YASUAKI OGAWA

Abstract. A model structure was originally introduced by Quillen as an abstraction of the category of topological spaces. However, it also plays an important role in the representation of algebras in connection with Grothendieck-Verdier’s derived category.

Afterwards, Hovey proved a one-to-one correspondence theorem between model struc- tures and cotorsion pairs in abelian categories. ˇSˇtov´ıˇcek provided an exact version of Hovey’s theorem which includes model structures whose homotopy categories are derived categories. In this talk, we focus an extriangulated version of Hovey’s theorem which was proved by Nakaoka-Palu. Finally, we will explain the theory behind Nakaoka-Palu’s approach, which serves a localization theory of extriangulated categories.

Contents

1. Introduction 1

2. Preliminary 3

2.1. Model categories 3

2.2. Extriangulated category 6

2.3. Cotorsion pairs 8

3. Model structures and Cotorsion pairs 9

3.1. Hovey’s correspondence theorem 9

3.2. Nakaoka-Palu’s correspondence theorem 11

4. Derived categories and model structures 13

4.1. Injective model structures 14

4.2. Lifting cotorsion pairs to categories of complexes 16

4.3. A tilting Quillen equivalence 18

5. Localization of extriangulated categories 18

5.1. Multiplicative systems compatible with the extriangulation 18

5.2. Localization via thick subcategories 19

5.3. The homotopy categories via extriangulated localizations 20

Acknowledgements 21

References 21

1. Introduction

The *Gabriel-Zisman localization* serves a foundation where one would like to consider
certain morphisms to be isomorphisms [GZ67]. It is deﬁned by formally inverting such
morphisms, e.g., *weak equivalences* and*quasi-isomorphisms*. But in this case, morphisms
in the obtained quotient category are rather diﬃcult to handle. Especially, given an

*Key words and phrases.* cotorsion pair, model structure, exact category, triangulated category, extrian-
gulated category.

2020*Mathematics Subject Classiﬁcation.* Primary 18E35; Secondary 18G80, 18E10.

1

additive category*C*, the quotient category*C*r*W*^{´1}swith respect to a class*W* of morphisms
is not necessarily additive. In this note, we focus on the two diﬀerent types of localization
which allow us to get a better description of morphisms.

(1) *Model structure*

——Model structures were introduced by Quillen as an abstraction of the category of topological spaces [Qui67, Qui69]. It forms the foundation of homotopy theory.

The*homotopy category* Hop*C*q of a model category*C*is the quotient category with
respect to the class of weak equivalences.

(2) Localization of*Extriangulated category*

——Grothendieck-Verdier introduced the derived category as the foundation of
the homological algebra, which is deﬁned to be a *Verdier quotient* of a *triangu-*
*lated category* [Ver67]. Nakaoka-Palu’s extriangulated category is a uniﬁcation of
triangulated categories and exact categories [NP19].

We shall present some relations between (1) and (2). We are particularly interested in
the *admissible* model strucutre which belongs to both (1) and (2). Basic examples of
admissible model structures can be found on the module category of a Frobenius algebra
and the category of (chain) complexes of modules.

(1) The module category Mod*A* over a Frobenius algebra *A* admits an admissible
model structure. Then, the homotopy category is equivalent to the (projectively)
stable categoryMod*A*.

(2) The category Cp*A*q of complexes over an algebra *A* admits an admissible model
structure. Then, the homotopy category is equivalent to the derived category
Dp*A*q.

These examples show that the model structure will play important roles in the representa-
tion theory of algebras. A central concept to deﬁne and understand the admissible model
structure is the *cotortion pair*, which were ﬁrstly introduced in [Sal79] to devide a given
abelian category in two smaller pieces. A cotorsion pair on an abelian category*C*is deﬁned
to be a pair p*U,V*q of subcategories with some conditions involving Ext^{1}_{C}p*U,V*q “0. Af-
terwards, a triangulated analog was considered in [IY08] and the uniﬁcation of them was
introduced by Nakaoka-Palu in terms of extriangulated categories. It was ﬁrstly proved by
Hovey that, in the level of abelian categories, admissible model structures bijectively cor-
respond to special cotorsion pairs. The ﬁrst aim of this note is to prove an extriangulated
version of Hovey’s correspondence theorem stated as below.

**Theorem 1.1.** [NP19, Section 5] *Let* *C* *be an extriangulated category satisfying* (WIC)*.*

*There is a one-to-one correspondence between admissible model structures on* *C* *and twin*
*cotorision pairs* pp*S,T*q*,*p*U,V*qq *satisfying* Conep*V,S*q “CoConep*V,S*q*.*

*In this case, the canonical functorC* ÑHop*C*q*induces an equivalence* *T* X*U*
r*S*X*V*s

ÝÑ„ Hop*C*q*.*

*Furthermore,* Hop*C*q *is naturally equipped with a triangulated structure.*

The second aim is to explain the triangulated structure on the homotopy category
Hop*C*q in a more conceptual framework. The extriangulated localization was formulated
in [NOS21], which is a simultaneous generalization of the Verdier localization and the
Serre localization. With this view in mind, the associated functor *Q*:*C* ÑHop*C*q can be
regarded as an extriangulated localization of*C*with respect to the biresolving subcategory
Conep*V,S*q “CoConep*V,S*q.

This note is organized as follows: Section 2 will be devoted to prepare basic results on model categories, extriangulated categories, and cotorsion pairs. Section 3 proves

Theorem 1.1. In Section 4, as an example of admissible model structures, we investigate the construction of the derived categories in terms of cotorsion pairs. Finally, in Section 5, we formulate the extriangulated localization which contains the Verdier localization and Serre localization. This new framework provides a good understanding of the homotopy category with respect to an admissible model structure.

**Convention 1.2.** *Throughout, all categories and functors are assumed to be additive.* For
a category *C*, we denote by Hom_{C}p*X, Y*q the morphism space from *X* to Y. This is also
denoted by *C*p*X, Y*q for short. All subcategories are always full. For subcategories *U* and
*V* of *C*, Hom_{C}p*U,V*q “ 0 means Hom_{C}p*U, V*q “ 0 for any *U* P *U* and *V* P *V*. If *C* is an
extriangulated category, the notion Ep*U,V*q is used in a similar meaning. We denote by
*U*^{K} the full subcategory of objects*X* withEp*U, X*q “0. The notion^{K}*U* is used in the dual
meaning. Morp*C*qis the category (or the class) of morphisms in*C*. All algebra is assumed
to be ﬁnite dimensional over a ﬁxed ﬁeld *k*.

2. Preliminary

This section is devoted to recall basic facts on model categories, extriangulated cate- gories and cotorsion pairs.

2.1. **Model categories.** A*model category* was introduced by Quillen [Qui67], and it has
been modiﬁed by some authors from various viewpoints, e.g. [Qui69, DS95, BR07]. A
model category is usually assumed to have all ﬁnite limits and colimits, and more often
to have all limits and colimits. However, to include model structures on triangulated and
exact categories, we need to drop such assumptions.

**Deﬁnition 2.1.** Let *C* be an additive category. A *model structure* on *C* is a triple
pCof*,W,*Fibqof classes of morphisms in*C* satisfying the following axioms. The morphisms
in Fib (resp. Cof*,W*) is called the *ﬁbrations* (resp. *coﬁbrations, weak equivalences*). We
denote by wFib“FibX*W* the class of *trivial ﬁbrations* and by wCof “CofX*W* the class
of *trivial coﬁbrations*.

(M1) [Two out of three axiom] For a composed morphism *gf* in *C*, two of *f, g* and *gf*
are weak equivalences, then so is the third.

(M2) [Retract axiom] For a morphism *f* : *X* Ñ *Y* in *C* together with the following
commutative diagram, *f* P*W* implies *g*P*W*.

*X*^{1}

id &&

//

*g*

*X*

*f* ////*X*^{1}

*g*

*Y*^{1}

id

99 //*Y* ^{//}^{//}*Y*^{1}

(M3) [Lifting axiom] For a commutative diagram in*C* of the following shape
*X*

*i*

*f* //*X*^{1}

*p*

*Y* ^{g} ^{//}*Y*^{1}

with*i*PCofand*p*PFib, if*i*or*p*belongs to*W*, there exists a morphism*h*:*Y* Ñ*X*^{1}
such that*h*˝*i*“*f* and *p*˝*h*“*g*.

(M4) [Factorization axiom] Morp*C*q “ wFib˝Cof “ Fib˝wCof holds. Precisely, any
morphism *f* PMorp*C*q admits factorizations*f* “ *f*2*f*1 “*f*_{2}^{1}*f*_{1}^{1} with *f*1 P Cof*, f*2 P
wFib*, f*_{1}^{1} PwCof and *f*_{2}^{1} PFib

We call an additive category equipped with a model structure a*model category*. Further-
more, we sometimes refer the following stronger version of (M4).

(M4’) [Functorial factorization axiom] A*functorial factorization* is a pair p*α, β*q of func-
tors Morp*C*q ÑMorp*C*q such that*f* “*β*p*f*q ˝*α*p*f*q for all*f* PMorp*C*q. There is two
functorial factorizationsp*α, β*qandp*γ, δ*q such that*α*p*f*q PCof*, β*p*f*q PwFib*, γ*p*f*q P
wCof and *δ*p*f*q PFibfor any*f* PMorp*C*q.

For a commutative diagram

*X*

*i*

*f* //*X*^{1}

*p*

*Y* ^{g} ^{//}*Y*^{1}

a morphism *h*:*Y* Ñ*X*^{1} which satisﬁes *h*˝*i*“*f* and *p*˝*h*“*g*is called a*lifting*. If such a
lifting exists, *i*is said to have the *left lifting property* against *p* and *p* is said to have the
*right lifting property* against*i*.

**Lemma 2.2.** (1) *A morphism is a ﬁbration iﬀ it has the right lifting property against*
*all trivial coﬁbrations.*

(2) *A morphism is a coﬁbration iﬀ it has the left lifting property against all trivial*
*ﬁbrations.*

(3) *A morphism is a trivial ﬁbration iﬀ it has the right lifting property against all*
*coﬁbrations.*

(4) *A morphism is a trivial coﬁbration iﬀ it has the left lifting property against all*
*ﬁbrations.*

(5) *W* “wFib˝wCof*.*

Although (co)limits do not necessarily exist, the existence of initial and terminal objects allow us to deﬁne coﬁbrant and ﬁbrant objects.

**Deﬁnition 2.3.** Let pCof*,W,*Fibq be a model structure. We associate the full subcate-
gories:

‚ *C*tcof :“ t*S* P*C*|0Ñ*S* is a trivial coﬁbrationu.

‚ *C*ﬁb:“ t*T* P*C*|*T* Ñ0 is a ﬁbrationu.

‚ *C*cof :“ t*U* P*C*|0Ñ*U* is a coﬁbrationu.

‚ *C*tﬁb:“ t*V* P*C*|*V* Ñ0 is a trivial ﬁbrationu.

The objects in *C*tcof (resp. *C*ﬁb*,C*tcof*,C*ﬁb) are called *trivially coﬁbrant* (resp. *ﬁbrant, coﬁ-*
*brant, trivially ﬁbrant*) objects. In addition, we put *C*cf :“*C*cofX*C*ﬁb. Note that, by the
retract axiom (M2), it follows that the subcategories *C*tcof*,C*ﬁb*,C*cof and *C*tﬁb are closed
under direct summands.

The following lemma shows that a model structure provides nice approximations.

**Deﬁnition 2.4.** Let*C* be a category and *D* a full subcategory of *C*.

(1) For an object *X* P *C*, a morphism *f* : *D*_{X} Ñ *X* from *D*_{X} P *D* is called a *right*
*D-approximation* of *X* if it induces a surjective morphism p*D, f*q. Dually, a left
*D*-approximation is deﬁned.

(2) If any object*X* P*C* admits a right*D*-approximation,*D*is called a*contravariantly*
*ﬁnite* subcategory of *C*. A *covariantly ﬁnite* subcategory is deﬁned dually.

**Lemma 2.5.** *For any object* *X:*

(1) *there exists a right* *C*tcof*-approximationS*_{X} Ý^{π}Ñ^{1} *X* *which is a ﬁbration;*

(2) *there exists a left* *C*ﬁb*-approximation* *X*ÝÑ^{ι} *T*^{X} *which is a trivial coﬁbration;*

(3) *there exists a right* *C*cof*-approximation* *U**X* *π*

ÝÑ*X* *which is a trivial ﬁbration;*

(4) *there exists a left* *C*tﬁb*-approximation* *X*ÝÑ^{ι}^{1} *V*^{X} *which is a coﬁbration.*

*A morphism* *ι* *(resp.* *ι*^{1}*) is called a (resp.* trivially*)* ﬁbrant replacement *of* *X* *and a*
*morphism* *π* *(resp.π*^{1}*) is called a (resp.* trivially*)* coﬁbrant replacement*of* *X.*

*Proof.* We only prove (1). The others are similar. For an object*X*, we consider a zero map
0 Ñ *X* and resolve it as 0 ÝÑ^{i} *S*_{X} ÝÑ^{p} *X* by the factorization system Morp*C*q “Fib˝wCof.

Since*i*: 0Ñ*S**X* is a trivial coﬁbration, by deﬁnition, we have that*S**X* is trivially coﬁbrant.

It remains to show that *p*:*S* Ñ *X* is a right *S**X*-approximation. To this end, consider a
morphism to *S*^{1} ÝÑ^{f} *X* with*S*^{1} P*S* and a commutative square

0 ^{//}

*S*_{X}

*p*

*S*^{1} ^{f} ^{//}*X*

Since 0Ñ*S*^{1}is a trivially coﬁbration and*p*is a ﬁbration, we have a lifting*h*:*S*^{1} Ñ*S*_{X}. □
**Remark 2.6.** The factorization axiom (M4) is not required to be functorial, so an assign-
ment *X* ÞÑ*S**X* in Lemma 2.5 does not give rise to a functor. Similar assertions hold for
other assignments in the lemma.

**Deﬁnition 2.7.** Let*C* be a model category. The*homotopy category* Hop*C*q is deﬁned to
be the Gabriel-Zisman localization of *C* with respect to the class*W* of weak equivalences.

To provide an alternative description of the homotopy category Hop*C*q, we recall the
homotopy relations.

**Deﬁnition 2.8.** Let*f, g*:*X*Ñ*Y* are maps in a model category *C*.

(1) A *cylinder object* for *X* is a factorization of the summation map p1_{X} 1_{X}q :
*X*ś

*X*Ñ *X* into a coﬁbration *X*ś

*X* ÝÝÝÝÑ^{pi}^{0} ^{i}^{1}^{q} *X*^{1} followed by a weak equivalence
*X*^{1} Ñ*X*.

(2) We say*f* and *g* are *left homotopic* if there is some cylinder object *X*^{1} and a map
*H* :*X*^{1} Ñ *Y* such that *Hi*_{0} “*f* and *Hi*_{1} “ *g*. This deﬁnes a relation called *left*
*hmotopy*, denoted „^{l}, on Hom_{C}p*X, Y*q.

The *path object* and the*right homotopy* „^{r} are deﬁned dually.

In fact, if *X* is coﬁbrant and *Y* is ﬁbrant, then „^{l}“„^{r}. In this case, we simply call
it *homotopy* and denote by „. Then, by putting *π*Hom_{C}p*X, Y*q “ Hom_{C}p*X, Y*q{ „, we
obtain the natural functor *π*:*C*cf Ñ*C*cf{ „ which is an identity on objects.

**Theorem 2.9.** *The composition* *C*cf ãÑ *C* ÝÑ^{Q} Hop*C*q *induces an equivalence* p*C*cf{ „q Ý^{„}Ñ
Hop*C*q*.*

**Example 2.10.** Let*A*be a ﬁnite dimensional self-injective*k*-algebra^{1}. Then the category
mod*A* of ﬁnite dimensional (right) *A*-modules admits a model strucutre pCof*,W,*Fibq:

‚ Cof“the set of all monomorphisms;

1An algebra*A*is said to be*self-injective* ifproj*A*“inj*A*. A Frobenius algebra is self-injective.

‚ Fib“the set of all epimorphisms;

‚ *W* “ t*f* |*f* factors through an object *P* Pproj*A*u.

In this case, the full subcategory*C*cf of coﬁbrant and ﬁbrant objects is the whole category
mod*A*. The homotopy relation on *C*cf deﬁnes the (two-sided) ideal rproj*A*s consisting of
all morphisms having a factorization through an object in proj*A*. Hence, the homotopy
category is equivalent to the stable category mod*A*“mod*A*{rproj*A*s.

We end this subsection by recalling the deﬁnition of the Quillen functors.

**Proposition 2.11.** *Let* *C* *andDbe model categories and consider an adjoint pair*p*F, G*q:
*C* Ñ*D, i.e.,F* :*C*Ñ*Dis a left adjoint toG*:*D*Ñ*C. Then, the following are equivalent.*

(1) *F* *preserves ﬁbrations and trivial ﬁbrations.*

(2) *G* *preserves coﬁbrations and trivial coﬁbrations.*

*Under the above equivalent conditions, we call* *F* *a* left Quillen functor *and* *G* *a* right
Quillen functor*. Moreover, the pair*p*F, G*q *is called a* Quillen adjunction*.*

2.2. **Extriangulated category.** An*extriangulated categories*was introduced by Nakaoka-
Palu [NP19] as a simultaneous generalization of triangulated categories and exact cate-
gories. It is deﬁned to be a triplep*C,*E*,*sq of

‚ an additive category*C*;

‚ an additive bifunctorE:*C*^{op}ˆ*C*ÑAb, whereAbis the category of abelian groups;

‚ a correspondence s which associates each equivalence class of a sequence of the
form *X*Ñ*Y* Ñ*Z* in*C* to an element in Ep*Z, X*q for any*Z, X* P*C*,

which satisﬁes some ‘additivity’ and ‘compatibility’. It is simply denoted by *C* if there is
no confusion. We refer to [NP19, Section 2] for its detailed deﬁnition. We only recall a
part of the ‘compatibility’ analogous to the octahedral axiom.

(ET4) Let*δ* PEp*D, A*q and*δ*^{1} PEp*F, B*q beE-extensions realized by
sp*δ*q “ r*A*ÝÑ^{f} *B* ^{f}

1

ÝÑ*D*s*,* sp*δ*^{1}q “ r*B* ÝÑ^{g} *C* ^{g}

1

ÝÑ*F*s*.*

Then there exist an object *E*P*C*, a commutative diagram
*A* ^{f} ^{//}*B* ^{f}

1 //

*g*

*D*

*d*

*A* ^{h} ^{//}*C*

*g*^{1}

*h*^{1} //*E*

*e*

*F* *F*

in*C*, and an E-extension*δ*^{2} PEp*E, A*q realized by*A*ÝÑ^{h} *C*Ý^{h}Ñ^{1} *E*, which satisfy the
following compatibilities:

(1) *D*ÝÑ^{d} *E* ÝÑ^{e} *F* realizes*f*_{˚}^{1}*δ*^{1};
(2) *d*^{˚}*δ*^{2} “*δ*;

(3) *f*_{˚}*δ*^{2}“*e*^{˚}*δ*^{1}.

where we put*f*_{˚}^{1}*δ*^{1} :“Ep*F, f*^{1}qp*δ*^{1}q*, d*^{˚}*δ*^{2} :“Ep*d, A*qp*δ*^{2}q and so on.

**Remark 2.12.** A triangulated category and an exact category are special cases of an
extriangulated category.

(1) By putting E :“ *C*p´*,*´r1sq, a triangulated category p*C,*r1s*,*∆q can be regarded
as an extriangulated category [NP19, Prop. 3.22]. In this case, we say that an
extriangulated category *corresponds to a triangulated category*.

(2) By puttingE:“Ext^{1}_{C}p´*,*´q, an exact category*C*can be regarded as an extriangu-
lated category [NP19, Example 2.13]. In this case, we say that an extriangulated
category *corresponds to an exact category*.

We shall use the following terminology in many places.

**Deﬁnition 2.13.** Let p*C,*E*,*sq be an extriangulated category.

(1) We call an element *δ* P Ep*Z, X*q an E*-extension*, for any *X, Z* P *C*; A sequence
*X*ÝÑ^{f} *Y* ÝÑ^{g} *Z*corresponding to anE-extension*δ* PEp*Z, X*qis called ans-*conﬂation*.

In addition,*f* and*g*are called an s-*inﬂation* and ans-*deﬂation*, respectively. The
pair x*X* ÝÑ^{f} *Y* ÝÑ^{g} *Z, δ*yis called an s-*triangle*.

(2) An object*P* P*C* is said to be*projective* if for any deﬂation*g*:*Y* Ñ*Z*, the induced
morphism *C*p*P, g*q : *C*p*P, Y*q Ñ *C*p*P, Z*q is surjective. We denote by projp*C*q the
subcategory of projectives in*C*. An*injective* object andinjp*C*q are deﬁned dually.

(3) We say that *C* *has enough projectives* if for any *Z* P *C*, there exists a conﬂation
*X*Ñ*P* Ñ*Z* with *P* projective. *Having enough injectives* are deﬁned dually.

(4) Let *C* have enough projectives and injectives. If the class of projectives coincides
with the class of injectives,*C* is said to be *Frobenius*.

**Remark 2.14.** (1) If an extriangulated category*C* corresponds to a triangulated cat-
egory, then*C*has enough projectives and injectives. The projectives and inejctives
are nothing but the zero objects.

(2) If an extriangulated category*C* corresponds to an exact category, projectives and
injectives agree with usual deﬁnitions.

Keeping in mind the triangulated case, we introduce the notions cone and cocone.

**Proposition 2.15.** *Let* *C* *be an extriangulated category. For an inﬂation* *f* P *C*p*X, Y*q*,*
*take a conﬂation* *X* ÝÑ^{f} *Y* ÝÑ^{g} *Z, and denote this* *Z* *by* Conep*f*q*. We call* Conep*f*q *a* cone
*of* *f. Then,* Conep*f*q *is uniquely determined up to isomorphism. The dual notion* cocone
CoConep*g*q *exists.*

Furthermore, for any subcategories*U* and*V*of*C*, we deﬁne a full subcategoryConep*V,U*q
to be the one consisting of objects*X*appearing in a conﬂation*V* Ñ*U* Ñ*X*with*U* P*U* and
*V* P*V*. A subcategoryCoConep*V,U*q is deﬁned dually. Next, we recall the notion of weak
pullback in extriangulated categories. Consider a conﬂation *X* ÝÑ^{f} *Y* ÝÑ^{g} *Z* corresponding
toE-extension*δ* P*E*p*Z, X*qand a morphism*z*:*Z*^{1} Ñ*Z*. Put*δ*^{1} :“Ep*z, X*qp*δ*qand consider
a corresponding conﬂation *X* Ñ *E* Ñ *Z*^{1}. Then, there exists a commutative diagram of
the following shape

*X* ^{//}*E*

(Pb)
*g*^{1} //

*z*^{1}

*Z*^{1}

*z*

*X* ^{f} ^{//}*Y* ^{g} ^{//}*Z*

The commutative square (Pb) is called a*weak pullback ofgalongz*which is a generalization
of the pullback in exact categories and the homotopy pullback in triangulated categories.

An induced sequence*E* p^{´g}_{z}1^{1}q

ÝÝÝÑ*Z*^{1}‘*Y* ÝÝÝÑ^{pz gq} *Z* is a coﬂation. The dual notion*weak pushout*
exists and it will be denoted by (Po).

We often refer the following condition, analogous to the weak idempotent completeness ([Buh, Prop. 7.6])

**Condition 2.16** (WIC)**.** For an extriangulated category *C*, we consider the following
conditions.

(1) Let*g*˝*f* be a composed morphism in *C*. If *g*˝*f* is an inﬂation, then so is*f*.
(2) Let*g*˝*f* be a composed morphism in *C*. If *g*˝*f* is an deﬂation, then so is *g*.

**Remark 2.17.** (1) If *C* corresponds to a triangulated category, (WIC) is automati-
cally satisﬁed.

(2) If*C* corresponds to an abelian category, (WIC) is automatically satisﬁed.

(3) If *C* corresponds to an exact category, (WIC) is equivalent to the usual condition
so that*C* is weakly idempotent complete.

We end this subsection by mentioning that the class of extriangulated categories is closed under certain operations.

**Proposition 2.18.** [NP19, Rem. 2.18, Prop. 3.30] *Let* *C* *be an extriangulated category.*

(1) *Any extension-closed subcategory admits an extriangulated structure induced from*
*that of* *C.*

(2) *Let* *I* *be a full additive subcategory, closed under isomorphisms which satisﬁes*
*I* Ďprojp*C*q Xinjp*C*q*, then the ideal quotientC*{r*I*s *has an extriangulated structure,*
*induced from that of* *C.*

2.3. **Cotorsion pairs.** The aim of this subsection is brieﬂy recalling the deﬁnition of
a cotorsion pair and explaining how it relates to the theory of derived category in the
representation of algebras.

**Deﬁnition 2.19.** Let*C*be an extriangulated category. A (complete)*cotorsion pair* p*U,V*q
is a pair of full subcategories on *C* which is closed under direct summands and satisﬁes
that Ep*U,V*q “0 and, for any*X*P*C*, there exist conﬂations

*X*Ñ*V*^{X} Ñ*U*^{X}*,* *V*_{X} Ñ*U*_{X} Ñ*X*

with *U*^{X}*, U*_{X} P *S, V*^{X}*, V*_{X} P *V*. The above deﬁning sequences are called *approximation*
*sequences*.

**Example 2.20.** (1) Let *C* be an exact category. Then, pproj*C,C*q forms a cotorsion
pair if and only if*C* has enough projectives.

(2) Denote by mod*A* the category of ﬁnite dimensional modules over a Gorenstein
algebra *A*, then pCM*A,*P^{ă8}q is a cotorsion pairs of mod*A*. Here, CM*A* is the
subcategory of Cohen-Macaulay modules, namely,

CM*A*:“ t*X*Pmod*A*|Ext^{i}_{A}p*X, A*q “0*, i*ą0u*,*

and P^{ă8} is the subcategory of modules of ﬁnite projective dimension.

In the representation theory of algebras, cotorsion pairs play a central role in connection
with derived categories, since, if a special cotorsion pair exists inmod*A*, we obtain a new
algebra *B* derived equivalent to *A*. In the rest of this subsection, the symbol *A* and *B*
always denote ﬁnite dimensional *k*-algebra over a ﬁxed ﬁeld *k*. Let us begin with some
preparations.

For a subcategory *C* of mod*A* we denote by *C*qthe subcategory of mod*A* consisting of
objects *X* for which there is an exact sequence 0 Ñ *X* Ñ *C*0 Ñ *C*1 Ñ ¨ ¨ ¨ Ñ *C**n* Ñ 0,
with*C*_{i}P*C*. Recall that *T* Pmod*A*is*basic* if*T* admits an indecomposable decomposition
*T* “ś*n*

*i*“1*T**i* with*T**i* ﬁ*T**j* for*i*‰*j*.

**Deﬁnition 2.21.** A ﬁnite dimensional *A*-module*T* Pmod*A* is called a *tilting* *A-module*,
if it satisﬁes the following conditions.

(1) pd*T* ă 8, wherepd*T* is the projective dimension of*T*.
(2) Ext^{i}_{C}p*T, T*q “0 for*i*ą0.

(3) There is an exact sequence 0Ñ*A*Ñ*T*0 Ñ ¨ ¨ ¨ Ñ*T**n*Ñ0 with *T**i* Padd*T*.

**Theorem 2.22.** [Hap88, Ch. III, Thm. 2.10] *Let* *T* Pmod*A* *be a tilting* *A-module and*
*put* *B* :“End_{A}p*T*q*. Then, the functor* Hom_{A}p*T,*´q:Mod*A*ÑMod*B* *induces a triangule*
*equivalence* Dp*A*qÝÑ^{„} Dp*B*q*.*

Thus, tilting modules are basic and powerful tools to construct derived equivalent alge- bras. The following theorem shows that tilting modules bijectively correspond to a class of cotorsion pairs.

**Deﬁnition 2.23.** A subcategory*U* of an exact category*C*is called a*resolving* subcategory
if, for any conﬂation 0Ñ*X*Ñ*Y* Ñ*Z* Ñ0,*Y, Z*P*U* implies*X* P*U*.

**Lemma 2.24.** *For a cotorsion pair* p*U,V*q*, the following are equivalent.*

(i) *U* *is resolving.*

(ii) Ext^{2}_{C}p*U, V*q “0 *for any* *U* P*U* *and* *V* P*V.*
(iii) Ext^{i}_{C}p*U, V*q “0 *for any* *U* P*U, V* P*V* *and* *i*ą0*.*

*Under the above equivalent conditions, we call* p*U,V*q *a* resolving *cotorsion pair*^{2}*.*

**Theorem 2.25.** [AR91, Thm. 5.5] *There is one-to-one correspondence between basic*
*tilting modules* *T* *and resolving cotorsion pairs* p*U,V*q *with* *V*q “ mod*A, given by* *T* ÞÑ
p}add*T,*p}add*T*q^{K}q *and* p*U,V*q ÞÑ *direct sum of the indecomposable modules in* *U* X*V.*

A trivial example of tilting *A*-module is *A* itself. The corresponding cotorsion pair is
pproj*A,*mod*A*q on mod*A*.

Krause-Solberg showed that a cotorsion pair p*U,V*q on mod*A* can give rise to that on
Mod*A*.

**Theorem 2.26.** [KS03, Thm. 2.4] *Let* p*U,V*q *be a resolving cotorsion pair on* mod*A.*

*Then* plim
ÝÑ*U,*lim

ÝÑ*V*q *is a resolving cotorsion pair on* Mod*A. Here,* lim

ÝÑ*U* *denotes the full*
*subcategory of all* *A-modules which are ﬁltered colimits of modules in* *U.*

3. Model structures and Cotorsion pairs

3.1. **Hovey’s correspondence theorem.** Let us start with brief observations on an
abelian category*C* equipped with a model structurepCof*,W,*Fibq. Following [BR07, VII],
we ﬁrstly sharpen Lemma 2.5 and show that the coﬁbrant/ﬁbrant replacements are closely
related to approximation sequences obtained from cotorsion pairs.

**Lemma 3.1.** [BR07, VII.2.1] *For any object* *X* P*C:*
(1) *there exists an exact sequence*0Ñ*T**X*

ÝÑ*f* *S**X* *π*^{1}

ÝÑ*X* *whereπ*^{1} *is a trivially coﬁbrant*
*replacement and* *T*_{X} P*C*ﬁb*;*

(2) *there exists an exact sequenceX* ÝÑ^{ι} *T*^{X} Ñ*S*^{X} Ñ0*whereιis a ﬁbrant replacement*
*and* *S*^{X} P*C*tcof*;*

(3) *there exists an exact sequence*0Ñ*V**X* Ñ*U**X* *π*

ÝÑ*X* *where* *π* *is a coﬁbrant replace-*
*ment and* *V*_{X} P*C*tﬁb*;*

2This is also called a*hereditary*cotorsion pair in the literature.

(4) *there exists an exact sequence* *X* ÝÑ^{ι}^{1} *V*^{X} Ñ *U*^{X} Ñ0 *where* *ι*^{1} *is a trivially ﬁbrant*
*replacement and* *U**X* P*C*cof*.*

*Proof.* We shall only show (1). Due to Lemma 2.5, it remains to show *T*_{X} P*C*ﬁb. To this
end, we consider a trivial coﬁbration *g*:*A*Ñ*B* together with the following commutative
squares.

*A*

*g* //*T*_{X} ^{f} ^{//}

*S*_{X}

*π*^{1}

*B* ^{//}0 ^{//}*X*

Since *π*^{1} is a ﬁbration, we have a lifting *h* : *B* Ñ *S*_{X} for the whole square consisting of
*A, B, S**X* and *X*. The lifting *h* factors through*T**X*, say *h* “*f* ˝*f*^{1}. Since *f* is monic, the
morphism *f*^{1} should be a lifting for the left square. Hence *T*_{X} Ñ0 is a ﬁbration. □
**Deﬁnition 3.2.** Let *C* be an abelian (more generally, extriangulated) category with a
model structure. For a given morphism *f*, we consider the following four conditions:

(1) *f* is an inﬂation with cone in *C*tcof iﬀ*f* PwCof.

(2) *f* is deﬂation with cocone in*C*ﬁb iﬀ*f* PFib;

(3) *f* is an inﬂation with cone in *C*cof iﬀ *f* PCof;

(4) *f* is a deﬂation with cocone in *C*tﬁb iﬀ*f* PwFib;

The model structure of*C*is said to be*right admissible* (resp. *left admissible* ) if the above
(3)(4) (resp. (1)(2)) are satisﬁed. A left and right admissible model structure is said to
be *admissible*.

The following shows that admissible model structure yields a pair of cotorsion pairs.

**Deﬁnition 3.3.** Letp*S,T*qandp*U,V*qbe cotorsion pairs on*C*. We call the pairpp*S,T*q*,*p*U,V*qq
of them a *twin cotorsion pair* if it satisﬁes*S* Ď*U* (or equivalently*T* Ě*V*).

**Proposition 3.4.** [BR07, VII.3.4] *Let* *C* *be an abelian category with a model structure*
pCof*,W,*Fibq*.*

(1) *If the model structure is left admissible, then* p*C*tcof*,C*ﬁbq *forms a cotorsion pair.*

(2) *If the model structure is right admissible, then* p*C*cof*,C*tﬁbq *forms a cotorsion pair.*

*In particular, if the model structure is admissible, we have a twin cotorsion pair*pp*C*tcof*,C*ﬁbq*,*p*C*cof*,C*tﬁbqq*.*

*Proof.* We shall prove only part (2) since the proof of (1) is dual. Due to the assumption
and Lemma 3.1, we have only to check Ext^{1}_{C}p*U, V*q “0 for*U* P*C*cof and*V* P*C*tﬁb. Consider
a conﬂation *V* Ñ*X* ÝÑ^{p} *U* and a commutative square

0 ^{//}

*X*

*p*

*U* *U*

Since 0 Ñ*U* is a coﬁbration and *p* is a trivial ﬁbration, we have a lifting, which forces *p*

splitting. □

It is natural to ask when a given twin cotorsion pairs induces a model structure^{3}. The
answer was essentially given by Hovey.

3Another natural question is when a single cotorsion pair corresponds to a model structure. An answer is given by Beligiannis-Reiten [BR07, VII. 4.2].

**Deﬁnition 3.5.** A twin cotorsion pairpp*S,T*q*,*p*U,V*qqon*C*is called*Hovey twin cotorsion*
*pair* if it satisﬁes Conep*V,S*q “CoConep*V,S*q.

**Theorem 3.6.** [Hov02, Thm. 2.2] *Let* *C* *be an abelian category. There exists one-to-one*
*correspondence between admissible model structures on* *C* *and Hovey twin cotorsion pairs*
*on* *C.*

tpCof*,W,*Fibq: admissible model strucutres on *C*u

Φ

tpp*S,T*q*,*p*U,V*qq: Hovey twin cotorsion pairs on *C*u

Ψ

OO

*Proof.* It is a special case of Theorem 1.1. □

**Remark 3.7.** Theorem 3.6 has been modiﬁed and generalized by some authors:

- Gillespie and ˇSˇtov´ıˇcek proved an exact version of the theorem [Gil11, Sto13];

- Yang proved a triangulated version of the theorem [Yan15].

- Nakaoka-Palu’ extriangulated version contains the above cases. [NP19].

In addition, we remark that the theorem is a bit modiﬁed formulation of the original one.

This type of formulations can be found in [NP19].

**Example 3.8.** The model structure in Example 2.10 is admissible and corresponds to a
Hovey twin cotorsion pair ppproj*A,*mod*A*q*,*pmod*A,*proj*A*qq.

3.2. **Nakaoka-Palu’s correspondence theorem.** This subsection is devoted to prove
Theorem 1.1, the extriangulated version of Hovey’s correspondence theorem.

3.2.1. *From admissible model strucutre to Hovey twin cotorsion pair.* Let *C* be an extri-
angulated category with an admissible model structure pCof*,W,*Fibq and put

pp*S,T*q*,*p*U,V*qq:“ pp*C*tcof*,C*ﬁbq*,*p*C*cof*,C*tﬁbqq*.*

**Proposition 3.9.** [NP19, Prop. 5.6] pp*S,T*q*,*p*U,V*qq*is a twin cotorsion pair.*

*Proof.* A similar discussion in the proof of Proposition 3.4 still works well. So we skip the

details. □

It directly follows from the next proposition thatpp*S,T*q*,*p*U,V*qqis a Hovey twin cotor-
sion pair.

**Proposition 3.10.** [NP19, Prop. 5.7] *The following are equivalent for any object* *N* P*C.*
(i) *N* PConep*V,S*q*.*

(ii) p0Ñ*N*q P*W.*

(iii) p*N* Ñ0q P*W.*
(iv) *N* PCoConep*V,S*q*.*

*Proof.* (i) ñ (ii): Consider a conﬂation*V* Ñ*S* Ñ *N* with*V* P*V, S* P*S*. Then, we have
a factorization of 0 Ñ *N* as a trivial coﬁbration followed by a trivial ﬁbration *S* Ñ *N*.
Hence p0Ñ*N*q P*W*.

(ii) ñ (i): Since p0 Ñ *N*q P *W* and *W* “ wFib˝wCof, we have a factorization of
p0Ñ*N*q which shows the condition (i).

(iii) ô (iv): It is a dual of (i)ô (ii).

(ii) ô(iii): It follows from the 2-out-of-3 axiom on *W*. □

3.2.2. *From Hovey twin cotorsion pair to admissible model strucutre.* In the rest of this
section, we ﬁx an extriangulated category *C* satisfying (WIC) together with a Hovey twin
cotorsion pair pp*S,T*q*,*p*U,V*qq.

To construct the corresponding model structure, we deﬁne the following classes of mor- phisms.

- wCof“ t*f* PMor*C*|*f* is an inﬂation with Conep*f*q P*S*u.

- Fib“ t*f* PMor*C*|*f* is a deﬂation with CoConep*f*q P*T*u.

- Cof“ t*f* PMor*C*|*f* is an inﬂation with Conep*f*q P*U*u.

- wFib“ t*f* PMor*C*|*f* is a deﬂation with CoConep*f*q P*V*u.

- *W* “wFib˝wCof.

**Lemma 3.11.** wCof*,*Fib*,*Cof *and* wFib *are closed under composition.*

*Proof.* Since the corresponding subcategories *S,T,U* and *V* are extension-closed, the as-

sertions directly follow from (ET4). □

**Proposition 3.12.** *The lifting axiom* (M3) *is satisﬁed, that is, we have the following.*

(1) wCof *satisﬁes the left lifting property against*Fib*.*

(2) wFib *satisﬁes the right lifting property against*Cof*.*

*Proof.* We only prove (1), because (2) is dual. We consider the following square with the
columns forming conﬂations.

*T*

*t*

*A* ^{a} ^{//}

*f*

*C*

*g*

*B*

*s*

*b* //*D*
*S*

Under the assumption *S* P*S* and *T* P*T*, we shall construct a lifting *h* :*B* Ñ *C* for the
above square. Since Ep*S, T*q “0, by a basic property of extriangulated categories, there
exists a map*c*:*B* Ñ*C* with*b*“*g*˝*c*. However, unlike the exact case, *a*“*c*˝*f* does not
necessarily hold. So we put *d*:“*a*´*c*˝*f*. Since*g*˝*d*“0, we have a map*c*^{1} :*A*Ñ*T* with
*d*“*t*˝*c*^{1}. Again, due toEp*S, T*q “0, we get *c*^{2} :*B* Ñ *T* with *c*^{2}˝*f* “*c*^{1}. Then the map

*h*:“*c*`*t*˝*c*^{2}:*B*Ñ*C* is a desired lifting. □

The next proposition shows that the factorization axiom (M4) is satisﬁed.

**Proposition 3.13.** Morp*C*q “wFib˝Cof“Fib˝wCof*.*

*Proof.* We only show Morp*C*q “wFib˝Cof. Let*f* P*C*p*A, B*q a map and resolve *A* to get
an approximation sequence *A*ÝÑ^{ι}^{A} *V*^{A}Ñ *U*^{A} with*U*^{A} P*U* and *V*^{A} P*V*. A weak pushout
of *ι*^{A} along *f* yields a conﬂation *A* ^{f}

1

ÝÑ *B*‘*V*^{A}Ñ *C* where we put *f*^{1} :“`_{f}

*ι*^{A}

˘. Resolve*C*
by p*U,V*q and obtain the following commutative diagram made of conﬂations.

*V*_{C}

*V*_{C}

*A* ^{i} ^{//}*M*

(Pb)

//

*p*

*U**C*

*A* ^{f}

1//*B*‘*V*^{A} ^{//}*C*

Since *i*PCof and*p*PwFib, a factorization*f* “ p1 0q ˝*p*˝*i*, the morphism*p*˝*i*followed by
the projection p1 0q P*B*‘*V*^{A}Ñ*B*, is a desired one.

The remaining axioms (M1) and (M2) follow from Corollary 5.22 and Propsition 5.24 in [NP19], repectively. A triangulated structure is given in [NP19, Thm. 6.20]. Although the detailed proof is not included here, we shall present more conceptual proofs in Section

5 via the localization of extriangulated categories. □

By the argument so far, admissible model structures and Hovey twin cotorsion pairs
correspond bijectively. Recall that, the homotopy category Hop*C*q admits another de-
scription *C*cf{ „ in terms of homotopy relations which should be better understood. We
summarize the essentials which shows homotopy relations can be interpreted in view of
ideal quotients.

**Proposition 3.14.** [Gil11, Prop. 4.4]*LetCbe an extriangulated category satisfying* (WIC)
*with an admissible model structure.*

(1) *Two morphismsf, g*:*X* Ñ*Y* *in* *C* *are right homotopic if and only ifg*´*f* *factors*
*through an object of* *C*tcof*.*

(2) *Two morphisms* *f, g* :*X* Ñ*Y* *in* *C* *are left homotopic if and only if* *g*´*f* *factors*
*through an object of* *C*tﬁb*.*

*In particular, the composition* *C*cf ãÑ *C* ÝÑ^{Q} Hop*C*q *induces an equivalence* _{rC} ^{C}^{cf}

tcofX*C*tﬁbs

Ý„Ñ
Hop*C*q*.*

*Proof.* We will prove (2). The part (1) is dual. We ﬁrst construct a cylinder object for*X*
by using the corresponding cotorsion pair p*U,V*q :“ p*C*cof*,C*tﬁbq. Resolving *X* by p*U,V*q,
we get a conﬂation *X* ÝÑ^{j} *V*^{X} Ñ *U*^{X} with *U*^{X} P *U* and *V*^{X} P *V*. Now we consider the
following factorization of the summation p1_{X} 1_{X}q:*X*ś

*X* Ñ*X*:

*X*ź

*X* ÝÑ^{i} *X*ź

*V*^{X p}ÝÑ*X*
with*i*“

´1_{X} 1_{X}

0 *j*

¯

and*p*“ p1*X* 0q. Since Ker*p*–*V**X* is trivially ﬁbrant, we have*p*PwFib.

To claim that *i*is a coﬁbration, we consider the equation
ˆ*j* 0

0 *j*

˙

“

ˆ*j* ´1_{V}_{X}
0 1_{V}_{X}

˙ ˆ1_{X} 1_{X}

0 *j*

˙
*.*
Since ´

*j* 0

0 *j*

¯

is an inﬂation, the property (WIC) guarantees *i* P Cof. Thus, *X*ś
*V*^{X} is
a cylinder object for *X*. By deﬁnition, *f* „^{l} *g* if and only if there is a map p*α β*q :
*X*ś

*V*^{X} Ñ *Y* such that the equation p*α β*q*i* “ p*f g*q holds. Since *i* “

´1_{X} 1_{X}

0 *j*

¯
, the
equation is equivalent to*f* `*βj* “*g*. Thus, *g*´*f* factors through*V*^{X} P*C*tﬁb.

The remaining assertion directly follows from Theorem 2.9. □ 4. Derived categories and model structures

In this section, we investigate a model structure on a category Cp*A*q of complexes of
*A*-modules the homotopy category of which coincides with the derived category Dp*A*q.

A classical example of such model structures is known as an *injective model structure*
which was ﬁrst constructed by Joyal and Beke [Bek00]. Based on an elegant approach in
[SS11, Sto13], we shall show that such model structures can be obtained from resolving
cotorsion pairs on Mod*A*. Throughout the section, ﬁx a ﬁnite dimensional *k*-algebra *A*
and the following symbols are used in many places:

- Cp*A*q - the category of complexes of*A*-modules;

- Cacp*A*q - the full subcategory of acyclic complexes ofCp*A*q;

- Kp*A*q - the homotopy category of complexes of*A*-modules;

- K_{ac}p*A*q- the full subcategory of acyclic complexes of Kp*A*q;

- Dp*A*q - the derived category ofMod*A*.

Moreover, for an extension-closed subcategory *U* of Mod*A*, letCp*U*q denote the category
of complexes in *U* and Cacp*U*q the full subcategory of acyclic complexes *X* P Cp*U*q such
that *Z*^{n}p*X*q P*U* for all *n*PZ.

4.1. **Injective model structures.** The derived category is, by deﬁnition, the Gabriel-
Zisman localization ofCp*A*qwith respect to the quasi-isomorphisms. Let us recall another
construction of the derived category which ﬁts into a well understood framework. The
homotopy category Kp*A*q is a triangulated category together with a thick subcategory
Kacp*A*q. The derived category is obtained as the Verdier localization ofKp*A*q with respect
to K_{ac}p*A*q. In addition, we have K_{ac}p*A*q ÑKp*A*qÝ^{Q}ÑDp*A*q with additional property that,
for any*X* PKp*A*q, there exists a triangle

*V**X* Ñ*U**X* Ñ*X*Ñ*V**X*r1s

with *U*_{X} PK_{ac}p*A*q and *V*_{X} PK_{ac}p*A*q^{K}. It turns out that pK_{ac}p*A*q*,*K_{ac}p*A*q^{K}q forms a cotor-
sion pair on Kp*A*q. Furthermore, it realizes the derived category as the full subcategory
K_{ac}p*A*q^{K} ofKp*A*q.

**Lemma 4.1.** *The composition*K_{ac}p*A*q^{K} ãÑKp*A*qÝ^{Q}ÑDp*A*q *is a triangle equivalence.*

Now, we are in position to state that one can realize the derived category as the homo- topy category with respect to an admissible model structure.

**Proposition 4.2.** *There exists a Hovey twin cotorsion pair*

`pCacp*A*q*,*Cacp*A*q^{K}q*,*pCp*A*q*,*CacpInj*A*qq˘

*the homotopy category of which is the derived category*Dp*A*q*. The corresponding admissible*
*model structure is called the* injective model structure*. Furthermore, the cotorsion pairs*
*are resolving.*

Before proving that, we recall the following well-known fact.

**Lemma 4.3.** *The following conditions are equivalent for* *X*PCp*A*q*.*

(i) Ext^{1}_{CpAq}p*E, X*q “0 *for any acyclic complexE* PC_{ac}p*A*q*.*

(ii) Hom_{KpAq}p*E, X*q “0 *for any acyclic complexE* PCacp*A*q*.*

*Under the above equivalent conditions, the object* *X* *in* Cp*A*q *are called an* injectively
ﬁbrant *objects*^{4} *and those in* Kp*A*q *are called* K-injectives*.*

*Proof of Proposition 4.2.* It is obvious thatpCp*A*q*,*C_{ac}pInj*A*qqforms a cotorsion pair, since
any acyclic complex in CacpInj*A*qis splitting.

Due the cotorsion pair pKacp*A*q*,*Kacp*A*q^{K}q, it is easy to check that pCacp*A*q*,*Cacp*A*q^{K}q is
also a cotorsion pair. In fact, for any *X*PCp*A*q, there exists a triangle*T*_{X} ÝÑ^{i} *S*_{X} Ñ*X* Ñ
*T*_{X}r1s with *S*_{X} P K_{ac}p*A*q and *T*_{X} PK_{ac}p*A*q^{K}. Any triangle in Kp*A*q comes from an exact

4Each term*X*^{i}of an injectively ﬁbrant object*X* belongs toInj*A*.

sequence in Cp*A*q. More precisely, by taking the pushout of*T**X* *i*

Ý

Ñ*S**X* along the injective
hull *T*_{X} Ñ*I*p*T*_{X}q, we get an exact sequence

0Ñ*T*_{X} Ñ*S*_{X} ‘*I*p*T*_{X}qÝÑ^{p} *X*^{1} Ñ0

which corresponds to the triangle. We may assume that there exists a contractible complex
*I* PC_{ac}p*A*q with an isomorphism *X*^{1}–*X*‘*I* inCp*A*q. Taking the pullback of*p* along the
section *X*Ñ*X*^{1} yields the following commutative diagram made of conﬂations.

0

0

*I*

*I*

0 ^{//}*T**X* //*S*_{X}^{1}

//

(Pb)

*X* ^{//}

0
0 ^{//}*T*_{X} ^{//}*S*_{X} ‘*I*p*T*_{X}q _{p} ^{//}

*X*^{1} ^{//}

0

0 0

The middle row is a desired sequence. Another approximation sequence can be obtained similarly.

Since CacpInj*A*q is the class of injectives inCp*A*q, it is easy to check the equiality
Cacp*A*q “ConepCacpInj*A*q*,*Cacp*A*qq “CoConepCacpInj*A*q*,*Cacp*A*qq*.*

Hence, the given pair forms a Hovey twin cotorsion pair.

It remains to show that *W* is the class of quasi-isomorphisms. Let *f* : *X* Ñ *Y* be
a quasi-isomorphism and consider a factorization *f* : *X* ÝÑ^{f}^{1} *Z* ÝÑ^{f}^{2} *Y* with *f*1 surjective
and *f*_{2} injective. Note that *K* :“ Ker*f*_{1}*,*Cok*f*_{2} belongs to C_{ac}p*A*q and, in particular,
*f*2 PwCof. Since Cp*A*qhas enough injectives formingCacpInj*A*q, we get an exact sequence
0 Ñ *K* Ñ*I* Ñ *K*^{1} Ñ 0 with *I* PC_{ac}pInj*A*q and *K*^{1} PC_{ac}p*A*q and construct the following
commutative diagram with all rows and columns exact:

0

0

0 ^{//}*K*

(Po)

//

*X* ^{f}^{1} ^{//}

*f*_{1}^{1}

*Z* ^{//}0
0 ^{//}*I* ^{//}

*X*^{1}

*f*_{1}^{2}

//

*Z* ^{//}0
*K*^{1}

*K*^{1}

0 0

Since *f*_{1}^{1} PwCof*, f*_{1}^{2}PwFib, we thus conclude*f* P*W*. Any weak equivalence is obviously a
quasi-isomorphism. Therefore, the homotopy category is the derived category. □

The injective model structure is explicitly given as follows:

- *W* “the class of quasi-isomorphisms;

- Cof“the class of monomorphisms;

- Fib“the class of epimorphisms whose kernels are injectively ﬁbrant.

Moreover, thanks to the description of the homotopy category in Theorem 1.1, we have an equivalence

Hop*C*q “Dp*A*q » Cacp*A*q^{K}X^{K}CacpInj*A*q

rCacp*A*q XCacpInj*A*qs “Kacp*A*q^{K}*.*

Dually, the *projective model structure* exists. The corresponding Hovey twin cotosion
pair is

`pC_{ac}pproj*A*q*,*Cp*A*qq*,*p^{K}C_{ac}p*A*q*,*C_{ac}p*A*qq˘
*,*
and the homotopy category is also the derived category.

4.2. **Lifting cotorsion pairs to categories of complexes.** In this subsection, we shall
show that a cotorsion pair p*U,V*q on Mod*A* gives rise to a model structure on Cp*A*q the
homotopy category is the derived category. Such model structures contain the injective
and projective model structures. Recall that C_{ac}p*A*q is a thick subcategory of Cp*A*q with
an inherited exact structure.

**Proposition 4.4.** [Sto13, Prop. 7.7]*A resolving cotorsion pair* p*U,V*q *on* Mod*A* *induces*
*a resolving cotorsion pair* pCacp*U*q*,*Cacp*V*qq*on the exact category* Cacp*A*q*.*

*Proof.* We denote for brievityC_{ac}p*U*q and C_{ac}p*V*q by *U*r and *V*r, respectively, and put *N* :“

Cacp*A*q. We ﬁrst show Ext^{1}_{N}p*U*r*,V*q “r 0. Every *U* P *U*r can be written as an extension of
stalk complexes^{5} *Z*^{i}p*U*q which is depicted as follows.

0

0

0

¨ ¨ ¨ ^{//}*Z*^{0}p*U*q

0 //*Z*^{1}p*U*q

0 //*Z*^{2}p*U*q

//¨ ¨ ¨

¨ ¨ ¨ ^{//}*U*^{0}

B^{0} //*U*^{1}

B^{1} //*U*^{2}

//¨ ¨ ¨

¨ ¨ ¨ ^{//}*Z*^{1}p*U*q

0 //*Z*^{2}p*U*q

0 //*Z*^{3}p*U*q

//¨ ¨ ¨

0 0 0

Therefore we have only to check Ext^{1}_{CpAq}p*U*^{1}*, V*q “0 for any*V* P*V*r and any stalk complex
*U*^{1} PCp*U*q concentrated in degree*n*. Any extension *δ* in Ext^{1}_{CpAq}p*U*^{1}*, V*q “0 is necessarily
degreewise splitting:

¨ ¨ ¨ ^{//}*V*^{n´1}

B^{n´1} //*V*^{n}
p^{1}_{0}q

B^{n} //*V*^{n`1}

//¨ ¨ ¨

¨ ¨ ¨ ^{//}*V*^{n´1}

*α*//*V*^{n}‘*U*^{1}

p0 1q

*β* //*V*^{n`1}

//¨ ¨ ¨

¨ ¨ ¨ ^{//}0 ^{0} ^{//}*U*^{1} ^{0} ^{//}0 ^{//}¨ ¨ ¨
where *α* “`_{B}^{n´1}

0

˘ and *β* “ pB^{n} *f*q for a morphism*f* :*U*^{1} Ñ*V*^{n`1}. Since B^{n`1}˝*f* “0,*f*
factors through *Z*^{n`1}p*V*q. Due to Ext^{1}_{A}p*U,V*q “0, *f* is factored as *f* :*U*^{1} ^{f}

1

ÝÑ *Z*^{n}p*V*q ^{B}

*n*

ÝÑ

5A stalk complex*X* is a complex satisfying*X*^{i}“0 for any*i*‰*n*for some*n*PZ.

*V*^{n`1}. Thus we have a splitting exact sequence 0 Ñ *U*^{1} p_{´f}^{1}1q

ÝÝÝÑ *V*^{n}‘*U*^{1} ^{pf}

1 1q

ÝÝÝÑ *V*^{n} Ñ 0
which shows that *δ* splits.

Clearly,*U*rand*V*rare closed under direct summands since*U* and*V* are. For any*X* P*N*,
we resolve the cocycle*Z*^{n}p*X*q PMod*A*by the cotorsion pairp*U,V*qto have approximation
sequences 0 Ñ *Z*^{n}p*X*q Ñ *V*_{n} Ñ *U*_{n} Ñ 0 and 0 Ñ *V*_{n}^{1} Ñ *U*_{n}^{1} Ñ *Z*^{n}p*X*q Ñ 0. As an
application of the Horseshoe lemma, we can construct desired approximation sequences
for*X*.

It is obvious that*U*ris closed under epikernels. Hence we have proved thatp*U*r*,V*rq forms

a resolving cotorsion pair Cacp*A*q. □

The following model structures were ﬁrst found in [SS11, Thm. 4.2] and streamlined in [Sto13, Thm. 7.16].

**Theorem 4.5.** *Let* p*U,V*q *be a resolving cotorsion pair on* Mod*A. Then, it induces a*
*Hovey twin cotorsion pair*

ppC_{ac}p*U*q*,*C_{ac}p*U*q^{K}q*,*p^{K}C_{ac}p*V*q*,*C_{ac}p*V*qqq

*the homotopy category of which is the derived category* Dp*A*q*. Furthermore, the cotorsion*
*pairs are resolving.*

*Proof.* We shall show that pC_{ac}p*U*q*,*C_{ac}p*U*q^{K}q is a cotorsion pair. Any *X* P Cp*A*q admits
an approximation sequence 0Ñ *X*^{1} Ñ*E* ÝÑ^{f}^{1} *X* Ñ0 with *E* PC_{ac}p*U*q and *X*^{1} PC_{ac}p*A*q^{K}.
Resolve *E* by the cotorsion pair pCacp*U*q*,*Cacp*V*qq to get an approximation sequence 0 Ñ
*V*r Ñ*U*r ÝÑ^{f}^{2} *E* Ñ0 with*U*r PC_{ac}p*U*qand*V*r PC_{ac}p*V*q. The composed map*f* :“*f*_{2}˝*f*_{1} gives
rise to the following commutative diagram of exact sequences.

0

0

*V*r

*V*r

0 ^{//}*X*^{2} ^{//}

*U*r ^{f} ^{//}

*f*2

*X* ^{//}0
0 ^{//}*X*^{1} ^{//}*E* ^{f}^{1} ^{//}*X* ^{//}0

Since *X*^{2} PCacp*A*q^{K}, the middle exact row provides a desired approximation sequence. A
similar method yields another approximation sequence. Thus the pair forms a resolving
cotorsion pair.

Dually we can show that the pair in the righthand side also forms a resolving cotorsion pair.

Proposition 4.4 shows Cacp*A*q “ ConepCacp*V*q*,*Cacp*U*qq “ CoConepCacp*V*q*,*Cacp*U*qq. We
have thus concluded that the given pair is a Hovey twin cotorsion pair.

A similar method given in the latter part of the proof of Proposition 4.2 shows that *W*

is the class of quasi-isomorphism. □

In the case of p*U,V*q “ pMod*A,*Inj*A*q, the obtained model structures in Theorem 4.5 is
nothing other than the injective model structures. Dually the projective model structures
correspond to the cotorsion pair pProj*A,*Mod*A*q.

**Remark 4.6.** (1) The admissible model structure given in Theorem 4.5, in practice,
satis