1 Preliminaries on model categories

We largely follow the conventions of Hovey’s book [Reference Hovey19]. A model category is a locally small and bicomplete category ${\mathcal {C}}$ endowed with a model structure consisting of three subcategories – whose morphisms are called the weak equivalences, cofibrations and fibrations, respectively – and two functorial factorizations as in [Reference Hovey19], Definition 1.1.3. As usual, a cofibration (resp. a fibration) is called trivial if it is also a weak equivalence.

An object $X\in {\mathcal {C}}$ is called cofibrant (resp. fibrant) if the map from the initial object to X (resp. from X to the terminal object) is a cofibration (resp. a fibration). We denote by ${\mathcal {C}}_c$ (resp. ${\mathcal {C}}_f$ , resp. ${\mathcal {C}}_{\mathit {cf}}$ ) the full subcategory of ${\mathcal {C}}$ consisting of all objects which are cofibrant (resp. fibrant, resp. cofibrant and fibrant). By

$$ \begin{align*} Q_c:{\mathcal{C}}\to{\mathcal{C}}_c\qquad\mbox{and}\qquad Q_f:{\mathcal{C}}\to{\mathcal{C}}_f \end{align*} $$

we denote the cofibrant replacement functor and the fibrant replacement functor, respectively (cf. [Reference Hovey19], page 5). The cofibrant replacement functor will often be abbreviated to $Q_c=Q$ .

A path object for an object $B\in {\mathcal {C}}$ is an object $P\in {\mathcal {C}}$ such that the diagonal morphism $\Delta :B\to B\times B$ admits a factorization $B\stackrel {i}{\longrightarrow }P\stackrel {p}{\longrightarrow }B\times B$ , where i is a weak equivalence and p is a fibration. We note that while path objects can be constructed via the functorial factorizations of $\Delta $ the above definition allows any factorization with the required properties.

If $f:A\to B$ and $g:A\to B$ are morphisms in ${\mathcal {C}}$ , then a right homotopy from f to g is a morphism $H:A\to P$ from A into a path object $B\stackrel {i}{\longrightarrow }P\stackrel {p}{\longrightarrow }B\times B$ for B such that $p_1 p H=f$ and $p_2 p H=g$ . Here, $p_1,p_2:B\times B\to B$ denote the two projections. The morphisms f and g are called right homotopic if there is a right homotopy from f to g.

There are also dual notions of cylinder objects and left homotopies as in [Reference Hovey19], Definition 1.2.4. Morphisms which are both right and left homotopic are called homotopic. If $X\in {\mathcal {C}}_c$ and $Y\in {\mathcal {C}}_f$ , then two morphisms $f,g:X\to Y$ are right homotopic if and only if they are left homotopic (cf. [Reference Hovey19], Proposition 1.2.5 (v)). Moreover, on the morphisms of ${\mathcal {C}}_{\mathit {cf}}$ the homotopy relation $\sim $ is an equivalence relation compatible with composition (cf. [Reference Hovey19], Corollary 1.2.7). Dividing out the homotopy relation on the level of morphisms leads to a category denoted ${\mathcal {C}}_{\mathit {cf}}/\!\sim $ .

The homotopy category ${\operatorname {Ho}}({\mathcal {C}})$ of a model category ${\mathcal {C}}$ is the localization of ${\mathcal {C}}$ with respect to the class of weak equivalences (cf. [Reference Hovey19], Definition 1.2.1). We will usually just write ${\mathcal {C}}\to {\operatorname {Ho}}({\mathcal {C}})$ for the canonical functor of ${\mathcal {C}}$ into its homotopy category. Any subcategory of ${\mathcal {C}}$ has an induced class of weak equivalences and the corresponding homotopy category is defined analogously. It is a fundamental result that the inclusion ${\mathcal {C}}_{\mathit {cf}}\to {\mathcal {C}}$ induces equivalences of categories

(3) $$ \begin{align} {\mathcal{C}}_{\mathit{cf}}/\!\sim\;\stackrel{\cong}{\longrightarrow}\;{\operatorname{Ho}}({\mathcal{C}}_{\mathit{cf}})\;\stackrel{\cong}{\longrightarrow}\;{\operatorname{Ho}}({\mathcal{C}}) \end{align} $$

(cf. [Reference Hovey19], Theorem 1.2.10 (i)). Given two objects $X,Y\in {\mathcal {C}}$ , the set of morphisms from X to Y in ${\operatorname {Ho}}({\mathcal {C}})$ is usually denoted by $[X,Y]$ .

Recall that a functor $F:{\mathcal {C}}\to {\mathcal {D}}$ (resp. $U:{\mathcal {D}}\to {\mathcal {C}}$ ) between model categories is called left Quillen (resp. right Quillen) if it admits a right (resp. left) adjoint and preserves cofibrations and trivial cofibrations (resp. fibrations and trivial fibrations). If $F:{\mathcal {C}}\rightleftarrows {\mathcal {D}}:U$ is an adjunction between model categories, then F is left Quillen if and only if U is right Quillen (cf. [Reference Hovey19], Lemma 1.3.4). In this case, we speak of a Quillen adjunction. A Quillen adjunction is called a Quillen equivalence if for all cofibrant $X\in {\mathcal {C}}$ and all fibrant $Y\in {\mathcal {D}}$ a morphism $FX\to Y$ is a weak equivalence in ${\mathcal {D}}$ if and only if the adjoint morphism $X\to UY$ is a weak equivalence in ${\mathcal {C}}$ (cf. [Reference Hovey19], Definition 1.3.12).

We refer to [Reference Hovey19], Proposition 1.3.13 and Corollary 1.3.16, for various characterizations of when a Quillen adjunction is a Quillen equivalence. At this point, we just recall how this can be seen on the level of homotopy categories. Note that the functors

$$ \begin{align*} {\mathcal{C}}\stackrel{Q_c}{\longrightarrow}{\mathcal{C}}_c\longrightarrow{\operatorname{Ho}}({\mathcal{C}}_c)\qquad\mbox{and}\qquad{\mathcal{C}}_c\stackrel{F}{\longrightarrow}{\mathcal{D}}\longrightarrow{\operatorname{Ho}}({\mathcal{D}}) \end{align*} $$

send weak equivalences to isomorphisms. Indeed, for the left functor this follows from the two-out-of-three property for weak equivalences. For the right functor, this follows from [Reference Hovey19], Lemma 1.1.12. By the universal property of localizations, there are induced functors ${\operatorname {Ho}}(Q_c):{\operatorname {Ho}}({\mathcal {C}})\to {\operatorname {Ho}}({\mathcal {C}}_c)$ and ${\operatorname {Ho}}(F):{\operatorname {Ho}}({\mathcal {C}}_c)\to {\operatorname {Ho}}({\mathcal {D}})$ . The left derived functor $LF:{\operatorname {Ho}}({\mathcal {C}})\to {\operatorname {Ho}}({\mathcal {D}})$ of F is then defined to be the composite

(4)

(cf. [Reference Hovey19], Definition 1.3.6). Similarly, one defines the right derived functor $RU:{\operatorname {Ho}}({\mathcal {D}})\to {\operatorname {Ho}}({\mathcal {C}})$ of U as the composite

(5)

In this situation, $LF$ is left adjoint to $RU$ (cf. [Reference Hovey19], Lemma 1.3.10). Moreover, F and U form a Quillen equivalence if and only if $LF$ and $RU$ are inverse equivalences of categories (cf. [Reference Hovey19], Proposition 1.3.13).

A powerful tool to construct model structures on an abelian category ${\mathcal {A}}$ was developed by Hovey in [Reference Hovey21]. It is based on a fixed proper class $\mathcal {P}$ of short exact sequences in ${\mathcal {A}}$ leading to relative extension groups ${\operatorname {Ext}}_{\mathcal {P}}^\bullet (X,Y)$ as in [Reference MacLane26], Chapter XII.4. An epimorphism $f:X\to Y$ in ${\mathcal {A}}$ is called a $\mathcal {P}$ -epimorphism if the short exact sequence

$$ \begin{align*} 0\longrightarrow\ker(f)\longrightarrow X\stackrel{f}{\longrightarrow}Y\longrightarrow 0 \end{align*} $$

belongs to $\mathcal {P}$ . The notion of a $\mathcal {P}$ -monomorphism is defined dually. Given classes ${\mathcal {C}}$ and ${\mathcal {F}}$ of objects of ${\mathcal {A}}$ consider the classes of objects

$$ \begin{align*} ^\perp{\mathcal{F}}=\{X\in{\mathcal{A}}\;|\;{\operatorname{Ext}}^1_{\mathcal{P}}(X,Y)=0\mbox{ for all }Y\in{\mathcal{F}}\} \end{align*} $$

and

$$ \begin{align*} {\mathcal{C}}^\perp=\{Y\in{\mathcal{A}}\;|\;{\operatorname{Ext}}^1_{\mathcal{P}}(X,Y)=0\mbox{ for all }X\in{\mathcal{C}}\}. \end{align*} $$

A cotorsion pair with respect to $\mathcal {P}$ is a pair $({\mathcal {C}},{\mathcal {F}})$ of classes of objects of ${\mathcal {A}}$ satisfying ${\mathcal {C}}={^\perp {\mathcal {F}}}$ and ${\mathcal {C}}^\perp ={\mathcal {F}}$ (cf. [Reference Hovey21], Definition 2.3). It is said to have enough functorial projectives if for any $X\in {\mathcal {A}}$ there is a $\mathcal {P}$ -epimorphism $f:Y\to X$ which is functorial in X and satisfies $Y\in {\mathcal {C}}$ and $\ker (f)\in {\mathcal {F}}$ . The existence of enough functorial injectives is defined dually. A cotorsion pair which has both enough functorial projectives and enough functorial injectives is called functorially complete.

For the definition of a model structure on ${\mathcal {A}}$ which is compatible with $\mathcal {P}$ , we refer to [Reference Hovey21], Definition 2.1. A subcategory of ${\mathcal {A}}$ is called $\mathcal {P}$ -thick if it is closed under retracts and satisfies the two-out-of-three property with respect to short exact sequences in $\mathcal {P}$ . If ${\mathcal {A}}$ is an abelian category endowed with the structure of a model category, then an object $X\in {\mathcal {A}}$ is called trivial if the morphism $0\to X$ (or equivalently the morphism $X\to 0$ ) is a weak equivalence. We denote by ${\mathcal {A}}_t$ the full subcategory of ${\mathcal {A}}$ consisting of all trivial objects. The following fundamental theorem is due to Hovey (cf. [Reference Hovey21], Theorem 2.2).

In the situation of Theorem 1.1 (ii) the definition of the corresponding cofibrations, fibrations and weak equivalences is given explicitly in [Reference Hovey21], Definition 5.1. If $\mathcal {P}$ is the class of all short exact sequences, then a model structure (resp. a model category) as in Theorem 1.1 is called abelian. An abelian model structure is called hereditary if the class of cofibrant objects is closed under taking kernels of epimorphisms and if the class of fibrant objects is closed under taking cokernels of monomorphisms.

For simplicity, we will often assume that all objects of our categories are small (cf. [Reference Hovey20], Definition A.1). For example, this holds in any Grothendieck category (cf. [Reference Hovey20], Proposition A.2) which is the main case of interest for us. We refer to [Reference Hovey21], Corollary 6.8, for a manageable criterion to check whether a given cotorsion pair on a Grothendieck category is functorially complete.

Let $\mathcal {I}$ be a class of morphisms in a category ${\mathcal {C}}$ . Recall that a morphism in ${\mathcal {C}}$ is called $\mathcal {I}$ -injective if it satisfies the right lifting property with respect to all morphisms in $\mathcal {I}$ in the sense of [Reference Hovey19], Definition 1.1.2. The class of these morphisms is denoted by $\mathcal {I}$ -inj. A morphism in ${\mathcal {C}}$ is called an $\mathcal {I}$ -cofibration if it satisfies the left lifting property with respect to all morphisms in $\mathcal {I}$ -inj. The class of these morphisms is denoted by $\mathcal {I}$ -cof.

For the definition of a model category ${\mathcal {C}}$ which is cofibrantly generated, we refer to [Reference Hovey19], Definition 2.1.17. If every object of ${\mathcal {C}}$ is small, then this means that there are sets $\mathcal {I}$ and $\mathcal {J}$ of morphisms of ${\mathcal {C}}$ such that the class of fibrations is equal to $\mathcal {J}$ -inj and the class of trivial fibrations is equal to $\mathcal {I}$ -inj. The elements of the sets $\mathcal {I}$ and $\mathcal {J}$ are called the generating cofibrations and the generating trivial cofibrations of ${\mathcal {C}}$ , respectively.

Assume that we have an adjunction $F:{\mathcal {C}}\rightleftarrows {\mathcal {D}}:U$ . If ${\mathcal {C}}$ is a model category and if ${\mathcal {D}}$ is bicomplete, then there is a natural candidate for an associated model structure on ${\mathcal {D}}$ . Namely, define a fibration (resp. a weak equivalence, resp. a trivial fibration) in ${\mathcal {D}}$ to be a morphism f in ${\mathcal {D}}$ such that $Uf$ is a fibration (resp. a weak equivalence, resp. a trivial fibration) in ${\mathcal {C}}$ . Moreover, define a cofibration in ${\mathcal {D}}$ to be a morphism which satisfies the left lifting property with respect to all trivial fibrations. If these three classes of morphisms form part of a model structure on ${\mathcal {D}}$ , then this is called the right transfer and we say that the right transfer exists. We note that the term right induction is common, as well.

In order to discuss the existence of the right transfer, we call a morphism in ${\mathcal {D}}$ anodyne if it satisfies the left lifting property with respect to all fibrations. If the right transfer exists, then an anodyne morphism is automatically a weak equivalence (cf. [Reference Hovey19], Lemma 1.1.10). In certain situations, this condition is also sufficient. Indeed, we have the following result which is a variant of [Reference Hirschhorn18], Theorem 11.3.2.

It might still be nontrivial to check that anodyne morphisms are weak equivalences. However, there is a simple condition under which this is true. Indeed, assume that ${\mathcal {C}}$ is a model category and that we have an adjunction $F:{\mathcal {C}}\rightleftarrows {\mathcal {D}}:U$ , where ${\mathcal {D}}$ is bicomplete. Since we have notions of fibrations and weak equivalences in ${\mathcal {D}}$ , we can make sense of path objects and right homotopies in ${\mathcal {D}}$ , as well. Moreover, since U preserves products, fibrations and weak equivalences it also preserves path objects and right homotopies. Due to a lack of reference, we include a proof of the following variant of [Reference Goerss and Schemmerhorn16], Theorem 3.8.

Proof. First, note that every object in ${\mathcal {D}}$ is fibrant because U preserves the terminal object $*$ . Let $j: A\to B$ be an anodyne morphism in ${\mathcal {D}}$ . By applying the left lifting property to the square

we get a morphism $u:B\to A$ such that $u j={\operatorname {id}}_A$ . Consider $f=j u:B\to B$ , and let P be a path object for B with factorization $\Delta =pi$ into a fibration p and a weak equivalence i. Since $u j={\operatorname {id}}_A$ , we have $f j=j$ . Therefore, the outer square in the diagram

commutes. The left lifting property implies the existence of $H:B\to P$ making the two triangles commute. By construction, H is a right homotopy from ${\operatorname {id}}_B$ to f. Since U preserves right homotopies, ${\operatorname {id}}_{UB}$ is right homotopic to $Uf=Uj Uu$ in ${\mathcal {C}}$ . Since right homotopic morphisms are identified in Ho $({\mathcal {C}})$ (cf. [Reference Hovey19], Theorem 1.2.10 (iii)) $Uu$ is right inverse to $Uj$ in Ho $({\mathcal {C}})$ . Of course, it is also left inverse because this already holds in ${\mathcal {C}}$ . Consequently, $Uj$ becomes an isomorphism in ${\operatorname {Ho}}({\mathcal {C}})$ , hence is a weak equivalence in ${\mathcal {C}}$ by [Reference Hovey19], Theorem 1.2.10 (iv). Thus, j is a weak equivalence in ${\mathcal {D}}$ .

A bicomplete category is called pointed if the morphism from the initial to the terminal object is an isomorphism. The homotopy category of any pointed model category ${\mathcal {C}}$ admits two endofunctors $\Sigma $ and $\Omega $ called the suspension functor and the loop functor, respectively. Given $X\in {\mathcal {C}}$ fibrant and P a path object for X, the loop $\Omega X$ of X is defined to be the pullback of the diagram

$$ \begin{align*}P\stackrel{p}{\longrightarrow}X\times X\longleftarrow *, \end{align*} $$

where $*$ denotes the initial/terminal object of ${\mathcal {C}}$ . Given $X\in {\mathcal {C}}$ cofibrant, its suspension $\Sigma X$ is defined dually using a cylinder object (cf. [Reference Barnes and Roitzheim1], Definition 3.1.2). The suspension functor $\Sigma $ is left adjoint to the loop functor $\Omega $ (cf. [Reference Barnes and Roitzheim1], Proposition 3.1.7). If they are equivalences of categories, then the model category ${\mathcal {C}}$ is called stable. If $F:{\mathcal {C}}\rightleftarrows \mathcal {D}:U$ is a Quillen adjunction, then $LF$ commutes with $\Sigma $ and $RU$ commutes with $\Omega $ (cf. [Reference Barnes and Roitzheim1], Corollary 3.1.4). If both model categories are stable, then also $RU$ commutes with $\Sigma $ and $LF$ commutes with $\Omega $ (cf. [Reference Barnes and Roitzheim1], Corollary 3.2.10).

To compute the suspension of an object X in an abelian model category ${\mathcal {A}}$ choose a trivial fibration $Y\to X$ with Y cofibrant. If $0\to Y\to Y'\to Y"\to 0$ is an exact sequence with $Y'$ trivially fibrant and $Y"$ cofibrant, then $\Sigma X\cong Y"$ in ${\operatorname {Ho}}({\mathcal {A}})$ . For the loop functor, one chooses a trivial cofibration $X\to Y$ with Y fibrant. If $0\to Y"\to Y'\to Y\to 0$ is an exact sequence in ${\mathcal {C}}$ with $Y'$ trivially cofibrant and $Y"$ fibrant, then $\Omega X\cong Y"$ in ${\operatorname {Ho}}({\mathcal {A}})$ . If an abelian model structure is hereditary, then it is also stable (cf. [Reference Becker3], Corollary 1.1.15).

Finally, we give two standard examples of a model category that will play a role in the following. Given an abelian category ${\mathcal {A}}$ , we denote by ${\operatorname {Ch}}({\mathcal {A}})$ the category of unbounded chain complexes

$$ \begin{align*} X=X_\bullet=(X_\bullet,d_\bullet)=(\ldots\stackrel{d_2}{\longrightarrow}A_1 \stackrel{d_1}{\longrightarrow}A_0\stackrel{d_0}{\longrightarrow}A_{-1} \stackrel{d_{-1}}{\longrightarrow}\ldots) \end{align*} $$

over ${\mathcal {A}}$ . Let $\iota _0:{\mathcal {A}}\to {\operatorname {Ch}}({\mathcal {A}})$ denote the functor obtained by viewing an object of ${\mathcal {A}}$ as a complex concentrated in degree zero. We will often use it to view ${\mathcal {A}}$ as a full subcategory of ${\operatorname {Ch}}({\mathcal {A}})$ and suppress it from the notation. Recall that $\iota _0$ admits both a left adjoint $Q_0$ and a right adjoint $Z_0$ given by

(6) $$ \begin{align} Q_0X={\operatorname{coker}}(d_1)\quad\mbox{and}\quad Z_0X=\ker(d_0). \end{align} $$

For any integer n, we denote by $X\mapsto B_nX={\operatorname {im}}(d_{n+1})$ the n-th boundary functor, by $X\mapsto Z_nX=\ker (d_n)$ the n-th cycle functor and by $X\mapsto H_nX=Z_nX/B_nX$ the n-th homology functor ${\operatorname {Ch}}({\mathcal {A}})\to {\mathcal {A}}$ . Let $[1]$ denote the shift functor on ${\operatorname {Ch}}({\mathcal {A}})$ defined by $(X_\bullet ,d_\bullet )[1]=(Y_\bullet ,e_\bullet )$ with $Y_n=X_{n+1}$ and $e_n=-d_{n+1}$ for all $n\in {\mathbb {Z}}$ . We denote by $[-1]$ the two-sided inverse of $[1]$ .

Note that together with ${\mathcal {A}}$ also the category ${\operatorname {Ch}}({\mathcal {A}})$ is abelian. Moreover, we have the following standard facts.

Proof. The abelian category ${\operatorname {Ch}}({\mathcal {A}})$ is cocomplete by defining colimits termwise. Filtered colimits are exact because kernels and cokernels are defined termwise, as well. If U is a generator of ${\mathcal {A}}$ consider the complex $D^nU=[0\to U\stackrel {{\operatorname {id}}}{\to }U\to 0]$ in ${\operatorname {Ch}}({\mathcal {A}})$ with U in degrees n and $n-1$ . Then $\bigoplus _{n\in {\mathbb {Z}}}D^nU$ is a generator of ${\operatorname {Ch}}({\mathcal {A}})$ . This shows (i).

As for (ii), let ${\mathcal {A}}^{\mathbb {Z}}$ denote the category of ${\mathbb {Z}}$ -graded objects of ${\mathcal {A}}$ and consider the functor $T:{\operatorname {Ch}}({\mathcal {A}})\to {\mathcal {A}}^{\mathbb {Z}}$ given by $X_\bullet \mapsto \bigoplus _{n\in {\mathbb {Z}}}X_n$ . It has the left adjoint D sending $\bigoplus _{n\in {\mathbb {Z}}}X_n$ to $\bigoplus _{n\in {\mathbb {Z}}}D^nX_n$ . In degree n, the map of complexes

$$ \begin{align*}D^nX_n\hookrightarrow DTX_\bullet\stackrel{\varepsilon_{X_\bullet}}{\longrightarrow}X_\bullet \end{align*} $$

is the identity on $X_n$ . This implies that the counit $\varepsilon _{X_\bullet }:DTX_\bullet \to X_\bullet $ is an epimorphism. Therefore, given $X_\bullet \in {\operatorname {Ch}}({\mathcal {A}})$ it suffices to construct an epimorphism $f:Q_\bullet \to DTX_\bullet $ , where $Q_\bullet \in {\operatorname {Ch}}({\mathcal {A}})$ is projective. For any n, we choose an epimorphism $Y_n\to X_n$ in ${\mathcal {A}}$ , where $Y_n$ is projective. Consider the epimorphism $g:\bigoplus _{n\in {\mathbb {Z}}}Y_n\to \bigoplus _{n\in {\mathbb {Z}}}X_n$ in ${\mathcal {A}}^{\mathbb {Z}}$ , and note that $\bigoplus _{n\in {\mathbb {Z}}}Y_n$ is a projective object in ${\mathcal {A}}^{\mathbb {Z}}$ . We claim that the induced map

$$ \begin{align*}f=Dg:Q_\bullet=D(\bigoplus_{n\in{\mathbb{Z}}}Y_n)\longrightarrow D(\bigoplus_{n\in{\mathbb{Z}}}X_n)=DTX_\bullet \end{align*} $$

in ${\operatorname {Ch}}({\mathcal {A}})$ is as required. Indeed, f is an epimorphism because D is a left adjoint and so preserves epimorphisms. Moreover, $Q_\bullet $ is projective because the right adjoint T is exact and hence D preserves projectives.

Recall that any Grothendieck category is automatically complete and therefore bicomplete. If ${\mathcal {A}}$ is a Grothendieck category, there is a cofibrantly generated model structure on ${\operatorname {Ch}}({\mathcal {A}})$ for which the cofibrations are the monomorphisms and the weak equivalences are the quasi-isomorphisms (cf. [Reference Beke4], Proposition 3.13). This is called the injective model structure on ${\operatorname {Ch}}({\mathcal {A}})$ . We write ${\operatorname {Ch}}({\mathcal {A}})^{\operatorname {Inj}}$ for the corresponding model category. Its homotopy category is equivalent to the unbounded derived category $D({\mathcal {A}})\cong {\operatorname {Ho}}({\operatorname {Ch}}({\mathcal {A}})^{\operatorname {Inj}})$ of ${\mathcal {A}}$ .

If ${\mathcal {A}}$ is the category of modules over a ring, then the injective model structure on ${\operatorname {Ch}}({\mathcal {A}})$ is also constructed in [Reference Hovey19], §2.3. It follows as in [Reference Hovey19], Proposition 2.3.20, that the fibrations are precisely the termwise split surjections with fibrant kernels. Looking at the long exact homology sequence, we see that the trivial fibrations are precisely the termwise split surjections with trivially fibrant kernels. Consequently, the injective model structure on ${\operatorname {Ch}}({\mathcal {A}})$ is compatible with the class $\mathcal {P}$ of all short exact sequences.

By Theorem 1.1, the injective model structure is related to two functorially complete cotorsion pairs on ${\operatorname {Ch}}({\mathcal {A}})$ given as follows. Note that every object of ${\operatorname {Ch}}({\mathcal {A}})^{\operatorname {Inj}}$ is cofibrant, that is, we have ${\mathcal {C}}={\operatorname {Ch}}({\mathcal {A}})^{\operatorname {Inj}}_c={\operatorname {Ch}}({\mathcal {A}})$ . Moreover, the class ${\mathcal {T}}={\operatorname {Ch}}({\mathcal {A}})^{\operatorname {Inj}}_t$ of trivial objects is precisely that of the acyclic complexes. If we denote by ${\mathcal {F}}={\operatorname {Ch}}({\mathcal {A}})^{\operatorname {Inj}}_f$ the class of fibrant objects, then we know from Theorem 1.1 that ${\mathcal {F}}={\mathcal {T}}^\perp $ . In other words, a complex Y is fibrant if and only if ${\operatorname {Ext}}_{{\operatorname {Ch}}({\mathcal {A}})}^1(X,Y)=0$ for any acyclic complex X. As in [Reference Enochs, Jenda and Xu11], Proposition 3.4, one shows that this condition is satisfied if and only if Y is a dg-injective complex, that is, Y is termwise injective and the internal hom complex $\underline {{\operatorname {Hom}}}(X,Y)$ is acyclic for any acyclic complex X.

Apart from the cotorsion pair $({\mathcal {C}}\cap {\mathcal {T}},{\mathcal {F}})=({\mathcal {T}},{\mathcal {T}}^\perp )$ , there is also the cotorsion pair $({\mathcal {C}},{\mathcal {T}}\cap {\mathcal {F}})=({\operatorname {Ch}}({\mathcal {A}}),{\mathcal {T}}\cap {\mathcal {T}}^\perp )$ . As in [Reference Enochs, Jenda and Xu11], Proposition 3.7, the class ${\mathcal {T}}\cap {\mathcal {T}}^\perp $ is that of the injective complexes, that is, of the injective objects of ${\operatorname {Ch}}({\mathcal {A}})$ . Recall that these are precisely the split acyclic complexes which are termwise injective.

Under suitable assumptions on ${\mathcal {A}}$ , there is a projective version of the above results. For our purposes, it will be sufficient to stick to the classical case that ${\mathcal {A}}={\operatorname {Mod}}(S)$ is the category of left modules over a ring S. In this case, we write ${\operatorname {Ch}}(S)={\operatorname {Ch}}({\operatorname {Mod}}(S))$ . According to [Reference Hovey19], Theorem 2.3.11, there is a cofibrantly generated model structure on ${\operatorname {Ch}}(S)$ for which the fibrations are the surjections and the weak equivalences are the quasi-isomorphisms. This is called the projective model structure on ${\operatorname {Ch}}(S)$ . We write ${\operatorname {Ch}}(S)^{\operatorname {Proj}}$ for the corresponding model category. Its homotopy category is again equivalent to the unbounded derived category $D(S)\cong {\operatorname {Ho}}({\operatorname {Ch}}(S)^{\operatorname {Proj}})$ of ${\operatorname {Mod}}(S)$ .

By [Reference Hovey19], Proposition 2.3.9, the cofibrations are precisely the termwise split monomorphisms with cofibrant cokernel. It follows that the projective model structure on ${\operatorname {Ch}}(S)$ is compatible with the class $\mathcal {P}$ of all short exact sequences. By Theorem 1.1 it is related to two functorially complete cotorsion pairs on ${\operatorname {Ch}}(S)$ . Note that every object of ${\operatorname {Ch}}(S)^{\operatorname {Proj}}$ is fibrant, that is, we have ${\mathcal {F}}={\operatorname {Ch}}(S)^{\operatorname {Proj}}_f={\operatorname {Ch}}(S)$ . Moreover, the class ${\mathcal {T}}={\operatorname {Ch}}(S)^{\operatorname {Proj}}_t$ of trivial objects is precisely that of the acyclic complexes. If we denote by ${\mathcal {C}}={\operatorname {Ch}}(S)^{\operatorname {Proj}}_c$ the class of cofibrant objects, then we know from Theorem 1.1 that ${\mathcal {C}}={^\perp {\mathcal {T}}}$ . In other words, a complex X is cofibrant if and only if ${\operatorname {Ext}}_{{\operatorname {Ch}}(S)}^1(X,Y)=0$ for any acyclic complex Y. As in [Reference Enochs, Jenda and Xu11], Proposition 3.5, one shows that this condition is satisfied if and only if X is a dg-projective complex, that is, X is termwise projective and the complex $\underline {{\operatorname {Hom}}}(X,Y)$ is acyclic for any acyclic complex Y.

Apart from the cotorsion pair $({\mathcal {C}},{\mathcal {T}}\cap {\mathcal {F}})=({^\perp {\mathcal {T}}},{\mathcal {T}})$ , there is also the cotorsion pair $({\mathcal {C}}\cap {\mathcal {T}},{\mathcal {F}})=({^\perp {\mathcal {T}}}\cap {\mathcal {T}},{\operatorname {Ch}}(S))$ . As in [Reference Enochs, Jenda and Xu11], Proposition 3.7, the class ${^\perp {\mathcal {T}}}\cap {\mathcal {T}}$ is that of the projective complexes, that is, of the projective objects of ${\operatorname {Ch}}(S)$ . Recall that these are precisely the split acyclic complexes which are termwise projective. In Spaltenstein’s terminology a complex is dg-projective if and only if it is K-projective and termwise projective.

Finally, it is clear from the above that ${\operatorname {id}}:{\operatorname {Ch}}(S)^{\operatorname {Proj}}\rightleftarrows {\operatorname {Ch}}(S)^{\operatorname {Inj}}:{\operatorname {id}}$ is a Quillen adjunction. Of course, it is even a Quillen equivalence because the classes of weak equivalences coincide. For both the projective and the injective model structure on ${\operatorname {Ch}}(S)$ , the suspension $\Sigma =L[-1]$ is the left derived functor of the shift functor $[-1]$ . Likewise, the loop functor $\Omega =R[1]$ is the right derived functor of the shift $[1]$ . In a different setting, an anologous result will be proven in Lemma 3.5 below.

2 Derived categories and relative homological algebra

Recall the following notions from relative homological algebra (cf. [Reference Christensen and Hovey8], §1). We continue to denote by ${\mathcal {A}}$ an abelian category and fix a class $\mathcal {P}$ of objects of ${\mathcal {A}}$ . Given any object P of ${\mathcal {A}}$ , we say that a morphism $f:A\to B$ in ${\mathcal {A}}$ is P-epic or a P-epimorphism if the induced map

$$ \begin{align*}{\operatorname{Hom}}_{{\mathcal{A}}}(P,A)\to {\operatorname{Hom}}_{{\mathcal{A}}}(P,B) \end{align*} $$

is surjective. Furthermore, we say that f is $\mathcal {P}$ -epic if it is P-epic for all $P\in \mathcal {P}$ . A sequence $A\overset {f}{\longrightarrow }B\overset {g}{\longrightarrow }C$ is called P-exact if $gf=0$ and if the induced sequence ${\operatorname {Hom}}_{{\mathcal {A}}}(P,A)\to {\operatorname {Hom}}_{{\mathcal {A}}}(P,B)\to {\operatorname {Hom}}_{{\mathcal {A}}}(P,C)$ of abelian groups is exact. The sequence $A\to B\to C$ is called $\mathcal {P}$ -exact if it is P-exact for all $P\in \mathcal {P}$ .

A $\mathcal {P}$ -resolution of an object $B\in {\mathcal {A}}$ is a $\mathcal {P}$ -exact sequence $P_\bullet \to B\to 0$ in ${\mathcal {A}}$ such that $P_n\in \mathcal {P}$ for all $n\geq 0$ . Given another object $C\in {\mathcal {A}}$ the group of relative extensions of C and B are defined by

(7) $$ \begin{align} {\operatorname{Ext}}_{\mathcal{P}}^n(B,C)=H^n{\operatorname{Hom}}_{\mathcal{A}}(P_\bullet,C)\quad\mbox{for all }n\geq 0. \end{align} $$

Up to isomorphism, they are independent of the chosen $\mathcal {P}$ -resolution of B and any short $\mathcal {P}$ -exact sequence $0\to B'\to B\to B"\to 0$ gives rise to a long exact sequence between relative extension groups, as usual.

Suppose now that we have an adjunction $F:\mathcal {B}\rightleftarrows {\mathcal {A}}:U$ between abelian categories. Moreover, assume that $(\mathcal {P}',\mathcal {E}')$ is a projective class on $\mathcal {B}$ . Recall that a retract of an object $B\in {\mathcal {A}}$ is an object $A\in {\mathcal {A}}$ admitting a right invertible morphism $B\to A$ . Since ${\mathcal {A}}$ is abelian, this is equivalent to A being isomorphic to a direct summand of B. We define the pullback of the pair $(\mathcal {P}',\mathcal {E}')$ along U to be the pair $(\mathcal {P},\mathcal {E})$ , where

• • $\mathcal {P}=\{\text {retracts of }FP\;|\;P\in \mathcal {P}'\}$ and

• • $\mathcal {E}=\{f:B\to C\;|\;Uf\in \mathcal {E}'\}$ .

The proof of the following result is straightforward and left to the reader.

For the rest of this article, we fix a field k of positive characteristic p and denote by G a topological group admitting an open subgroup I which is pro-p. Let ${\operatorname {Rep}}^\infty _k(G)$ denote the category of smooth k-linear representations of G, that is, of k-vector spaces endowed with a k-linear action of G such that the stabilizer of any vector is open in G. Its morphisms are the k-linear and G-equivariant maps. Given $V,W\in {\operatorname {Rep}}^\infty _k(G)$ , we will write ${\operatorname {Hom}}_G(V,W)$ for the set of morphisms from V to W in ${\operatorname {Rep}}^\infty _k(G)$ . We explicitly allow G to be discrete. In this case, the category ${\operatorname {Rep}}^\infty _k(G)={\operatorname {Mod}}(k[G])$ is that of all k-linear representations of G.

Given $V\in {\operatorname {Rep}}^\infty _k(I)$ , we denote by ${\operatorname {ind}}_I^G(V)\in {\operatorname {Rep}}^\infty _k(G)$ the compact induction of V from I to G, that is, the k-vector space of all compactly supported maps $f:G\to V$ satisfying $f(gi)=i^{-1}f(g)$ for all $i\in I$ and $g\in G$ . The G-action on ${\operatorname {ind}}_I^G(V)$ is given by $(gf)(g')=f(g^{-1}g')$ for all $g,g'\in G$ . We endow k with the trivial action of I and set ${\mathbb {X}} ={\operatorname {ind}}_I^G(k)\in {\operatorname {Rep}}^\infty _k(G)$ .

Setting $H={\operatorname {End}}_G({\mathbb {X}} )^{\operatorname {op}}$ , Frobenius reciprocity shows that we have adjoint functors

(8) $$ \begin{align} F:{\operatorname{Mod}}(H)\rightleftarrows {\operatorname{Rep}}_k^\infty(G):U \end{align} $$

given by $FM={\mathbb {X}} \otimes _HM$ and $UV=V^I\cong {\operatorname {Hom}}_G({\mathbb {X}} ,V)$ . Since k has characteristic p, the functor U has the property that $UV\neq 0$ for all $0\neq V\in {\operatorname {Rep}}_k^\infty (G)$ (cf. [Reference Schneider34], Lemma 1 (v)). Since U is also left exact, it reflects monomorphisms.

For $V\in {\operatorname {Rep}}_k^\infty (G)$ , the counit of the adjunction (8) is the G-equivariant map

(9) $$ \begin{align} \varepsilon_V: FUV={\mathbb{X}} \otimes_H V^I \to V,\quad 1_{gI}\otimes v \mapsto g v, \end{align} $$

where $1_{gI}$ denotes the characteristic function of the coset $gI$ . Moreover, given $M\in {\operatorname {Mod}}(H)$ , the unit $\eta _M$ is the H-linear map

(10) $$ \begin{align} \eta_M:M\to UFM=({\mathbb{X}} \otimes_H M)^I,\quad m\mapsto 1_I\otimes m. \end{align} $$

As seen in Example 2.2, we have the standard projective class $(\mathcal {P}',\mathcal {E}')$ on ${\operatorname {Mod}}(H)$ , where $\mathcal {P}'$ is the class of all projective H-modules and $\mathcal {E}'$ is the class of all surjective H-linear maps.

We use the unit $\eta _H$ to identify $H\overset {\cong }{\longrightarrow }UFH\cong {\mathbb {X}} ^I$ as $(H,H)$ -bimodules. Explicitly, this map is given by $h\mapsto 1_I \cdot h$ . The terminology in Definition 2.6 is justified by the following result.

Given $V\in {\operatorname {Rep}}^\infty _k(G)$ , Lemma 2.4 guarantees the existence of an I-epimorphism $W\to V$ , where W is I-projective. This can be constructed in both an explicit and functorial way by considering the H-linear surjection $H\otimes _kUV\to UV$ . Applying the functor $F={\mathbb {X}} \otimes _H(\cdot )$ and composing with the counit $\varepsilon _V$ gives the functorial G-equivariant map ${\mathbb {X}} \otimes _kUV\to V$ . Here, the left-hand side is even I-free. Moreover, on I-invariants, we get back the surjection $H\otimes _kUV\to UV$ we started with. Thus, the map ${\mathbb {X}} \otimes _kUV\to V$ is an I-epimorphism. Its image is the G-subrepresentation of V generated by $UV=V^I$ . Iterating this process with the kernel of ${\mathbb {X}} \otimes _kUV\to V$ , we see that any representation admits a functorial I-resolution.

Let $(\mathcal {P}, \mathcal {E})$ be a projective class on an abelian category ${\mathcal {A}}$ . As in [Reference Christensen and Hovey8], Definition 2.1, there is a candidate for an associated model structure on the category ${\operatorname {Ch}}({\mathcal {A}})$ of unbounded chain complexes over ${\mathcal {A}}$ . If this model structure is well defined, we will call it the $\mathcal {P}$ -projective model structure. For the I-projective class on ${\operatorname {Rep}}^\infty _k(G)$ , the definition is as follows. For ease of notation, we set ${\operatorname {Ch}}(G)={\operatorname {Ch}}({\operatorname {Rep}}^\infty _k(G))$ and write ${\operatorname {Hom}}_G(V_\bullet ,W_\bullet )$ for the set of morphisms in this category.

Note that the adjunction $F:{\operatorname {Mod}}(H)\rightleftarrows {\operatorname {Rep}}^\infty _k(G):U$ extends to an adjunction $F:{\operatorname {Ch}}(H)\rightleftarrows {\operatorname {Ch}}(G):U$ between the respective categories of complexes by working termwise.

Our first aim is to show that Definition 2.10 indeed gives a model structure on ${\operatorname {Ch}}(G)$ whose weak equivalences (resp. cofibrations, resp. fibrations) are the I-equivalences (resp. I-cofibrations, resp. I-fibrations). To this end, we apply the following result of Christensen and Hovey (cf. [Reference Christensen and Hovey8], Theorem 5.1).

As an application, we obtain the following result.

We will call the above model structure on ${\operatorname {Ch}}(G)$ the I-projective model structure. Its existence can also be deduced from the general results in the previous section. From Lemma 2.11, we see that the I-projective model structure on ${\operatorname {Ch}}(G)$ is the right transfer of the projective model structure on ${\operatorname {Ch}}(H)$ . Instead of Theorem 2.12, we could thus have referred to Theorem 1.2 and Proposition 1.4 by constructing suitable path objects in ${\operatorname {Ch}}(G)$ .

Proposition 2.14. The adjoint functors $F:{\operatorname {Ch}}(H)\rightleftarrows {\operatorname {Ch}}(G):U$ given by $FM_\bullet ={\mathbb {X}} \otimes _H M_\bullet $ and $UV_\bullet =V_\bullet ^I$ form a Quillen equivalence with respect to the I-projective model structure on ${\operatorname {Ch}}(G)$ and the projective model structure on ${\operatorname {Ch}}(H)$ . In particular, the derived adjunction

$$ \begin{align*} LF:D(H)\rightleftarrows {\operatorname{Ho}}({\operatorname{Ch}}(G)):RU \end{align*} $$

is an equivalence between the unbounded derived category of ${\operatorname {Mod}}(H)$ and the homotopy category of the I-projective model structure on ${\operatorname {Ch}}(G)$ .

In order to compute the left derived functor $LF$ in Proposition 2.14, let $M_\bullet \in {\operatorname {Ch}}(H)$ and let $Q_\bullet $ be any dg-projective complex of H-modules which is quasi-isomorphic to $M_\bullet $ . Then $LFM_\bullet \cong {\mathbb {X}} \otimes _HQ_\bullet $ in ${\operatorname {Ho}}({\operatorname {Ch}}(G))$ for the I-projective model structure on ${\operatorname {Ch}}(G)$ . To compute the right derived functor $RU$ , let $V_\bullet \in {\operatorname {Ch}}(G)$ . Since every object of ${\operatorname {Ch}}(G)$ is fibrant for the I-projective model structure we have $RUV_\bullet \cong UV_\bullet $ in $D(H)$ .

The results in Proposition 2.14 admit important improvements. To see this, we start with the following characterization of the I-cofibrant objects in ${\operatorname {Ch}}(G)$ . Using [Reference Christensen and Hovey8], Proposition 2.5, this gives a description of the I-cofibrations in general. Note that even if a Quillen adjunction $F:{\mathcal {C}}\rightleftarrows {\mathcal {D}}:U$ is a Quillen equivalence the right adjoint U will generally not preserve cofibrant objects. The adjunction ${\operatorname {id}}:{\operatorname {Ch}}(S)^{\operatorname {Proj}}\rightleftarrows {\operatorname {Ch}}(S)^{\operatorname {Inj}}:{\operatorname {id}}$ gives an easy example. However, our situation is much more special.

Proof. Let us put $Y=V_\bullet $ and denote by $q_{UY}:QUY\to UY$ the cofibrant replacement of $UY=V_\bullet ^I$ . The adjoint morphism $\varepsilon _Y Fq_{UY}:FQUY\to Y$ is an I-equivalence by Proposition 2.14. We claim it is a trivial I-fibration. To see this, we need to check that we get a trivial fibration in ${\operatorname {Ch}}(H)$ if we apply the functor U. Consider the commutative diagram

As seen in the proof of Proposition 2.14, the left vertical arrow is an isomorphism because $QUY$ is cofibrant. Since $q_{UY}$ is a trivial fibration, the claim follows.

Assuming that Y is I-cofibrant it follows that the map $FQUY\to Y$ splits. Thus, Y is a direct summand of $FQUY$ and $UY$ is a direct summand of $UFQUY\cong QUY$ . This implies that $UY$ is a cofibrant object of ${\operatorname {Ch}}(H)$ , that is, is dg-projective. Moreover, Lemma 2.8 shows that $FQUY$ is termwise I-projective. The same is then true of its direct summand Y. In particular, Y is termwise generated by its I-invariants.

Conversely, assume that Y satisfies the conditions in (ii). Then $UY$ is dg-projective and hence is termwise projective. It follows from Lemma 2.8 that Y is termwise I-projective and that the counit $\varepsilon _Y:FUY\to Y$ is an isomorphism. Since $FUY$ is I-cofibrant by definition of a left Quillen functor it follows that $Y\cong FUY$ is I-cofibrant.

We obtain the following significant strengthening of Proposition 2.14. It says that the functors F and U restrict to equivalences between the cofibrant-fibrant objects even before dividing out the homotopy relation. Note that in the situation of Proposition 2.14 all objects are fibrant. We may therefore work with the respective classes of cofibrant objects.

For complexes concentrated in degree zero, this reflects the fact that F and U restrict to inverse equivalences between the category of projective H-modules and the category of I-projective G-representations, respectively (cf. Corollary 2.9). Given $V,W\in {\operatorname {Rep}}^\infty _k(G)$ , we denote by ${\operatorname {Ext}}^n_{G,I}(V,W)$ the relative extension groups defined in (7) formed with respect to the I-projective class in Definition 2.6. The following result is a formal consequence of Proposition 2.14.

Proof. Denote by $\mathcal {D}={\operatorname {Ho}}({\operatorname {Ch}}(G))$ the homotopy category of ${\operatorname {Ch}}(G)$ endowed with the I-projective model structure. By Proposition 2.14 and [Reference Christensen and Hovey8], Corollary 2.14, we have

$$ \begin{align*} {\operatorname{Ext}}^n_{G,I}(V,W)&\cong{\operatorname{Hom}}_{\mathcal{D}}(V,W[-n])\cong{\operatorname{Hom}}_{D(H)}(UV,UW[-n])\\ &\cong{\operatorname{Ext}}^n_H(UV,UW) \end{align*} $$

functorially in V and W for all $n\geq 0$ .

The existence of the I-cofibrant replacement shows that if $V_\bullet \in {\operatorname {Ch}}(G)$ , then there is a trivial I-fibration $f:Q_\bullet \to V_\bullet $ where $Q_\bullet $ is I-cofibrant. By Lemma 2.11 and Lemma 2.15, this means that $Uf:UQ_\bullet \to U V_\bullet $ is a surjective quasi-isomorphism where $UQ_\bullet \in {\operatorname {Ch}}(H)$ is dg-projective. It will be useful to have the following slight variant of this fact generalizing the existence of enough I-projectives in ${\operatorname {Rep}}^\infty _k(G)$ discussed earlier.

Proof. By Lemma 1.7 (ii), there is a surjection $g:P_\bullet \to UV_\bullet $ in ${\operatorname {Ch}}(H)$ where $P_\bullet \in {\operatorname {Ch}}(H)$ is projective. Setting $Q_\bullet =FP_\bullet \in {\operatorname {Ch}}(G)$ , the unit of the adjunction $\eta _{P_\bullet }:P_\bullet \to UQ_\bullet =UFP_\bullet $ is an isomorphism by Lemma 2.5. Therefore, $UQ_\bullet \in {\operatorname {Ch}}(H)$ is projective. Moreover, the map

$$ \begin{align*}f:Q_\bullet=FP_\bullet\stackrel{Fg}{\longrightarrow}FUV_\bullet\stackrel{\varepsilon_{V_\bullet}}{\longrightarrow}V_\bullet \end{align*} $$

is an I-fibration by Lemma 2.11 because $Uf\eta _{P_\bullet }=U\varepsilon _{V_\bullet }UFg\eta _{P_\bullet }=g$ is surjective.

For the sake of completeness, we note that it is also possible to start with a model structure on ${\operatorname {Ch}}(G)$ and apply a left transfer construction to get a model structure on ${\operatorname {Ch}}(H)$ . However, in general it does not make sense to talk about a projective model structure on ${\operatorname {Ch}}(G)$ unless G is discrete.

Instead, we consider the injective model structure on ${\operatorname {Ch}}(G)$ introduced in §1 and show that it admits a left transfer to ${\operatorname {Ch}}(H)$ in the following sense.

In analogy to the right transfer, the left transfer has an apparent ambiguity concerning the notion of a trivial fibration. This could either be a morphism satisfying the right lifting property with respect to all cofibrations or it could be a fibration which is also a weak equivalence. We need to see that these two classes of morphisms coincide. To distinguish them, we call a map f in ${\operatorname {Ch}}(H)$ coanodyne if it satisfies the right lifting property with respect to all cofibrations in the sense of Proposition 2.21.

Remark 2.22. Note that any coanodyne map $f: X\to Y$ in ${\operatorname {Ch}}(H)$ is a split epimorphism. Indeed, the map $0\to Y$ is a cofibration for the left transfer. By applying the right lifting property to

we obtain the desired splitting.

If the left transfer is a model structure, then the coanodyne morphisms are precisely the fibrations which are also weak equivalences. Moreover, one needs to show that functorial factorizations exist. In general, this involves serious set-theoretic issues. However, the following result shows that under certain assumptions on the categories these problems are easy to solve (cf. [Reference Bayeh, Hess, Karpova, Kędziorek, Riehl and Shipley2], Theorem 2.23).

We note that together with ${\operatorname {Mod}}(H)$ and ${\operatorname {Rep}}^\infty _k(G)$ also the categories ${\operatorname {Ch}}(H)$ and ${\operatorname {Ch}}(G)$ are Grothendieck categories (cf. Lemma 1.7) and hence are locally presentable (cf. [Reference Beke4], Proposition 3.10).

Proof. Let $T_\bullet $ be the mapping cone of ${\operatorname {id}}_{K_\bullet }$ defined by $T_n=K_n\oplus K_{n-1}$ with differential $d(x,y)=(dx+y, -dy)$ . Consider the canonical map $\iota _\bullet :K_\bullet \to T_\bullet $ given by $\iota (x)=(x,0)$ . Since this is a termwise split injection so is $F\iota _\bullet $ . In particular, $F\iota _\bullet $ is a cofibration and so is $\iota _\bullet $ by definition of the left transfer. Since $f_\bullet $ is coanodyne, we may apply the right lifting property to the square

and obtain a map $g_\bullet :T_\bullet \to P_\bullet $ making the two triangles commute. This implies $f_\bullet g_\bullet =0$ whence $g_\bullet $ has image in $K_\bullet $ . We may therefore view $g_\bullet $ as a map $T_\bullet \to K_\bullet $ with the property that $g_\bullet \iota _\bullet ={\operatorname {id}}_{K_\bullet }$ . For any $n\in {\mathbb {Z}}$ , this allows us to write $g_n(x,y)=x+s_ny$ for some H-linear map $s_n:K_{n-1}\to K_n$ . Let $x\in K_{n+1}$ and $y\in K_n$ . Since $g_\bullet $ is a chain map, we have

$$ \begin{align*} d_{n+1}x+y-s_nd_ny&=g_n(d_{n+1}x+y, -d_ny)\\ &=g_nd_{n+1}(x,y)\\ &=d_ng_{n+1}(x,y)\\ &=d_{n+1}x+d_ns_{n+1}y \end{align*} $$

and hence $y=s_nd_ny+d_ns_{n+1}y$ . Thus, $s_\bullet $ defines a chain homotopy between the zero map and the identity map on $K_\bullet $ . By functoriality, $F s_\bullet $ is then a chain homotopy between the zero map and the identity on $FK_\bullet $ .

The model structure on ${\operatorname {Ch}}(H)$ constructed in Proposition 2.21 will be called the G-injective model structure. It makes $F:{\operatorname {Ch}}(H)\rightleftarrows {\operatorname {Ch}}(G):U$ a Quillen adjunction. We hope to give an in-depth study of this adjunction elsewhere.

3 Gorenstein projective model structures

Let $\mathcal {A}$ be an abelian category with enough projectives. An object $A\in \mathcal {A}$ is called Gorenstein projective if there is an acyclic complex

$$ \begin{align*} Y = \cdots\longrightarrow Y_1\stackrel{d_1}{\longrightarrow}Y_0\stackrel{d_0}{\longrightarrow} Y_{-1}\longrightarrow\cdots \end{align*} $$

of projective objects of $\mathcal {A}$ for which $A=Z_0Y=B_0Y$ and which remains exact upon applying ${\operatorname {Hom}}_{\mathcal {A}}(\cdot ,P)$ for any projective object P of $\mathcal {A}$ .

A ring S is called Gorenstein if it is both left and right noetherian and if it has finite injective dimension both as a left and as a right S-module. In this case, the left and right selfinjective dimensions of S coincide (cf. [Reference Enochs and Jenda9], Proposition 9.1.8). If n denotes their common value, then S is called an n-Gorenstein ring.

If S is an n-Gorenstein ring, then an S-module M has finite projective dimension if and only if it has finite injective dimension. In this case, both dimensions are bounded above by n (cf. [Reference Enochs and Jenda9], Theorem 9.1.10). Moreover, M is Gorenstein projective as an object of ${\operatorname {Mod}}(S)$ if and only if ${\operatorname {Ext}}^i_S(M,P)=0$ for all $i\geq 1$ and all projective S-modules P (cf. [Reference Enochs and Jenda9], Corollary 11.5.3). This is true if and only if ${\operatorname {Ext}}^i_S(M,P)=0$ for all $i\geq 1$ and all modules P of finite projective dimension.

The following fundamental theorem is due to Hovey (cf. [Reference Hovey21], Theorem 8.6).

We call this the Gorenstein projective model structure on ${\operatorname {Mod}}(S)$ . The class ${\mathcal {C}}$ (resp. ${\mathcal {F}}$ , resp. ${\mathcal {T}}$ ) of cofibrant (resp. fibrant, resp. trivial) objects is that of Gorenstein projective modules (resp. all modules, resp. modules of finite projective dimension). By [Reference Hovey21], Corollary 8.5, the class ${\mathcal {C}}\cap {\mathcal {T}}$ is that of projective S-modules. The subcategory ${\mathcal {T}}$ of ${\operatorname {Mod}}(S)$ is thick and both $({\mathcal {C}}\cap {\mathcal {T}},{\mathcal {F}})$ and $({\mathcal {C}},{\mathcal {F}}\cap {\mathcal {T}})$ are functorially complete cotorsion pairs. Letting $\mathcal {P}$ denote the class of all short exact sequences in ${\operatorname {Mod}}(S)$ the Gorenstein projective model structure on ${\operatorname {Mod}}(S)$ is obtained from these data as in Theorem 1.1 (ii). Since all objects are fibrant and since the class of Gorenstein projective modules is closed under kernels of epimorphisms, the Gorenstein projective model structure on ${\operatorname {Mod}}(S)$ is abelian and hereditary. In particular, it is stable which is also proven directly in [Reference Hovey21], Theorem 9.3.

Assume that S is Gorenstein. Given an S-module M the cofibrant replacement functor Q for the Gorenstein projective model structure on ${\operatorname {Mod}}(S)$ gives a functorial exact sequence

$$ \begin{align*} 0\longrightarrow KM\longrightarrow QM\longrightarrow M\longrightarrow 0 \end{align*} $$

in which $QM$ is Gorenstein projective and $KM$ has finite projective dimension. Let us recall from [Reference Enochs and Jenda9], Theorem 11.5.1, or [Reference Zhang39], Theorem 3.5, how to construct such an exact sequence at least nonfunctorially.

Denote by n the self-injective dimension of S. One first chooses a projective resolution $Q_\bullet \to M\to 0$ of M and sets $G={\operatorname {coker}}(Q_{n+1}\to Q_n)$ . Then G is Gorenstein projective by [Reference Enochs and Jenda9], Theorem 10.2.14. By definition, G admits a complete projective resolution $P_\bullet $ as above. By shifting, we see that also $N={\operatorname {im}}(P_{-n}\to P_{-n-1})=\ker (P_{-n-1}\to P_{-n-2})$ is Gorenstein projective. Moreover, the exact sequence $0\to G\to P_{-1}\to \cdots \to P_{-n}\to N\to 0$ is ${\operatorname {Hom}}_S(\cdot ,Q)$ -acyclic for any projective S-module Q. Therefore, we can inductively choose S-linear maps $P_i\to Q_{n+i}$ to obtain a commutative diagram

The mapping cone of the vertical homomorphism yields an exact sequence $0\to P_{-1}\to P_{-2}\oplus Q_{n-1}\to \cdots \to P_{-n}\oplus Q_1\to N\oplus Q_0\to M\to 0$ . Here, $N\oplus Q_0$ is Gorenstein projective and the kernel of the epimorphism to M has projective dimension at most $n-1$ .

We point out that Theorem 3.2 admits a variant which is based on the dual notion of a Gorenstein injective module (cf. [Reference Hovey21], Theorem 8.4). The corresponding homotopy category is equivalent to the quotient $\mathrm {GInj}(S)/\mathrm {Inj}(S)$ of the category $\mathrm {GInj}(S)$ of Gorenstein injective modules by the category $\mathrm {Inj}(S)$ of injective S-modules (cf. [Reference Hovey21], Proposition 9.1 and Proposition 9.2). It was shown by Krause in [Reference Krause24], Proposition 7.13, that the latter is equivalent to the singularity category $K_{\operatorname {ac}}(\mathrm {Inj}(S))$ of S, that is, to the category of acyclic complexes of injective S-modules up to chain homotopy. From the model categorical point of view this was taken up by Becker in [Reference Becker3] who again treats the projective and the injective situation simultaneously. We point out that results of this form have their origin in the seminal but unpublished work [Reference Buchweitz5] of Buchweitz. We refer to the end of [Reference Krause24], §7, for a more comprehensive list of historical remarks.

We call this the singular projective model structure on ${\operatorname {Ch}}(S)$ . Its existence can also be deduced from the general result [Reference Enochs and Jenda10], Theorem 7.2.15, of Enochs and Jenda. The class ${\mathcal {C}}$ (resp. ${\mathcal {F}}$ ) of cofibrant (resp. fibrant) objects consists of the acyclic and termwise projective (resp. all) complexes. The class ${\mathcal {T}}$ of trivial objects is given by the condition ${\mathcal {T}}={\mathcal {C}}^\perp $ as in Theorem 3.4 (i). The singular projective model structure is obtained from ${\mathcal {C}}$ , ${\mathcal {F}}$ and ${\mathcal {T}}$ as in Theorem 1.1 (ii) where we let $\mathcal {P}$ denote the class of all short exact sequences in ${\operatorname {Ch}}(S)$ . Again, this is an abelian hereditary model structure and the associated model category is stable (cf. [Reference Becker3], Corollary 1.1.15). The loop and suspension functors can be made explicit as follows.

Note that if S is a Gorenstein ring, then the Gorenstein projective model structure on ${\operatorname {Mod}}(S)$ is Quillen equivalent to its Gorenstein injective counterpart (cf. [Reference Hovey21], page 583). It follows from the results of Becker that the corresponding homotopy category is also equivalent to the category $K_{\operatorname {ac}}(\mathrm {Inj}(S))$ of complexes of injective S-modules up to chain homotopy (cf. [Reference Becker3], Proposition 2.2.1 (2) and Proposition 3.1.5).

By a result of Krause, the full subcategory of compact objects of $K_{\operatorname {ac}}(\mathrm {Inj}(S))$ is equivalent to the quotient of the bounded derived category $D^b(\mathrm {noeth}(S))$ of noetherian S-modules by the subcategory of perfect complexes (cf. [Reference Krause24], Corollary 5.4 and [36], 07LT). Quotients of this form are usually called singularity categories in algebraic geometry. This is where the name for the above model structure derives from. We would also like to emphasize that the results of [Reference Becker3] and [Reference Krause24] are much more precise in the sense that they are derived from the existence of various recollements between triangulated categories.

4 The right transfer to smooth G-representations

We continue to denote by k a field of characteristic p and by G a topological group admitting an open subgroup I which is pro-p. We wish to apply the results of §3 to the ring $S=H={\operatorname {End}}_G({\mathbb {X}})^{\operatorname {op}}$ . In the following, we will therefore often make the assumption that the ring H is Gorenstein.

In all of the above examples, the characteristic of k may in fact be arbitrary. However, we will continue to assume that k is of characteristic p. In the situation of Example 4.1 (iv), the group $I=I_C$ is a called a pro-p Iwahori subgroup of $G=\mathbb {G}(K)$ and H is called the corresponding pro-p Iwahori-Hecke algebra over k. Since this is the motivating example for our work, we decided to stick to this terminology in general.

For the following result, recall that we denote by $B_0$ the $0$ -th boundary functor introduced in §1.

Proof. Factorize $d_1$ into $Y_1\twoheadrightarrow B_0Y\hookrightarrow Y_0$ , and apply the functor F. This gives a factorization $FY_1\stackrel {g}{\twoheadrightarrow }FB_0Y\stackrel {h}{\to }FY_0$ of $Fd_1$ with ${\operatorname {im}}(h)={\operatorname {im}}(hg)={\operatorname {im}}(Fd_1)=B_0FY$ . Since g is surjective, we have $\ker (h)=g(\ker (hg))=g(\ker (Fd_1))=g(Z_1FY)$ . However, the exact sequence $Y_2\to Y_1\to B_0Y\to 0$ gives the exact sequence $FY_2\stackrel {Fd_2}{\to }FY_1\stackrel {g}{\to }FB_0Y\to 0$ . This shows that $\ker (g)={\operatorname {im}}(Fd_2)=B_1FY$ . Therefore, g induces an isomorphism $H_1FY=Z_1FY/B_1FY=Z_1FY/\ker (g)\cong g(Z_1FY)=\ker (h)$ , and we get the required exact sequence

$$ \begin{align*} 0\to H_1FY\to FB_0Y\to B_0FY\to 0. \end{align*} $$

If Y is termwise projective, then the sequence $FY_1\to FY_0\to FY_{-1}$ is I-exact. In fact, the unit $\eta _Y:Y\to UFY$ of the adjunction is an isomorphism of complexes (cf. Lemma 2.5). Therefore, Lemma 2.7 shows that the canonical map $B_0Y\cong B_0UFY\to UB_0FY$ is bijective. Consider the composed map $FY_1\twoheadrightarrow FB_0Y\twoheadrightarrow B_0FY$ . If we apply the functor U, then the resulting map $Y_1\cong UFY_1\to UFB_0Y\to UB_0FY\cong B_0Y={\operatorname {im}}(d_1)$ is given by $d_1$ and hence is surjective. This implies that the map $FB_0Y\to B_0FY$ is an I-epimorphism. Now, apply U to the exact sequence in (i) to get a map $UFB_0Y\to UB_0FY\cong B_0Y$ with kernel $UH_1FY$ . One checks directly that it is a left inverse of the unit $\eta _{B_0Y}$ .

The previous results can be interpreted in terms of Quillen adjunctions for suitable model structures on ${\operatorname {Rep}}^\infty _k(G)$ and ${\operatorname {Ch}}(G)$ . For the rest of this section, we endow ${\operatorname {Ch}}(H)$ with the singular projective model structure from Theorem 3.4 (i). If H is Gorenstein, we will always endow ${\operatorname {Mod}}(H)$ with the Gorenstein projective model structure from Theorem 3.2. As before, let $F:{\operatorname {Mod}}(H)\rightleftarrows {\operatorname {Rep}}^\infty _k(G):U$ be the adjunction given by $FM={\mathbb {X}}\otimes _HM$ and $UV=V^I\cong {\operatorname {Hom}}_G({\mathbb {X}},V)$ . By the same symbols, we denote its termwise extension to an adjunction $F:{\operatorname {Ch}}(H)\rightleftarrows {\operatorname {Ch}}(G):U$ .

Proposition 4.5. The right transfer of the singular projective model structure exists along the adjunction $F:{\operatorname {Ch}}(H)\rightleftarrows {\operatorname {Ch}}(G):U$ and makes it a Quillen equivalence. Endowing ${\operatorname {Ch}}(G)$ with the right transfer, there is an equivalence of categories

$$ \begin{align*} K_{\operatorname{ac}}({\operatorname{Proj}}(H))\cong {\operatorname{Ho}}({\operatorname{Ch}}(G)). \end{align*} $$

Proof. Together with ${\operatorname {Rep}}^\infty _k(G)$ also ${\operatorname {Ch}}(G)$ is a Grothendieck category (cf. Lemma 1.7 (i)). In particular, all objects of ${\operatorname {Ch}}(G)$ are small. Since ${\operatorname {Ch}}(H)$ is cofibrantly generated, the first result will follow from Theorem 1.2 and Proposition 1.4 if we can show that path objects exist in ${\operatorname {Ch}}(G)$ . To construct them, let $V\in {\operatorname {Ch}}(G)$ . We need a factorisation

$$ \begin{align*}\Delta_V: V\stackrel{\alpha}{\longrightarrow} P \stackrel{\beta}{\longrightarrow}V\times V \end{align*} $$

of the diagonal morphism such that $U\alpha $ is a weak equivalence and $U\beta $ is a fibration, that is, an epimorphism.

Let $C={\operatorname {coker}} (\Delta _V)$ , and let $\pi : V\times V\to C$ denote the canonical projection. Since the abelian category ${\operatorname {Ch}}(H)$ has enough projectives (cf. Lemma 1.7 (ii)), there is a surjection $f:Q\to UC$ where Q is a projective complex. In terms of the singular projective model structure, this is a fibration in which Q is trivially cofibrant (cf. the proof of Theorem 3.4). Since Q is termwise projective, the unit of the adjunction $\eta _Q:Q \to UFQ$ is an isomorphism (cf. Lemma 2.5). Let $g:FQ\to C$ be the adjoint of f. Then $f=Ug\circ \eta _Q$ whence $Ug$ is surjective.

We now define $P=\{(v, q)\in (V\times V)\oplus FQ \;|\;\pi (v)=g(q)\}$ so that the projections $p:P\to V\times V$ and $h:P\to FQ$ make the square

Cartesian. Let $i: V\to P$ be given by $i(v)=(\Delta (v), 0)$ . This gives a factorization $\Delta _V=pi$ , and we claim that it has the desired properties. As a right adjoint, the functor U preserves fibre products. Therefore, the square

is Cartesian. Since the class of epimorphisms in an abelian category is stable under pullbacks and since $Ug$ is an epimorphism, it follows that so is $Up$ .

Note that $\Delta _V$ is a split monomorphism with splitting the first projection. Therefore, the image of the sequence $0\longrightarrow V\stackrel {\Delta _V}{\longrightarrow }V\times V\stackrel {\pi }{\longrightarrow } C\longrightarrow 0$ under U remains exact. In particular, $U\pi $ is surjective and so is its pullback $Uh$ . Since fiber products in abelian categories preserve kernels, the sequence

$$ \begin{align*}0\longrightarrow UV\stackrel{Ui}{\longrightarrow} UP\stackrel{Uh}{\longrightarrow} UFQ\longrightarrow 0 \end{align*} $$

is exact. Thus, ${\operatorname {coker}} (Ui)\cong UFQ\cong Q$ is trivially cofibrant. This shows that $Ui$ is a trivial cofibration and hence a weak equivalence, as required.

We have now shown that the right transfer exists. To get that the adjunction is a Quillen equivalence, we simply apply Lemma 1.6 and Lemma 2.5. Recall that the cofibrant objects in ${\operatorname {Ch}}(H)$ are the acyclic complexes which are termwise projective (cf. Theorem 3.4 (i)). The final statement follows from Theorem 3.4 (i) and [Reference Hovey19], Proposition 1.3.13.

The model structure on ${\operatorname {Ch}}(G)$ constructed in Proposition 4.5 will be referred to as the I-singular projective model structure.

Proof. The statement about the Quillen equivalence follows formally from Proposition 4.5 and Lemma 3.5 because U and F commute with the shift functors. Note that $RU$ commutes with the loop functor by [Reference Barnes and Roitzheim1], Corollary 3.1.4. Therefore, Lemma 3.5 and the right derived version of [Reference Hovey19], Theorem 1.3.7, imply

$$ \begin{align*} RU\Omega\cong\Omega RU\cong R[1]RU\cong R([1]U)\cong R(U[1])\cong RUR[1]. \end{align*} $$

Composing with $LF$ gives the isomorphism $\Omega \cong R[1]$ on ${\operatorname {Ch}}(G)$ . The uniqueness of adjoints then also gives $L[-1]\cong \Sigma $ .

As in Lemma 2.15, we are in a situation where the right adjoint U preserves cofibrant objects.

As a formal consequence, we get the following analog of Corollary 2.16.

Moreover, we obtain the following description of the cofibrations in ${\operatorname {Ch}}(G)$ .

Proof. Assume that i is a cofibration. For any $n\in \mathbb {Z}$ consider the complex $D^{n+1}V_n=[0\to V_n\overset {{\operatorname {id}}}{\to }V_n\to 0]$ concentrated in degrees $n+1$ and n. Then $D^{n+1}V_n\to 0$ is an I-epimorphism and hence is a fibration. In fact, this is a trivial fibration because applying U gives the bounded acyclic complex $[0\to UV_n\overset {{\operatorname {id}}}{\to }UV_n\to 0]$ which is trivial in the singular projective model structure by [Reference Becker3], Proposition 3.1.1. Consider the map $V_\bullet \to D^{n+1}V_n$ given by the differential of $V_\bullet $ in degree $n+1$ and the identity of $V_n$ in degree n. By the left lifting property of i applied to the diagram

there is a map $j:W_\bullet \to D^{n+1}V_n$ such that $j_n$ is a splitting of $i_n$ . Since the cokernel of a cofibration is cofibrant (cf. [Reference Hovey19], Corollary 1.1.11), it follows together with Lemma 4.7 that i is a termwise split monomorphism whose cokernel is an I-exact sequence of I-projectives.

For the converse the arguments given in [Reference Hovey19], Proposition 2.3.9, carry over mutatis mutandis.

We now pass back from complexes to the categories ${\operatorname {Mod}}(H)$ and ${\operatorname {Rep}}^\infty _k(G)$ . For the rest of this section, we will therefore assume that H is a Gorenstein ring and that ${\operatorname {Mod}}(H)$ carries the Gorenstein projective model structure from Theorem 3.2.

Proof. Since ${\operatorname {Mod}}(H)$ is cofibrantly generated, this will follow from Theorem 1.2 and Proposition 1.4 if we can show that path objects exist in ${\operatorname {Rep}}^\infty _k(G)$ . To construct them, we follow the argument in the proof of [Reference Hovey21], Proposition 9.1. Given $W\in {\operatorname {Rep}}^\infty _k(G)$ , we choose an I-epimorphism $q:Y\to W$ , where Y is I-projective. Then the maps

$$ \begin{align*} W\stackrel{i}{\longrightarrow}W\times Y\stackrel{p}{\longrightarrow}W\times W \end{align*} $$

given by $i(w)=(w,0)$ and $p(w,y)=(w,w+q(y))$ give a factorization $\Delta _W=p i$ of the diagonal $\Delta _W:W\to W\times W$ . Moreover, $Ui$ is an injection such that ${\operatorname {coker}}(Ui)\cong UY$ is a projective H-module. Thus, $Ui$ is a trivial cofibration in ${\operatorname {Mod}}(H)$ . This implies that i is a weak equivalence. Moreover, the map $Up$ is surjective because the map $Uq$ is. Thus, $Up$ is a fibration in ${\operatorname {Mod}}(H)$ . This implies that p is a fibration.

Using Lemma 2.18, one can follow the above arguments to give a second proof of the existence of path objects in the situation of Proposition 4.5.

The model structure on ${\operatorname {Rep}}^\infty _k(G)$ constructed in Proposition 4.10 will be called the I-Gorenstein projective model structure. This is the model structure on ${\operatorname {Rep}}^\infty _k(G)$ we will consider for the rest of this section. To describe it more concretely, call an object $V\in {\operatorname {Rep}}^\infty _k(G)\ I$ -trivial if the object $UV\in {\operatorname {Mod}}(H)$ is trivial, that is, has finite projective dimension. With this terminology, the fibrations in ${\operatorname {Rep}}^\infty _k(G)$ are the I-epimorphisms and the trivial fibrations are the I-epimorphisms with I-trivial kernel.

We now give a different characterization of the I-trivial objects. Recall from the discussion following Corollary 2.9 that every $V\in {\operatorname {Rep}}^\infty _k(G)$ admits an I-resolution $X_\bullet \to V\to 0$ . By Lemma 2.4 and Lemma 2.7, this is a complex $X_\bullet $ of I-projective G-representations such that $UX_\bullet \to UV\to 0$ is exact.

Proof. Assume that V has finite I-projective dimension. Applying U to a finite I-resolution of V, we obtain a finite projective resolution of $UV$ . Thus, V is I-trivial. Conversely, suppose that V is I-trivial and choose any I-resolution $X_\bullet =(X_\bullet ,d_\bullet )$ of V. Let n denote the projective dimension of $UV$ . Since $UX_\bullet $ is a projective resolution of $UV$ , the H-module $\ker (Ud_{n-1})$ is projective by [36], 00O5. By Lemma 2.7, we have $\ker (Ud_{n-1})=U{\operatorname {im}}(d_n)$ . Moreover, ${\operatorname {im}} (d_n)$ is generated by its I-invariants because it is a quotient of the I-projective object $X_n$ (cf. Lemma 2.8). Applying Lemma 2.8 again, we see that ${\operatorname {im}}(d_n)$ is I-projective and that

$$ \begin{align*} 0\to {\operatorname{im}} (d_n) \to X_{n-1}\to\ldots\to X_0\to V\to 0 \end{align*} $$

is a finite I-resolution of V.

In a next step, we characterize the trivial cofibrations in ${\operatorname {Rep}}^\infty _k(G)$ .

Proof. If i is a split injection with an I-projective cokernel, we may assume $W=V\oplus C$ where C is I-projective. If we have a commutative diagram

where p is an I-epimorphism, then the map $C\to W'$ lifts to a map $g:C\to V'$ because C is I-projective. The map $(f,g):V\oplus C=W\to V'$ is then a lift of i.

Conversely, suppose that $i:V\to W$ is a trivial cofibration. Its cokernel C is trivially cofibrant by [Reference Hovey19], Corollary 1.1.11. This means that the functor ${\operatorname {Hom}}_G(C,\cdot )$ sends I-epimorphisms to surjections which is equivalent to C being I-projective. It remains to show that i is a split injection. Factorize i as $V\stackrel {\alpha }{\to }V\oplus W\stackrel {p_2}{\to } W$ , where $\alpha (v)=(v, i(v))$ and $p_2$ is the projection onto the second summand. Since $p_2$ is an I-epimorphism and since i is a trivial cofibration, we obtain a lift h in the commutative diagram

The first projection $p_1:V\oplus W\to V$ then gives the splitting $p_1 h$ of i.

The same argument shows that if $V\in {\operatorname {Rep}}^\infty _k(G)$ is I-trivial, then any cofibration $i:V\to W$ is a split injection with a cofibrant cokernel. In general, however, cofibrations will not be monomorphisms. Indeed, the adjunction in Proposition 4.10 is Quillen by Lemma 1.6. Therefore, the functor F preserves cofibrations, that is, the image under F of any monomorphism with a Gorenstein projective cokernel is a cofibration in ${\operatorname {Rep}}^\infty _k(G)$ . By the lack of exactness of F, this will not be a monomorphism in general. However, we have the following result.

Proof. For the first part, note that

is a pushout square and that the class of cofibrations is closed under pushouts (cf. [Reference Hovey19], Corollary 1.1.11). For the second part, let $W'={\operatorname {im}}(i)+k[G]\cdot UW$ . The inclusion $W'\hookrightarrow W$ is an isomorphism on I-invariants, hence is a trivial fibration. Thus, we may apply the lifting property of i to the diagram

and get $W=W'$ . The last statement follows from taking $V=0$ .

In order to characterize the cofibrant objects, we first need to define a certain class of short exact sequences in ${\operatorname {Rep}}^\infty _k(G)$ .

A morphism $p:W\to X$ in ${\operatorname {Rep}}^\infty _k(G)$ is called a $\mathcal {P}$ -epimorphism if the sequence $0\to \ker (p)\to W\stackrel {p}{\to }X\to 0$ belongs to $\mathcal {P}$ . Note that the $\mathcal {P}$ -epimorphisms are precisely the epimorphisms which are also I-epimorphisms. The notion of a $\mathcal {P}$ -monomorphism is defined dually. In fact, the class $\mathcal {P}$ is proper in the sense of [Reference MacLane26], Chapter XII.4.