Affine strict polynomial functors and formality

Advances in Mathematics 320 (2017) 652–673

Contents lists available at ScienceDirect

Advances in Mathematics www.elsevier.com/locate/aim

Affine strict polynomial functors and formality Marcin Chałupnik 1 Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

a r t i c l e

i n f o

Article history: Received 23 January 2015 Received in revised form 30 August 2017 Accepted 1 September 2017 Available online 12 September 2017 Communicated by Roman Bezrukavnikov

a b s t r a c t We introduce the notion of an affine strict polynomial functor. We show how this concept helps to understand homological behavior of the operation of Frobenius twist in the category of strict polynomial functors over a field of positive characteristic. We also point out for an analogy between our category and the category of representations of the group of algebraic loops on GLn . © 2017 Elsevier Inc. All rights reserved.

MSC: 18A25 18A40 18G15 20G15 Keywords: Strict polynomial functor DG category Ext-group Formality

1. Introduction In the present paper we study homological algebra in the category Pd of strict polynomial functors of degree d over a field of characteristic p > 0. We introduce a new type of E-mail address: [email protected]. The author was supported by the Narodowe Centrum Nauki grants no. 2011/01/B/ST1/06184 and 2015/19/B/ST1/01150. 1

http://dx.doi.org/10.1016/j.aim.2017.09.006 0001-8708/© 2017 Elsevier Inc. All rights reserved.

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

653

strict polynomial functors we call the affine strict polynomial functors. It was motivated by the observation that the Ext-groups of the form Ext∗Ppd (F (1) , G) where F (1) is the Frobenius twist of F ∈ Pd are equipped with certain extra structure. It has become more and more apparent that for needs of the program started in [2], [3] aiming at computing the Ext groups between strict polynomial functors important in representation theory, understanding this extra structure is necessary. This structure comes, roughly speaking, from the fact that the Frobenius twist functor I (1) ∈ Pp has non-trivial endo-Ext-groups and these Ext-groups act (nonlinearly) on F (1) := F ◦ I (1) . However, for technical reasons, it is more convenient to look at the operation adjoint to the twisting. Namely, precomposing with I (1) is an exact operation, hence it gives rise to a functor between derived categories: C : DP d −→ DP pd . It was shown in [4] that this functor has a right adjoint Kr : DP pd −→ DP d called there the derived right Kan extension. Our idea is to factorize Kr through certain richer triangulated category DP af d which should be thought of as the derived category of the category Pdaf of affine strict polynomial functors of degree d. Now since Ext∗Ppd (F (1) , G)  HExt∗Pd (F, Kr (G)) and Kr (G) comes from this richer category we see where extra structure comes from. In fact this enriched category Pdaf has quite transparent interpretation: it is the category of strict polynomial functors but from (certain subcategory of) the category of (graded free finitely generated) A-modules where A := Ext∗Pp (I (1) , I (1) ) (see details in Section 2). However, since our construction is technically a bit involved, let us present here some informal review of it. We first recall from [4] definition of Kr . Namely we have Kr (F )(V ) = RHomPpd (Γd (V ∗ ⊗ I (1) ), F ) where Γd is the divided power functor. Since the contravariant argument in the above RHom is also a (contravariant) strict polynomial functor in V , our construction produces a strict polynomial functor. Now we can see some extra structure on H ∗ (Kr (F )): the Ext-groups Ext∗Ppd (Γd (V ∗ ⊗ I (1) ), Γd (V ∗ ⊗ I (1) )) act on it. These Ext-groups will turn out to be Hom-spaces in the source of our functor category Pdaf . However, the situation is still not satisfactory since we want these Extgroups acting on Kr (F ), not merely on its cohomology. In order to lift this action to the

654

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

complexes we need some sort of formality. Let us make this idea more precise. We fix a projective resolution X of Γd (I ∗ ⊗ I (1) ) meant as a strict polynomial functor in two variables in the sense of [7] and introduce an auxiliary differential graded (=DG) cateop gory (denoted by Γd VX ) whose Hom spaces are complexes HomPpd (X, X). This category r genuinely acts on K (F ) and our Ext-groups form the cohomology category of it. Now op the main point is Theorem 4.3 which says that the DG category Γd VX is formal. This allows us to endow Kr (F ) with a structure which a priori existed only on cohomology. The article is organized as follows. In Section 2 we introduce the category Pdaf and establish its basic properties. Section 3 recalls generalities on derived categories of DG categories and discusses certain finiteness assumption which is useful in our situation. In op Section 4 we introduce the category Γd VX and prove its formality. Finally, in Section 5 we construct the affine derived right Kan extension Kaf and relate it to Kr . Theorem 5.1 which establishes the fundamental properties of Kaf is the main result of the paper. In the last section we put our work into a wider context and discuss some possible further developments. In particular we observe a formal analogy between Pdaf and the category of representations of the groups of algebraic loops on GLn . Acknowledgments I am grateful to Julian Külshammer for turning my attention to [14] and to Stanisław Betley for many remarks on the earlier version of the article. 2. The category of affine strict polynomial functors We fix a field k of characteristic p > 0. Let V stand for the category of finite dimensional k-spaces. Let A denote the graded k-algebra A := k[x]/xp for x of degree 2. By the classical computation [10, Th. 4.5]: A  Ext∗Pp (I (1) , I (1) ). We recall that a graded k-linear category is a category whose Hom sets are graded k-linear spaces and composition preserves these structures. Then we introduce the k-linear graded category VA which is the full subcategory of the graded category of graded A-modules consisting of objects of the form V ⊗ A for V ∈ V. Hence HomVA (V ⊗ A, W ⊗ A) = HomA (V ⊗ A, W ⊗ A)  Hom(V, W ) ⊗ A (unless otherwise stated all (graded) linear and tensor operation are taken over k). Also, since all (graded) categories considered in our article are k-linear over our ground field k, from now on we skip referring to k whenever no confusion is possible. We shall introduce certain version of the Friedlander–Suslin strict polynomial functors which, roughly speaking, correspond to the functors from A-modules to k-modules. Technically, it will be convenient to adopt the approach to the strict polynomial functors due to Pirashvili which allows one to interpret strict polynomial functors as genuine functors. We recall (see e.g. [9, Sect. 3]) that one considers the category Γd V whose objects are finite dimensional k-spaces but HomΓd V (V, W ) := Γd (Hom(V, W ))

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

655

where Γd (X) := (X ⊗d )Σd is the space of symmetric d-tensors on a k-space X. Then it is easy to see that a strict polynomial functor homogeneous of degree d in the sense of [10] is nothing but a genuine k-linear functor from Γd V to V. Now we introduce the graded category Γd VA with the objects the same as in VA and HomΓd VA (V ⊗ A, W ⊗ A) := Γd (Hom(V, W ) ⊗ A). The composition of morphisms is defined in the spirit of [9]. Namely: by using the natural map Γd (X) ⊗ Γd (Y ) −→ Γd (X ⊗ Y ) and the composition in VA we obtain Γd (HomA (V ⊗ A, W ⊗ A)) ⊗ Γd (HomA (W ⊗ A, U ⊗ A)) −→ Γd (HomA (V ⊗ A, W ⊗ A) ⊗ HomA (W ⊗ A, U ⊗ A)) −→ Γd (HomA (V ⊗ A, U ⊗ A)). The grading on the Hom-spaces comes from the standard grading on tensor product i.e. d |x1 ⊗ . . . ⊗ xd | = j=1 |xj | and is clearly preserved by the composition. We call functor between graded categories a graded functor if its action on the Homspaces preserves grading. Let A be a k-linear graded category. We consider its subcategory A0 with the same objects but HomA0 (X, Y ) := Hom0A (X, Y ). Then we say that A is a graded abelian category when the underlying k-linear category of A0 is a k-linear abelian category. In that case for φ ∈ Hom0A (X, Y ), ker(φ), im(φ) etc. can be naturally regarded as objects of our graded category A. Also, we say that X ∈ A is projective, injective etc. when it is projective, injective etc. in A0 . Since we shall use in many places objects with underlying graded vector spaces, in order to keep formulations of theorems and definitions possibly concise we take the following abbreviation: Definition/Notation 2.1. We say that a graded vector space satisfies f+ condition if it is finite dimensional in each degree and bounded below. We say that an (e.g. strict polynomial) functor satisfies f+ condition if all its values are graded spaces satisfying f+ condition. In accordance with this convention we denote by V f + the graded category of Z-graded bounded below k-linear spaces, finite dimensional in each degree. Now we are ready for defining our functor category. Definition/Proposition 2.2. An affine strict polynomial functor F homogeneous of degree d is a graded k-linear functor F : Γd VA −→ V f + .

656

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

The affine strict polynomial functors homogeneous of degree d with morphisms being natural transformations form a graded abelian category Pdaf . Let me at this point comment on our notational conventions. In fact the objects of VA and Γd VA are indexed just by vector spaces. Nevertheless, we prefer to use a label V ⊗ A instead of V . The reason is that when we construct functors on Γd VA we usually somehow use A-structure on Hom-spaces. Even if not, like in the forgetful functor I af ∈ P1af which sends V ⊗ A to itself, the apparently simpler notation would produce quite strange formula: I af (V ) := V ⊗A. Hence forgetting would rather look like inducing which would be quite confusing. Like in any functor category, for any U ∈ V we have the representable functor hU ⊗A ∈ af Pd given by the formula V ⊗ A → HomΓd VA (U ⊗ A, V ⊗ A) and the co-representable functor c∗U ⊗A ∈ Pdaf given by the formula V ⊗ A → HomΓd VA (V ⊗ A, U ⊗ A)∗ where (−)∗ stands for the graded k-linear dual (i.e. (V ∗ )j = Hom(V −j , k)). Now by applying the Yoneda lemma to our situation we obtain Proposition 2.3. There are natural in U ⊗ A isomorphisms HomP af (hU ⊗A , F )  F (U ⊗ A) d

HomP af (F, c∗U ⊗A )  (F (U ⊗ A))∗ d

for any F ∈ Pdaf . Since VA is sort of scalar extension of V, one can expect some inducing/forgetting adjunction between respective functor categories. Indeed, we have the functors z : VA −→ V,

t : V −→ VA ,

where, strictly speaking, we identify the graded category VA with the underlying ordinary category. Our functors are given on the objects by formulae: z(V ⊗ A) := V ⊗ A, t(V ) := V ⊗ A. The action on Hom-spaces is the following: zV,W : Hom(V, W ) ⊗ A  HomA (V ⊗ A, W ⊗ A) −→ Hom(V ⊗ A, W ⊗ A) is the natural embedding; tV,W : Hom(V, W ) −→ Hom(V, W ) ⊗ A

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

657

is the embedding on the degree 0 part. These functors are nothing but forgetful and induction functors. These functors extend naturally to our d-tensor categories. We have Γd z : Γd VA −→ Γd V, whose action on the morphisms: (Γd z)V,W : Γd (Hom(V, W ) ⊗ A) −→ Γd (Hom(V ⊗ A, W ⊗ A)) is just Γd (zV,W ). Analogously we have Γd t : Γd V −→ Γd VA . Now we consider the graded category Pdf + consisting of graded k-linear functors from Γd V regarded as graded category with the trivial grading to V f +. In other words: an object of Pdf + is a collection {F j }j≥j0 of objects of Pd and Homj f + (F • , G• ) := Pd



HomPd (F s , Gs+j ).

s

Then we shall show that the assigning F → F ◦ Γd z produces the graded functor which, to simplify notation, will be denoted by z ∗ : Pdf + −→ Pdaf This is not entirely obvious as can already be seen for F concentrated in degree 0. Then taking z ∗ (F )(V ) := F (V ⊗ A) as concentrated in degree 0 does not produce a graded functor. The correct approach relies on the fact specific to strict polynomial functors, that any F ∈ Pd can be naturally extended to the functor F gr : Γd V f + −→ V f + (see e.g.  j [17, Sect. 2.5]). We recall that for the graded space V = V we have a decomposition    F (V ) = F where for γ = (γ1 , . . . , γs ) with γj = d, Fγ ( V j ) is the sub-s-functor  j γ  of F ( V ) of degree γj in V j . Then we assign to Fγ degree jγj . In general, for  s  F = F ∈ Pdf + we assign to F s (V ⊗ A) degree s + j jγj . Now it is easy to see that z ∗ is well defined and we analogously define the graded functor t∗ : Pdaf −→ Pdf + as precomposition with t. Now, as one can expect, we have Proposition 2.4. 1. z ∗ preserves representable objects i.e. z ∗ (Γd,U ) = hU ⊗A , where Γd,U ∈ Pd ⊂ Pdf + is defined as V → HomΓd V (U, V ) = Γd (U ∗ ⊗ V ).

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

658

2. The functor t∗ is right adjoint to z ∗ . Proof. We have z ∗ (Γd,U )(V ⊗ A) = Γd,U (V ⊗ A) = Γd (Hom(U, V ⊗ A))  Γd (Hom(U, V ) ⊗ A)  hU ⊗A (V ⊗ A), thus getting the first part. Since Γd,U are projective generators of Pd , any F ∈ Pdf + has a projective resolution by objects having in each degree a finite sum of the functors of the form Γd,U . Therefore in order to get the second part, it suffices to obtain a natural in U ∈ Γd V and F ∈ Pdaf isomorphism HomP f + (Γd,U , t∗ (F ))  HomP af (z ∗ (Γd,U ), F ). d

d

Now by the Yoneda lemma we have HomP f + (Γd,U , t∗ (F ))  t∗ (F )(U ) = F (U ⊗ A). d

By using the first part we get HomP af (z ∗ (Γd,U ), F )  HomP af (hU ⊗A , F )  F (U ⊗ A). d

d

2

The functor z ∗ provides a lot of examples of affine strict polynomial functors. We shall occasionally denote z ∗ (F ) as F af . E.g. we have affine versions of tensor functors: (I d )af (V ⊗ A) := (V ⊗ A)⊗d , (S d )af (V ⊗ A) := S d (V ⊗ A) etc. Perhaps more interesting are objects in Pdaf which do not come from Pd . The most fundamental examples are χj ∈ P1af for 0 ≤ j ≤ p − 1 given by χj (V ⊗ A) := (xj · (V ⊗ A))/(xj+1 · (V ⊗ A))  V [−2j]. We finish reviewing basic properties of the category Pdaf by finding a projective generator of it and investigating properties of the functor evn sending F to F (An )  F (kn ⊗ A). Proposition 2.5. Let U ∈ V and dim(U ) ≥ d. The transformation Θ : hU ⊗A ⊗ F (U ⊗ A) −→ F adjoint to the map FU ⊗A,V ⊗A giving the action of F on morphisms is surjective, provided that dim(U ) ≥ d. Therefore hU ⊗A is a projective generator of Pdaf and c∗U ⊗A is an injective generator of Pdaf . Proof. We start by showing the surjectivity of Θ : hU ⊗A ⊗ F (U ⊗ A) −→ F

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

659

for dim(U ) ≥ d. This can be derived from the analogous fact proved for strict polynomial functors in [10, Th. 2.10]. Namely, let  : Γd,U ⊗ t∗ (F )(U ) −→ F Θ be the analogous map for t∗ (F ) ∈ Pd . Then, explicitly, for any V ∈ V we have  ) : Γd (Hom(U, V )) ⊗ F (U ⊗ A) −→ F (V ⊗ A) Θ(V  ) = Θ(V ) ◦ (tU,V ⊗ id). Hence, since Θ(V  ) is epimorphic and we have a factorization Θ(V by [10, Th. 2.10], so is Θ(V ). Now, since the projectivity of hU ⊗A follows from Proposition 2.3, the fact that it is a generator follows. The statement about c∗U ⊗A can be proved analogously. 2 af Let Sd,n denote the graded k-algebra Γd (EndA (An ))  Γd (End(kn ) ⊗ A). Then by n

n

af Proposition 2.3 HomP af (hA , hA )  Sd,n . Therefore, since by Proposition 2.3 again d n

F (An )  HomP af (hA , F ), F (An ) is endowed with a natural structure of a graded af Sd,n -module.

d

Proposition 2.6. If n ≥ d then af evn : Pdaf −→ Sd,n -modf + , af where Sd,n -modf + is the category of bounded below finite dimensional in each degree af graded Sd,n -modules, is an equivalence of graded abelian categories. af Proof. It is analogous to that of [10, Th. 3.2]. We assign to M ∈ Sd,n -modf + the affine functor Φ(M ) given by the formula

V ⊗ A → Γd (HomA (An , V ⊗ A)) ⊗S af M. d,n

n

n

It is easy to see that evn ◦ Φ  IdS af -modf + and that Φ ◦ evn (hA )  hA . This, since d,n n hA generates Pdaf , shows that Φ is quasi-inverse of evn . 2 3. Deriving Pdaf In this section we introduce the derived category of Pdaf . Since Pdaf is graded category, it is natural to look at it as a DG category (with trivial differentials) and use the formalism of derived categories of DG categories. Since DG homological algebra is not as well known as its abelian counterpart we start with recalling some standard constructions concerning DG functor categories and their derived categories. Our main reference in this section is a classical paper [12], in particular we borrow notation from there.

660

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

By DG category we mean a k-linear category whose Hom-sets are naturally (possibly unbounded) complexes. Thus a DG category has an underlying graded category. The simplest example of DG category is the category Dif(k) of complexes of k-vector spaces with internal Hom complexes as morphisms. By DG functor we mean a k-linear functor between DG categories whose action on Hom-complexes preserves grading and commutes with differentials. Let A be a small DG category. We call a (right) A-module a DG functor M : Aop −→ Dif(k). We introduce a category Dif(A) whose objects are A-modules and morphisms are complexes: HomDif(A) (M, N ) :=



Nat(M, N [j])

j∈Z

where Nat stands for the set of natural transformations of degree 0 between underlying graded functors, i.e. similarly to the definition of internal Hom we do not assume that transformations commute with differentials. Thus Dif(A) is a DG category enjoying some extra features like existing representable objects, cohomology objects (see [12, pp. 67, 69]). A natural environment for developing homological algebra in Dif(A) is its derived category DA. Let CA be the category with the same objects as Dif(A) but HomCA (M, N ) := Z 0 (HomDif(A) (M, N )) i.e. we consider only morphisms of degree 0 commuting with differentials. Then the quickest way of defining DA is just by saying that it is the localization of CA with respect to the class of quasi-isomorphisms, where f ∈ HomCA (M, N ) is called a quasi-isomorphism when it induces an isomorphism on cohomology objects. However, in order to get a more concrete description of DA it is convenient to use the formalism of Quillen model categories. There are two Quillen model structures on Dif(A): the projective and injective one, in both structures the class of weak equivalences is the class of quasi-isomorphisms. We focus here on the projective structure. In this structure every object is fibrant while the cofibrant objects are those satisfying “property (P)” [12, Sect. 3], [13, Sect. 3.2]. In order to describe this property explicitly we recall that for A ∈ A we have the repre ∈ Dif(A) given as sentable functor A   ) := HomA (A , A). A(A We call an object in Dif(A) relatively projective if it is isomorphic in CA to a direct  summand in a direct sum of some A[n]. Then we say that an A-module has property (P) if it is chain homotopic to an A-module P admitting a filtration 0 = F−1 ⊂ F0 ⊂ . . . ⊂ Fj ⊂ Fj+1 ⊂ . . . ⊂ P in CA such that

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

661

 1. j Fj = P . 2. Fj ⊂ Fj+1 splits in Dif(A). 3. Fj+1 /Fj is relatively projective. Of course, like in any model category, every object is weakly equivalent to a cofibrant one. In our situation however, this can be made functorially. Namely, let HA stand for the category with the same objects as Dif(A) but HomHA (M, N ) := H 0 (HomDif(A) (M, N )) i.e. this time we consider morphisms of degree 0 commuting with differentials modulo chain homotopy. Let HAp be the full subcategory of HA consisting of cofibrant objects. Then for any M ∈ Dif(A) we can choose a quasi-isomorphic cofibrant A-module p(M ) in such a way that we get a functor p : HA −→ HAp which is right adjoint to the forgetful functor [13, Prop. 3.1]. Now we can describe DA more explicitly, since the natural projection induces an equivalence of triangulated categories: HAp  DA. In fact, for practical computations in DA the following basic properties of representable and cofibrant A-modules are usually sufficient: Fact 3.1. Let M, N ∈ Dif(A) and A ∈ A.  is cofibrant. 1. A 2. There is a natural in A and N isomorphism  N )  H 0 (N (A)). HomDA (A, 3. If M is cofibrant then HomDA (M, N )  HomHA (M, N ) The formalism of Quillen model categories also provides convenient tools for constructing derived functors. We shall need this construction in a very special case of “standard functors” produced by bimodules [12, Sect. 6]. Let B be another small DG category. Then an A–B bimodule X is an object of the category Dif(A ⊗ Bop ). With this data one can associate a pair of adjoint functors HX : Dif(A) −→ Dif(B), TX : Dif(B) −→ Dif(A) given by the formulae: HX (M )(B) := HomDif(A) (X(−, B), M ),   ν TX (N )(A) := coker( N (C) ⊗ HomB (B, C) ⊗ X(A, B) −→ N (B) ⊗ X(A, B)), B,C∈B

where ν(n ⊗ f ⊗ x) = N (n)(f ) ⊗ x − n ⊗ X(A, f )(x).

B∈B

662

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

For a future use we shall provide an explicit description of the unit μ of {TX , HX }  )) is a quotient of 

 adjunction applied to representable object. Since TX ((B B∈B B (B)⊗ X(−, B), it suffices to construct an arrow

 )(B  ) : B

 (B  ) −→ HomDif(A) (X(−, B  ), μ (B



 (B) ⊗ X(−, B)). B

B∈B

In fact, since it is a DG-version of tensor/hom adjunction, for any N ∈ Dif(B) the arrow μ (N )(B  ) : N (B  ) −→ HomDif(A) (X(−, B  ),



N (B) ⊗ X(−, B))

B∈B

simply maps N (B  ) to the transformation sending X(−, B  ) to the summand N (B  ) ⊗ X(−, B  ) in 

N (B) ⊗ X(−, B).

B∈B

Then we recall from [12, p. 82] that we have a natural in B  ∈ B isomorphism α :

 ). Explicitly, it can be factorized through the map X(−, B  )  TX (B α  : X(−, B  ) −→



 (B) ⊗ X(−, B) B

B∈B

sending X(−, B  ) to

 (B  ) ⊗ X(−, B  ) ⊂ idB  ⊗ X(−, B  ) ∈ B



 (B) ⊗ X(−, B). B

B∈B

Under these identifications we finally describe the unit map

 )(B  ) : B

 (B  ) = HomB (B  , B  ) −→ HomDif(A) (X(−, B  ), X(−, B  )) μ(B as the map coming from the action of X on the morphisms. Now if X(−, B) is a cofibrant object in Dif(A) for any B ∈ B then TX preserves

 )  X(−, B  )), while HX preserves quasi-isomorphisms. cofibrant objects (since TX (B Thus the functors RHX : DA −→ DB, LTX : DB −→ DA given by the formulae: RHX (M ) := HX (M ), LTX (N ) := TX (p(N )) form the pair of adjoint functors between triangulated categories.

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

663

We will use this formalism in Sections 4 and 5 in two special cases. The first is when we have a quasi-isomorphism of small DG categories φ : B −→ A (i.e. φ is a DG functor which induces an equivalence of the cohomology categories). Then by applying this machinery to the bimodule X(A, B) := HomA (A, φ(B)) we get (Example after [12, Lemma 6.1]) Fact 3.2. Let φ : B −→ A be a quasi-isomorphism of small DG categories. Then the functor RHX = φ∗ : DA −→ DB given by the formula φ∗ (M )(B) := M (φ(B)) is an equivalence of triangulated categories. Moreover, its inverse (φ∗ )−1 satisfies the property    φ(B) (φ∗ )−1 (B) for any B ∈ B. Another instance of this construction will be crucial in Section 5. Let X be an A–B-bimodule. We introduce the DG category BX . Its objects are those of B but HomBX (B, B  ) := HomDif(A) (X(−, B), X(−, B  )). This is a k-linear DG category and any A–B-bimodule X is automatically an A–BX -bimodule. Indeed, any transformation α ∈ HomDif(A) (X(−, B), X(−, B  )) produces for any A ∈ A the map α(A) : X(A, B) −→ X(A, B  ), which shows that X(A, −) is a covariant functor on BX . In fact our construction is just a DG categorification of the fact that a (bi)module is a module over the ring of its endomorphisms. Hence, if X(−, B) is cofibrant for any B ∈ B, we have the pair of adjoint functors LTX , RHX between the derived categories DBX and DA. The special feature of this bimodule is the following Proposition 3.3. Let X be an A–B-bimodule regarded as A–BX -bimodule. Assume that for any B ∈ B, X(−, B) is cofibrant and small A-module. Then the unit map μ : IdDBX −→ RHX ◦ LTX is an isomorphism. Proof. Observe that, since X(−, B  ) is small for all B  ∈ B, RHX commutes with  is an infinite sums, hence so does RHX ◦ LTX . Thus it suffices to show that μ(B)   



isomorphism. Moreover, since B is cofibrant, we have LTX (B ) = TX (B ). Thus it suffices

664

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

to show that the unit of the non-derived adjunction applied to a representable object is an isomorphism. To this end we shall use an explicit description of this map worked out after the definition of the {T X, HX } adjunction. In our situation we get the map

 )(B  ) : B

 (B  ) = HomB (B  , B  ) −→ HomDif(A) (X(−, B  ), X(−, B  )) μ(B X coming from the action of X on the morphisms which is an isomorphism by the definition of the category BX . 2 Now we would like to apply this machinery to our graded category Pdaf . We recall that in Keller’s terminology Dif(A) stands for the category of contravariant functors op on A, hence it is natural to consider a DG category Diff + (Γd VA ) which consists of DG functors from Γd VA (regarded as a DG category with trivial differentials) to the category Diff + (k) of bounded below complexes of finite dimensional vector spaces over k (we recall op our Definition 2.1). Now in order to make sure that the category Diff + (Γd VA ) still has the (projective) Quillen model structure we need the fact that cofibrant resolutions exist op inside this subcategory of Dif(Γd VA ). op Theorem 3.4. Let F ∈ Diff + (Γd VA ). Then a cofibrant resolution p(F ) can be chosen so f+ d op that p(F ) ∈ Dif (Γ VA ).

Before we start the proof let us explain the reason for which this theorem holds, since at first sight it is strange that we have bounded below “projective” resolutions. To this end let us look at the simplest case of d = 1. In this case by Proposition 2.6, P1af is equivalent to the category of finitely generated bounded below graded A-modules. Now the projective periodic resolution of the trivial A-module k can be written as P• for • ≤ 0 where P−2j = A[−2pj], P−(2j+1) = A[−(2pj+2)] and d−2j = ·x, d−(2j+1) = ·xp−1 . Hence we see that, essentially because A is connected and positively graded, P• as DG module is bounded below. The same phenomenon occurs with the normalized bar resolution which can be applied in a more general situation. Another important point which is crucial for extending this result to d > 1 is that Pd has finite homological dimension [6]. These observations will guide the proof of Theorem 3.4. af -module M := F (Ad ) which is Proof. It will be more transparent to work with a DG Sd,d equivalent by Proposition 2.6. Let us start with the special case of finite dimensional M . Then M has a finite filtration with subquotients concentrated in a single degree. By the horseshoe lemma it suffices to find resolutions for all modules concentrated in a single degree. Hence it suffices to find a resolution for any finite dimensional module over the af Schur algebra Sd,d := Γd (End(kd )) regarded as Sd,d -module through the projection r : af −→ Sd,d onto the degree 0 part. Since the category Sd,d -modf has finite homological Sd,d dimension [6], any such Sd,d -module has a finite projective resolution. Hence it suffices to find bounded below cofibrant resolutions for finite dimensional projective Sd,d-modules.

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

665

Let N be such a module. To this end we take the normalized bar resolution P• associated af to the ring extension Sd,d ⊂ Sd,d of N (see e.g. [18, Ex. 8.6.4]). We have  af af ⊗j P−j = Sd,d ⊗ (Sd,d ) ⊗N   af af where Sd,d := ker(r) and all tensor products are taken over Sd,d . Now, since Sd,d starts in degree 2, P−j starts in degree 2j. Hence P• regarded as a DG module is bounded below. In order to show that P• is a cofibrant resolution of N we recall that a (differential) graded module is called relatively projective if it is homotopic to a summand in a direct sum of shifted free modules [12, p. 69]. Then we have af Lemma 3.5. Sd,d is relatively projective graded Sd,d -module. af = evd (hA ) (see Section 2) and t∗ commutes with the evaluation Proof. Since Sd,d d

functor, our claim is reduced to showing that t∗ (hA ) is a sum of shifted projective objects in Pd . But d

t∗ (hA )  d



Γμ0 ⊗ Γμ1 ⊗ . . . ⊗ Γμp−1 [−h(μ)]

|μ|=d

p−1 where h(μ) = j=0 2jμj , which, since tensor products of divided powers are projective, completes the proof. 2 Now, by general properties of bar construction (see e.g. [18, Sect. 8.6]), it follows af that H • (P )  N and that each P−j is relatively projective Sd,d -module. Therefore P• has property (P), hence is cofibrant. Moreover we note that if N is concentrated in degree 0 then P• is concentrated in nonnegative degrees. Coming back to a general finite dimensional M we see that one can find an f+ (i.e. bounded below finite dimensional in each degree by Definition 2.1) cofibrant resolution P such that: inf{k : P k = 0} ≥ inf{k : M k = 0} − d0 where d0 is the homological dimension of Sd,d -modf . af Then we turn to the general case. For any f+ DG Sd,d -module M , there is a countable generating set {cj }j≥0 such that there is only a finite set cj ’s in each degree. Let Mj :=  < ck >k≤j . Then M = j Mj and in each degree the filtration Mj stabilizes. Now, for each j we construct an f+ cofibrant resolution Pj of Mj satisfying three properties: 1. Pj embeds into Pj+1 , 2. The quotient Pj+1 /Pj is cofibrant. 3. inf{k : Pjk = 0} ≥ inf{k : Mjk = 0} − d0 .

666

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

This may be achieved by taking the construction of resolution described in the preceding paragraph to Mj+1 and Mj+1 /Mj and applying the horseshoe lemma to the extension Mj −→ Mj+1 −→ Mj+1 /Mj  for successive j’s. We claim that P := j Pj is an f+ cofibrant resolution of M . The acyclicity and f+ condition for P are obvious. It remains to show that it has property ij (P) i.e. find a filtration with relatively projective subquotients. Let {Fji }i=0 be such −1 a filtration on Pj /Pj−1 and let Gji := πj (Fji ) where πj : Pj −→ Pj /Pj−1 is the projection. Then {Gji } ordered lexicographically is the desired filtration on P . 2 op This theorem allows us to equip Diff + (Γd VA ) with the projective model structure and apply to it all the constructions described earlier in this section. In particular we will op op heavily use the triangulated category Df + Γd VA obtained from C f + Γd VA by inverting af quasi-isomorphisms. We shall denote this category as DP d , for it should be thought of as the derived category of Pdaf . However, in the next sections when constructing derived functors on DP af d we will have to check carefully that they preserve our extra finiteness and boundedness below assumptions. We finish this section by observing that the fact that z ∗ extends to the graded functor on Pdf + allows one to further extend it to the functor (denoted by the same letter): op z ∗ : Diff + (Γd V op ) −→ Diff + (Γd VA ).

Now, since z ∗ is exact with respect to the structures of abelian categories mentioned in Section 2, it follows that it preserves acyclic objects, hence it automatically factors to the functor (still denoted by z ∗ ): z ∗ : DP d −→ DP af d . Analogously we obtain the functor t∗ : DP af d −→ DP d . Since both z ∗ and t∗ preserve weak equivalences, they remain adjoint at the level of derived categories. 4. Formality op In this section we introduce and study the DG category Γd VX whose cohomology d category is Γ VA . The main result of this section, Theorem 4.3, says that this category is formal i.e. quasi-isomorphic to its cohomology category Γd VA . This gives rise to a derived equivalence of respective functor categories (Corollary 4.4). This derived equivalence will

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

667

be used in the next section to lifting the action of Γd VA from cohomology to complexes which is the crucial part of the construction of the affine derived right Kan extension. op We shall construct our category Γd VX by applying the machinery of Section 3 to pd op d op d DG categories: A := Γ V , B := Γ V . We will also use the category Ppd of strict polynomial bifunctors of bi-degree (d, pd) in the sense of [7]. Readily, this category admits description in the spirit of [9] as the category of k-linear functors from Γpd V ⊗ Γd V op d to V. Thus we see that any complex of objects of Ppd is an A–B-bimodule. The bimodule X which we will use in our construction is a projective resolution of certain bifunctor. Since we want X to be bounded we need a not entirely obvious fact d that Ppd has finite homological dimension. Proposition 4.1. For any d, e ≥ 0, the category Ped has finite homological dimension. Proof. We should show that ExtjP d (F, G) vanishes for j > N , for some constant N e independent of F, G. We have constructed in [4, Prop. 3.5] a spectral sequence converging to Ext-groups in Ped with the uniformly bounded second page consisting of the groups ExtjP d (Γd (I ∗ ⊗ I), H) d

for various H ∈ Pdd , where Γd (I ∗ ⊗I) stands for bifunctor given by the formula (V, W ) → Γd (V ∗ ⊗ W ). Hence our task is reduced to the special case when the first variable of the Ext is the bifunctor Γd (I ∗ ⊗ I). Then we recall that by the Cauchy Decomposition Formula [1, Th. III.1.4], this bifunctor has filtration with the associated graded object being the sum of separable bifunctors (i.e. bifunctors of the form (V, W ) → K(V ∗ ) ⊗ L(W ) for K, L ∈ Pd ). Hence we have further reduced our task to the special case when the first variable is a separable bifunctor. Now, by tensoring finite projective resolutions of K and L we obtain a uniformly bounded projective resolution of K ∗ ⊗ L by [7, Prop. 1.2]. 2 d denote the bifunctor given by the formula (V, W ) → Now let Γd (I ∗ ⊗ I (1) ) ∈ Pdp ∗ (1) Γ (V ⊗ W ). We take as X a bounded projective resolution of Γd (I ∗ ⊗ I (1) ) and op consider the DG category BX = Γd VX . We recall from Section 3 that the objects of this category are finite dimensional k-vector spaces and d

HomΓd VXop (V, V  ) = HomPdp (X(V  , −), X(V, −)). Now it follows from the standard Ext-computations that Proposition 4.2. The assignment V ⊗ A → V extends to an equivalence of graded categories op Γd VA  H ∗ (Γd VX ).

668

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

Proof. Indeed, all we need is a natural in V, V  calculation of Ext-groups Ext∗Pdp (Γd (V  ∗ ⊗ (−)(1) ), Γd (V ∗ ⊗ (−)(1) ))  Γd (Hom(V, V  ) ⊗ A). This isomorphism is the “parameterized version” of [8, Th. 5.4] and can be easily proved by the methods used there. We leave the straightforward details to the reader. 2 The next result is much deeper and is an incarnation of formality phenomena observed in [4]. Theorem 4.3. The assignment V ⊗ A → V extends to a quasi-isomorphism of small DG op categories φ : Γd VA  Γd VX . Proof. We define φV,V  as the composite of several operations. First we dualize: Γd (Hom(V, V  ) ⊗ A)  Γd (Hom(V  ∗ ⊗ A∗ , V ∗ )) and apply the Yoneda lemma: Γd (Hom(V  ∗ ⊗ A∗ , V ∗ ))  HomPd (Γd (V  ∗ ⊗ A∗ ⊗ −), Γd (V ∗ ⊗ −)) which clearly preserves composing morphisms. Then we precompose with I (1) HomPd (Γd (V  ∗ ⊗A∗ ⊗−), Γd (V ∗ ⊗−)) −→ HomPpd (Γd (V  ∗ ⊗A∗ ⊗(−)(1) ), Γd (V ∗ ⊗(−)(1) )) which also commutes with composing. Next we lift morphisms to resolutions HomPpd (Γd (V  ∗ ⊗ A∗ ⊗ (−)(1) ), Γd (V ∗ ⊗ (−)(1) )) −→ HomPpd (X(V  ⊗ A, −), X(V, −)). The commuting of the lift with composition follows from the functoriality of X (or rather its extension to the graded spaces) with respect to the first variable. Now we recall from [4, Prop. 3.2] that there is an element ed ∈ HomDP ddp (Γd (I ∗ ⊗I (1) ), Γd (I ∗ ⊗A∗ ⊗I (1) )) with certain special properties. We can realize ed as an element in HomPdp d (X(−, −), X(− ⊗  A, −)). Hence precomposing with ed evaluated on V produces for any V, V  the arrow HomPpd (X(V  ⊗ A, −), X(V, −)) −→ HomPpd (X(V  , −), X(V, −)) which by the naturality of ed , commutes with composition. The composite of these four arrows is a natural with respect to V, V  arrow φ: φV,V  : Γd (Hom(V, V  ) ⊗ A) −→ HomPpd (X(V  , −), X(V, −)).

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

669

The reader of [4] will recognize that our φV,V  is nothing but Φ(Γd,V )(V  ). It was showed in the proof of [4, Th. 3.2] that this arrow is a quasi-isomorphism. We repeated this construction here, since for our Theorem 4.3 we need strict commuting of our arrows with composition. 2 Now we would like to deduce from Theorem 4.3 the derived equivalence of the respective functor categories by using Fact 3.2. However, when we apply Fact 3.2 literally we obtain the equivalence op φ∗ : DΓd VX  DΓd VA .

Therefore we should check whether φ∗ preserves f+ conditions. To this end we recall that by Fact 3.2 again, (φ∗ )−1 preserves representable objects. Hence, since Hom-complexes op in Γd VX are totally finite dimensional (we use here the fact that X is bounded), (φ∗ )−1 preserve finite dimensionality and boundedness below conditions. Hence we obtain Corollary 4.4. The functor op φ∗ : DP X := Df + Γd VX −→ Df + Γd VA = DP af d

is an equivalence of triangulated categories. Moreover (φ∗ )−1 (hV ⊗A )  V where V is the op object of Dif(Γd VX ) represented by V ∈ V. 5. The affine derived right Kan extension op In this section we use the complex X regarded as Γpd V op –Γd VX bimodule to construct an “affine” version of the derived right Kan extension introduced in [4]. First we observe that since X is totally finite dimensional, the “standard functors” from Section 3: op HX : Dif(Γpd V op ) −→ Dif(Γd VX ),

op TX : Dif(Γd VX ) −→ Dif(Γpd V op )

preserve the subcategories Diff + . Hence we pass to the derived categories and (slightly abusing notation by using the same letters) consider the derived functors RHX : DP pd −→ DP X

LTX : DP X −→ DP pd .

Now we are ready for defining our “affine” adjunction. We define the affine precomposition with the Frobenius twist: Caf : DP af d −→ DP pd as Caf := LTX ◦ (φ∗ )−1 , and the affine derived right Kan extension:

670

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

Kaf : DP pd −→ DP af d as Kaf := φ∗ ◦ RHX . Our terminology is justified by the first part of the following theorem in which C : DP d −→ DP pd stands for the precomposing with the Frobenius twist I (1) and Kr : DP pd −→ DP d for its right adjoint functor introduced in [4]. Theorem 5.1. The functors Caf , Kaf have the following properties: 1. Caf ◦ z ∗  C, t∗ ◦ Kaf  Kr . 2. Kaf is right adjoint to Caf . 3. Kaf ◦ Caf  IdDP af . d

4. Caf is fully faithful. 5. The triangulated quotient category DP pd /DP af d is equivalent to the Verdier localization of DP pd with respect to the essential image of Caf (see e.g. [15]). Proof. In order to get the first part we evaluate Caf ◦ z ∗ on the projective generator Γd,V . We obtain natural in V ∈ V isomorphisms: Caf ◦ z ∗ (Γd,V ) = Caf (hV ⊗A ) = LTX ◦ (φ∗ )−1 (hV ⊗A ) = LTX (V ) = X(V, −)  Γd (V ∗ ⊗ (−)(1) ) which gives the first isomorphism. For the proofs of parts 2 and 3 it will be more convenient to prove analogous statements for larger triangulated categories coming from the complexes without f+ conditions (see Definition 2.1). Then the original statements will automatically follow from the fact, which we have already checked, that all the functors under consideration preserve f+ conditions. Since φ∗ is an equivalence, we are left with the statements about LTX and RHX . Then the adjunction is immediate while part 3 follows from Proposition 3.3. Parts 4, 5 are formal consequences of parts 2, 3 ([15, Prop. 2.3.1]). 2 Theorem 5.1 has an important computational application: the so called Collapsing Conjecture formulated (in somewhat weaker form) in [17, Conj. 8.1] and proved in [4]. Corollary 5.2 ( The Collapsing Conjecture [4, Cor. 3.7]). For any F, G ∈ Pd and i > 0, there is a natural in F, G isomorphism

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

671

Ext∗Ppi d (F (i) , G(i) )  Ext∗Pd (F, GAi ) where Ai := A ⊗ A(1) ⊗ . . . A(i−1) = k[x1 , . . . , xi ]/(xp1 , . . . , xpi ) for |xj | = 2pj−1 and GAi (V ) := G(V ⊗ Ai ). Proof. We start with i = 1. We get Ext∗Ppd (F (1) , G(1) ) = Ext∗Ppd (C(F ), C(G))  HExt∗Pd (F, Kr C(G))  HExt∗Pd (F, t∗ Kaf Caf z ∗ (G))  HExt∗Pd (F, t∗ z ∗ (G)) = Ext∗Pd (F, GA ). The case i > 1 is obtained by iterating this computation. An important point is that t∗ z ∗ (GAi ) = GAi+1 which follows from the fact (observed already in [2]) that the Frobenius twist extended to the graded spaces multiplies degrees by p (see also the end of the proof of [4, Th. 3.2]). 2 In fact, part of motivation for the present work was to put the Collapsing Conjecture into a more abstract context. The Collapsing Conjecture is essentially a statement about the unit of the {C, Kr } adjunction. Our Theorem 5.1 allows one to divide its construction into two steps. The first is sort of scalar extension from Pd to Pdaf and we see that the unit here is the precomposition with the graded space A. The second is the affine derived right Kan extension whose unit is just the identity. This point of view offers somewhat more conceptual picture of the Collapsing Conjecture. 6. Concluding remarks In this section we briefly discuss various implications and ramifications of our work and sketch possible further developments. As we have explained in the previous section the affine strict polynomial functors help to better understand phenomena surrounding the Collapsing Conjecture. However, my original motivation was more general. Namely, as I have mentioned in the Introduction, the category Pdaf provides conceptual explanation of various homological phenomena in Pd which were observed empirically on many occasions. For example, in many calculations of the groups Ext∗Ppd (F (1) , G) some extra structure seemed to emerge. To put it simply: all these computations were given in terms of the graded space A. What is important, this phenomenon is not restricted to the case of G = G (1) which is covered by the Collapsing Conjecture but can also be observed e.g. (1) in Ext∗Ppd (Wμ , Sλ ) (the Ext groups between twisted Weyl and Schur functors) whose computing is crucial for understanding the structure of DP pd . This was already apparent (1) in [3] where the groups Ext∗Ppd (Wμ , Sλ ) were computed in certain special case, but it will be seen much more vividly in [5] where the case of “p-quotient consisting of several diagrams” is considered. With our factorization Kr = t∗ ◦ Kaf this becomes quite natural, since we have

672

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

Ext∗Ppd (F (1) , G)  HExt∗Pd (F, Kr (G))  HExt∗Pd (F, t∗ Kaf (G)) and we recall that t∗ Kaf (G)(V ) := Kaf (G)(V ⊗ A). In the case of Ext-groups between Weyl and Schur functors it becomes a central point, since it was shown in [4, Prop. 4.1] (1) that the problem of computing Ext∗Ppd (Wμ , Sλ ) essentially reduces to that of finding of Kr (Sλ ). We will see in [5] that indeed in practice we describe Kr (Sλ ) as t∗ Kaf (Sλ ) for certain explicitly described affine functor Kaf (Sλ ). Another mysterious fact emerging from the Ext-computations for Weyl and Schur functors was that they were governed by the combinatorics of p-quotients of Young diagrams. Here too, the category Pdaf provides sort of heuristic explanation. It is best seen in terms of representations instead of functors. Namely, since EndA (An ) acts on F (An ) for any F ∈ Pdaf , affine strict polynomial functors produce graded (polynomial) representations of the graded algebraic group GLn (A). Since A is the group algebra for the cyclic group Z/p, our group is closely related to the group GLn with coefficients in the group algebra for the infinite cyclic group. But this group: GLn (k[x, x−1 ]) is nothing but the group of algebraic loops on GLn (k). The latter group is (up to central extension) the affine Kac–Moody group of type An , in particular its simple representations are labeled by n-tuples of Young diagrams. This analogy explains why we call our functors affine and also suggest that the combinatorics of tuples of Young diagrams should somehow organize the structure of Pdaf . More precisely: the relevant combinatorial structure is the set of p-tuples of Young diagrams with the total weight d. One possible way of incorporating combinatorics into a structure of our category would be by showing that it is an “A-highest weight category” in the sense of [14] but we postpone a more systematic study of the structure of Pdaf to a future work. At the time being we only announce an explicit construction (which will be described in detail in [5]) which explains how combinatorics of p-tuples of Young diagrams appears in Ext-computations. Namely there exists in the category Pdaf a construction analogous to that of Schur functor. It associates af af to a p-tuple of Young diagrams {λ0 , . . . , λp−1 } the affine functor S{λ by 0 ,...,λp−1 } ∈ Pd means of certain symmetrizations, antisymmetrizations etc. Then, it will be shown in af [5] that if p-core of λ is trivial then Kaf (Sλ ) = Sq(λ) where q(λ) is p-quotient of λ (for definitions and basic properties of p-cores and p-quotients consult e.g. [11, pp. 75–76]). At last we remark that if p-quotient of λ consists of a single diagram τ then af Sq(λ) (V ⊗ A)  Sτ (V )

up to shift. This will allow one to see results of [3] as special case of those of [5]. The reader of [4] will see two obvious differences with the present work. Firstly, in contrast to [4] I have decided not to consider multiple Frobenius twists. The reason was mainly just not to overload notation by adding another index i while usually in applications like Corollary 5.2 the case of multiple twists can be deduced from the case of a single twist just by iteration. On the other hand, in the next work I plan to extend the construction of affine category to obtain a stratification of DP pd by triangulated

M. Chałupnik / Advances in Mathematics 320 (2017) 652–673

673

categories analogous to a stratification considered by Kuhn [16] in the abelian case. Thus I have decided to postpone this, rather straightforward, generalization to this future work. Yet another difference with [4] is of more fundamental nature. Namely we do not consider the affine derived left Kan extension. The source of problems is an infinite homological dimension of Pdaf . This implies that we have no reason to expect that Caf preserves infinite limits which is a necessary condition for possessing left adjoint. In fact I was able to obtain a two-sided adjunction (hence a recollement of triangulated categories) only between bounded derived categories. Unfortunately, it is not very useful since affine functors typically do not have bounded (co)fibrant resolutions. References [1] K. Akin, D. Buchsbaum, J. Weyman, Schur functors and Schur complexes, Adv. Math. 44 (1982) 207–278. [2] M. Chałupnik, Extensions of strict polynomial functors, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005) 773–792. [3] M. Chałupnik, Extensions of Weyl and Schur functors, Homology, Homotopy Appl. 11 (2) (2009) 27–48. [4] M. Chałupnik, Derived Kan extension for strict polynomial functors, Int. Math. Res. Not. IMRN 20 (2015) 10017–10040. [5] M. Chałupnik, P-quotients of Young diagrams and extensions of Schur functors, in preparation. [6] S. Donkin, On Schur algebras and related algebras I, J. Algebra 104 (1986) 310–328. [7] V. Franjou, E. Friedlander, Cohomology of bifunctors, Proc. Lond. Math. Soc. 97 (2008) 514–544. [8] V. Franjou, E. Friedlander, A. Scorichenko, A. Suslin, General linear and functor cohomology over finite fields, Ann. of Math. 150 (2) (1999) 663–728. [9] V. Franjou, T. Pirashvili, Strict polynomial functors and coherent functors, Doc. Math. 127 (2008) 23–52. [10] E. Friedlander, A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997) 209–270. [11] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl., Addison–Wesley P. C., 1981. [12] B. Keller, Deriving DG categories, Ann. Sci. Éc. Norm. Supér. (4) 27 (1) (1994) 63–102. [13] B. Keller, On Differential Graded Categories, ICM, vol. II, Eur. Math. Soc., Zurich, 2006, pp. 151–190. [14] A. Kleshchev, Affine highest weight categories and affine quasihereditary algebras, Proc. Lond. Math. Soc. (3) 110 (2015) 841–882. [15] H. Krause, Localization theory for triangulated categories, in: T. Holm, P. Jørgensen, R. Rouquier (Eds.), Triangulated Categories, in: LMS Lect. Notes Series, vol. 375, Cambridge University Press, 2010, pp. 161–235. [16] N. Kuhn, A stratification of generic representation theory and generalized Schur algebras, K-Theory 26 (2002) 15–49. [17] A. Touzé, Troesch complexes and extensions of strict polynomial functors, Ann. Sci. Éc. Norm. Supér. (1) 45 (2012) 53–99. [18] Ch. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1997.