3].

Henceforth, we will consider
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ and
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ as extriangulated
categories with bivariant $\delta$-functors
$G^{i}(-,-):=\mathbb{E}^{i}(-,-)=\mathrm{Hom}(-,\Sigma^{i}-)$, where $\Sigma$
is the suspension functor of the triangulated categories
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ and
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ respectively.

The notion of cotorsion pairs was first defined in [28] for abelian
categories. These were then generalised to exact categories [17, 29],
triangulated categories [24], and extriangulated categories [25]. We will use
the following definitions in this work.

######  Definition 3.10

Let $\mathcal{X},\mathcal{Y}$ be full subcategories of $\mathcal{C}$.

  1. (1)

The pair $(\mathcal{X},\mathcal{Y})$ is called a _cotorsion pair_ if

    1. (a)

$\mathbb{E}(x,\mathcal{Y})=0$ if and only if $x\in\mathcal{X}$;

    2. (b)

$\mathbb{E}(\mathcal{X},y)=0$ if and only if $y\in\mathcal{Y}$.

  2. (2)

A cotorsion pair $(\mathcal{X},\mathcal{Y})$ is called _hereditary_ if
$G^{k}(\mathcal{X},\mathcal{Y})=0$ for all $k\geq 2$.

  3. (3)

A cotorsion pair $(\mathcal{X},\mathcal{Y})$ is called _complete_ if for each
$c\in\mathcal{C}$

    1. (Ca)

there exists a conflation $c\to y\to x$ with $x\in\mathcal{X}$ and
$y\in\mathcal{Y}$;

    2. (Cb)

there exists a conflation $y^{\prime}\to x^{\prime}\to c$ with
$x^{\prime}\in\mathcal{X}$ and $y^{\prime}\in\mathcal{Y}$.

######  Remark 3.11

Using [25, Remark 4.4], to check if a pair of full subcategories
$(\mathcal{X},\mathcal{Y})$ is complete, it is enough to check the weaker
conditions $\mathbb{E}(X,Y)=0$, (Ca), and (Cb).

######  Remark 3.12

The definition of cotorsion pairs for extriangulated categories was first
introduced in [25], and that of hereditary cotorsion pairs in [3]. However, we
would like to point out that cotorsion pairs in the sense of [25, 3] are
complete cotorsion pairs in our sense. Moreover, the authors in [3] also
define the notion of $s$-cotorsion pairs which, in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, turns out to be
equivalent to the notion of hereditary cotorsion pairs (Lemma 3.13).

Following [28], for a cotorsion pair $(\mathcal{X},\mathcal{Y})$, we will call
$\mathcal{Y}$ a _cotorsion class_ , and $\mathcal{X}$ a _cotorsion-free
class_. We will call a cotorsion class _hereditary_ (resp. _complete_) if the
associated cotorsion pair is hereditary (resp. complete). We will denote the
poset of cotorsion classes in $\mathcal{C}$ under inclusion by
$\operatorname{cotors}\mathcal{C}$. This is isomorphic to the poset of
cotorsion pairs in $\mathcal{C}$ under the order

$(X_{1},Y_{1})\preceq(X_{2},Y_{2}):\iff X_{1}\supseteq X_{2}\iff
Y_{1}\subseteq Y_{2}.$

Furthermore, we will denote the subposets of hereditary, complete, and
complete and hereditary cotorsion classes/pairs by
$\operatorname{c-cotors}\mathcal{C}$, $\operatorname{h-cotors}\mathcal{C}$,
and $\operatorname{(c,h)-cotors}\mathcal{C}$, respectively.

######  Lemma 3.13

Let $(\mathcal{X},\mathcal{Y})$ be a cotorsion pair in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Then the following
are equivalent.

  1. (1)

$(\mathcal{X},\mathcal{Y})$ is a hereditary cotorsion pair.

  2. (2)

$\mathbb{E}^{2}(\mathcal{X},\mathcal{Y})=0$.

  3. (3)

$\mathcal{X}$ is closed under cocones.

  4. (4)

$\mathcal{Y}$ is closed under cones.

###### Proof.

($1\implies 2$) Clear.
($2\implies 3$, $2\implies 4$) The proof of [3, Lemma 3.2] does not use the
completeness axioms assumed by the authors.
($3\implies 1$) Let $X\in\mathcal{X}$. Then there is a conflation
$\Sigma^{-1}t_{\leq-1}X\to t_{\geq 0}X\to X$ with $t_{\geq
0}X,X\in\mathcal{X}$. Since $\mathcal{X}$ is closed under cocones, $t_{\leq
0}\Sigma^{-1}X\cong\Sigma^{-1}t_{\leq-1}X\in\mathcal{X}$. Repeating the above
argument, we conclude that $t_{\leq 0}\Sigma^{-k}X\in\mathcal{X}$ for all
$k\geq 1$.

Using the triangle $t_{\geq 1}\Sigma^{-k}X\to\Sigma^{-k}X\to t_{\leq
0}\Sigma^{-k}X$, we get that
$\mathbb{E}^{k+1}(X,\mathcal{Y})\cong\mathbb{E}(\Sigma^{-k}X,\mathcal{Y})=0$.
($4\implies 1$) Dual to the above proof.∎

Similar to cotorsion pairs, the notion of torsion pairs was first introduced
for abelian categories in [12]. This was then generalised to triangulated
categories in [8], where they were proven to be equivalent to $t$-structures.
A weaker definition of torsion pairs in triangulated categories was used in
[19], which we generalise here to extriangulated categories equipped with a
$\delta$-functor.

######  Definition 3.14

Let $\mathcal{T},\mathcal{F}$ be full subcategories of $\mathcal{C}$.

  1. (1)

The pair $(\mathcal{T},\mathcal{F})$ is called a _torsion pair_ if

    1. (a)

$\mathrm{Hom}(t,\mathcal{F})=0$ if and only if $t\in\mathcal{T}$;

    2. (b)

$\mathrm{Hom}(\mathcal{T},f)=0$ if and only if $f\in\mathcal{F}$.

  2. (2)

A torsion pair $(\mathcal{T},\mathcal{F})$ is called _positive_ if
$G^{k}(\mathcal{T},\mathcal{F})=0$ for all $k\leq-1$.

  3. (3)

[1, Definition 3.1] The pair $(\mathcal{T},\mathcal{F})$ is called an
$s$_-torsion pair_ if

    1. (Sa)

$\mathrm{Hom}(\mathcal{T},\mathcal{F})=0$;

    2. (Sb)

$G^{-1}(\mathcal{T},\mathcal{F})=0$;

    3. (Sc)

For all $C\in\mathcal{C}$, there exists a conflation $T\to C\to F$ with $T\in
T$ and $F\in\mathcal{F}$.

######  Remark 3.15

Using [1, Proposition 3.2], we know that every $s$-torsion pair is a torsion
pair.

For a torsion pair $(\mathcal{T},\mathcal{F})$, we will call $\mathcal{T}$ a
_torsion class_ , and $\mathcal{F}$ a _torsion-free class_. We will call a
torsion class _positive_ (resp. an $s$-torsion class) if the associated
torsion pair is positive (resp. an $s$-torsion pair). We will denote the poset
of torsion classes in $\mathcal{C}$ under inclusion by
$\operatorname{tors}\mathcal{C}$. This is isomorphic to the poset of torsion
pairs in $\mathcal{C}$ under the order

$(T_{1},F_{1})\preceq(T_{2},F_{2})\iff T_{1}\subseteq T_{2}\iff F_{1}\supseteq
F_{2}.$

We will denote the subposets of positive, contravariantly finite, covariantly
finite, and functorially finite torsion classes/pairs by
$\operatorname{p-tors}\mathcal{C}$, $\operatorname{contr-tors}\mathcal{C}$,
$\operatorname{cov-tors}\mathcal{C}$, $\operatorname{f-tors}\mathcal{C}$,
respectively. The subposet of $s$-torsion pairs will be denoted by
$\operatorname{stors}\mathcal{C}$. Finally, we will denote the intersection of
$\operatorname{x-tors}\mathcal{C}$ and $\operatorname{y-tors}\mathcal{C}$ by
$\operatorname{(x,y)-tors}\mathcal{C}$, where $\mbox{x, y}\in\\{\mbox{p,
contr, cov, f, s}\\}$.

######  Lemma 3.16

Let $(\mathcal{T},\mathcal{F})$ be a torsion pair in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Then the following are
equivalent.

  1. (1)

$(\mathcal{T},\mathcal{F})$ is a positive torsion pair.

  2. (2)

$\mathbb{E}^{-1}(\mathcal{T},\mathcal{F})=0$.

  3. (3)

For all morphisms $f\in\mathcal{T}$, $\tau_{\geq-d+2}C(f)\in\mathcal{T}$.

  4. (4)

For all morphisms $g\in\mathcal{F}$, $\tau_{\leq
0}\Sigma^{-1}C(f)\in\mathcal{F}$.

###### Proof.

$(1\implies 2)$ Clear.
$(2\implies 3)$ Let $f:X\to Y\in\mathcal{T}$. For $F\in\mathcal{F}$, the
triangle $X\to Y\to C(f)$ induces the exact sequence

$\mathbb{E}^{-1}(X,F)\to\mathrm{Hom}(C(f),F)\to\mathrm{Hom}(Y,F).$

By assumption, $\mathbb{E}^{-1}(X,F)$ and $\mathrm{Hom}(Y,F)$ vanish, which
gives that $\mathrm{Hom}(C(f),F)=0$. Using the triangle
$\tau_{\leq-d+1}C(f)\to C(f)\to\tau_{\geq-d+2}C(f)$, we get the exact sequence

$\mathrm{Hom}(\tau_{\leq-d+1}C(f)[1],F)\to\mathrm{Hom}(\tau_{\geq-d+2}C(f),F)\to\mathrm{Hom}(C(f),F)=0.$

Comparing the support of the cohomology of $\tau_{\leq-d+1}C(f)[1]$ and $F$,
we get that $\mathrm{Hom}(\tau_{\leq-d+1}C(f)[1],F)=0$ which gives that
$\mathrm{Hom}(\tau_{\geq-d+2}C(f),F)=0$. Thus,
$\tau_{\geq-d+2}C(f)\in\prescript{\perp}{}{\mathcal{F}}=\mathcal{T}$.
$(2\implies 4)$ Dual to the above proof.
$(3\implies 1)$ Let $T\in\mathcal{T}$. Using the triangle $T\to 0\to\Sigma T$,
we get that $\tau_{\geq-d+2}\Sigma T\in\mathcal{T}$. Repeating this argument,
we conclude that $\tau_{\geq-d+2}\Sigma^{k}T\in\mathcal{T}$ for all $k\geq 0$.

Using the triangle
$\tau_{\leq-d+1}\Sigma^{k}T\to\Sigma^{k}T\to\tau_{\geq-d+2}\Sigma^{k}T$, for
all $F\in\mathcal{F}$, we get the exact sequence

$0=\mathrm{Hom}(\tau_{\geq-d+2}\Sigma^{k}T,F)\to\mathrm{Hom}(\Sigma^{k}T,F)\to\mathrm{Hom}(\tau_{\leq-d+1}\Sigma^{k}T,F).$

Comparing the support of the cohomology of $\tau_{\leq-d+1}\Sigma^{k}T$ and
$F$, we get that $\mathrm{Hom}(\tau_{\leq-d+1}\Sigma^{k}T,F)=0$. Thus
$\mathbb{E}^{-k}(T,F)\cong\mathrm{Hom}(\Sigma^{k}T,F)=0$ for all $k\geq 0$,
and $(\mathcal{T},\mathcal{F})$ is a positive torsion pair.
$(4\implies 1)$ Dual to the above proof. ∎

######  Corollary 3.17

The class of $s$-torsion classes in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ is contained in the class
of positive torsion classes in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

###### Proof.

This follows from (2) $\implies$ (1) in Lemma 3.16. ∎

The next theorem provides us with a way to link cotorsion pairs to torsion
pairs using the truncation functor defined in the previous section. We will
need the following generalization of Wakamatsu Lemma to extriangulated
categories.

######  Lemma 3.18

[22, Lemma 3.1] Let $\mathcal{A}$ be an extension closed subcategory of an
extriangulated category $\mathcal{C}$ and $t\xrightarrow{f}a\to b$ be a
conflation such that $f$ is the minimal left $\mathcal{A}$-approximation of
$t\in T$. Then $b\in\prescript{\perp_{1}}{}{\mathcal{A}}$.

The following theorems 3.19 and 3.20 generalise the existing bijection between
torsion classes in $\operatorname{mod}\Lambda$ and cotorsion classes in
$\operatorname{K}^{[-1,0]}(\operatorname{proj}\Lambda)$ ([27, Proposition
3.2], [13]).

######  Theorem 3.19

Let $\mathcal{Y}$ be a cotorsion class in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Then
$\tau_{\geq-d+2}\mathcal{Y}$ is a torsion class in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Moreover, if
$\mathcal{Y}$ is hereditary (resp. complete, resp. hereditary and
contravariantly finite), then $\tau_{\geq-d+2}\mathcal{Y}$ is positive (resp.
covariantly finite, resp. an $s$-torsion class).

###### Proof.

Set $\mathcal{Y}^{\prime}:=\tau_{\geq-d+2}\mathcal{Y}$ and
$\mathcal{X}^{\prime}=(\tau_{\geq-d+2}\mathcal{Y})^{\perp}$. We need to show
that $\prescript{\perp}{}{\mathcal{X}^{\prime}}=\mathcal{Y}^{\prime}$. By
definition,
$\mathcal{Y}^{\prime}\subseteq\prescript{\perp}{}{\mathcal{X}^{\prime}}$. Let
$Y^{\prime}\in\prescript{\perp}{}{\mathcal{X}^{\prime}}$. Then there exists
$Y\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ such that
$\tau_{\geq-d+2}(Y)\cong Y^{\prime}$. We want to show that $Y\in\mathcal{Y}$.

Note that, since $(\prescript{\perp_{1}}{}{\mathcal{Y}},\mathcal{Y})$ is a
cotorsion pair,

$\begin{split}\prescript{\perp_{1}}{}{\mathcal{Y}}&=\\{X\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\mathbb{E}(X,\mathcal{Y})=0\\}\\\
&=\\{X\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\mathrm{Hom}(\tau_{\geq-d+2}\mathcal{Y},\Sigma^{-1}\tau_{\leq-1}\nu
X)=0\\}\ \ \ \ \ \mbox{(by Lemma \ref{Extlem})}\\\
&=\\{X\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\Sigma^{-1}\tau_{\leq-1}\nu
X\in\mathcal{X}^{\prime}\\},\end{split}$

and

$\begin{split}\mathcal{Y}&=\\{Z\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\mathbb{E}(\prescript{\perp_{1}}{}{\mathcal{Y}},Z)=0\\}\\\
&=\\{Z\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\mathrm{Hom}(\tau_{\geq-d+2}Z,\Sigma^{-1}\tau_{\leq-1}\nu\prescript{\perp_{1}}{}{\mathcal{Y}})=0\\}\
\ \ \ \mbox{(by Lemma \ref{Extlem})}.\end{split}$

Since, for all $X\in\prescript{\perp_{1}}{}{\mathcal{Y}}$,
$\Sigma^{-1}\tau_{\leq-1}\nu X\in\mathcal{X}^{\prime}$, and
$Y^{\prime}\in\prescript{\perp}{}{\mathcal{X}^{\prime}}$, we get that
$Y\in\mathcal{Y}$. Thus, $(\mathcal{Y}^{\prime},\mathcal{X}^{\prime})$ is a
torsion pair in $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

Now, suppose that $\mathcal{Y}$ is hereditary, which is equivalent to
$\mathcal{Y}$ being stable under cones (Lemma 3.13). Let $g:Z\to
Z^{\prime}\in\mathcal{Y}^{\prime}$. Then there exists $f:Y\to
Y^{\prime}\in\mathcal{Y}$ such that $\tau_{\geq-d+2}f\cong g$. Using Lemma
3.9, we can assume that
$C(f)\in\operatorname{K}^{[-d+1]}(\operatorname{proj}\Lambda)$. Since
$\mathcal{Y}$ is closed under cones, $C(f)\in\mathcal{Y}$. Using Lemma 3.7,
$\tau_{\geq-d+2}C(g)\cong\tau_{\geq-d+2}C(f)\in\mathcal{Y}^{\prime}$. Using
Lemma 3.16, we conclude that $\mathcal{Y}^{\prime}$ is a positive torsion
class.

Next, if $\mathcal{Y}$ is complete, then it is covariantly finite in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. This implies that
$\mathcal{Y}^{\prime}$ is covariantly finite in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ (Lemma 3.8).

Finally, suppose $\mathcal{Y}$ is hereditary and contravariantly finite. Then
$\mathcal{Y}^{\prime}$ is positive and contravariantly finite in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Let
$Z\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ and $f:Y^{\prime}\to
Z$ a minimal right $\mathcal{Y}^{\prime}$-approximation of $Z$. Consider the
triangles $Y^{\prime}\xrightarrow{f}Z\xrightarrow{g}C(f)$ and
$\tau_{\leq-d+1}C(f)\to C(f)\xrightarrow{\pi}\tau_{\geq-d+2}C(f)$ in
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$. Then $\mathrm{H}^{0}(g)$ is an
epimorphism and $\mathrm{H}^{0}(\pi)$ is an isomorphism, which implies that
$\mathrm{H}^{0}(\pi\circ g)$ is an epimorphism. Thus, the cocone of $\pi\circ
g$, say $Y$, is in $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$, and we
get a conflation $Y\xrightarrow{f^{\prime}}Z\xrightarrow{\pi\circ
g}\tau_{\geq-d+2}C(f)$ in $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.
We will show that $Y\in\mathcal{Y}^{\prime}$.

Using the octahedral axiom for the triangles shown below,

${\Sigma Y}$${\Sigma
Y^{\prime}}$${\Sigma\tau_{\leq-d+1}C(f)}$${Z}$${\tau_{\geq-d+2}C(f)}$${C(f)}$$\scriptstyle{f^{\prime}[1]}$$\scriptstyle{[1]}$$\scriptstyle{\Sigma
h}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{g}$$\scriptstyle{\pi\circ
g}$$\scriptstyle{\pi}$

we get a triangle $Y^{\prime}\xrightarrow{h}Y\to\tau_{\leq-d+1}C(f)$. Let
$F\in\mathcal{X}^{\prime}$. Then we have an exact sequence
$\mathrm{Hom}(\tau_{\leq-d+1}C(f),F)\to\mathrm{Hom}(Y,F)\to\mathrm{Hom}(Y^{\prime},F)=0$.
Since $F\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$,
$\mathrm{Hom}(\tau_{\leq-d+1}C(f),F)=0$. Thus, $\mathrm{Hom}(Y,F)=0$ and
$Y\in\prescript{\perp}{}{\mathcal{X}^{\prime}}=\mathcal{Y}^{\prime}$.

Thus, there exists a map $h^{\prime}:Y\to Y^{\prime}$ such that
$fh^{\prime}=f^{\prime}$, which implies that $fh^{\prime}h=f^{\prime}h=f$.
Since $f$ is minimal, we get that $f^{\prime}:Y\to
Z\cong[0,f]:Y^{\prime\prime}\oplus Y^{\prime}\to Z$ for some
$Y^{\prime\prime}\in\mathcal{D}^{b}(\operatorname{mod}\Lambda)$. This implies
that $C(f^{\prime})\cong\tau_{\geq-d+2}C(f)\cong C(f)\oplus\Sigma
Y^{\prime\prime}$. This gives that
$C(f)\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Using Lemma 3.18
for the conflation $Y^{\prime}\xrightarrow{f}Z\xrightarrow{g}C(f)$ in the
extriangulated category $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$,
we get that $C(f)\in\mathcal{X}^{\prime}$. Hence,
$(\mathcal{Y}^{\prime},\mathcal{X}^{\prime})$ is an $s$-torsion pair in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. ∎

The above theorem gives us a poset homomorphism
$\Phi:\operatorname{cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\to\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

###  3.3. The bijections

We now show that the above map $\Phi$ is a poset isomorphism.

######  Theorem 3.20

The map
$\Phi:\operatorname{cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\to\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$
defined as $\mathcal{Y}\mapsto\tau_{\geq-d+2}\mathcal{Y}$ is a poset
isomorphism. Moreover, it restricts to an isomorphism between the following
subposets.

  1. (1)

The set of hereditary cotorsion classes and the set of positive torsion
classes.

  2. (2)

The set of complete cotorsion classes and the set of covariantly finite
torsion classes.

  3. (3)

The set of hereditary contravariantly finite cotorsion classes and the set of
$s$-torsion classes.

###### Proof.

We claim that the poset homomorphism

$\begin{split}\Psi:\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)&\to\operatorname{cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\\\
\mathcal{T}&\mapsto\\{Y\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\tau_{\geq-d+2}Y\in\mathcal{T}\\}\end{split}$

is the inverse of $\Phi$. Let
$\mathcal{T}\in\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.
Set
$\mathcal{Y}=\\{Y\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\tau_{\geq-d+2}Y\in\mathcal{T}\\}$,
and $\mathcal{X}:=\prescript{\perp_{1}}{}{\mathcal{Y}}$. Step 1. The map
$\Psi$ is well-defined: For this, we need to prove that
$\mathcal{X}^{\perp_{1}}=\mathcal{Y}$. By definition,
$\mathcal{Y}\subseteq\mathcal{X}^{\perp_{1}}$. Using Lemma 3.6, we know that
$\mathcal{X}=\\{X\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\Sigma^{-1}\tau_{\leq-1}\nu
X\in\mathcal{T}^{\perp}\\}$, and
$\mathcal{X}^{\perp_{1}}=\\{Z\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\mid\mathrm{Hom}(\tau_{\geq-d+2}Z,\Sigma^{-1}\tau_{\leq-1}\nu\mathcal{X})=0\\}$.
Let $Y\in\mathcal{X}^{\perp_{1}}$. In order to show that $Y\in\mathcal{Y}$, we
need to show that
$\tau_{\geq-d+2}Y\in\mathcal{T}=\prescript{\perp}{}{(\mathcal{T}^{\perp})}$.
Let $F\in\mathcal{T}^{\perp}$. Then $\Sigma
F\in\mathcal{D}^{[-d+1,-1]}(\operatorname{mod}\Lambda)$. Using Corollary 3.4,
there exists $I\in\operatorname{K}^{[-d+2,1]}(\operatorname{inj}\Lambda)$ such
that $\tau_{\leq 0}I\cong F$. Thus, $\tau_{\leq-1}\Sigma I\cong\Sigma F$. This
implies that $\Sigma^{-1}\tau_{\leq-1}\nu(\nu^{-1}\Sigma
I)=F\in\mathcal{T}^{\perp}$. Thus, $\nu^{-1}\Sigma I\in\mathcal{X}$. Since
$\mathcal{Y}\in\mathcal{X}^{\perp_{1}}$,
$\mathrm{Hom}(\tau_{\geq-d+2}Y,\Sigma^{-1}\tau_{\leq-1}\nu(\nu^{-1}\Sigma
I))=\mathrm{Hom}(\tau_{\geq-d+2}Y,F)=0$. This implies that
$\tau_{\geq-d+2}Y\in\mathcal{T}$, and we are done. Step 2. The maps $\Phi$ and
$\Psi$ are mutual inverses: This follows from the fact that $\tau_{\geq-d+2}$
is full and essentially surjective with the kernel
$\operatorname{add}\Lambda[d-1]$ contained in any cotorsion class, and that
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ is a
$\mathrm{Hom}$-finite Krull-Schmidt category. Step 3. The map $\Psi$ sends
positive torsion classes to hereditary cotorsion classes: Suppose
$\mathcal{T}$ is positive. Then, using Lemma 3.16, for all morphisms
$g\in\mathcal{T}$, $\tau_{\geq-d+2}C(g)\in\mathcal{T}$. Let
$y\xrightarrow{f}y^{\prime}\to y^{\prime\prime}$ be a conflation in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ with
$y,y^{\prime}\in\mathcal{Y}$. By definition, $\tau_{\geq-d+2}y$,
$\tau_{\geq-d+2}y^{\prime}\in\mathcal{T}$. Moreover,
$\tau_{\geq-d+2}(C(\tau_{\geq-d+2}f))\in\mathcal{T}$. Thus, it follows from
Lemma 3.7 that
$\tau_{\geq-d+2}y^{\prime\prime}\cong\tau_{\geq-d+2}(C(\tau_{\geq-d+2}f))\in\mathcal{T}$
and $y^{\prime\prime}\in\mathcal{Y}$. This shows that $\mathcal{Y}$ is closed
under cones, and hence, by Lemma 3.13, $\mathcal{Y}$ is a hereditary cotorsion
class. Step 4. The map $\Psi$ sends covariantly finite torsion classes to
complete cotorsion classes: Suppose that $\mathcal{T}$ is covariantly finite.
Then using Lemma 3.8, $\mathcal{Y}$ is also covariantly finite. We now show
that $\mathcal{Y}$ is extension closed. Let $y\xrightarrow{f}y^{\prime}\to
y^{\prime\prime}$ be a conflation in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ with
$y,y^{\prime\prime}\in\mathcal{Y}$. By definition, this implies that
$\tau_{\geq-d+2}y$, $\tau_{\geq-d+2}y^{\prime\prime}\in\mathcal{T}$. Moreover,
using Lemma 3.7, we know that
$\tau_{\geq-d+2}(C(\tau_{\geq-d+2}f))\cong\tau_{\geq-d+2}y^{\prime\prime}$.
Using the triangle

$\tau_{\leq-d+1}(C(\tau_{\geq-d+2}f))\to
C(\tau_{\geq-d+2}f)\to\tau_{\geq-d+2}(C(\tau_{\geq-d+2}f)),$

we get that for all $F\in\mathcal{F}$,
$\mathrm{Hom}(C(\tau_{\geq-d+2}f),F)=0$. Thus, using the triangle

$\tau_{\geq-d+2}y\xrightarrow{\tau_{\geq-d+2}f}\tau_{\geq-d+2}y^{\prime}\to
C(\tau_{\geq-d+2}f),$

we get that $\mathrm{Hom}(\tau_{\geq-d+2}y^{\prime},F)=0$ for all
$F\in\mathcal{F}$. Thus,
$\tau_{\geq-d+2}y^{\prime}\in\prescript{\perp}{}{F}=\mathcal{T}$ and
$y^{\prime}\in\mathcal{Y}$.

Let $C\in\operatorname{K}^{[-d+2,0]}(\operatorname{proj}\Lambda)$. Let $f:C\to
Y$ be the minimal $\mathcal{Y}$-approximation of $C$. This gives a conflation
$C\xrightarrow{f}Y\to C(f)$ in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Since $\mathcal{Y}$
is closed under extensions using Lemma 3.18, we get that
$C(f)\in\prescript{\perp_{1}}{}{\mathcal{Y}}=\mathcal{X}$. Thus, for every
$C\in\operatorname{K}^{[-d+2,0]}(\operatorname{proj}\Lambda)$, there exists a
conflation $C\to Y\to C(f)$ with $Y\in\mathcal{Y}$ and $C(f)\in\mathcal{X}$.

Finally, let $C\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. We
have triangles

$\Sigma^{-1}t_{\leq-d+1}C\to t_{\geq-d+2}C\to C$

and

$\Sigma^{-1}t_{\leq-1}C\to t_{\geq 0}C\to C$

with
$\Sigma^{-1}t_{\leq-1}C,t_{\geq-d+2}C\in\operatorname{K}^{[-d+2,0]}(\operatorname{proj}\Lambda)$.
Using the above argument, we know that there exist conflations
$t_{\geq-d+2}C\to y\to x$ and $\Sigma^{-1}t_{\leq-1}C\to y^{\prime}\to
x^{\prime}$ in $\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ with
$y,y^{\prime}\in\mathcal{Y}$ and $x,x^{\prime}\in\mathcal{X}$. Using the
octahedral axiom for the following diagrams,

${\Sigma^{-1}x}$${\Sigma^{-1}x}$${\Sigma^{-1}t_{\leq-d+1}C}$${t_{\geq-d+2}C}$${C}$${t_{\leq-d+1}C}$${\Sigma^{-1}t_{\leq-d+1}C}$${y}$${d}$${t_{\leq-d+1}C}$${x}$${x}$${\Sigma^{-1}x^{\prime}}$${\Sigma^{-1}x^{\prime}}$${\Sigma^{-1}C}$${\Sigma^{-1}t_{\leq-1}C}$${t_{\geq
0}C}$${C}$${\Sigma^{-1}C}$${y^{\prime}}$${e}$${C}$${x^{\prime}}$${x^{\prime}}$

we get conflations $C\to d\to x$ and $y^{\prime}\to e\to C$. Since
$y,t_{\leq-d+1}C\in\mathcal{Y}$, and $\mathcal{Y}$ is closed under extensions,
we get that $d\in\mathcal{Y}$. Similarly, since $x^{\prime},t_{\geq
0}C\in\mathcal{X}$, and $\mathcal{X}$ is closed under extensions, we get that
$e\in\mathcal{X}$. Thus, using Remark 3.11$,(\mathcal{X},\mathcal{Y})$ is a
complete cotorsion pair. Step 5. The map $\Psi$ sends $s$-torsion classes to
hereditary contravariantly finite cotorsion classes: Suppose $\mathcal{T}$ is
an $s$-torsion class. Since an $s$-torsion class is positive and
contravariantly finite, by Step 3 and Lemma 3.8, $\mathcal{Y}$ is hereditary
and contravariantly finite in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. ∎

##  4\. Silting objects, cotorsion pairs, and torsion pairs

The main goal of this section is to prove the following theorem.

######  Theorem 4.1

The map $\Phi$ defined in Theorem 3.19 and the maps $\psi$ and $\psi^{\prime}$
given in Definition 4.2 and Definition 4.5 restrict to give the following
commutative triangle of poset isomorphisms.

(3)
${d\mbox{-}\operatorname{silt}\Lambda}$${\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)}$${\operatorname{(f,p)-tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)}$$\scriptstyle{\psi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\Phi}$

###  4.1. Definition of $\psi$

######  Definition 4.2

Let $M$ be a $d$-term silting object in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$. Define

$\displaystyle\mathcal{X}_{M}:=$ $\displaystyle\mbox{ the smallest full
subcategory of }\operatorname{K}^{b}(\operatorname{proj}\Lambda)\mbox{
containing }\\{\Sigma^{m}M\mid m\leq 0\\}$ and closed under extensions  and
summands, $\displaystyle\mathcal{Y}_{M}:=$ $\displaystyle\mbox{ the smallest
full subcategory of }\operatorname{K}^{b}(\operatorname{proj}\Lambda)\mbox{
containing }\\{\Sigma^{m}M\mid m\geq 0\\}$ and closed under extensions  and
summands.

Set
$\psi(M):=(\mathcal{X}_{M}\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda),\mathcal{Y}_{M}\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda))$.

Using [21, Theorem 6.1], we know that the map
$M\mapsto(\mathcal{X}_{M},\mathcal{Y}_{M})=:\Theta(M)$ gives a poset
isomorphism from
$\operatorname{silt}\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ to the
poset of bounded co-$t$-structures on
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$. Moreover, the inverse of
this map is given by taking the additive generator of the co-heart of a
bounded co-$t$-structure. Set
$\mathcal{X}^{\prime}_{M}:=\mathcal{X}_{M}\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$
and
$\mathcal{Y}^{\prime}_{M}=\mathcal{Y}_{M}\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.

######  Lemma 4.3

The map $\psi$ given in Definition 4.2 induces an injective poset homomorphism
$\psi:d\mbox{-}\operatorname{silt}\Lambda\to\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.

###### Proof.

We first show that $\psi$ is well-defined. Since
$(\mathcal{X}_{M},\mathcal{Y}_{M})$ is a co-$t$-structure on
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$,
$\mathbb{E}(\mathcal{X}^{\prime}_{M},\mathcal{Y}^{\prime}_{M})=0$.

Moreover, since $M$ is a $d$-term silting object, and $\operatorname{K}^{\leq
0}(\operatorname{proj}\Lambda)$ and
$\operatorname{K}^{\geq-d+1}(\operatorname{proj}\Lambda)$ are full
subcategories of $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ closed
under extensions and summands containing $\\{\Sigma^{m}M\mid m\geq 0\\}$ and
$\\{\Sigma^{m}M\mid m\leq 0\\}$ respectively, we get that
$\mathcal{Y}_{M}\subseteq\operatorname{K}^{\leq
0}(\operatorname{proj}\Lambda)$ and
$\mathcal{X}_{M}\subseteq\operatorname{K}^{\geq-d+1}(\operatorname{proj}\Lambda)$.

Let $C\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Then there
exists a triangle $T\xrightarrow{f}\Sigma C\xrightarrow{g}F$ with
$T\in\mathcal{X}_{M}$ and $F\in\Sigma\mathcal{Y}_{M}$. Thus,
$F\in\operatorname{K}^{\leq-1}(\operatorname{proj}\Lambda)$ and
$T\in\operatorname{K}^{\geq-d+1}(\operatorname{proj}\Lambda)$. Since
$\operatorname{K}^{\leq 0}(\operatorname{proj}\Lambda)$ is extension closed,
using the triangle $\Sigma^{-1}F\to T\to\Sigma C$, we get that
$T\in\operatorname{K}^{\leq 0}(\operatorname{proj}\Lambda)$. Similarly, using
the triangle $\Sigma C\to F\to\Sigma T$, we get that
$F\in\operatorname{K}^{\geq-d}(\operatorname{proj}\Lambda)$. Thus, we get a
triangle $C\to\Sigma^{-1}F\to T$ with
$\Sigma^{-1}F\in\mathcal{Y}^{\prime}_{M}$ and $T\in\mathcal{X}^{\prime}_{M}$.

Let $C\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Then there
exists a triangle $T\xrightarrow{f}C\xrightarrow{g}F$ with
$T\in\mathcal{X}_{M}$ and $F\in\Sigma\mathcal{Y}_{M}$. Thus,
$F\in\operatorname{K}^{\leq-1}(\operatorname{proj}\Lambda)$ and
$T\in\operatorname{K}^{\geq-d+1}(\operatorname{proj}\Lambda)$. Since
$\operatorname{K}^{\geq-d}(\operatorname{proj}\Lambda)$ is extension closed,
using the triangle $X\to F\to\Sigma T$, we get that
$F\in\operatorname{K}^{\geq-d}(\operatorname{proj}\Lambda)$. Thus,
$\Sigma^{-1}F\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.
Similarly, using the triangle $\Sigma^{-1}F\to T\to C$, we get that
$T\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Thus, we get a
triangle $\Sigma^{-1}F\to T\to C$ with
$\Sigma^{-1}F\in\mathcal{Y}^{\prime}_{M}$ and $T\in\mathcal{X}^{\prime}_{M}$.

Thus, using Remark 3.11$,(\mathcal{X}^{\prime}_{M},\mathcal{Y}^{\prime}_{M})$
is a complete cotorsion pair in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Since
$\mathcal{X}_{M}$ and $\mathcal{Y}_{M}$ are closed under cocones and cones
respectively, so are $\mathcal{X}^{\prime}_{M}$ and
$\mathcal{Y}^{\prime}_{M}$, which implies that
$(\mathcal{X}^{\prime}_{M},\mathcal{Y}^{\prime}_{M})$ is hereditary by Lemma
3.13.

Now, suppose $M\leq N$ in $d$-$\operatorname{silt}\Lambda$. Then
$\mathcal{X}_{M}\supseteq\mathcal{X}_{N}$ which implies that
$\mathcal{X}^{\prime}_{M}\supseteq\mathcal{X}^{\prime}_{N}$. Thus, $\psi$ is a
poset homomorphism.

We finally show that $\psi$ is injective. Let $M,M^{\prime}\in
d\mbox{-}\operatorname{silt}\Lambda$ such that
$\mathcal{Y}^{\prime}_{M}=\mathcal{Y}^{\prime}_{M^{\prime}}$. Using the
bijection between silting objects and co-$t$-structures [21, Theorem 6.1], it
is enough to show that $\mathcal{Y}_{M}=\mathcal{Y}_{M^{\prime}}$. Suppose
not. Without loss of generality, let
$Y\in\mathcal{Y}_{M}\setminus\mathcal{Y}_{M^{\prime}}$. Since
$\mathcal{Y}_{M}\subseteq\operatorname{K}^{\leq
0}(\operatorname{proj}\Lambda)$, $Y\in\operatorname{K}^{\leq
0}(\operatorname{proj}\Lambda)$. We have a triangle $\Sigma^{-1}t_{\leq-d}Y\to
t_{\geq-d+1}Y\to Y$. Since
$\mathcal{X}_{M},\mathcal{X}_{M^{\prime}}\subseteq\operatorname{K}^{\geq-d+1}(\operatorname{proj}\Lambda)$,
we get that
$\Sigma^{-1}t_{\leq-d}Y\in\mathcal{X}_{M}^{\perp_{1}}\cap\mathcal{X}_{M^{\prime}}^{\perp_{1}}=\mathcal{Y}_{M}\cap\mathcal{Y}_{M^{\prime}}$.
Thus, $t_{\geq-d+1}Y\in\mathcal{Y}_{M}$, since $\mathcal{Y}_{M}$ is closed
under extensions. Since
$t_{\geq-d+1}Y\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, we
get that
$t_{\geq-d+1}Y\in\mathcal{Y}^{\prime}_{M}=\mathcal{Y}^{\prime}_{M^{\prime}}$.
Since $\mathcal{Y}_{M^{\prime}}$ is closed under cones, we get that
$Y\in\mathcal{Y}_{M^{\prime}}$, a contradiction. Hence
$\mathcal{Y}_{M}=\mathcal{Y}_{M^{\prime}}$ which implies that $M\cong
M^{\prime}$.

∎

######  Theorem 4.4

The map $\psi$ is an isomorphism of posets.

###### Proof.

We will show this by constructing an inverse poset homomorphism
$\chi:\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\to
d\mbox{-}\operatorname{silt}\Lambda$ of $\psi$.

Let $\mathcal{Y}^{\prime}$ be a complete hereditary cotorsion class in
$\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$. Set
$\mathcal{X}^{\prime}=\prescript{\perp_{1}}{}{\mathcal{Y}^{\prime}}$. Define
$C(\mathcal{X}^{\prime})$ to be the smallest full subcategory of
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ closed under summands and
extensions containing $\mathcal{X}^{\prime}$ and $\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda)$ and $C(\mathcal{Y}^{\prime})$ to be the
smallest full subcategory of
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ closed under summands and
extensions containing $\mathcal{Y}^{\prime}$ and
$\operatorname{K}^{\leq-d+1}(\operatorname{proj}\Lambda)$. Note that
$C(\mathcal{X}^{\prime})\subseteq\operatorname{K}^{\geq-d+1}(\operatorname{proj}\Lambda)$
and $C(\mathcal{Y}^{\prime})\subseteq\operatorname{K}^{\leq
0}(\operatorname{proj}\Lambda)$. We will show that
$(C(\mathcal{X}^{\prime}),C(\mathcal{Y}^{\prime}))$ is a bounded
co-$t$-structure on $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ whose
co-heart lies in $\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$,
and set
$\chi(\mathcal{Y}^{\prime}):=\Theta^{-1}(C(\mathcal{X}^{\prime}),C(\mathcal{Y}^{\prime}))$.

Using the fact that both
$\mathbb{E}(\mathcal{X}^{\prime},\mathcal{Y}^{\prime})$ and
$\mathbb{E}(\mathcal{X}^{\prime},\operatorname{K}^{\leq-d+1}(\operatorname{proj}\Lambda))$
vanish, and that $\mathcal{X}^{\prime\perp_{1}}$ in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ is closed under extensions
and summands, we get that

$\mathbb{E}(\mathcal{X}^{\prime},C(\mathcal{Y}^{\prime}))=0.$

Similarly, $\mathbb{E}(\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda),C(\mathcal{Y}^{\prime}))=0$. Thus,
$\mathcal{X}^{\prime},\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda)\subseteq\prescript{\perp_{1}}{}{C(\mathcal{Y}^{\prime})}$.
Since $\prescript{\perp_{1}}{}{C(\mathcal{Y}^{\prime})}$ is also closed under
extensions and summands, we get that

$\mathbb{E}(C(\mathcal{X}^{\prime}),C(\mathcal{Y}^{\prime}))=0.$

We next show that $\Sigma^{-1}C(\mathcal{X}^{\prime})\subseteq
C(\mathcal{X}^{\prime})$. Note that it is enough to show that
$\Sigma^{-1}\mathcal{X}^{\prime}\subseteq C(\mathcal{X}^{\prime})$ and
$\Sigma^{-1}\operatorname{K}^{\geq 0}(\operatorname{proj}\Lambda)\subseteq
C(\mathcal{X}^{\prime})$. Clearly,

$\Sigma^{-1}\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda)=\operatorname{K}^{\geq
1}(\operatorname{proj}\Lambda)\subseteq\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda)\subseteq C(\mathcal{X}^{\prime}).$

Let $X\in\mathcal{X}^{\prime}$. We have the following triangle in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.

$t_{\geq 1}\Sigma^{-1}X\to\Sigma^{-1}X\to t_{\leq 0}\Sigma^{-1}X\to t_{\geq
0}X\to X$

Since $X\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, $t_{\geq
0}X\in\prescript{\perp_{1}}{}{\mathcal{Y}^{\prime}}=\mathcal{X}^{\prime}$.
Since $\mathcal{X}^{\prime}$ is closed under cocones, we get that $t_{\leq
0}\Sigma^{-1}X\in\mathcal{X}^{\prime}$. Since $\Sigma^{-1}X$ is an extension
of $t_{\geq 1}\Sigma^{-1}X\in\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda)$ and $t_{\leq
0}\Sigma^{-1}X\in\mathcal{X}^{\prime}$, we get that $\Sigma^{-1}X\in
C(\mathcal{X}^{\prime})$. Thus $\Sigma^{-1}C(\mathcal{X}^{\prime})\subseteq
C(\mathcal{X}^{\prime})$.

Finally, let $C\in\operatorname{K}^{b}(\operatorname{proj}\Lambda)$. Set
$C^{\prime}:=t_{\geq-d+1}C$ and $C^{\prime\prime}:=t_{\leq 0}C^{\prime}$.
Since $t_{\leq
0}t_{\geq-d+1}C\in\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$,
using the completeness of the cotorsion pair
$(\mathcal{X}^{\prime},\mathcal{Y}^{\prime})$, we get that there exists a
triangle $Y\to X\xrightarrow{f}t_{\leq 0}t_{\geq-d+1}C$ with
$Y\in\mathcal{Y}^{\prime}$ and $X\in\mathcal{X}^{\prime}$.

Now, using the octahedral axiom for the triangles $X\xrightarrow{f}t_{\leq
0}t_{\geq-d+1}C\to\Sigma Y$, $t_{\leq 0}t_{\geq-d+1}C=t_{\geq-d+1}t_{\leq
0}C\xrightarrow{j}t_{\leq 0}C\to t_{\leq-d}t_{\leq 0}C=t_{\leq-d}C$, and
$X\xrightarrow{j\circ f}t_{\leq 0}C\to C(j\circ f)$, we get a triangle $\Sigma
Y\to C(j\circ f)\to t_{\leq-d}C$. Since $\Sigma Y,t_{\leq-d}C\in\Sigma
C(\mathcal{Y}^{\prime})$, we get that $C(j\circ f)\in\Sigma
C(\mathcal{Y}^{\prime})$.

${C(j\circ f)}$${\Sigma Y}$${t_{\leq-d}C}$${X}$${t_{\leq 0}C}$${t_{\leq
0}t_{\geq-d+1}C}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{f}$$\scriptstyle{j\circ
f}$$\scriptstyle{l^{\prime}}$$\scriptstyle{j}$

Finally, using the octahedral axiom for the triangles $C\xrightarrow{l}t_{\leq
0}C\to\Sigma t_{\geq 1}C$, $t_{\leq 0}C\xrightarrow{l^{\prime}}C(j\circ
f)\to\Sigma X$, and $C\xrightarrow{l^{\prime}\circ l}C(j\circ f)\to
C(l^{\prime}\circ l)$, we get a triangle $\Sigma t_{\geq 1}C^{\prime}\to
C(l^{\prime}\circ l)\to\Sigma X$.

${C(l^{\prime}\circ l)}$${\Sigma t_{\geq 1}C}$${\Sigma X}$${C}$${C(j\circ
f)}$${t_{\leq
0}C}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{[1]}$$\scriptstyle{l}$$\scriptstyle{l^{\prime}\circ
l}$$\scriptstyle{l^{\prime}}$

Since $\Sigma t_{\geq 1}C,\Sigma X\in\Sigma C(\mathcal{X}^{\prime})$, we get
that $C(l^{\prime}\circ l)\in\Sigma C(\mathcal{X}^{\prime})$. Thus, we have a
triangle $\Sigma^{-1}C(l^{\prime}\circ l)\to C\to C(j\circ f)$ with
$\Sigma^{-1}C(l^{\prime}\circ l)\in C(\mathcal{X}^{\prime})$ and $C(j\circ
f)\in\Sigma C(\mathcal{Y}^{\prime})$. Therefore,
$(C(\mathcal{X}^{\prime}),C(\mathcal{Y}^{\prime}))$ is a co-$t$-structure.
Moreover, since $\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda)\subseteq C(\mathcal{X}^{\prime})$, we get that
$\bigcup_{i\in\mathbb{Z}}\Sigma^{i}C(\mathcal{X}^{\prime})\supseteq\bigcup_{i\in\mathbb{Z}}\Sigma^{i}\operatorname{K}^{\geq
0}(\operatorname{proj}\Lambda)=\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.
Similarly,
$\bigcup_{i\in\mathbb{Z}}\Sigma^{i}C(\mathcal{Y}^{\prime})=\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.
Hence, $(C(\mathcal{X}^{\prime}),C(\mathcal{Y}^{\prime}))$ is a bounded
co-$t$-structure on $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$.
Moreover, the co-heart $C(\mathcal{X}^{\prime})\cap C(\mathcal{Y}^{\prime})$
is contained in $\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$, and
hence its additive generator is a $d$-term silting object.

Note that if
$\mathcal{Y}\subseteq\mathcal{Y}^{\prime}\in\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$,
then $C(\mathcal{Y})\subseteq C(\mathcal{Y}^{\prime})$, which implies that
$\Theta^{-1}(C(\prescript{\perp_{1}}{}{\mathcal{Y}}),C(\mathcal{Y}))\leq\Theta^{-1}(C(\prescript{\perp_{1}}{}{\mathcal{Y}^{\prime}}),C(\mathcal{Y}^{\prime}))$.
Thus, $\chi$ is a poset homomorphism.

Since $\mathcal{X}^{\prime}\subseteq
C(\mathcal{X}^{\prime})\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$
and $\mathcal{Y}^{\prime}\subseteq
C(\mathcal{Y}^{\prime})\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$,
we get that $\mathcal{Y}^{\prime}=(\mathcal{X}^{\prime})^{\perp_{1}}\supseteq
C(\mathcal{X}^{\prime})^{\perp_{1}}\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)=C(\mathcal{Y}^{\prime})\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.
The last equality follows from the fact that for a co-$t$-structure
$(\mathcal{A},\mathcal{B})$, $\mathcal{A}^{\perp_{1}}=\mathcal{B}$ and
$\mathcal{A}=\prescript{\perp_{1}}{}{\mathcal{B}}$ [26, Proposition 1.6].
Thus,
$\mathcal{Y}^{\prime}=C(\mathcal{Y}^{\prime})\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.
Similarly,
$\mathcal{X}^{\prime}=C(\mathcal{X}^{\prime})\cap\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$.
Thus, $\psi\circ\chi=\mbox{Id}$. This implies that
$\psi\circ\chi\circ\psi=\psi$. Since $\psi$ is injective by Lemma 4.3, we get
that $\chi\circ\psi=\mbox{Id}$. Thus, $\chi$ is the inverse of $\psi$. ∎

###  4.2. Definition of $\psi^{\prime}$

######  Definition 4.5

Let $M$ be a $d$-term silting object in
$\operatorname{K}^{b}(\operatorname{proj}\Lambda)$. Define

$\displaystyle\mathcal{U}_{M}$
$\displaystyle=\\{N\in\mathcal{D}^{b}(\operatorname{mod}\Lambda)\mid\mathrm{Hom}(M,\Sigma^{m}N)=0,\
\forall\ m>0\\},$ $\displaystyle\mathcal{V}_{M}$
$\displaystyle=\\{N\in\mathcal{D}^{b}(\operatorname{mod}\Lambda)\mid\mathrm{Hom}(M,\Sigma^{m}N)=0,\
\forall\ m<0\\}.$

Set
$\psi^{\prime}(M):=(\mathcal{U}_{M}\cap\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda),\Sigma^{-1}\mathcal{V}_{M}\cap\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda))$.

Using [21, Theorem 6.1] we know that the poset of equivalence classes of
silting objects in $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ is
isomorphic to the poset of bounded $t$-structures with length heart on
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$ under the map
$M\mapsto(\mathcal{U}_{M},\Sigma^{-1}\mathcal{V}_{M})$. Set
$\mathcal{U}^{\prime}_{M}:=\mathcal{U}_{M}\cap\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$
and
$\mathcal{V}^{\prime}_{M}=\Sigma^{-1}\mathcal{V}_{M}\cap\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

######  Lemma 4.6

The map
$\psi^{\prime}:d\mbox{-}\operatorname{silt}\Lambda\to\operatorname{p-tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$
given by $M\mapsto\mathcal{U}^{\prime}_{M}$ is a poset homomorphism, with the
image contained in the set of $s$-torsion classes. In particular, the image is
contained in the set of contravariantly finite positive torsion classes.

###### Proof.

We start by showing that for $M\in d\mbox{-}\operatorname{silt}\Lambda$,
$(\mathcal{U}^{\prime}_{M},\mathcal{V}^{\prime}_{M})$ is an $s$-torsion pair
in $\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

Since $(\mathcal{U}_{M},\Sigma^{-1}\mathcal{V}_{M})$ is a $t$-structure in
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$,
$\mathrm{Hom}(\mathcal{U}^{\prime}_{M},\mathcal{V}^{\prime}_{M})=0$. Moreover,
since $\mathcal{U}_{M}$ is closed under positive shifts,
$\mathbb{E}^{-1}(\mathcal{U}^{\prime}_{M},\mathcal{V}^{\prime}_{M})=0$.

In order to prove condition (Sc), we claim that
$\mathcal{D}^{\leq(-d+1)}(\operatorname{mod}\Lambda)\subseteq\mathcal{U}_{M}$.
This is because if $N\in\mathcal{D}^{\leq(-d+1)}(\operatorname{mod}\Lambda)$,
then $N\cong\tau^{\leq-d+1}N$, which implies that

$\begin{split}\mathrm{Hom}_{\mathcal{D}^{b}(\operatorname{mod}\Lambda)}(\Sigma^{-m}M,N)&\cong\mathrm{Hom}_{\mathcal{D}^{b}(\operatorname{mod}\Lambda)}(\Sigma^{-m}M,\tau^{\leq-d+1}N)\\\
&\cong\mathrm{Hom}_{K(\operatorname{mod}\Lambda)}(\Sigma^{-m}M,\tau^{\leq-d+1}N)\\\
&=0\end{split}$

for all $m>0$, since $M$ is a complex of projectives concentrated in
$[-d+1,0]$. Dually, we have that $\mathcal{D}^{\geq
1}(\operatorname{mod}\Lambda)\subseteq\Sigma^{-1}\mathcal{V}_{M}$.

Finally, let $Z\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Since
$(\mathcal{U}_{M},\Sigma^{-1}\mathcal{V}_{M})$ is a $t$-structure in
$\mathcal{D}^{b}(\operatorname{mod}\Lambda)$, there exists a triangle

$U\xrightarrow{u}Z\xrightarrow{v}V$

with $U\in\mathcal{U}_{M}$ and $V\in\Sigma^{-1}\mathcal{V}_{M}$. Note that we
have a triangle

$\tau^{\leq 0}U\to U\to\tau^{\geq 1}U.$

Using the previous paragraph, we know that $\tau^{\geq
1}U\in\Sigma^{-1}\mathcal{V}_{M}$, which implies that the map from
$U\to\tau^{\geq 1}U$ is $0$. Thus, $\tau^{\leq 0}U\cong
U\oplus\Sigma^{-1}\tau^{\geq 1}U$. This implies that $\tau^{\geq 1}U\cong 0$
and $\tau^{\leq 0}U\cong U$. Thus $U\in\mathcal{D}^{\leq
0}(\operatorname{mod}\Lambda)$. Using the triangle $Z\to V\to\Sigma U$ and the
fact that $\mathcal{D}^{\leq 0}(\operatorname{mod}\Lambda)$ is closed under
extensions, we get that $V\in\mathcal{D}^{\leq 0}(\operatorname{mod}\Lambda)$.
Similarly, we have a triangle

$\tau^{\leq-d+1}V\to V\to\tau^{\geq-d+2}V.$

Since $\tau^{\leq-d+1}V\in\mathcal{U}_{M}$, we get that the map from
$\tau^{\leq-d+1}V\to V$ is $0$, which implies that $\tau^{\geq-d+2}V\cong
V\oplus\Sigma\tau^{\leq-d+1}V$. This gives that $\tau^{\leq-d+1}V=0$ and
$\tau^{\geq-d+2}V\cong V$. Thus,
$V\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Using the triangle
$\Sigma^{-1}V\to U\to Z$ and the fact that
$\mathcal{D}^{\geq-d+2}(\operatorname{mod}\Lambda)$ is closed under
extensions, we get that
$U\in\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$. Hence,
$U\in\mathcal{U}^{\prime}_{M}$ and $V\in\mathcal{V}^{\prime}_{M}$. Thus,
$(\mathcal{U}_{M}^{\prime},\mathcal{V}_{M}^{\prime})$ is an $s$-torsion pair.
Using [1] and Corollary 3.17 we know that $s$-torsion classes are positive and
contravariantly finite. Thus, the image of $\psi^{\prime}$ is contained in the
set of contravariantly finite positive torsion classes.

Now, suppose $M\leq N$. Since $M\mapsto(\mathcal{U}_{M},\Sigma^{-1}V_{M})$ is
a map of posets, $\mathcal{U}_{M}\subseteq\mathcal{U}_{N}$. And hence
$\mathcal{U}^{\prime}_{M}\subseteq\mathcal{U}^{\prime}_{N}$. Thus,
$\psi^{\prime}$ is a poset homomorphism. ∎

We now have the following triangle of maps.

(4)
${d\mbox{-}\operatorname{silt}\Lambda}$${\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)}$${\operatorname{p-tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)}$$\scriptstyle{\psi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\Phi}$

######  Proposition 4.7

The above triangle commutes.

###### Proof.

We need to show that for all $M\in d\mbox{-}\operatorname{silt}\Lambda$,
$\mathcal{U}_{M}^{\prime}=\tau_{\geq-d+2}\mathcal{Y}_{M}^{\prime}$.

Let $N\in\mathcal{U}_{M}^{\prime}$ and $\cdots\to N^{-2}\to N^{-1}\to N^{0}\to
0\in\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ quasi-isomorphic to $N$.
Then $\tau_{\geq-d+2}(N^{-d+1}\to\cdots\to N^{0})\cong N$. We claim that
$N^{-d+1}\to\cdots\to N^{0}\in\mathcal{Y}_{M}$.

Consider the triangle $t_{\geq-d+1}N\to N\to t_{\leq-d}N$. Let $j\leq 0$. Then
$\Sigma^{j}M\in\mathcal{X}_{M}$. We have the following exact sequence.

$\mathrm{Hom}(\Sigma^{j}M,t_{\leq-d}N)\to\mathbb{E}(\Sigma^{j}M,t_{\geq-d+1}N)\to\mathbb{E}(\Sigma^{j}M,N)$

Since $N\in\mathcal{U}_{M}^{\prime}$,
$\mathbb{E}(\Sigma^{j}M,N)\cong\mathrm{Hom}(M,\Sigma^{-j+1}N)=0$. Moreover,
since $M$ is concentrated in degrees $-d+1,\cdots,0$,
$\mathrm{Hom}(\Sigma^{j}M,t_{\leq-d}N)=0$. Therefore,

$\mathbb{E}(\Sigma^{j}M,t_{\geq-d+1}N)=0.$

This implies that $\Sigma^{j}M\in\prescript{\perp_{1}}{}{(t_{\geq-d+1}N)}$ for
all $j\leq 0$. Since, $\prescript{\perp_{1}}{}{(t_{\geq-d+1}N)}$ is closed
under extensions and summands, we get that
$\mathcal{X}_{M}\subseteq\prescript{\perp_{1}}{}{(t_{\geq-d+1}N)}$. Thus,
$t_{\geq-d+1}N\in\mathcal{X}_{M}^{\perp_{1}}=\mathcal{Y}_{M}$. Thus,
$\mathcal{U}_{M}^{\prime}\subseteq\tau_{\geq-d+2}\mathcal{Y}_{M}^{\prime}$.

For the other inclusion, we first claim that
$\mathcal{Y}_{M}^{\prime}=t_{\geq-d+1}\mathcal{Y}_{M}$. It is enough to show
that $\mathcal{Y}_{M}$ is closed under $t_{\geq-d+1}$. Let
$Y\in\mathcal{Y}_{M}$. Consider the triangle $t_{\geq-d+1}Y\to Y\to
t_{\leq-d}Y$. Since $\mathcal{X}_{M}$ is contained in
$\operatorname{K}^{\geq-d+1}(\operatorname{proj}\Lambda)$, we get that for all
$X\in\mathcal{X}_{M}$, $\mathbb{E}(X,t_{\geq-d+1}Y)=0$, which implies that
$t_{\geq-d+1}Y\in\mathcal{X}_{M}^{\perp_{1}}=\mathcal{Y}_{M}$.

Thus,
$\tau_{\geq-d+2}\mathcal{Y}_{M}^{\prime}=\tau_{\geq-d+2}t_{\geq-d+1}\mathcal{Y}_{M}=\tau_{\geq-d+2}\mathcal{Y}_{M}$.

Let $m>0$ and $j\geq 0$. Using the triangle
$\tau_{\leq-d+1}\Sigma^{j}M\to\Sigma^{j}M\to\tau_{\geq-d+1}\Sigma^{j}M$, we
get the exact sequence

$\mathrm{Hom}(\Sigma^{-m}M,\Sigma^{j}M)\to\mathrm{Hom}(\Sigma^{-m}M,\tau_{\geq-d+1}\Sigma^{j}M)\to\mathbb{E}(\Sigma^{-m}M,\tau_{\leq-d+1}\Sigma^{j}M).$

Since $M$ is a silting object, $\mathrm{Hom}(\Sigma^{-m}M,\Sigma^{j}M)=0$.
Moreover, since $M$ is concentrated in degrees $-d+1,\cdots,0$ as a complex of
projectives, $\mathbb{E}(\Sigma^{-m}M,\tau_{\leq-d+1}\Sigma^{j}M)=0$. Thus,
$\mathrm{Hom}(M,\Sigma^{m}\tau_{\geq-d+1}\Sigma^{j}M)=0$ for all $m>0$, and
$\tau_{\geq-d+1}\Sigma^{j}M\in\mathcal{U}_{M}^{\prime}$ for all $j\geq 0$.

Finally, to show that
$\tau_{\geq-d+2}\mathcal{Y}_{M}\subseteq\mathcal{U}_{M}^{\prime}$, it is
enough to show that the category

$S:=\\{X\in\operatorname{K}^{b}(\operatorname{proj}\Lambda)\mid\mathrm{Hom}(M,\Sigma^{m}\tau_{\geq-d+2}X)=0\
\forall\ m>0\\}$

is closed under extensions and summands.

Clearly, it is closed under summands. Let $X\xrightarrow{f}Z\to Y$ be a
triangle in $\operatorname{K}^{b}(\operatorname{proj}\Lambda)$ with $X,Y\in
S$. Set $C=C(\tau_{\geq-d+2}f$. Then $\tau_{\geq-d+2}C\cong\tau_{\geq-d+2}Y$.
Let $m>0$. Using the triangle $\tau_{\leq-d+1}C\to C\to\tau_{\geq-d+2}C$, we
get the exact sequence

$\mathrm{Hom}(M,\Sigma^{m}\tau_{\leq-d+1}C)\to\mathrm{Hom}(M,\Sigma^{m}C)\to\mathrm{Hom}(M,\Sigma^{m}\tau_{\geq-d+2}C),$

such that
$\mathrm{Hom}(M,\Sigma^{m}\tau_{\leq-d+1}C)=\mathrm{Hom}(M,\Sigma^{m}\tau_{\geq-d+2}C)=0$.
Thus,

$\mathrm{Hom}(\Sigma^{-m}M,C)\cong\mathrm{Hom}(M,\Sigma^{m}C)=0.$

Using the triangle $\tau_{\geq-d+2}X\to\tau_{\geq-d+2}Z\to C$ and the fact
that $(\Sigma^{-m}M)^{\perp}$ is closed under extensions, we get that
$\mathrm{Hom}(M,\Sigma^{m}\tau_{\geq-d+2}Z)=0$ for all $m>0$. Thus, $Z\in S$.
∎

###  4.3. Proof of Theorem 4.1

###### Proof.

Using Lemma 4.6, we know that the image of $\psi^{\prime}$ is contained in the
class of contravariantly finite torsion pairs. Using Proposition 4.7, this
image is also contained in the image of $\Phi$, which by Theorem 3.19 is
contained in the class of covariantly finite torsion pairs. Thus, we get that
the following commutative triangle is well-defined.

${d\mbox{-}\operatorname{silt}\Lambda}$${\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)}$${\operatorname{(f,p)-tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)}$$\scriptstyle{\psi}$$\scriptstyle{\psi^{\prime}}$$\scriptstyle{\Phi}$

Using Theorem 3.20, we know that the inverse
$\Psi:\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)\to\operatorname{cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$
of $\Phi$ sends functorially finite torsion pairs to complete hereditary
cotorsion pairs. Hence,

$\Phi:\operatorname{(c,h)-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)\to\operatorname{(f,p)-tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$

is a poset isomorphism.

Thus, we conclude that all the maps in triangle (3) are isomorphisms of
posets. ∎

######  Proposition 4.8

The class of functorially finite positive torsion classes in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ coincides with the class
of functorially finite $s$-torsion classes in
$\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$.

###### Proof.

Suppose $\mathcal{T}$ is a functorially finite positive torsion class. Then,
using Theorem 3.20, we get that $\Phi^{-1}(\mathcal{T})$ is a contravariantly
finite hereditary cotorsion class. The same theorem then implies that
$\mathcal{T}$ is a functorially finite $s$-torsion class. On the other hand,
the class of functorially finite $s$-torsion classes is contained in the class
of functorially finite positive torsion classes by Corollary 3.17. ∎

###  4.4. Lattices

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category.

######  Theorem 4.9

The poset of torsion pairs in $\mathcal{C}$ is a lattice.

###### Proof.

Let $(\mathcal{T}_{1},\mathcal{F}_{1})$, $(\mathcal{T}_{2},\mathcal{F}_{2})$
be two torsion pairs. To show that their meet exists, it is enough to show
that
$(\mathcal{T}_{1}\cap\mathcal{T}_{2},(\mathcal{T}_{1}\cap\mathcal{T}_{2})^{\perp})$
is a torsion pair, i.e.,
$\mathcal{T}_{1}\cap\mathcal{T}_{2}=\prescript{\perp}{}{((\mathcal{T}_{1}\cap\mathcal{T}_{2})^{\perp})}$.
By definition,
$\mathcal{T}_{1}\cap\mathcal{T}_{2}\subseteq\prescript{\perp}{}{((\mathcal{T}_{1}\cap\mathcal{T}_{2})^{\perp})}$.
On the other hand, for $i=1,2$, since
$\mathcal{T}_{1}\cap\mathcal{T}_{2}\subseteq\mathcal{T}_{i}$, we get that
$\prescript{\perp}{}{((\mathcal{T}_{1}\cap\mathcal{T}_{2})^{\perp})}\subseteq\prescript{\perp}{}{(\mathcal{T}_{i}^{\perp})}=\mathcal{T}_{i}$.
Thus,
$\prescript{\perp}{}{((\mathcal{T}_{1}\cap\mathcal{T}_{2})^{\perp})}\subseteq\mathcal{T}_{1}\cap\mathcal{T}_{2}$.
Hence
$\mathcal{T}_{1}\cap\mathcal{T}_{2}=\prescript{\perp}{}{((\mathcal{T}_{1}\cap\mathcal{T}_{2})^{\perp})}$.

Dually, the join of $(\mathcal{T}_{1},\mathcal{F}_{1})$,
$(\mathcal{T}_{2},\mathcal{F}_{2})$ is the torsion pair
$(\prescript{\perp}{}{(\mathcal{F}_{1}\cap\mathcal{F}_{2})},\mathcal{F}_{1}\cap\mathcal{F}_{2})$.
∎

A similar proof can be applied to obtain the corresponding result for
cotorsion pairs in $\mathcal{C}$.

######  Theorem 4.10

The poset of cotorsion pairs in $\mathcal{C}$ is a lattice.

###### Proof.

Let $(\mathcal{X}_{1},\mathcal{Y}_{1})$, $(\mathcal{X}_{2},\mathcal{Y}_{2})$
be two cotorsion pairs in $\mathcal{C}$. To show that their join exists, it is
enough to show that
$(\mathcal{X}_{1}\cap\mathcal{X}_{2},(\mathcal{X}_{1}\cap\mathcal{X}_{2})^{\perp_{1}})$
is a cotorsion pair, i.e.,
$\mathcal{X}_{1}\cap\mathcal{X}_{2}=\prescript{\perp_{1}}{}{((\mathcal{X}_{1}\cap\mathcal{X}_{2})^{\perp_{1}})}$.
By definition,
$\mathcal{X}_{1}\cap\mathcal{X}_{2}\subseteq\prescript{\perp_{1}}{}{((\mathcal{X}_{1}\cap\mathcal{X}_{2})^{\perp_{1}})}$.
On the other hand, for $i=1,2$, since
$\mathcal{X}_{1}\cap\mathcal{X}_{2}\subseteq\mathcal{X}_{i}$, we get that
$\prescript{\perp_{1}}{}{((\mathcal{X}_{1}\cap\mathcal{X}_{2})^{\perp_{1}})}\subseteq\prescript{\perp_{1}}{}{(\mathcal{X}_{i}^{\perp_{1}})}=\mathcal{X}_{i}$.
Thus,
$\prescript{\perp_{1}}{}{((\mathcal{X}_{1}\cap\mathcal{X}_{2})^{\perp_{1}})}\subseteq\mathcal{X}_{1}\cap\mathcal{X}_{2}$.
Hence
$\mathcal{X}_{1}\cap\mathcal{X}_{2}=\prescript{\perp_{1}}{}{((\mathcal{X}_{1}\cap\mathcal{X}_{2})^{\perp_{1}})}$.

Dually, the meet of $(\mathcal{X}_{1},\mathcal{Y}_{1})$,
$(\mathcal{X}_{2},\mathcal{Y}_{2})$ is the cotorsion pair
$(\prescript{\perp_{1}}{}{(\mathcal{Y}_{1}\cap\mathcal{Y}_{2})},\mathcal{Y}_{1}\cap\mathcal{Y}_{2})$.
∎

Moreover, in the case of
$\mathcal{C}=\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$, we get that
the subposet of positive torsion pairs in $\mathcal{C}$ is also a lattice.

######  Theorem 4.11

The subposet
$\operatorname{p-tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ of
$\operatorname{tors}\mathcal{D}^{[-d+2,0]}(\operatorname{mod}\Lambda)$ is
closed under meets and joins, and hence, a lattice.

###### Proof.

Using the characterization of positive torsion pairs in Lemma 3.16, we get
that positive torsion classes and positive torsion-free classes are stable
under intersections. ∎

######  Corollary 4.12

The subposet
$\operatorname{h-cotors}\operatorname{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$
is a lattice.

##  5\. Examples

######  Example 5.1

Let $Q=1\to 2$ and $\Lambda=kQ$. Then the Hasse diagram of the poset
$3$-$\operatorname{silt}\Lambda$ is shown below.

${P_{1}\oplus P_{2}}$${S_{1}\oplus P_{1}}$${\Sigma S_{1}\oplus
P_{1}}$${S_{1}\oplus\Sigma P_{2}}$${P_{2}\oplus\Sigma P_{1}}$${\Sigma
P_{2}\oplus\Sigma P_{1}}$${\Sigma^{2}P_{2}\oplus S_{1}}$${\Sigma
S_{1}\oplus\Sigma
P_{1}}$${P_{2}\oplus\Sigma^{2}P_{1}}$${\Sigma^{2}P_{2}\oplus\Sigma
S_{1}}$${\Sigma
P_{2}\oplus\Sigma^{2}P_{1}}$${\Sigma^{2}P_{2}\oplus\Sigma^{2}P_{1}}$

Since $\mathcal{D}^{[-1,0]}(\operatorname{mod}\Lambda)$ has finitely many
indecomposables, this is also the lattice of positive torsion classes in
$\mathcal{D}^{[-1,0]}(\operatorname{mod}\Lambda)$. This example also shows
that, unlike the case of (positive) torsion classes in a module category ([11,
Theorem 3.1]), the lattice of positive torsion classes in
$\mathcal{D}^{[-d,0]}(\operatorname{mod}\Lambda)$ is not necessarily
semidistributive.

######  Example 5.2

Let $Q$ be the quiver ${1}$${2}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$
and $I=\langle\alpha\beta,\beta\alpha\rangle$. Then the Hasse diagram of the
poset $3\mbox{-}\operatorname{silt}kQ/I$ is as follows.

${\begin{subarray}{c}0\to 0\to P_{1}\\\ \oplus\\\ 0\to 0\to
P_{2}\end{subarray}}$${\begin{subarray}{c}0\to 0\to P_{1}\\\ \oplus\\\ 0\to
P_{2}\to P_{2}\end{subarray}}$${\begin{subarray}{c}0\to P_{1}\to P_{2}\\\
\oplus\\\ 0\to 0\to P_{2}\end{subarray}}$${\begin{subarray}{c}P_{2}\to
P_{1}\to 0\\\ \oplus\\\ 0\to 0\to
P_{1}\end{subarray}}$${\begin{subarray}{c}0\to P_{2}\to 0\\\ \oplus\\\ 0\to
P_{2}\to P_{1}\end{subarray}}$${\begin{subarray}{c}0\to P_{1}\to 0\\\
\oplus\\\ 0\to P_{1}\to P_{2}\end{subarray}}$${\begin{subarray}{c}P_{1}\to
P_{2}\to 0\\\ \oplus\\\ 0\to 0\to
P_{2}\end{subarray}}$${\begin{subarray}{c}0\to P_{1}\to 0\\\ \oplus\\\ 0\to
P_{2}\to 0\end{subarray}}$${\begin{subarray}{c}0\to P_{1}\to 0\\\ \oplus\\\
P_{2}\to P_{1}\to 0\end{subarray}}$${\begin{subarray}{c}P_{2}\to 0\to 0\\\
\oplus\\\ 0\to P_{2}\to P_{1}\end{subarray}}$${\begin{subarray}{c}P_{1}\to
0\to 0\\\ \oplus\\\ 0\to P_{1}\to
P_{2}\end{subarray}}$${\begin{subarray}{c}0\to P_{2}\to 0\\\ \oplus\\\
P_{1}\to P_{2}\to 0\end{subarray}}$${\begin{subarray}{c}P_{2}\to 0\to 0\\\
\oplus\\\ P_{2}\to P_{1}\to 0\end{subarray}}$${\begin{subarray}{c}P_{1}\to
0\to 0\\\ \oplus\\\ P_{1}\to P_{2}\to
0\end{subarray}}$${\begin{subarray}{c}P_{1}\to 0\to 0\\\ \oplus\\\ P_{2}\to
0\to 0\end{subarray}}$

We also show that the class of $s$-torsion pairs can be strictly contained in
the class of positive torsion pairs.

######  Example 5.3

Let $Q=\leavevmode\hbox to51.22pt{\vbox
to16.27pt{\pgfpicture\makeatletter\hbox{\hskip 25.61107pt\lower-9.39287pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.61107pt}{-6.88193pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
6.80554pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${1}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
6.80554pt\hfil&\hfil\hskip 30.80551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${2}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
6.80554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.79999pt}{-1.97083pt}\pgfsys@lineto{11.40002pt}{-1.97083pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.6pt}{-1.97083pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-11.79999pt}{-6.79303pt}\pgfsys@lineto{11.40002pt}{-6.79303pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.6pt}{-6.79303pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ be the Kronecker
quiver. Let $\mathcal{P}$ and $\mathcal{I}$ denote the preprojective and the
postinjective component of the AR-quiver of $kQ$. The positive torsion classes
in $\mathcal{C}=\mathcal{D}^{[-1,0]}(\operatorname{mod}\Lambda)$ are given by
the additive subcategories generated by the following sets.

  1. (1)

Any final part of $\Sigma\mathcal{I}$.

  2. (2)

Shift of a subset of the tubes and $\Sigma\mathcal{I}$.

  3. (3)

Any final part of $\mathcal{I}\cup\Sigma\mathcal{P}$, the shift of all tubes,
and $\Sigma\mathcal{I}$.

  4. (4)

$\Sigma S_{2}$.

  5. (5)

$M$ and the final part of $\Sigma\mathcal{I}$ starting at $\Sigma M$,
$M\in\mathcal{I}$.

  6. (6)

Any subset of the tubes, the shift of this subset, and $\mathcal{I}$.

  7. (7)

Any subset of the tubes, shifts of all tubes, and
$\mathcal{I}\cup\Sigma\mathcal{P}\cup\Sigma\mathcal{I}$.

  8. (8)

Any final subset of $\mathcal{P}$, all tubes and their shifts,
$\Sigma\mathcal{P}\cup\mathcal{I}\cup\Sigma\mathcal{I}$.

  9. (9)

$S_{2},\Sigma S_{2}$.

  10. (10)

$M$ and the final part of $\Sigma\operatorname{mod}\Lambda$ starting at
$\Sigma M$, $M\in\mathcal{P}$.

The Hasse diagram of the lattice $\operatorname{p-tors}\mathcal{C}$ is
depicted in the following picture.

${\mathcal{C}}$${{\langle\mathrm{ind}\
\mathcal{C}\setminus[01]\rangle}}$${\vdots}$${\framebox{$\mathbb{P}(K\cup\infty)$}}$${{\langle[01],\overline{\Sigma[01]}\rangle}}$${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet\
\bullet\
\bullet}}$${\vdots}$${{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ\
\circ\
\circ}}$${{\langle[10],\overline{\Sigma[10]}\rangle}}$${\langle\Sigma\operatorname{mod}\Lambda\rangle}$${\framebox{$\mathbb{P}(K\cup\infty)$}}$${{\langle[01],\Sigma[01]\rangle}}$${\vdots}$${{\langle\Sigma[01]\rangle}}$${\framebox{$\mathbb{P}(K\cup\infty)$}}$${\vdots}$${{\langle\Sigma[21],\Sigma[10]\rangle}}$${{\langle\Sigma[10]\rangle}}$${0}$

Here the
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$
points are indexed by modules in the preprojective component, and the
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\circ$
points are indexed by modules in the postinjective component. For an object
$M\in\mathcal{P}\cup\mathcal{I}$, the notation $\overline{\Sigma M}$ is used
to denote the set of objects in the final part of the AR quiver starting at
$\Sigma M$. Finally, $\mathbb{P}(K\cup\infty)$ represents the boolean lattice
of subsets of $K\cup\infty$.

Note that the positive torsion class
$\mathcal{T}=\mathrm{add}\\{\mathcal{S}\cup\Sigma\mathcal{S}\cup\Sigma\mathcal{I}\\}$,
where $\mathcal{S}$ is a proper subset of the tubes, is not an $s$-torsion
class as the objects in $\mathcal{I}$ do not admit a right
$\mathcal{T}$-approximation.

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E. Gupta, Université Paris-Saclay, UVSQ, CNRS, Laboratoire de Mathématiques de
Versailles, 78000, Versailles, France.

E-mail address<EMAIL_ADDRESS>