MODEL CATEGORIES AND PRO-p IWAHORI–HECKE MODULES

Abstract Let G denote a possibly discrete topological group admitting an open subgroup I which is pro-p. If H denotes the corresponding Hecke algebra over a field k of characteristic p, then we study the adjunction between H-modules and k-linear smooth G-representations in terms of various model structures. If H is a Gorenstein ring, we single out a full subcategory of smooth G-representations which is equivalent to the category of all Gorenstein projective H-modules via the functor of I-invariants. This applies to groups of rational points of split connected reductive groups over finite and over non-Archimedean local fields, thus generalizing a theorem of Cabanes. Moreover, we show that the Gorenstein projective model structure on the category of H-modules admits a right transfer. On the homotopy level, the right derived functor of I-invariants then admits a right inverse and becomes an equivalence when restricted to a suitable subcategory.


Introduction
Let p denote a prime number, and let G denote a topological group admitting an open subgroup I which is pro-p.If k is a field of characteristic p, we let X = ind G I (k) denote the G-representation compactly induced from the trivial I -representation k.The opposite endomorphism ring H = End G (X) op can also be realized as the double coset algebra H = k[I\G/I] with respect to the usual convolution product.
We let Rep ∞ k (G) denote the category of k -linear G-representations which are smooth in the sense that the stabilizer of any vector is open in G.If G is discrete, then Rep ∞ k (G) is the category of all k -linear G-representations.Let Mod(H) denote the category of left H -modules.There is an adjunction given by F M = X ⊗ H M and UV = V I ∼ = Hom G (X,V ).The study of G-representations in terms of this adjunction plays a prominent role in various situations.If G is the group of rational points of a split connected reductive group over a finite field of characteristic p, for example, and if I denotes a p-Sylow subgroup of G, then this strategy has a long history.
Our motivating example, however, concerns the case that G is the group of rational points of a split connected reductive group over a non-Archimedean local field of residue characteristic p.If I denotes a pro-p Iwahori subgroup of G in the sense of Bruhat-Tits theory, then the algebra H is known as the corresponding pro-p Iwahori-Hecke algebra.In this situation, the category Rep ∞ k (G) plays a central role in the mod-p local Langlands program.Although the category of H -modules is rather well understood, the properties of the category Rep ∞ k (G) remain obscure.In fact, the behavior of the adjunction (1) is not sufficiently clear beyond G = GL 2 (Q p ) and a few related cases (cf. the results of Ollivier in [27]).
Since k is of characteristic p, the functor U is generally not exact.It is therefore natural to study the situation from a homological point of view.For reductive groups over non-Archimedean local fields, important work in this direction has been done by Ollivier, Schneider and Koziol (cf.[28], [29] and [23], for example).Moreover, Schneider showed that the adjunction (1) induces an equivalence on a derived level if H is replaced by a suitable dg-variant H • (cf.[34], Theorem 9).Unfortunately, the structure of this algebra seems very hard to understand although some recent progress has been made by Ollivier and Schneider (cf.[30] and [31]).
In this article, we take a different approach.Our aim is to shed some light on the nature of the adjunction (1) using the language and methods of model categories.We stay in the very general situation described at the beginning.For the finer results in §4 and §5, we will assume, however, that the ring H is Gorenstein.In Example 4.1, we give a list of important situations to which this assumption applies.Due to results of Tinberg, Ollivier and Schneider, this includes groups of rational points of split connected reductive groups over finite and non-Archimedean local fields.
In §1, we give a brief exposition of the general theory of model categories.Any model category C has an associated homotopy category Ho(C) that may be thought of as a generalized derived category.The model structures relevant for our purposes will either be constructed using cotorsion pairs in the sense of Hovey (cf.Theorem 1.1) or via the left/right transfer along an adjunction.We emphasize that cotorsion pairs differ from the torsion theories advertised in [29].In rather special situations, however, there is a correspondence between localizing cotorsion pairs and torsion theories in the homotopy category (cf.[33], Proposition 3.3).In Proposition 1.4, we give a criterion for the existence of the right transfer based on path objects.
Let Ch(H) and Ch(G) denote the categories of unbounded chain complexes over Mod(H) and Rep ∞ k (G), respectively.By working termwise, there is an induced adjunction F : Ch(H) Ch(G) : U .In Proposition 2.13 and Proposition 2.14, we show that the projective model structure on Ch(H) admits a right transfer to Ch(G) yielding a Quillen equivalence.In particular, the derived adjunction LF : D(H) Ho(Ch(G)) : RU is an equivalence of categories.Here, D(H) denotes the usual unbounded derived category of the ring H.Although this result is somewhat formal, the necessary input from relative homological algebra plays a major role in the sequel.In particular, this concerns the notion of an I-exact sequence of G-representations.This is an object X ∈ Ch(G) such that the complex UX ∈ Ch(H) of I -invariants is exact (cf.Lemma 2.4).Moreover, pulling back the notion of a projective H -module yields the relative notion of an I-projective G-representation.These are characterized in Lemma 2.8.It lies at the heart of many of our arguments that the adjunction (1) restricts to an equivalence between the full subcategories of projective H -modules and I -projective G-representations, respectively (cf.Corollary 2.9).
We point out that if G is compact, then the category Rep ∞ k (G) does not have any nonzero projective objects unless the group G is discrete (cf.Remark 2.20).However, we show that the injective model structure on Ch(G) admits a left transfer to Ch(H) (cf.Proposition 2.23).A detailed analysis of the corresponding Quillen adjunction will be given elsewhere.
In §3, we collect known results about model structures over Gorenstein rings.We start with the notion of a Gorenstein projective object in an abelian category with enough projectives.If S is a Gorenstein ring, then Hovey constructed a model structure on Mod(S) for which the cofibrant objects are the Gorenstein projective S -modules and the trivial objects are the S -modules of finite projective dimension (cf.Theorem 3.2).This is called the Gorenstein projective model structure.If S = H comes from a split connected reductive group over a non-Archimedean local field as in Example 4.1 (iv), then the corresponding homotopy category seems particularly well suited to study the supersingular H -modules (cf.Remark 4.2).Returning to the general case, there is a Quillen equivalent model structure on Ch(S) constructed by Becker.This is called the singular projective model structure (cf.Theorem 3.4).Its homotopy category is related to the singularity category of S studied by Krause (cf.[24]).We emphasize that there is an extensive literature on homological algebra over Gorenstein rings starting with the seminal article [5] of Buchweitz.In §4, we specialize to the case S = H assuming that the ring H is Gorenstein.In Proposition 4.5, we show that the singular projective model structure on Ch(H) admits a right transfer along the adjunction F : Ch(H) Ch(G) : U and yields a Quillen equivalence.On both sides, the loop and suspension functors are induced by the usual shift functors (cf.Lemma 3.5 and Corollary 4.6).Moreover, the derived adjunction is an equivalence of categories LF : K ac (Proj(H)) Ho(Ch(G)) : RU, where the left-hand side denotes the category of acyclic complexes of projective Hmodules up to chain homotopy.The cofibrant objects of Ch(G) are the I -exact complexes of I -projective G-representations (cf.Lemma 4.7), and the cofibrations are made explicit in Corollary 4.9.
In Proposition 4.10, we construct suitable path objects to show that the Gorenstein projective model structure on Mod(H) admits a right transfer along the adjunction (1).The analysis of the derived adjunction is much less formal than in the previous situations because the cofibrant H -modules are only Gorenstein projective.Our main observation is that for a Gorenstein projective module the unit of the adjunction ( 1) is a split monomorphism with an explicit cokernel (cf.Proposition 4.3).Moreover, we can make explicit the homotopy relation on morphisms between cofibrant objects (cf.Lemma 4.21).As a consequence, the left derived functor LF is faithful and the right derived functor RU is essentially surjective (cf.Theorem 4.22).We also give a detailed analysis of the cofibrations in Rep ∞ k (G).Although this model category is not constructed from a cotorsion pair directly, the cofibrant and the trivial objects satisfy a suitable orthogonality relation (cf.Lemma 4.17).Finally, in Proposition 4.24, we give several equivalent obstructions to the derived adjunction (2) being an equivalence.The most interesting of these is related to the fact that the right transfer need no longer be stable.The necessary computation of the loop and suspension functors is described in Lemma 4. 23.
However, the situation can be improved significantly.As a major result, we show in §5 that the functor U : Rep ∞ k (G) → Mod(H) restricts to an equivalence between suitable full subcategories.On the one hand, we let GProj(H) denote the full subcategory of Mod(H) consisting of all Gorenstein projective H -modules.On the other hand, we define the relative notion of an I-Gorenstein projective G-representation (cf.Definition 5.3).This yields a full subcategory C(G) of Rep ∞ k (G) such that the functor U restricts to an equivalence C(G) ∼ = GProj(H) (cf.Theorem 5.4).There is an explicit inverse functor F whose relation to the functor F is spelled out in Lemma 5.8.If the ring H is selfinjective, then GProj(H) = Mod(H) and C(G) is the category of representations which are both a quotient and a subobject of an I -projective G-representation.In this case, we recover a result of Cabanes concerning the case of finite groups with a split BN -pair of characteristic p (cf. Remark 5.6).
Endowed with the class of short I -exact sequences, the category C(G) turns out to be a weakly idempotent complete Frobenius category (cf.Corollary 5.9).Thus, C(G) has a canonical exact model structure and an associated homotopy category (cf.Proposition 5.1).If Rep ∞ k (G) is endowed with the right transfer studied in §4 and if i : denotes the inclusion functor, then the composition Ho(C(G)) is an equivalence of categories (cf.Theorem 5.11).In particular, RU admits a right inverse and Ho(i) allows us to view Ho(C(G)) as a (not necessarily full) subcategory of Ho(Rep ∞ k (G)) on which RU becomes an equivalence.We hope this result amply demonstrates the strength of our approach.
At the end of our article, we specialize to the case that H is the pro-p Iwahori-Hecke algebra of a split connected reductive group G over a non-Archimedean local field of residue characteristic p.If G is semisimple we consider the canonical Gorenstein projective resolution GP • (M ) → M → 0 of an H -module M constructed by Ollivier and Schneider (cf.[28], §6.4).On the other hand, we consider the complex C or c (X (•) ,F(M )) of oriented chains of the G-equivariant coefficient system F(M ) on the Bruhat-Tits building X of G associated with M by the second author (cf.[22], §3.2).In Proposition 5.13, we show that there is a functorial isomorphism of complexes In particular, the chain complex of F(M ) consists of objects of C(G).For the chain complex of a fixed point system as studied in [28], §3.1, this is generally not true (cf.Example 5.15).

Notation and conventions.
A class of objects of a category C will usually be identified with the corresponding full subcategory.If we denote an adjunction by F : C D : U , then F is always assumed to be left adjoint to U. The unit (resp.the counit) of an adjunction will always be denoted by η (resp.ε).We say that a functor F : C → D preserves (resp.reflects) a property (P) if F * (resp.* ) has property (P) whenever * (resp.F * ) does.For any unital ring S we denote by Mod(S) the category of left Smodules.Given M,N ∈ Mod(S), we write Hom S (M,N ) for the set of S -linear maps from M to N.

Preliminaries on model categories
We largely follow the conventions of Hovey's book [19].A model category is a locally small and bicomplete category 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 [19], Definition 1.1.3.As usual, a cofibration (resp.a fibration) is called trivial if it is also a weak equivalence.
An object X ∈ 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 C c (resp.C f , resp.C cf ) the full subcategory of C consisting of all objects which are cofibrant (resp.fibrant, resp.cofibrant and fibrant).By we denote the cofibrant replacement functor and the fibrant replacement functor, respectively (cf.[19], page 5).The cofibrant replacement functor will often be abbreviated to 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 Δ the above definition allows any factorization with the required properties.The homotopy category Ho(C) of a model category C is the localization of C with respect to the class of weak equivalences (cf.[19], Definition 1.2.1).We will usually just write C → Ho(C) for the canonical functor of C into its homotopy category.Any subcategory of C has an induced class of weak equivalences and the corresponding homotopy category is defined analogously.It is a fundamental result that the inclusion C cf → C induces equivalences of categories Recall that a functor F : C → D (resp.U : D → 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 : C D : U is an adjunction between model categories, then F is left Quillen if and only if U is right Quillen (cf.[19], 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 ∈ C and all fibrant Y ∈ D a morphism F X → Y is a weak equivalence in D if and only if the adjoint morphism X → UY is a weak equivalence in C (cf. [19], Definition 1.3.12).
We refer to [19], 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 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 [19], Lemma 1.1.12.By the universal property of localizations, there are induced functors Ho(Q In this situation, LF is left adjoint to RU (cf.[19], 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.[19], Proposition 1.3.13).
A powerful tool to construct model structures on an abelian category A was developed by Hovey in [21].It is based on a fixed proper class P of short exact sequences in A leading to relative extension groups Ext • P (X,Y ) as in [26], Chapter XII.4.An epimorphism belongs to P. The notion of a P-monomorphism is defined dually.Given classes C and F of objects of A consider the classes of objects and A cotorsion pair with respect to P is a pair (C,F) of classes of objects of A satisfying C = ⊥ F and C ⊥ = F (cf. [21], Definition 2.3).It is said to have enough functorial projectives if for any X ∈ A there is a P-epimorphism f : Y → X which is functorial in X and satisfies Y ∈ C and ker(f ) ∈ 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 A which is compatible with P, we refer to [21], Definition 2.1.A subcategory of A is called P-thick if it is closed under retracts and satisfies the two-out-of-three property with respect to short exact sequences in P. If A is an abelian category endowed with the structure of a model category, then an object X ∈ A is called trivial if the morphism 0 → X (or equivalently the morphism X → 0) is a weak equivalence.We denote by A t the full subcategory of A consisting of all trivial objects.The following fundamental theorem is due to Hovey (cf.[21], Theorem 2.2).

Theorem 1.1 (Hovey
In the situation of Theorem 1.1 (ii) the definition of the corresponding cofibrations, fibrations and weak equivalences is given explicitly in [21], Definition 5.1.If 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.[20], Definition A.1).For example, this holds in any Grothendieck category (cf.[20], Proposition A.2) which is the main case of interest for us.We refer to [21], Corollary 6.8, for a manageable criterion to check whether a given cotorsion pair on a Grothendieck category is functorially complete.
Let I be a class of morphisms in a category C. Recall that a morphism in C is called I-injective if it satisfies the right lifting property with respect to all morphisms in I in the sense of [19], Definition 1.1.2.The class of these morphisms is denoted by I-inj.A morphism in C is called an I-cofibration if it satisfies the left lifting property with respect to all morphisms in I-inj.The class of these morphisms is denoted by I-cof.
For the definition of a model category C which is cofibrantly generated, we refer to [19], Definition 2.1.17.If every object of C is small, then this means that there are sets I and J of morphisms of C such that the class of fibrations is equal to J -inj and the class of trivial fibrations is equal to I-inj.The elements of the sets I and J are called the generating cofibrations and the generating trivial cofibrations of C, respectively.
Assume that we have an adjunction F : C D : U .If C is a model category and if D is bicomplete, then there is a natural candidate for an associated model structure on D. Namely, define a fibration (resp.a weak equivalence, resp.a trivial fibration) in D to be a morphism f in D such that Uf is a fibration (resp.a weak equivalence, resp.a trivial fibration) in C.Moreover, define a cofibration in 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 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 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.[19], Lemma 1.1.10).In certain situations, this condition is also sufficient.Indeed, we have the following result which is a variant of [18] Proof.If f is a morphism in D, then Uf ∈ I-inj (resp.Uf ∈ J -inj) if and only if f ∈ F Iinj (resp.f ∈ F J -inj) by the adjunction.Therefore, the class of fibrations in D is F J -inj, the class of trivial fibrations is F I-inj, the class of cofibrations is F I-cof and the class of anodyne morphisms is F J -cof.
It now suffices to check that the conditions in [18], Theorem 11.3.1, are satisfied for F I, F J and the class of weak equivalences.Note first that the latter has the two-out-of-three property and is closed under retracts because U is a functor.Conditions (1)-( 3) and (4)(b) in [18], Theorem 11.3.1,follow from the fact that every object in D is small, that every anodyne morphism is a weak equivalence and that the morphisms in F I-inj are exactly the trivial fibrations.
Remark 1.3.The idea of the right transfer goes back to the original work of Quillen (cf.[32], section II.4).There are more general versions of the above theorem replacing the smallness assumptions on D by suitable properties of the functor F. See [18], Theorem 11.3.2, for instance.
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 C is a model category and that we have an adjunction F : C D : U , where D is bicomplete.Since we have notions of fibrations and weak equivalences in D, we can make sense of path objects and right homotopies in 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 [16] Consequently, Uj becomes an isomorphism in Ho(C), hence is a weak equivalence in C by [19], Theorem 1.2.10 (iv).Thus, j is a weak equivalence in D.
Remark 1.5.The result in Proposition 1.4 is again implicit in the work of Quillen [32] who works in the simplicial setting.The condition that every object is fibrant can be replaced by the more general condition that D admits a functorial fibrant replacement and functorial path objects (cf.[16], Theorem 3.8).The above proof is a rewording of its simplicial version in [15], Lemma II.6.1.
Lemma 1.6.Suppose that we have an adjunction F : C D : U , where D is bicomplete and C is a model category whose right transfer exists.Then F and U form a Quillen adjunction.This is a Quillen equivalence if and only if the unit η X : X → UF X is a weak equivalence for all cofibrant X in C.
Proof.It follows immediately from the definition of the right transfer that the functor U is right Quillen.Now, assume that η X is a weak equivalence for all cofibrant X in C. Consider a morphism f : F X → Y , where X ∈ C is cofibrant and Y ∈ D is fibrant.By definition of the right transfer, f : F X → Y is a weak equivalence if and only if so is Uf.By the two-out-of-three property this is true if and only if Ufη X is a weak equivalence.However, the latter is the adjoint of f.Thus, we have a Quillen equivalence.
Conversely, if we have a Quillen equivalence and if X is cofibrant in C consider the fibrant replacement f : Since this is a weak equivalence, so are Uf and the adjoint morphism Ufη X .The two-out-of-three property implies that also η X is a weak equivalence.
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 C admits two endofunctors Σ and Ω called the suspension functor and the loop functor, respectively.Given X ∈ C fibrant and P a path object for X, the loop ΩX of X is defined to be the pullback of the diagram where * denotes the initial/terminal object of C. Given X ∈ C cofibrant, its suspension ΣX is defined dually using a cylinder object (cf.[1], Definition 3.1.2).The suspension functor Σ is left adjoint to the loop functor Ω (cf.[1], Proposition 3.1.7).If they are equivalences of categories, then the model category C is called stable.If F : C D : U is a Quillen adjunction, then LF commutes with Σ and RU commutes with Ω (cf.[1], Corollary 3.1.4).If both model categories are stable, then also RU commutes with Σ and LF commutes with Ω (cf.[1], Corollary 3.2.10).
To compute the suspension of an object X in an abelian model category If an abelian model structure is hereditary, then it is also stable (cf.[3], 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 A, we denote by Ch(A) the category of unbounded chain complexes over A. Let ι 0 : A → Ch(A) denote the functor obtained by viewing an object of A as a complex concentrated in degree zero.We will often use it to view A as a full subcategory of Ch(A) and suppress it from the notation.Recall that ι 0 admits both a left adjoint Q 0 and a right adjoint Z 0 given by For any integer n, we denote by the two-sided inverse of [1].
Note that together with A also the category Ch(A) is abelian.Moreover, we have the following standard facts.

Lemma 1.7. Let A be an abelian category. (i) If A is a Grothendieck category, then so is Ch(A). (ii) If A has enough projectives, then so does Ch(A).
Proof.The abelian category Ch(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 A consider the complex

generator of Ch(A). This shows (i).
As for (ii), let A Z denote the category of Z-graded objects of A and consider the functor in Ch(A) is as required.Indeed, f is an epimorphism because D is a left adjoint and so preserves epimorphisms.Moreover, Q • is projective because the right adjoint T is exact and hence D preserves projectives.

N. Dupré and J. Kohlhaase
Recall that any Grothendieck category is automatically complete and therefore bicomplete.If A is a Grothendieck category, there is a cofibrantly generated model structure on Ch(A) for which the cofibrations are the monomorphisms and the weak equivalences are the quasi-isomorphisms (cf.[4], Proposition 3.13).This is called the injective model structure on Ch(A).We write Ch(A) Inj for the corresponding model category.Its homotopy category is equivalent to the unbounded derived category D(A) ∼ = Ho(Ch(A) Inj ) of A.
If A is the category of modules over a ring, then the injective model structure on Ch(A) is also constructed in [19], §2.3.It follows as in [19], 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 Ch(A) is compatible with the class P of all short exact sequences.
By Theorem 1.1, the injective model structure is related to two functorially complete cotorsion pairs on Ch(A) given as follows.Note that every object of Ch(A) Inj is cofibrant, that is, we have C = Ch(A) Inj c = Ch(A).Moreover, the class T = Ch(A) Inj t of trivial objects is precisely that of the acyclic complexes.If we denote by F = Ch(A) Inj f the class of fibrant objects, then we know from Theorem 1.1 that F = T ⊥ .In other words, a complex Y is fibrant if and only if Ext 1 Ch(A) (X,Y ) = 0 for any acyclic complex X.As in [11], 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 Hom(X,Y ) is acyclic for any acyclic complex X.
Apart from the cotorsion pair (C ∩ T ,F) = (T ,T ⊥ ), there is also the cotorsion pair (C,T ∩ F) = (Ch(A),T ∩ T ⊥ ).As in [11], Proposition 3.7, the class T ∩ T ⊥ is that of the injective complexes, that is, of the injective objects of Ch(A).Recall that these are precisely the split acyclic complexes which are termwise injective.

Remark 1.8.
There is also the notion of a K-injective complex in the sense of Spaltenstein (cf.[35], page 124).Recall that a complex Y ∈ Ch(A) is K -injective if and only if the complex Hom(X,Y ) is acyclic for any acyclic complex X.As mentioned above, a complex is dg-injective if and only if it is K -injective and termwise injective.Consequently, many of Spaltenstein's results in [35] can be reproved using the injective model structure on Ch(A).For example, the fibrant replacement Under suitable assumptions on A, there is a projective version of the above results.For our purposes, it will be sufficient to stick to the classical case that A = Mod(S) is the category of left modules over a ring S. In this case, we write Ch(S) = Ch(Mod(S)).
According to [19], Theorem 2.3.11,there is a cofibrantly generated model structure on Ch(S) for which the fibrations are the surjections and the weak equivalences are the quasiisomorphisms.This is called the projective model structure on Ch(S).We write Ch(S) Proj for the corresponding model category.Its homotopy category is again equivalent to the unbounded derived category D(S) ∼ = Ho(Ch(S) Proj ) of Mod(S).
By [19], Proposition 2.3.9, the cofibrations are precisely the termwise split monomorphisms with cofibrant cokernel.It follows that the projective model structure on Ch(S) is compatible with the class P of all short exact sequences.By Theorem 1.1 it is related to two functorially complete cotorsion pairs on Ch(S).Note that every object of Ch(S) Proj is fibrant, that is, we have F = Ch(S) Proj f = Ch(S).Moreover, the class T = Ch(S) Proj t of trivial objects is precisely that of the acyclic complexes.If we denote by C = Ch(S) Proj c the class of cofibrant objects, then we know from Theorem 1.1 that C = ⊥ T .In other words, a complex X is cofibrant if and only if Ext 1 Ch(S) (X,Y ) = 0 for any acyclic complex Y.As in [11], 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 Hom(X,Y ) is acyclic for any acyclic complex Y.
Apart from the cotorsion pair (C,T ∩ F) = ( ⊥ T ,T ), there is also the cotorsion pair As in [11], Proposition 3.7, the class ⊥ T ∩ T is that of the projective complexes, that is, of the projective objects of 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 id : Ch(S) Proj Ch(S) Inj : 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 Ch(S), the suspension Σ = L[−1] is the left derived functor of the shift functor [−1].Likewise, the loop functor Ω = 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.

Derived categories and relative homological algebra
Recall the following notions from relative homological algebra (cf.[8], §1).We continue to denote by A an abelian category and fix a class P of objects of A. Given any object P of A, we say that a morphism f : A → B in A is P-epic or a P-epimorphism if the induced map Hom A (P,A) → Hom A (P,B) is surjective.Furthermore, we say that f is P-epic if it is P -epic for all P ∈ P. A sequence Definition 2.1.A projective class on A is a pair (P,E), where P is a class of objects of A and E is a class of morphisms in A such that the following three conditions are satisfied.
(i) E is the class of all P-epimorphisms.
(ii) P is the class of all objects P of A such that every morphism in E is P -epic.
(iii) For each object B in A, there is a morphism f : P → B in A with P ∈ P and f ∈ E. Example 2.2.For any ring S, let P be the class of all projective S -modules and let E be the class of all surjective S -linear maps.Then the pair (P,E) is a projective class on A = Mod(S) called the standard projective class.
A P-resolution of an object B ∈ A is a P-exact sequence P • → B → 0 in A such that P n ∈ P for all n ≥ 0. Given another object C ∈ A the group of relative extensions of C and B are defined by Up to isomorphism, they are independent of the chosen P-resolution of B and any short P-exact sequence 0 → B → B → B → 0 gives rise to a long exact sequence between relative extension groups, as usual.
Remark 2.3.In spite of the notational similarities, the relative extension groups in (7) are not to be confused with those defining cotorsion pairs.In fact, we shall see in examples below that short P-exact sequences are generally not exact in the usual sense and hence do not belong to any proper class of short exact sequences in A.
Suppose now that we have an adjunction F : B A : U between abelian categories.Moreover, assume that (P ,E ) is a projective class on B. Recall that a retract of an object B ∈ A is an object A ∈ A admitting a right invertible morphism B → A. Since A is abelian, this is equivalent to A being isomorphic to a direct summand of B. We define the pullback of the pair (P ,E ) along U to be the pair (P,E), where The proof of the following result is straightforward and left to the reader.

Lemma 2.4. With the above notation, the pullback (P,E) is a projective class on A. Moreover, a sequence
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 Rep ∞ 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 ∈ Rep ∞ k (G), we will write Hom G (V ,W ) for the set of morphisms from V to W in Rep ∞ k (G).We explicitly allow G to be discrete.

In this case, the category Rep
) for all g,g ∈ G.We endow k with the trivial action of I and set Setting H = End G (X) op , Frobenius reciprocity shows that we have adjoint functors given by F M = X ⊗ H M and UV = V I ∼ = Hom G (X,V ).Since k has characteristic p, the functor U has the property that UV = 0 for all 0 = V ∈ Rep ∞ k (G) (cf.[34], Lemma 1 (v)).Since U is also left exact, it reflects monomorphisms. For where 1 gI denotes the characteristic function of the coset gI.Moreover, given M ∈ Mod(H), the unit η M is the H -linear map Lemma 2.5.For any projective H-module P the unit η P is an isomorphism.
Proof.It is immediate that η H is an isomorphism.Since U and F are additive functors and preserve arbitrary direct sums, η P is an isomorphism for any projective H -module P.
As seen in Example 2.2, we have the standard projective class (P ,E ) on Mod(H), where P is the class of all projective H -modules and E is the class of all surjective H -linear maps.Definition 2.6.We define the I-projective class on Rep ∞ k (G) to be the pullback (P,E) of the standard projective class (P ,E ) along U. Explicitly, (i) V ∈ P if and only if there is a split surjection F P → V for some projective H -module P and Lemma 2.4).In this context, P-resolutions will also be called I-resolutions.Finally, we say that Proof.Since im(f ) ⊆ ker(g) and the right adjoint U commutes with kernels, the I -exactness gives ker(Ug) = im(Uf) ⊆ U im(f ) ⊆ U ker(g) = ker(Ug).
We use the unit η H to identify H ∼ = −→ UF H ∼ = X I as (H,H)-bimodules.Explicitly, this map is given by h → 1 I • h.The terminology in Definition 2.6 is justified by the following result.
(i) V is I-projective.
(ii) V is a direct summand of an I-free G-representation.
(iii) UV is a projective H-module and V is generated by its I-invariants.
Under these conditions, the counit ε V : F UV → V is an isomorphism.
Proof.If V is I -projective, then by definition V is a direct summand of F P for some projective H -module P. Since P is a direct summand of a free H -module, it follows that F P and hence V are direct summands of an I -free representation.That (ii) implies (iii) follows from Lemma 2.5 and because the property of being generated by I -invariants is stable under quotients.Finally, if UV is projective, then F UV is I -projective by definition.By Lemma 2.5, the unit η UV is an isomorphism.It always has the splitting Uε V .Thus, we see that the counit F UV → V induces an isomorphism on I -invariants.Since the functor U reflects monomorphisms, it follows that F UV → V is injective with image equal to the subrepresentation of V generated by UV .This shows that (iii) implies (i) and that in this case the counit is an isomorphism.

Corollary 2.9. The functors F and U induce inverse equivalences of categories between the full subcategories of projective H-modules and of I-projective G-representations, respectively.
Proof.Let V ∈ Rep ∞ k (G) be I -projective, and let P ∈ Mod(H) be projective.By definition, F P is I -projective.Moreover, UV = V I is projective by Lemma 2.8.Thus, the two functors give a well-defined adjunction between the two full subcategories.Since the unit η P and the counit ε V are isomorphisms by Lemma 2.5 and Lemma 2.8, the statement follows.Lemma 2.4 guarantees the existence of an I -epimorphism W → V , where W is I -projective.This can be constructed in both an explicit and functorial way by considering the H -linear surjection H ⊗ k UV → UV .Applying the functor F = X ⊗ H (•) and composing with the counit ε V gives the functorial G-equivariant map X ⊗ k UV → V .Here, the left-hand side is even I -free.Moreover, on I -invariants, we get back the surjection H ⊗ k UV → UV we started with.Thus, the map X ⊗ k UV → V is an Iepimorphism.Its image is the G-subrepresentation of V generated by UV = V I .Iterating this process with the kernel of X ⊗ k UV → V , we see that any representation admits a functorial I -resolution.
Let (P,E) be a projective class on an abelian category A. As in [8], Definition 2.1, there is a candidate for an associated model structure on the category Ch(A) of unbounded chain complexes over A. If this model structure is well defined, we will call it the P-projective model structure.For the I -projective class on Rep ∞ k (G), the definition is as follows.For ease of notation, we set Ch(G) = Ch(Rep ∞ k (G)) and write Hom G (V • ,W • ) for the set of morphisms in this category.

Definition 2.10. A map
of complexes of abelian groups for any I -projective P. Similarly, f is an I-fibration if Hom G (P,V • ) → Hom G (P,W • ) is an epimorphism of complexes of abelian groups for any I -projective P. Finally, f is an I-cofibration if f has the left lifting property with respect to all I -trivial fibrations, that is, with respect to all maps that are both I -fibrations and I -equivalences.
Note that the adjunction F : Mod(H) Rep ∞ k (G) : U extends to an adjunction F : Ch(H) Ch(G) : U between the respective categories of complexes by working termwise.

an epimorphism).
Proof.Since U ∼ = Hom G (X,•) and since X is I -projective it follows that Uf is a quasi-isomorphism for every I -equivalence f.Conversely, for any indexing set J , we have an isomorphism of functors Hom G (X ⊕J ,•) ∼ = J Hom G (X,•) and taking homology of complexes commutes with direct products.It follows that if Uf = Hom G (X,f ) is a quasi-isomorphism (resp.an epimorphism), then so is Hom G (V ,f) for any I -free representation V.By taking direct summands, the same holds for any I -projective V (cf.Lemma 2.8 (ii)).
Our first aim is to show that Definition 2.10 indeed gives a model structure on 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.[8], Theorem 5.1).Theorem 2.12 (Christensen-Hovey).Let A be a bicomplete abelian category all of whose objects are small, and let (P,E) be a projective class on A. Suppose that P is determined by a set, that is, that there is a set S of objects of P such that E is precisely the class of S-epimorphisms.Then the P-projective model structure on Ch(A) is well defined and cofibrantly generated.
As an application, we obtain the following result.

Proposition 2.13.
There is a cofibrantly generated model structure on Ch(G) where the fibrations are the chain maps with surjective I-invariants and the weak equivalences are the chain maps whose I-invariants are quasi-isomorphisms.
Proof.In view of Lemma 2.11, we just need to check that the hypotheses in Theorem 2.12 are satisfied.Note that Rep ∞ k (G) is a Grothendieck category by [34], Lemma 1, and so all of its objects are small by [20], Proposition A.2. Since E is precisely the class of morphisms which are X-epimorphisms, the projective class (P,E) is determined by the singleton {X}.
We will call the above model structure on 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 Ch(G) is the right transfer of the projective model structure on 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 Ch(G).Proof.As we observed above, the I -projective model structure on Ch(G) is the right transfer of the projective model structure on Ch(H).By Lemma 1.6, it suffices to see that the unit η X• is a weak equivalence at all cofibrant (i.e.dg-projective) X • ∈ Ch(H).Since X • is cofibrant, it is termwise projective (cf.[19], Lemma 2.3.6).Thus, it follows from Lemma 2.5 that η X• is an isomorphism and hence a weak equivalence as required.For the final statement, see [19], Proposition 1.3.13.
In order to compute the left derived functor LF in Proposition 2.14, let M • ∈ Ch(H) and let Q • be any dg-projective complex of H -modules which is quasi-isomorphic to The results in Proposition 2.14 admit important improvements.To see this, we start with the following characterization of the I -cofibrant objects in Ch(G).Using [8], Proposition 2.5, this gives a description of the I -cofibrations in general.Note that even if a Quillen adjunction F : C D : U is a Quillen equivalence the right adjoint U will generally not preserve cofibrant objects.The adjunction id : Ch(S) Proj Ch(S) Inj : id gives an easy example.However, our situation is much more special.Lemma 2.15.For an object V • of Ch(G), the following are equivalent.
(ii) V • is termwise generated by its I-invariants, and the complex UV • in Ch(H) is cofibrant in the projective model structure, that is, dg-projective.
If these conditions are satisfied, then the counit Proof.Let us put Y = V • and denote by q UY : QU Y → UY the cofibrant replacement of UY = V I • .The adjoint morphism ε Y F q UY : F QUY → Y is an I -equivalence by Proposition 2.14.We claim it is a trivial I -fibration.As seen in the proof of Proposition 2.14, the left vertical arrow is an isomorphism because QU Y is cofibrant.Since q UY is a trivial fibration, the claim follows.
Assuming that Y is I -cofibrant it follows that the map F QUY → Y splits.Thus, Y is a direct summand of F QUY and UY is a direct summand of UF QUY ∼ = QU Y .This implies that UY is a cofibrant object of Ch(H), that is, is dg-projective.Moreover, Lemma 2.8 shows that F QUY 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 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 ∈ Rep ∞ k (G), we denote by 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.
The existence of the I -cofibrant replacement shows that if Proof.By Lemma 1.7 (ii), there is a surjection g : , the unit of the adjunction η P• : is an I -fibration by Lemma 2.11 because Remark 2.19.The above result can also be deduced from the axioms of a model category directly.Indeed, for any V • ∈ Ch(G), we can factorize the map 0 → V • into a trivial I -cofibration followed by an I -fibration, that is, there is an I -fibration f : For the sake of completeness, we note that it is also possible to start with a model structure on Ch(G) and apply a left transfer construction to get a model structure on Ch(H).However, in general it does not make sense to talk about a projective model structure on Ch(G) unless G is discrete.U .Consider the surjection ind U U (P U ) → P U .Since P is projective, the canonical map P → P U lifts to a map P → ind U U (P U ). Therefore, the canonical map P U → P U factors through ind U U (P U ) U → ind U U (P U ) U → P U , hence is zero as seen above.This also applies to U ⊆ U so that the map P U → P U → P U is zero.Letting U ⊆ U vary, we obtain that the canonical map P = U P U → P U is zero.Now, fix an open pro-p subgroup U of G. Letting U run through the open normal subgroups of U the canonical surjection V = U P U → P splits.For any open normal subgroup U ⊆ U and any W ∈ Rep ∞ k (U ), we let W (U ) = ker(W → W U ). Since U acts trivially on P U , we have (P U )(U ) = 0. Thus, V (U ) = U (P U )(U ) ⊆ U =U P U .As seen above, we have P = P (U ) for any open normal subgroup U of U and hence

Remark 2.20. Let G be a nondiscrete profinite group admitting an open subgroup which is pro-p. If k is a field of characteristic p, then the only projective object of Rep
Instead, we consider the injective model structure on Ch(G) introduced in §1 and show that it admits a left transfer to Ch(H) in the following sense.

Proposition 2.21.
There is a model structure on Ch(H) where the cofibrations are the maps g such that F g is a monomorphism and the weak equivalences are the maps g such that F g is a quasi-isomorphism.The fibrations are the maps satisfying the right lifting property with respect to all trivial cofibrations.
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 Ch(H) coanodyne if it satisfies the right lifting property with respect to all cofibrations in the sense of Proposition 2.21.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.[ Proof.Let T • be the mapping cone of id K• defined by Since this is a termwise split injection so is F ι • .In particular, F ι • is a cofibration and so is ι • by definition of the left transfer.Since f • is coanodyne, we may apply the right lifting property to the square and obtain a map g • : T • → P • making the two triangles commute.This implies f • g • = 0 whence g • has image in K • .We may therefore view g • as a map . By Remark 2.22 the map f • is a split epimorphism.Therefore, also F f • is a split epimorphism and we have ker ), the long exact homology sequence shows that F f • is a quasiisomorphism.
The model structure on Ch(H) constructed in Proposition 2.21 will be called the Ginjective model structure.It makes F : Ch(H) Ch(G) : U a Quillen adjunction.We hope to give an in-depth study of this adjunction elsewhere.

Gorenstein projective model structures
Let A be an abelian category with enough projectives.An object A ∈ A is called Gorenstein projective if there is an acyclic complex of projective objects of A for which A = Z 0 Y = B 0 Y and which remains exact upon applying Hom A (•,P ) for any projective object P of 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.[9], 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.[9], Theorem 9.1.10).Moreover, M is Gorenstein projective as an object of Mod(S) if and only if Ext i S (M,P ) = 0 for all i ≥ 1 and all projective S -modules P (cf.[9], Corollary 11.5.3).This is true if and only if Ext i S (M,P ) = 0 for all i ≥ 1 and all modules P of finite projective dimension.Remark 3.1.If S is an n-Gorenstein ring, then an S -module M is Gorenstein projective if and only if M = Z 0 Y for an acyclic complex Y ∈ Ch(S) of projective S -modules.Indeed, for any projective S -module P we then have Ext i S (M,P ) = Ext i S (Z 0 Y ,P ) = Ext i+n S (Z −n Y ,P ) = 0 for any i > 0 because P has injective dimension at most n.The same argument shows that all cycles of Y are Gorenstein projective.That the complex Y remains acyclic upon applying Hom S (•,P ) is then automatic.To see this, one simply applies Hom S (•,P ) to the short exact sequences 0 The following fundamental theorem is due to Hovey (cf.[21], Theorem 8.6).Theorem 3.2 (Hovey).Let S be a Gorenstein ring.On Mod(S), there is a cofibrantly generated model structure for which

• The cofibrations are the monomorphisms with Gorenstein projective cokernel, • The fibrations are the epimorphisms, • The trivial objects are the modules of finite projective dimension.
We call this the Gorenstein projective model structure on Mod(S).The class C (resp.F, resp.T ) of cofibrant (resp.fibrant, resp.trivial) objects is that of Gorenstein projective modules (resp.all modules, resp.modules of finite projective dimension).By [21], Corollary 8.5, the class C ∩ T is that of projective S -modules.The subcategory T of Mod(S) is thick and both (C ∩ T ,F) and (C,F ∩ T ) are functorially complete cotorsion pairs.Letting P denote the class of all short exact sequences in Mod(S) the Gorenstein projective model structure on 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 Mod(S) is abelian and hereditary.In particular, it is stable which is also proven directly in [21], 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 Mod(S) gives a functorial exact sequence in which QM is Gorenstein projective and KM has finite projective dimension.Let us recall from [9], Theorem 11.5.1, or [39], 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 Then G is Gorenstein projective by [9], Theorem 10.2.14.By definition, G admits a complete projective resolution P • as above.By shifting, we see that also acyclic for any projective S -module Q.Therefore, we can inductively choose S -linear maps The mapping cone of the vertical homomorphism yields an exact sequence 0 Here, N ⊕ Q 0 is Gorenstein projective and the kernel of the epimorphism to M has projective dimension at most n − 1.
Remark 3.3.Denote by Proj(S) and GProj(S) the full subcategory of Mod(S) consisting of all projective and all Gorenstein projective S -modules, respectively.The results of [21], Proposition 9.1 and Proposition 9.2, show that the homotopy category Ho(Mod(S)) can be viewed as the full subcategory GProj(S)/Proj(S) of the stable module category Mod(S)/Proj(S) of S consisting of all Gorenstein projective modules.
We point out that Theorem 3.2 admits a variant which is based on the dual notion of a Gorenstein injective module (cf.[21], Theorem 8.4).The corresponding homotopy category is equivalent to the quotient GInj(S)/Inj(S) of the category GInj(S) of Gorenstein injective modules by the category Inj(S) of injective S -modules (cf.[21], Proposition 9.1 and Proposition 9.2).It was shown by Krause in [24], Proposition 7.13, that the latter is equivalent to the singularity category K ac (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 [3] 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 [5] of Buchweitz.We refer to the end of [24], §7, for a more comprehensive list of historical remarks.Proof.Part (i) is [3], Proposition 2.2.1 (1), which is applied as in [3], §3.The description of the corresponding homotopy category follows from [3], Proposition 2.2.1 (1) and Example 1.4.7.The main point is that the right homotopy relation between cofibrant objects is the usual chain homotopy of complexes.To see this, note that the trivially cofibrant objects are the projective complexes as is shown in the proof of [3], Proposition 2.2.1 (1).Therefore, one can argue as in [21], Proposition 9.1.Finally, part (ii) is [3], Proposition 3.1.3.
We call this the singular projective model structure on Ch(S).Its existence can also be deduced from the general result [10], Theorem 7.2.15, of Enochs and Jenda.The class C (resp.F) of cofibrant (resp.fibrant) objects consists of the acyclic and termwise projective (resp.all) complexes.The class T of trivial objects is given by the condition T = C ⊥ as in Theorem 3.4 (i).The singular projective model structure is obtained from C, F and T as in Theorem 1.1 (ii) where we let P denote the class of all short exact sequences in Ch(S).Again, this is an abelian hereditary model structure and the associated model category is stable (cf.[3], Corollary 1.1.15).The loop and suspension functors can be made explicit as follows.[1] coincides with the loop functor Ω.

Proof. There are isomorphisms Ext
) for all X,Y ∈ Ch(S).This implies that both functors preserve the class of trivial objects.By exactness, they preserve all weak equivalences.Since they are inverse to each other, this implies that they form a Quillen equivalence in both directions.
In order to compute the suspension functor, let X ∈ Ch(S) be cofibrant and factorize X → 0 into a cofibration followed by a trivial fibration.This gives an exact sequence 0 → X → Y → Z → 0 where Y is trivial and Z is cofibrant.Then Z is the suspension of X in Ho(Ch(S)).Since Z is termwise projective, the above sequence is termwise split.As usual, we obtain a map of complexes Z → X[−1] which is unique up to chain homotopy.However, together with X also Y is cofibrant since X → Y is a cofibration.As remarked in the proof of Theorem 3.4, this implies that Y is a projective complex, hence is contractible.It follows that the map Z → X[−1] is a homotopy equivalence in the usual sense of chain complexes.As remarked earlier, this implies that Note that if S is a Gorenstein ring, then the Gorenstein projective model structure on Mod(S) is Quillen equivalent to its Gorenstein injective counterpart (cf.[21], page 583).It follows from the results of Becker that the corresponding homotopy category is also equivalent to the category K ac (Inj(S)) of complexes of injective S -modules up to chain homotopy (cf.[3], Proposition 2.2.1 (2) and Proposition 3.1.5).
By a result of Krause, the full subcategory of compact objects of K ac (Inj(S)) is equivalent to the quotient of the bounded derived category D b (noeth(S)) of noetherian Smodules by the subcategory of perfect complexes (cf.[24], 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 [3] and [24] are much more precise in the sense that they are derived from the existence of various recollements between triangulated categories.

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 = End G (X) op .In the following, we will therefore often make the assumption that the ring H is Gorenstein.If G = P σ and if I = I C , then the ring H is isomorphic to the Hecke algebra of a finite group with a split BN -pair of characteristic p as in (i).Therefore, H is a Frobenius algebra and hence is self-injective.
(iii) We keep the notation of (ii).If P † σ denotes the stabilizer of σ in X then P † σ is an open subgroup of G(K) containing P σ .If G = P † σ and if I = I C , then H is a Gorenstein ring by a result of Ollivier and Schneider (cf.[28], Proposition 5.5).Its self-injective dimension is equal to the rank of the center of G.
(iv) We keep the notation of (ii).If G = G(K) and if ring by a result of Ollivier and Schneider (cf.[28], Theorem 0.1).Its self-injective dimension is bounded above by the rank of G.
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 = 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.Remark 4.2.If the Gorenstein ring H has finite global dimension, then all H -modules have finite projective dimension and Ho(Mod(H)) = 0.In the above examples, this happens only in exceptional cases.In the situation of Example 4.1 (iv) for instance, the global dimension of H was studied in [28], §7.If the semisimple rank of G is positive and if the residue class field of K is not F 2 , then the global dimension of H is infinite (cf.[28], Corollary 7.2 and Lemma 7.3).Moreover, the supersingular H -modules tend to have infinite projective dimension (cf.[23], §1, for a more precise result).
For the following result, recall that we denote by B 0 the 0-th boundary functor introduced in §1.

and apply the functor F. This gives a factorization
ker(h), and we get the required exact sequence the adjunction is an isomorphism of complexes (cf.Lemma 2.5).Therefore, Lemma 2.7 shows that the canonical map ) is given by d 1 and hence is surjective.This implies that the map Proof.As for the first statement, it suffices to show the injectivity of the map Hom H (N,M ) → Hom H (N,U F M ) obtained by composing with the adjunction isomorphism Hom G (F N,F M) ∼ = Hom H (N,U F M ).Explicitly, this composition is given by sending g : Proposition 4.3 (ii).This proves the first statement.
In the homotopy category Ho(Mod(H)), any H -module becomes isomorphic to its cofibrant replacement.For the second statement, it therefore suffices to show that any Gorenstein projective H -module is contained in the essential image of the functor U.This follows from Proposition 4.3 (ii) because The previous results can be interpreted in terms of Quillen adjunctions for suitable model structures on Rep ∞ k (G) and Ch(G).For the rest of this section, we endow Ch(H) with the singular projective model structure from Theorem 3.4 (i).If H is Gorenstein, we will always endow Mod(H) with the Gorenstein projective model structure from Theorem 3.2.As before, let F : Mod(H) Rep ∞ k (G) : U be the adjunction given by F M = X ⊗ H M and UV = V I ∼ = Hom G (X,V ).By the same symbols, we denote its termwise extension to an adjunction F : Ch(H) Ch(G) : U .

Proof. Together with Rep ∞
k (G) also Ch(G) is a Grothendieck category (cf.Lemma 1.7 (i)).In particular, all objects of Ch(G) are small.Since 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 Ch(G).To construct them, let V ∈ Ch(G).We need a factorisation of the diagonal morphism such that Uα is a weak equivalence and Uβ is a fibration, that is, an epimorphism.
Let C = coker(Δ V ), and let π : V × V → C denote the canonical projection.Since the abelian category Ch(H) has enough projectives (cf.Lemma 1.7 (ii)), there is a surjection f : Q → 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 η } so that the projections p : P → V × V and h : P → F Q make the square Cartesian.Let i : V → P be given by i(v) = (Δ(v),0).This gives a factorization Δ V = pi, and we claim that it has the desired properties.As a right adjoint, the functor U preserves fibre products.Therefore, the square Uπ 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 Δ V is a split monomorphism with splitting the first projection.Therefore, the image of the sequence 0 In particular, Uπ is surjective and so is its pullback Uh.Since fiber products in abelian categories preserve kernels, the sequence 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 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 [19], Proposition 1.3.13.
The model structure on Ch(G) constructed in Proposition 4.5 will be referred to as the I-singular projective model structure.
Corollary 4.6.The I-singular projective model structure on Ch(G) is stable.More precisely, the adjunction [−1] : Ch(G) Ch(G) : [1] is a Quillen equivalence whose derived adjunction is given by the loop and suspension functors.

N. Dupré and J. Kohlhaase
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 [1], Corollary 3.1.4.Therefore, Lemma 3.5 and the right derived version of [19], Theorem 1.3.7,imply Composing with LF gives the isomorphism Ω ∼ = R [1] on Ch(G).The uniqueness of adjoints then also gives As in Lemma 2.15, we are in a situation where the right adjoint U preserves cofibrant objects.
Lemma 4.7.Endow Ch(G) with the I-singular projective model structure.For an object V • ∈ Ch(G), the following are equivalent.
• is an acyclic complex of projective H-modules.

If these conditions are satisfied, then the counit ε
Proof.Any cofibrant object Z ∈ Ch(H) is termwise projective.By Lemma 2.5, the unit η Z : Z → UF Z is an isomorphism.With these observations and using the characterization of I -projectives in Lemma 2.8 (iii) the proof of Lemma 2.15 carries over almost verbatim.
As a formal consequence, we get the following analog of Corollary 2.16.Proof.Assume that i is a cofibration.For any n ∈ Z consider the complex concentrated in degrees n+1 and n.Then D n+1 V n → 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 → UV n id → UV n → 0] which is trivial in the singular projective model structure by [3], Proposition 3.1.1.Consider the map V • → D n+1 V n given by the differential of V • 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 • → D n+1 V n such that j n is a splitting of i n .Since the cokernel of a cofibration is cofibrant (cf.[19], 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 [19], Proposition 2.3.9, carry over mutatis mutandis.
We now pass back from complexes to the categories Mod(H) and Rep ∞ k (G).For the rest of this section, we will therefore assume that H is a Gorenstein ring and that Mod(H) carries the Gorenstein projective model structure from Theorem 3.2.Proof.Since 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 Rep ∞ k (G).To construct them, we follow the argument in the proof of [21], Proposition 9.1.Given W ∈ Rep ∞ k (G), we choose an I -epimorphism q : Y → W , where Y is I -projective.Then the maps given by i(w) = (w,0) and p(w,y) = (w,w + q(y)) give a factorization Δ W = pi of the diagonal Δ W : W → W × W .Moreover, Ui is an injection such that coker(Ui) ∼ = UY is a projective H -module. Thus, Ui is a trivial cofibration in 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 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 Rep ∞ k (G) constructed in Proposition 4.10 will be called the I-Gorenstein projective model structure.This is the model structure on Rep ∞ k (G) we will consider for the rest of this section.To describe it more concretely, call an object V ∈ Rep ∞ k (G) I-trivial if the object UV ∈ Mod(H) is trivial, that is, has finite projective dimension.With this terminology, the fibrations in Rep ∞ 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 ∈ Rep ∞ k (G) admits an I -resolution X • → V → 0. By Lemma 2.4 and Lemma 2.7, this is a complex X • of I -projective G-representations such that UX • → UV → 0 is exact.Definition 4.11.We say that a representation V ∈ Rep ∞ k (G) has finite I-projective dimension if it admits an I -resolution of finite length, that is, if there exists an I -exact sequence 0 where X i is I -projective for any i.Proof.Assume that V has finite I -projective dimension.Applying U to a finite Iresolution 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 Let n denote the projective dimension of UV .Since UX • 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 im(d n ).Moreover, 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 im(d n ) is I -projective and that In a next step, we characterize the trivial cofibrations in Rep ∞ k (G).

trivial cofibration if and only if it is a split injection with an I-projective cokernel.
Proof.If i is a split injection with an I -projective cokernel, we may assume W = V ⊕ C where C is I -projective.If we have a commutative diagram where p is an I -epimorphism, then the map C → W lifts to a map g : Conversely, suppose that i : V → W is a trivial cofibration.Its cokernel C is trivially cofibrant by [19], Corollary 1.1.11.This means that the functor Hom G (C,•) sends Iepimorphisms to surjections which is equivalent to C being I -projective.It remains to show that i is a split injection.Factorize i as 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 ⊕ W → V then gives the splitting p 1 h of i.
The same argument shows that if V ∈ Rep ∞ k (G) is I -trivial, then any cofibration i : V → 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 Rep ∞ k (G).By the lack of exactness of F, this will not be a monomorphism in general.However, we have the following result.Lemma 4.14.If i : V → W is a cofibration, then so is the inclusion im(i) → W and we have W = im(i) + k[G] • UW .In particular, any cofibrant object is generated by its I-invariants.
Proof.For the first part, note that is a pushout square and that the class of cofibrations is closed under pushouts (cf.
→ X → 0 belongs to P. Note that the P-epimorphisms are precisely the epimorphisms which are also I -epimorphisms.The notion of a P-monomorphism is defined dually.In fact, the class P is proper in the sense of [26], Chapter XII.4.Proof.Clearly, P is closed under isomorphisms of short exact sequences and contains all split exact sequences.Now, let f : V → V and g : V → V be a composable pair of morphisms.Assume first that f and g are epimorphisms.We must show that if gf is an I -epimorphism, then so is g and that if f and g are I -epimorphisms, then so is gf .Both cases follow immediately by applying U to gf .Now, assume that f and g are Proposition 4.19.Any P-monomorphism i : V → W with a cofibrant cokernel is a cofibration.
Proof.Let p : V → V be a trivial fibration in a commutative square is an I -epimorphism and coker(i) is generated by its I -invariants (cf.Lemma 4.14).Since p is an I -epimorphism, we get im(g) ⊆ im(p).As in the previous proof, we may therefore assume that V = im(p) and that p is a P-epimorphism.Write K = ker(p) and C = coker(i).Since i is a Pmonomorphism, we can consider long exact sequences to obtain the following commutative diagram whose rows and columns are exact.
Since C is cofibrant and K has finite I -projective dimension (cf.Lemma 4.12), we have Ext 1 P (C,K) = 0 by Lemma 4.17.Now, the maps f and g satisfy p * f = i * g and therefore p * δf = δp * f = δi * g = 0.This implies δf = 0 whence there exists h : W → V with f = hi.
By construction, g − ph : W → V is zero on im(i) and therefore factors through a map α : C → V .Since C is cofibrant and p is a trivial fibration, α lifts to a map β : C → V such that pβ = α.Precomposing β with the quotient map W → C, we obtain a map j : W → V satisfying pj = g − ph.Therefore, the sum h + j : W → V gives our desired lift of i. Remark 4.20.By definition of the Gorenstein projective model structure (cf.Theorem 3.2), the cofibrations in Mod(H) are precisely the inclusions ker(q) → M , where M is arbitrary and q : M → N is a fibration onto a cofibrant object N. Using the above result, we see that the analogous maps in Rep ∞ k (G) are also cofibrations.Indeed, if X ∈ Rep ∞ k (G) is cofibrant and if p : W → X is an I -epimorphism, then p is a P-epimorphism because X is generated by its I -invariants (cf.Lemma 4.14).Therefore, the inclusion ker(p) → W is a P-monomorphism with cokernel X and so is a cofibration.
Recall that the adjunction in Proposition 4.10 is Quillen.We now investigate how far it is from being a Quillen equivalence.In a first step, we make explicit the homotopy relation in Rep ∞ k (G) following [21], Proposition 9.1.

then f and g are homotopic if and only if g − f factors through an I-projective.
Proof.Since W is fibrant, it follows from [19], Proposition 1.2.5 (v), that f and g are homotopic if and only if they are right homotopic.So assume that f and g are right homotopic.By [19], Corollary 1.2.6, there is a homotopy from f to g through any path object of W. In particular, we may take the path object Conversely, assume that g − f admits a factorization V → Z j → W where Z is Iprojective.Then j factors as Z → Y q → W because q is an I -epimorphism.Consequently, g − f = qh where h denotes the composition V → Z → Y .Since g = f + qh, we see that f and g are right homotopic by means of H = (f,h) : As a consequence, we obtain that the statements in Corollary 4. 4 Since η N admits a left inverse (cf.Proposition 4.3), it follows that f − g factors through UY .However, UY is a projective H -module (cf.Lemma 2.8).Therefore, f and g are homotopic by [21], Proposition 9.1.
the fibrant replacement of F X in Rep ∞ k (G).By the proof of [19], Proposition 1.3.13, the map X ηX → UF X Uf → UZ is a weak equivalence.However, since f is a weak equivalence so is Uf by definition of the right transfer.By the 2-out-of-3 property η X is a weak equivalence for any cofibrant object X.By Lemma 1.6, this implies (i).
If a cofibrant object X is realized as X = B 0 Y for some acyclic complex Y of projective H -modules, then η X is a monomorphism with cokernel UH 1 F Y (cf.Proposition 4.3).By [21], Lemma 5.8, the map η X is a weak equivalence if and only if the H -module UH 1 F Y has finite projective dimension.Therefore, the above arguments show that (ii) is equivalent to (iii).Assuming (i), the functors LF and RU are equivalences of categories commuting with Σ and Ω, respectively (cf.[1], Corollary 3.1.4).Since Mod(H) is stable (cf.[21], Theorem 9.3), it follows formally that so is Rep ∞ k (G).Note that the unit id → ΩΣ of the adjunction in Ho can then be identified with the aforementioned natural transformation LF ΩM → ΩLF M .Assuming (iv), the map ϕ is an isomorphism.We will show this implies (iii).
By the proof of Lemma 4.23 (i), the isomorphism ΣF B 0 Y ∼ = F M is induced by the surjection g : F B 0 Y ⊕ F Y 0 → F M sending (v,w) to F d 0 (w).In order to compute the loop of F M, choose an I -epimorphism q : Z → F M where Z is I -projective.By Lemma 4.23 (ii), we have ΩF M = ker(p), where p : F M × Z → F M × F M is given by p(v,w) = (v,v + q (w)).Consider the commutative square where ι is the canonical map.Since ι is a trivial cofibration (cf.Lemma 4.13) and since p is a fibration there is a map . Since F Y is a complex, the map Hι factors through a map f : F B 0 Y → ker(p).By the proof of [1], Proposition 3.1.7,its homotopy class is the adjoint morphism ϕ.Note that together with ι also f factors through the canonical map , it follows from [19], Theorem 1.2.10 (iv), that f is a weak equivalence.By definition of the right transfer, Uf is a weak equivalence in Mod(H).As seen above, it factors through the natural map UF B 0 Y → UB 0 F Y .However, the proof of Proposition 4.3 shows that the latter is a split surjection with kernel UH 1 F Y .Setting M 1 = UB 0 F Y and M 2 = UH 1 F Y , we identify it with the first projection M 1 × M 2 → M 1 .Write Uf = rj, where j is a trivial cofibration and r is a trivial fibration.Then j is a monomorphism with a projective cokernel and may be identified with a canonical inclusion of the form given by RU on objects and sending the class of f ∈ [Ω n X,Ω n Y ] to Σ n RU f .However, we currently do not know if this improves the properties of RU further.

Frobenius categories
Recall that a sequence X i → Y p → Z in an additive category A is called exact if i is a kernel of p and if p is a cokernel of i.An exact category is an additive category A endowed with a class of exact sequences satisfying the axioms in [6], Definition 2.1.For the sake of clarity these exact sequences are sometimes called admissible and the morphism i (resp.p) in an admissible exact sequence X i → Y p → Z is called an admissible monomorphism (resp.an admissible epimorphism).
An object Y of an exact category A is called projective (resp.injective) if the functor Hom A (Y ,•) (resp.Hom A (•,Y )) is exact, that is, if it transforms admissible exact sequences into exact sequences of abelian groups.An exact category A is said to have enough projectives (resp.injectives) if for any object X there is an admissible epimorphism Y → X (resp.an admissible monomorphism X → Y ) where Y is projective (resp.injective).An exact category A with enough projectives and enough injectives is called a Frobenius category if the classes of projective and injective objects of A coincide.
In the following, we will often assume that the exact category A is weakly idempotent complete in the sense of [6], Definition 7.2.By [12], Proposition 2.4, this is equivalent to the classes of admissible monomorphisms and admissible epimorphisms being closed under retracts.Hovey's correspondence in Theorem 1.1 was generalized to weakly idempotent complete exact categories by Gillespie in [12], Corollary 3.4.The corresponding model structures are called exact (cf.[12], Definition 3.1).We note that the completeness of the cotorsion pairs in [12], Definition 2.1, is not required to be functorial and that the bicompleteness assumption on A is relaxed significantly (cf. the beginning of [12], §4).
then V ∼ = B 0 F Y for some acyclic complex Y of projective H -modules by Corollary 4.8.It then follows from Proposition 4.3 (ii) and Remark 3.1 that UV ∼ = UB 0 F Y ∼ = B 0 Y is a Gorenstein projective H -module and that U F is isomorphic to the identity functor on GProj(H).In particular, the functor U : C(G) → GProj(H) is essentially surjective and the functor F : GProj(H) → C(G) is faithful.However, U on C(G) is faithful, too.This follows from the fact that the objects of C(G) are quotients of I -projective G-representations, hence are generated by their I -invariants (cf.Lemma 2.8).
It then follows from U F ∼ = id that U is fully faithful on the essential image of F : GProj(H) → C(G).However, this functor is essentially surjective.Indeed, if V ∈ C(G) we may assume V = B 0 F Y for some acyclic complex Y of projective H -modules.Set M = B 0 Y .By Remark 3.1 and [14], Theorem 4.1, there is a map f : Y → Y M of complexes such that B 0 f is the identity on M. We claim that the map B 0 F f : Since U reflects monomorphisms and since FM is generated by its I -invariants, it suffices to see that UB 0 F f is an isomorphism in Mod(H).This follows from Proposition 4.3 (ii) because under the isomorphisms UB 0 F Y ∼ = B 0 Y = M ∼ = U FM the map UB 0 F f is the identity on M.
We have the following alternative characterization of the objects of C(G).(ii) The representation V is generated by its I-invariants and there is a map i : V → W in Rep ∞ k (G) such that W is I-projective and Ui is a cofibration in Mod(H).In particular, any object V of C(G) is both a quotient and a subobject of an I-projective G-representation and UV is a Gorenstein projective H-module.
Proof.That any object V ∈ C(G) is both a quotient and a subobject of an I -projective G-representation is true by definition.Moreover, UV is Gorenstein projective by the proof of Theorem 5.4.Therefore, we only need to show that (i) and (ii) are equivalent.Assume that V satisfies (i).By definition, V ∼ = B 0 F Y for some acyclic complex of projectives Y ∈ Ch(H).This gives an embedding i : V → F Y 0 where F Y 0 is I -projective.Moreover, V is generated by its I -invariants as seen in the proof of Theorem 5.4.Under the isomorphism UV ∼ = UB 0 F Y ∼ = B 0 Y in Proposition 4.3 (ii), the H -module coker(Ui) gets identified with B 1 Y .Thus, coker(Ui) is Gorenstein projective and Ui is a cofibration.This shows (ii).Conversely, assume that V satisfies (ii).Since Ui is a cofibration we have coker(Ui) ∈ GProj(H).By the functorial factorisations in Mod(H), we can embed coker(Ui) into a projective H -module via a cofibration.This way we inductively construct an acyclic complex where Y i is a projective H -module for all i ≤ 0. If we set X i = F Y i , then Lemma 2.5 and Lemma 2.8 allow us to identify this complex with the I -invariants of the complex which is therefore I -exact.Composing it with an I -resolution of V gives an I -exact complex X of I -projective G-representations with V = B 0 X.This shows V ∈ C(G).Remark 5.6.Assume that H comes from a finite group G with a split BN -pair of characteristic p as in Example 4.1 (i).Then H is self-injective and GProj(H) = Mod(H).Moreover, any monomorphism in Mod(H) is a cofibration.It follows from Proposition 5.5 that C(G) is the full subcategory of representations which are both a quotient and a subobject of an I -projective G-representation.In this case, the finite-dimensional objects of C(G) were first studied by Cabanes (cf.[7], §1).Moreover, Theorem 5.4 recovers [7], Theorem 2. Our arguments differ only gradually.Everything relies on the fact that the adjunction (F,U ) induces an equivalence between projective H -modules and I -projective G-representations (cf.Corollary 2.9).where ε V is surjective since V is generated by its I -invariants.Moreover, ε W is an isomorphism by Lemma 2.8.Therefore, the composition ε −1 W i induces an isomorphism f : V → im(F Ui).Since the inclusion g : im(F Ui) → F UW is a cofibration by Lemma 4.14 so is i = ε W gf .
The difference between the functors F and F can be made explicit as follows.Proof.The first statement follows from Proposition 4.3 (i).Since F preserves surjections, it follows that so does F. The final statement follows from Theorem 5.4.
For a suitable exact structure on C(G), the functor F actually becomes exact.G-representations.In this case, the equivalence U : C(G) → GProj(H) = Mod(k) is given by the forgetful functor.In the situations of Example 4.1, however, the category Mod(H) is comparatively easy to understand and yet strongly linked to Rep ∞ k (G).Passing to Ho(Mod(H)) ∼ = GProj(H)/Proj(H) simplifies the situation further by getting rid of the objects of finite projective dimension.Finally, we explain in which way the objects of C(G) appear in the theory of equivariant coefficient systems on Bruhat-Tits buildings as studied in [22] and [28].To this aim, we assume that H is associated to G = G(K) and I = I C as in Example 4.1 (iv).Given M ∈ Mod(H), Ollivier and Schneider construct in [28], §6.4, a functorial exact sequence of H -modules 0 → GP(M ) d → . . .→ GP(M ) 0 → M → 0, where d is the semisimple rank of G.If G is semisimple, then all GP(M ) i are Gorenstein projective by [28], Lemma 6.4.On the other hand, the second author functorially associates with M ∈ Mod(H) a G-equivariant coefficient system F(M ) on the semisimple Bruhat-Tits building X of G (cf. [22], §3.2).The corresponding complex C or c (X (•) ,F(M )) of oriented chains is I -exact (cf.[22], Proposition 2.9).In fact, it is concentrated in degrees 0 ≤ i ≤ d and admits a functorial isomorphism C or c (X (•) ,F(M )) I ∼ = GP(M ) • (11) in Ch(H) (cf.[22], Remark 3.24).In particular, there is a functorial isomorphism H 0 (C or c (X (•) ,F(M )) I ) ∼ = M of H -modules (cf.[22], Theorem 3.21).Proof.This will follow from Theorem 5.4 and (11) if we can show that the complex C or c (X (•) ,F(M )) consists of objects of C(G).To see this, let σ be a facet of X contained in the closure of C and denote by P † σ the stabilizer of σ in G. Setting X σ = ind P † σ I (k), we have the Gorenstein ring H † σ = End P † σ (X σ ) op associated to P † σ and I = I C as in Example 4.1 (iii).Note that we may view H † σ as a subalgebra of H. Denote by U σ , F σ and F σ our usual functors corresponding to P † σ and H † σ .In [22], Theorem 3.12, the second author constructs a fully faithful functor t σ : Mod(H † σ ) → Rep ∞ k (P † σ ) which is right inverse to U σ .Given N ∈ Mod(H † σ ) the canonical complete resolution Y N • provides an embedding N → Y N 0 where Y N 0 is an H † σ -module of finite projective dimension.By our semisimplicity assumption, the ring H † σ is selfinjective (cf.[28], Proposition 5.5).It follows that Y N 0 is a projective H † σ -module.The arguments given in the proof of [22], Theorem 3.12, then show that there is a functorial H † σ -linear isomorphism t σ N ∼ = F σ N .Thus, t σ ∼ = F σ as functors.
Consider the automorphism j σ of the k -algebra H † σ introduced in [28], §3.3.1.Via scalar restriction it induces an automorphism of the category Mod(H † σ ) denoted by N → N ( σ ).For M ∈ Mod(H), denote by M σ ∈ Mod(H † σ ) the scalar restriction of M along the inclusion If f : A → B and g : A → B are morphisms in C, then a right homotopy from f to g is a morphism H : A → P from A into a path object B i −→ P p −→ B × B for B such that p 1 pH = f and p 2 pH = g.Here, p 1 ,p 2 : B × B → 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 [19], Definition 1.2.4.Morphisms which are both right and left homotopic are called homotopic.If X ∈ C c and Y ∈ C f , then two morphisms f,g : X → Y are right homotopic if and only if they are left homotopic (cf.[19], Proposition 1.2.5 (v)).Moreover, on the morphisms of C cf the homotopy relation ∼ is an equivalence relation compatible with composition (cf.[19], Corollary 1.2.7).Dividing out the homotopy relation on the level of morphisms leads to a category denoted C cf / ∼.

Proposition 2 . 14 .
The adjoint functors F : Ch(H) Ch(G) : U given by F M • = X ⊗ H M • and UV • = V I • form a Quillen equivalence with respect to the I-projective model structure on Ch(G) and the projective model structure on Ch(H).In particular, the derived adjunction LF : D(H) Ho(Ch(G)) : RU is an equivalence between the unbounded derived category of Mod(H) and the homotopy category of the I-projective model structure on Ch(G).
Ho(Ch(G)) for the I -projective model structure on Ch(G).To compute the right derived functor RU , let V • ∈ Ch(G).Since every object of Ch(G) is fibrant for the I -projective model structure we have RU

Corollary 2 . 16 .
Endow Ch(H) with the projective model structure and Ch(G) with the I-projective model structure of Proposition 2.13.The functors U = (•) I and F= X ⊗ H (•) restrict to inverse equivalences of categories Ch(G) c Ch(H) c .Proof.If Y ∈ Ch(G) is I -cofibrant, then UY is cofibrant in Ch(H) by Lemma 2.15.If Z ∈ Ch(H) is cofibrant then F Z is I -cofibrantby definition of a left Quillen functor.The counit ε Y : F UY → Y is an isomorphism by Lemma 2.15, and the unit η Z : Z → UF Z is an isomorphism by the proof of Proposition 2.14.

Remark 2 . 22 .
Note that any coanodyne map f : X → Y in Ch(H) is a split epimorphism.Indeed, the map 0 → Y is a cofibration for the left transfer.By applying the right lifting property

Theorem 3 . 4 (
Becker).(i)For any ring S, there is a cofibrantly generated model structure on Ch(S) for which • The cofibrations are the monomorphisms with acyclic and termwise projective cokernel, • The fibrations are the epimorphisms,• The trivial objects are the complexes Y satisfying Ext 1 Ch(S) (X,Y ) = 0 for all termwise projective acyclic complexes X.Its homotopy category is equivalent to the category K ac (Proj(S)) of acyclic complexes of projective S-modules up to chain homotopy.(ii)If S is a Gorenstein ring, then the adjoint functors Q 0 : Ch(S) Mod(S) : ι 0 defined in(6) form a Quillen equivalence for the model structure on Ch(S) as in (i) and the Gorenstein projective model structure on Mod(S).

Example 4 . 1 .
The ring H is Gorenstein in the following situations.(i)Let G be a finite group with a split BN -pair of characteristic p. Recall that then B admits a normal p-Sylow subgroup U such that B = U T with T = B ∩ N .Setting I = U and endowing G with the discrete topology the Hecke algebra H is a Frobenius algebra by a result of Tinberg (cf.[38],Proposition 3.7).In this case, H is even self-injective.As an example one may take the group G = G(K) of rational points of a split connected reductive group G over a finite field K of characteristic p and I = U the group of rational points of the unipotent radical of a Borel subgroup of G.(ii) Let G(K) denote the group of rational points of a split connected reductive group G over a non-Archimedean local field K of residue characteristic p.The group G(K) carries a natural locally profinite topology.Denote by X the semisimple Bruhat-Tits building of G(K), by C a chamber of X and by σ a facet of X contained in the closure of C. Denote by P σ the parahoric subgroup of σ and by I C the pro-p radical of P C .Both of them are compact open subgroups of G(K) and we have I C ⊆ P σ .

Proposition 4 . 5 .
The right transfer of the singular projective model structure exists along the adjunction F : Ch(H) Ch(G) : U and makes it a Quillen equivalence.Endowing Ch(G) with the right transfer, there is an equivalence of categories K ac (Proj(H)) ∼ = Ho(Ch(G)).

Corollary 4 . 8 .
Endow Ch(H) and Ch(G) with the singular projective model structure and the I-singular projective model structure, respectively.Then U and F restrict to inverse equivalences of categories Ch(G) c Ch(H) c .Moreover, we obtain the following description of the cofibrations in Ch(G).

Corollary 4 . 9 .
A morphism i : V • → W • in Ch(G)is a cofibration for the I-singular projective model structure if and only if it is a termwise split monomorphism whose cokernel is an I-exact sequence of I-projectives.

Proposition 4 . 10 .
The right transfer of the Gorenstein projective model structure exists along the adjunction F : Mod(H) Rep ∞ k (G) : U .

Lemma 4 . 12 .
A representation V ∈ Rep ∞ k (G) is I-trivial ifand only if it has finite I-projective dimension.

W
[19], Corollary 1.1.11).For the second part, let W = im(i)+k[G] • UW .The inclusion W → W is an isomorphism on I -invariants, hence is a trivial fibration.Thus, we may apply the lifting property of i to the diagram V 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 Rep ∞ k (G).Definition 4.15.Let P be the class of sequences 0

Lemma 4 . 16 .
The class P is proper.
the proof of Proposition 4.10.Recall that this involves an I -epimorphism q : Y → W where Y is I -projective.There is then a map H = (r,s) : V → W × Y with pH = (f,g).By construction of p, this means f = r and g = r + qs.Thus, g − f = qs factors through the I -projective Y.
(Rep ∞ k (G)) gives rise to a natural transformation LF Ω → ΩΣLF Ω ∼ = ΩLF ΣΩ ∼ = ΩLF using the stability of Mod(H).To describe this more explicitly, let Y = (Y • ,d • ) be an acyclic complex of projective Hmodules, and set M = B −1 Y .Then ΩM ∼ = B 0 Y in Ho(Mod(H)) by the construction of the loop and suspension functors on abelian model categories.Since B 0 Y is cofibrant, this gives LF ΩM ∼ = F B 0 Y .Similarly, LF M ∼ = F M and therefore ΩLF M ∼ = ΩF M. Note that the canonical map q

Proposition 5 . 5 .
For any object V ∈ Rep ∞ k (G), the following are equivalent.(i) We have V ∈ C(G).

Proposition 5 .Lemma 5 . 7 .
5 yields the following class of cofibrations in Rep∞ k (G).If V ∈ Rep ∞ k (G)is generated by its I-invariants, then any map i : V → W as in Proposition 5.5 (ii) is a cofibration.Proof.By assumption, the mapUi : UV → UW is a cofibration in Mod(H).Since F is left Quillen, the map F Ui : F UV → F UW is a cofibration in Rep ∞ k (G).We have a commutative square

Corollary 5 . 9 .
With respect to the short I-exact sequences, the category C(G) is a weakly idempotent complete Frobenius category and the functors F : GProj(H)C(G) : U are exact inverse equivalences.
).Let A be a bicomplete abelian category with a fixed proper class P of short exact sequences.(i) If A carries a model structure which is compatible with P, then A t is P-thick and (A c ,A t ∩ A f ) and (A c ∩ A t ,A f ) are functorially complete cotorsion pairs.

Proposition 1.4. Assume
, Theorem 3.8.that C is a model category and that we have an adjunction F : C D : U , where D is bicomplete.If every object in C is fibrant and if every object of D admits a path object, then every anodyne morphism in D is a weak equivalence.
To see this, we need to check that we get a trivial fibration in Ch(H) if we apply the functor U. Consider the commutative diagram Denote by D = Ho(Ch(G)) the homotopy category of Ch(G) endowed with the I -projective model structure.By Proposition 2.14 and [8], Corollary 2.14, we have Ext then there are functorial bijections Ext n G,I (V ,W ) ∼ = Ext n H (UV ,UW ) for all n ≥ 0.Proof.
By Lemma 2.11 and Lemma 2.15, this means that Uf : UQ • → UV • is a surjective quasi-isomorphism where UQ • ∈ 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 Rep ∞ k (G) discussed earlier.
Lemma 2.18.For any spanned by u∈U /U u.
U because p divides (U : U ) and k has characteristic p.It follows that the canonical map ind U U (W ) U → ind U U (W ) U is zero for any trivial U -representation W. Now, assume that P ∈ Rep ∞ k (G) is a projective object and that U is an open pro-p subgroup of G.The restriction functor Rep ∞ k (G) → Rep ∞ k (U ) preserves projective objects Dupré and J. Kohlhaase to C exists if and only if F f is a weak equivalence in D for every coanodyne morphism f in C.
2], Theorem 2.23).Theorem 2.23.Suppose we have an adjunction F : C D : U between locally presentable categories where D carries a cofibrantly generated model structure.Then the left transfer 1050 N.
[19] to the homotopy level.We continue to endow Mod(H) with the Gorenstein projective model structure and Rep ∞ Note that F preserves cofibrant objects because it is left Quillen.Since all objects of Mod(H) and Rep ∞ k (G) are fibrant, it suffices to see that if M,N ∈ Mod(H) are Gorenstein projective, then the map Hom H (M,N ) → Hom G (F M,F N) induced by F reflects the respective homotopy relations (cf.[19],Theorem 1.2.10 (ii)).Let f,g ∈ Mod H (M,N ), and assume that F f is homotopic to F g.By Lemma 4.21, there is an I [19]) with the I -Gorenstein projective model structure.Theorem 4.22.In the derived adjunction LF : Ho(Mod(H)) Ho(Rep ∞ k (G)) : RU the functor LF is faithful and the functor RU is essentially surjective.Proof.That RU is essentially surjective follows directly from Corollary 4.4.In order to show that LF is faithful, let Q c denote the cofibrant replacement functor of Mod(H).Recall from (4) that LF = Ho(F )Ho(Q c ).By the proof of[19], Proposition 1.2.3, the functor Ho(Q c ) is always an equivalence.Therefore, it suffices to show that Ho(F ) is faithful.
contained in the kernel of Uf it follows that M 2 is a direct summand of ker(r).Since ker(r) has finite projective dimension so doesM 2 = UH 1 F Y .The transition maps in this direct limit are induced by Ω, and [Ω n X,Ω n Y ] denotes the set of morphisms in the usual homotopy category.The loop functor induces a fully faithful functor Ω on Ho(Rep ∞ k (G)).Moreover, since RU commutes with Ω and since Mod(H) is stable, there is an induced functor