KOSZUL, RINGEL, AND SERRE DUALITY FOR STRICT POLYNOMIAL FUNCTORS

HENNING KRAUSE

Abstract. This is a report on recent work of Chałupnik and Touzé. We explain the Koszul duality for the category of strict polynomial functors and make explicit the underlying monoidal structure which seems to be of independent interest. Then we connect this to Ringel duality for Schur algebras and describe Serre duality for strict polynomial functors.

Contents

1. Introduction  1
2. Divided powers and strict polynomial functors  2
3. Exterior powers and Koszul duality  9
4. Derived Koszul duality  11
5. Koszul versus Ringel and Serre duality  18
References  20

1. Introduction

A Koszul duality for strict polynomial functors has been introduced in recent work of Chałupnik [8] and Touzé [30]. In this note we give a detailed report. Our aim is:

– to make explicit the underlying monoidal structure for the category of strict polynomial functors,

– to explain the connection with Ringel duality for Schur algebras,

– to describe Serre duality for strict polynomial functors in terms of Koszul duality, and

– to remove assumptions on the ring of coefficients. The category of strict polynomial functors was introduced by Friedlander and

Suslin [15] in their work on the cohomology of finite group schemes. We use an equivalent description in terms of representations of divided powers, following expositions of Kuhn [20] and Pirashvili [22]; this idea goes back to Bousfield [7] and Eilenberg–MacLane [13].

The construction of the Koszul duality can be summarised as follows.

Theorem 1.1. Let k be a commutative ring and d a non-negative integer. Denote by Rep Γ_{k}^{d} the category of k-linear representations of the category of divided powers of degree d over k.

(1) The category Rep Γ_{k}^{d} has a tensor product

− ⊗_{Γ_{k}^{d}} −: Rep Γ_{k}^{d} × Rep Γ_{k}^{d} ⟶ Rep Γ_{k}^{d}

Version from March 7, 2012.

which is induced by the tensor product for the category of divided powers.

(2) The left derived tensor functor given by the exterior power Λ^{d} of degree d induces for the derived category of Rep Γ_{k}^{d} an equivalence

Λ^{d} ⊗_{Γ_{k}^{d}}^{L} −: D(Rep Γ_{k}^{d}) ⟶^{∼} D(Rep Γ_{k}^{d}).  □

The crucial observation for proving the second part of this theorem is that classical Koszul duality provides resolutions of the exterior power Λ^{d} and the symmetric power S^{d} which give an isomorphism

(1.1)  Λ^{d} ⊗_{Γ_{k}^{d}}^{L} Λ^{d} ≅ S^{d}.

Strict polynomial functors are closely related to modules over Schur algebras. In fact, Rep Γ_{k}^{d} is equivalent to the category of modules over the Schur algebra S_{k}(n, d) for n ≥ d, by a result of Friedlander and Suslin [15]. Schur algebras were introduced by Green [16] and provide a link between representations of the general linear groups and the symmetric groups. Theorem 1.1 establishes via transport of structure a tensor product for the category of modules over the Schur algebra S_{k}(n, d); it seems that this tensor product has not been noticed before.

A Koszul type duality for modules over Schur algebras seems to appear first in work of Akin and Buchsbaum [2]. For instance, they construct resolutions of Schur and Weyl modules, and it is Koszul duality which maps one to the other; see Proposition 4.16. A few years later, Ringel [25] introduced characteristic tilting modules for quasi-hereditary algebras. These tilting modules give rise to derived equivalences, relating a quasi-hereditary algebra and its Ringel dual. Donkin [11] described those tilting modules for Schur algebras, and it turns out that Koszul duality as introduced by Chałupnik [8] and Touzé [30] is equivalent to Ringel duality for modules over Schur algebras; this is the content of Theorem 5.1.

The Koszul duality Λ^{d} ⊗_{Γ_{k}^{d}}^{L} − is an endofunctor and it would be interesting to have a description of its powers. It is somewhat surprising that its square is a Serre duality functor in the sense of [24]; this is another consequence of the isomorphism (1.1) and the content of Corollary 5.4.

Acknowledgements. My interest in the subject of these notes originates from lectures of Greg Stevenson (at a summer school in Bad Driburg in August 2011) and of Antoine Touzé (at a conference in Strasbourg in November 2011). I am grateful to both of them for making me aware of the beautiful work of Chałupnik and Touzé. In addition, I would like to thank Dave Benson, Ivo Dell’Ambrogio, Greg Stevenson, and Dieter Vossieck for numerous helpful comments on this subject.

Convention. Throughout this work we fix a commutative ring k.

2. Divided powers and strict polynomial functors

The category of strict polynomial functors was introduced by Friedlander and Suslin [15]. We we use an equivalent description in terms of representations of divided powers, following expositions of Kuhn [20] and Pirashvili [22].

Finitely generated projective modules. Let P_{k} denote the category of finitely generated projective k-modules. Given V, W in P_{k}, we write V ⊗W for their tensor product over k and Hom(V, W) for the group of k-linear maps V → W. This provides two bifunctors

− ⊗ −: P_{k} × P_{k} ⟶ P_{k}
Hom(−, −): (P_{k})^{op} × P_{k} ⟶ P_{k}

with a natural isomorphism

Hom_{P_{k}}(U ⊗ V, W) ≅ Hom_{P_{k}}(U, Hom(V, W)).

The functor sending V to V^{∗} = Hom(V, k) yields a duality

(P_{k})^{op} ⟶^{∼} P_{k}.

Note that for U, V, W, V′, W′ in P_{k} there are natural isomorphisms

(2.1)  V^{∗} ⊗ W ≅ Hom(V, W)
(2.2)  Hom(U, V) ⊗ W ≅ Hom(U, V ⊗ W)
(2.3)  Hom(V, W) ⊗ Hom(V′, W′) ≅ Hom(V ⊗ V′, W ⊗ W′).

Divided and symmetric powers. Fix a positive integer d and denote by 𝔖_{d} the symmetric group permuting d elements. For each V ∈ P_{k}, the group 𝔖_{d} acts on V^{⊗d} by permuting the factors of the tensor product. Denote by Γ^{d}V the submodule (V^{⊗d})^{𝔖_{d}} of V^{⊗d} consisting of the elements which are invariant under the action of 𝔖_{d}; it is called the module of divided powers of degree d.^{1} The maximal quotient of V^{⊗d} on which 𝔖_{d} acts trivially is denoted by S^{d}V and is called the module of symmetric powers of degree d. Set Γ^{0}V = k and S^{0}V = k.

From the definition, it follows that (Γ^{d}V)^{∗} ≅ S^{d}(V^{∗}). Note that S^{d}V is a free k-module provided that V is free; see [5, III.6.6]. Thus Γ^{d}V and S^{d}V belong to P_{k} for all V ∈ P_{k}, and we obtain functors Γ^{d}, S^{d}: P_{k} → P_{k}.

For further material on divided and symmetric powers, see [3, 6, 26].

The category of divided powers. We consider the category Γ^{d}P_{k} which is defined as follows. The objects are the finitely generated projective k-modules and for two objects V, W set

Hom_{Γ^{d}P_{k}}(V, W) = Γ^{d} Hom(V, W).

Note that this identifies with Hom(V^{⊗d}, W^{⊗d})^{𝔖_{d}} via the isomorphism (2.3), where 𝔖_{d} acts on Hom(V^{⊗d}, W^{⊗d}) via

(fσ)(v_{1} ⊗ ⋯ ⊗ v_{d}) = f(v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(d)}).

Using this identification one defines the composition of morphisms in Γ^{d}P_{k}.

Example 2.1. Let n be a positive integer and set V = k^{n}. Then End_{Γ^{d}P_{k}}(V) is isomorphic to the Schur algebra S_{k}(n, d) as defined by Green [16, Theorem 2.6c]. We view this isomorphism as an identification.

The bifunctors − ⊗ − and Hom(−, −) for P_{k} induce bifunctors

− ⊗ −: Γ^{d}P_{k} × Γ^{d}P_{k} ⟶ Γ^{d}P_{k}
Hom(−, −): (Γ^{d}P_{k})^{op} × Γ^{d}P_{k} ⟶ Γ^{d}P_{k}

with a natural isomorphism

Hom_{Γ^{d}P_{k}}(U ⊗ V, W) ≅ Hom_{Γ^{d}P_{k}}(U, Hom(V, W)).

More precisely, the tensor product for Γ^{d}P_{k} coincides on objects with the one for P_{k}, while on morphisms it is for pairs V, V′ and W, W′ of objects the composite

Γ^{d} Hom(V, V′) × Γ^{d} Hom(W, W′) → Γ^{d}(Hom(V, V′) ⊗ Hom(W, W′))

⟶^{∼} Γ^{d} Hom(V ⊗ W, V′ ⊗ W′)

1The original definition of the module Γ^{d}V of divided powers is different; it is, however, isomorphic to the module of symmetric tensors which is used here; see [6, IV.5, Exercise 8].

where the second map is induced by (2.3).

The duality for P_{k} induces a duality

(Γ^{d}P_{k})^{op} ⟶^{∼} Γ^{d}P_{k}.

Strict polynomial functors. Let M_{k} denote the category of k-modules. We study the category of k-linear representations of Γ^{d}P_{k}. This is by definition the category of k-linear functors Γ^{d}P_{k} → M_{k} and we write

Rep Γ_{k}^{d} = Fun_{k}(Γ^{d}P_{k}, M_{k}).

The set of morphisms between two representations X, Y in Rep Γ_{k}^{d} is denoted by Hom_{Γ_{k}^{d}}(X, Y).

The representations of Γ^{d}P_{k} form an abelian category, where (co)kernels and (co)products are computed pointwise over k.

A representation X is called finite when X(V) is finitely generated projective for each V ∈ Γ^{d}P_{k}. It is sometimes convenient to restrict to the full subcategory

rep Γ_{k}^{d} = Fun_{k}(Γ^{d}P_{k}, P_{k})

of finite representations; it is an extension closed subcategory of Rep Γ_{k}^{d} and therefore a Quillen exact category [23].

Remark 2.2. A representation X ∈ Rep Γ_{k}^{d} is by definition a pair of functions, the first of which assigns to each V ∈ P_{k} a k-module X(V) and the second assigns to each pair V, W ∈ P_{k} a k-linear map

X_{V,W} : Γ^{d} Hom(V, W) ⟶ Hom(X(V), X(W)).

The map X_{V,W} admits an interpretation which is based on classical properties of symmetric tensors and divided powers, to be found in [6, 26]. Given a pair of k-modules M, N with M ∈ P_{k}, there is a canonical k-linear map

Hom(Γ^{d}M, N) ⟶ Pol^{d}(M, N)

where Pol^{d}(M, N) denotes the k-module consisting of homogeneous polynomial maps of degree d from M to N; it is an isomorphism when k is an infinite integral domain and N is torsion-free [6, IV.5.9]. On the other hand, there is a canonical bijection

Hom(Γ^{d}M, N) ⟶ P^{d}(M, N)

where P^{d}(M, N) denotes the set of homogeneous polynomial laws of degree d of M in N, by [6, IV.5, Exercise 10]. These observations explain the term strict polynomial functor used by Friedlander and Suslin in [15, §2].

The Yoneda embedding. The Yoneda embedding

(Γ^{d}P_{k})^{op} ⟶ Rep Γ_{k}^{d}, V ↦ Hom_{Γ^{d}P_{k}}(V, −)

identifies Γ^{d}P_{k} with the full subcategory consisting of the representable functors. For V ∈ Γ^{d}P_{k} we write

Γ^{d,V} = Hom_{Γ^{d}P_{k}}(V, −).

For X ∈ Rep Γ_{k}^{d} there is the Yoneda isomorphism

Hom_{Γ_{k}^{d}}(Γ^{d,V} , X) ⟶^{∼} X(V)

and it follows that Γ^{d,V} is a projective object in Rep Γ_{k}^{d}. A representation X is finitely generated if there are objects V_{1}, …, V_{r} ∈ Γ^{d}P_{k} and an epimorphism

Γ^{d,V_{1}} ⊕ ⋯ ⊕ Γ^{d,V_{r}} ⟶ X

Note that each X in Rep Γ_{k}^{d} can be written canonically as a colimit of representable functors

colim_{Γ^{d,V} →X} Γ^{d,V} ⟶^{∼} X

where the colimit is taken over the category of morphisms Γ^{d,V} → X and V runs through the objects of Γ^{d}P_{k}.

Duality. Given a representation X ∈ Rep Γ_{k}^{d}, its dual X^{∘} is defined by

X^{∘}(V) = X(V^{∗})^{∗}.

For each pair of k-modules V, W there is a natural isomorphism

Hom(V, W^{∗}) ≅ Hom(W, V^{∗}).

This induces for all X, Y ∈ Rep Γ_{k}^{d} a natural isomorphism

(2.4)  Hom_{Γ_{k}^{d}}(X, Y^{∘}) ≅ Hom_{Γ_{k}^{d}}(Y, X^{∘})

and the evaluation morphism X → X^{∘∘}, which is an isomorphism when X is finite.

Example 2.3. The divided power functor Γ^{d} and the symmetric power functor S^{d} belong to Rep Γ_{k}^{d}. In fact

Γ^{d} = Hom_{Γ^{d}P_{k}}(k, −) and S^{d} = (Γ^{d})^{∘}.

The internal tensor product. The category of representations of Γ^{d}P_{k} inherits a tensor product from the tensor product for Γ^{d}P_{k}. We provide a construction in terms of Kan extensions which is also known as Day convolution [12, 17]. The internal Hom functor appears in work of Touzé [29, §2].

Proposition 2.4. The bifunctors − ⊗ − and Hom(−, −) for Γ^{d}P_{k} induce via the Yoneda embedding bifunctors

− ⊗_{Γ_{k}^{d}} −: Rep Γ_{k}^{d} × Rep Γ_{k}^{d} ⟶ Rep Γ_{k}^{d}
ℋom_{Γ_{k}^{d}}(−, −): (Rep Γ_{k}^{d})^{op} × Rep Γ_{k}^{d} ⟶ Rep Γ_{k}^{d}

with a natural isomorphism

Hom_{Γ_{k}^{d}}(X ⊗_{Γ_{k}^{d}} Y, Z) ≅ Hom_{Γ_{k}^{d}}(X, ℋom_{Γ_{k}^{d}}(Y, Z)).

To be precise, one requires for V, W ∈ Γ^{d}P_{k} that

(2.5)  Γ^{d,V} ⊗_{Γ_{k}^{d}} Γ^{d,W} = Γ^{d,V ⊗W}
(2.6)  ℋom_{Γ_{k}^{d}}(Γ^{d,V} , Γ^{d,W}) = Γ^{d,Hom(V,W)}

and this determines both bifunctors.

Proof. Given X, Y ∈ Rep Γ_{k}^{d} and V ∈ Γ^{d}P_{k}, one defines

Γ^{d,V} ⊗_{Γ_{k}^{d}} Y = colim_{Γ^{d,W} →Y} Γ^{d,V ⊗W}
ℋom_{Γ_{k}^{d}}(Γ^{d,V} , Y) = colim_{Γ^{d,W} →Y} Γ^{d,Hom(V,W)}

and then

X ⊗_{Γ_{k}^{d}} Y = colim_{Γ^{d,V} →X} Γ^{d,V} ⊗_{Γ_{k}^{d}} Y
ℋom_{Γ_{k}^{d}}(X, Y) = lim_{Γ^{d,V} →X} ℋom_{Γ_{k}^{d}}(Γ^{d,V} , Y).

For the adjunction isomorphism, write X = colim_{α} X_{α}, Y = colim_{β} Y_{β}, and Z = colim_{γ} Z_{γ} as colimits of representable functors as before. Then we have

Hom_{Γ_{k}^{d}}(X ⊗_{Γ_{k}^{d}} Y, Z) ≅ Hom_{Γ_{k}^{d}}(colim_{α} colim_{β} X_{α} ⊗_{Γ_{k}^{d}} Y_{β}, colim_{γ} Z_{γ})
≅ lim_{α} lim_{β} colim_{γ} Hom_{Γ_{k}^{d}}(X_{α} ⊗_{Γ_{k}^{d}} Y_{β}, Z_{γ})
≅ lim_{α} lim_{β} colim_{γ} Hom_{Γ_{k}^{d}}(X_{α}, ℋom_{Γ_{k}^{d}}(Y_{β}, Z_{γ}))
≅ Hom_{Γ_{k}^{d}}(colim_{α} X_{α}, lim_{β} colim_{γ} ℋom_{Γ_{k}^{d}}(Y_{β}, Z_{γ}))
≅ Hom_{Γ_{k}^{d}}(X, ℋom_{Γ_{k}^{d}}(Y, Z)).  □

The tensor product and the internal Hom functor enjoy the usual categorical properties. For instance, the tensor product is right exact and can be computed using projective presentations.

Lemma 2.5. Given X ∈ Rep Γ_{k}^{d} and V ∈ Γ^{d}P_{k}, there are natural isomorphisms

Γ^{d,V} ⊗_{Γ_{k}^{d}} X ≅ X ∘ Hom(V, −)
ℋom_{Γ_{k}^{d}}(Γ^{d,V} , X) ≅ X ∘ V ⊗ −.

Proof. The isomorphisms are clear from (2.5) and (2.6) when X is representable. The general case follows since Γ^{d,V} ⊗_{Γ_{k}^{d}} − and ℋom_{Γ_{k}^{d}}(Γ^{d,V} , −) preserve colimits.

□

Hom-tensor identities. We collect some basic identities for the internal products − ⊗_{Γ_{k}^{d}} − and ℋom_{Γ_{k}^{d}}(−, −).

Lemma 2.6. Given X, Y, Z ∈ Rep Γ_{k}^{d}, the natural morphism

ℋom_{Γ_{k}^{d}}(X, Y) ⊗_{Γ_{k}^{d}} Z ⟶ ℋom_{Γ_{k}^{d}}(X, Y ⊗_{Γ_{k}^{d}} Z)

is an isomorphism provided that X or Z is finitely generated projective.^{2}

Proof. The isomorphism is clear from (2.2) when all objects are representable functors. The general case follows by writing functors as colimits of representable functors, keeping in mind that Γ^{d,V} ⊗_{Γ_{k}^{d}} − and ℋom_{Γ_{k}^{d}}(Γ^{d,V} , −) preserve colimits. □

Lemma 2.7. Let X, Y be in Rep Γ_{k}^{d} and suppose that X is finitely presented. Then there are natural isomorphisms

X ⊗_{Γ_{k}^{d}} Y^{∘} ≅ ℋom_{Γ_{k}^{d}}(X, Y)^{∘}
(X ⊗_{Γ_{k}^{d}} Y)^{∘} ≅ ℋom_{Γ_{k}^{d}}(X, Y^{∘}).

Proof. Suppose first that X is representable. Then the isomorphism follows from Lemma 2.5, using (2.1). For a representation X that admits a presentation

Γ^{d,V_{1}} ⟶ Γ^{d,V_{0}} ⟶ X ⟶ 0

one uses exactness. □

Lemma 2.8. Given X, Y ∈ Rep Γ_{k}^{d}, there is a natural isomorphism

ℋom_{Γ_{k}^{d}}(X, Y^{∘}) ≅ ℋom_{Γ_{k}^{d}}(Y, X^{∘}).

2This means that finitely generated projective objects are strongly dualisable in the sense of [21, III.1]. We refer to these notes for a discussion of further tensor categorical properties.

Proof. Choose a representable functor P. Using (2.4) and Lemma 2.7, we have

Hom_{Γ_{k}^{d}}(P, ℋom_{Γ_{k}^{d}}(X, Y^{∘})) ≅ Hom_{Γ_{k}^{d}}(P ⊗_{Γ_{k}^{d}} X, Y^{∘})
≅ Hom_{Γ_{k}^{d}}(Y, (P ⊗_{Γ_{k}^{d}} X)^{∘})
≅ Hom_{Γ_{k}^{d}}(Y, ℋom_{Γ_{k}^{d}}(P, X^{∘}))
≅ Hom_{Γ_{k}^{d}}(P, ℋom_{Γ_{k}^{d}}(Y, X^{∘})).

The assertion now follows from Yoneda’s lemma. □

The external tensor product. Let d, e be non-negative integers and denote by (Γ^{d} ⊗ Γ^{e})P_{k} the category having as objects the finitely generated projective kmodules and as morphisms V → W

Γ^{d} Hom(V, W) ⊗ Γ^{e} Hom(V, W).

The inclusion 𝔖_{d} × 𝔖_{e} ⊆ 𝔖_{d+e} induces for each V ∈ P_{k} an inclusion

Γ^{d+e}V = (V^{⊗d+e})^{𝔖_{d+e}} ⊆ (V^{⊗d+e})^{𝔖_{d}×𝔖_{e}} ≅ (V^{⊗d})^{𝔖_{d}} ⊗ (V^{⊗e})^{𝔖_{e}} = Γ^{d}V ⊗ Γ^{e}V.

This yields a k-linear functor

G^{d,e}: Γ^{d+e}P_{k} ⟶ (Γ^{d} ⊗ Γ^{e})P_{k}

which is the identity on objects. It induces a tensor product

− ⊗ −: Rep Γ_{k}^{d} × Rep Γ_{k}^{e} ⟶ Rep Γ_{k}^{d+e}.

To be precise, one defines for X ∈ Rep Γ_{k}^{d}, Y ∈ Rep Γ_{k}^{e}, and V ∈ P_{k}

(X ⊗ Y)(V) = X(V) ⊗ Y (V)

and X ⊗ Y acts on morphisms via G^{d,e}.

Note that the external tensor product is exact and that

(X ⊗ Y)^{∘} ≅ X^{∘} ⊗ Y^{∘}

for finite representations X, Y.

For a sequence λ = (λ_{1}, λ_{2}, …, λ_{n}) of non-negative integers we write

Γ^{λ} = Γ^{λ_{1}} ⊗ ⋯ ⊗ Γ^{λ_{n}} and S^{λ} = S^{λ_{1}} ⊗ ⋯ ⊗ S^{λ_{n}}.

Note that

(2.7)  Γ^{(1,...,1)} ≅ ⊗^{n} ≅ S^{(1,...,1)}.

Graded representations. It is sometimes convenient to consider the category

∏_{d≥0} Rep Γ_{k}^{d}

consisting of graded representations X = (X^{0}, X^{1}, X^{2}, …). An example is for each V ∈ P_{k} the representation

Γ^{V} = (Γ^{0,V} , Γ^{1,V} , Γ^{2,V} , …).

There are two tensor products for graded representations. Given representations X, Y, they are defined in degree d by

(X ⊗ Y)^{d} = ⨁_{i+j=d} X^{i} ⊗ Y^{j} and (X ⊗_{Γ_{k}} Y)^{d} = X^{d} ⊗_{Γ_{k}^{d}} Y^{d}.

Decomposing divided powers. Given V ∈ P_{k}, let SV = ⨁_{d≥0} S^{d}V denote the symmetric algebra. This gives a functor from P_{k} to the category of commutative k-algebras which preserves coproducts. Thus

SV ⊗ SW ≅ S(V ⊕ W)

and therefore by duality

ΓV ⊗ ΓW ≅ Γ(V ⊕ W).

This yields an isomorphism of graded representations

Γ^{V} ⊗ Γ^{W} ≅ Γ^{V ⊕W} .

For each positive integer n, one obtains a decomposition

Γ^{d,k^{n}} = ⨁_{i=0}^{d}(Γ^{d−i,k^{n−1}} ⊗ Γ^{i})

and using induction a canonical decomposition

(2.8)  Γ^{d,k^{n}} = ⨁_{λ∈Λ(n,d)} Γ^{λ}

where Λ(n, d) denotes the set of sequences λ = (λ_{1}, …, λ_{n}) of non-negative integers satisfying ∑ λ_{i} = d.

This decomposition of divided powers has the following immediate consequence.

Proposition 2.9. The category of finitely generated projective objects of Rep Γ_{k}^{d} is equivalent to the category of direct summands of finite direct sums of representations of the form Γ^{λ}, where λ = (λ_{1}, …, λ_{n}) is a sequence of non-negative integers satisfying ∑ λ_{i} = d and n is a positive integer.

Proof. This follows from Yoneda’s lemma and the fact that each representable functor admits a decomposition (2.8) into representations of the form Γ^{λ}. □

Representations of Schur algebras. Strict polynomial functors and modules over Schur algebras are closely related by a result due to Friedlander and Suslin [15, Theorem 3.2]; it is an immediate consequence of Proposition 2.9. Given any ring A, we denote by Mod A the category of A-modules.

Theorem 2.10. Let d, n be positive integers. Evaluation at k^{n} induces a functor Rep Γ_{k}^{d} → Mod S_{k}(n, d) which is an equivalence if n ≥ d.

Proof. Let P ∈ Rep Γ_{k}^{d} be a small projective generator, that is, P is a projective object and the functor Hom_{Γ_{k}^{d}}(P, −) is faithful and preserves set-indexed direct sums. Then

Hom_{Γ_{k}^{d}}(P, −): Rep Γ_{k}^{d} ⟶ Mod End_{Γ_{k}^{d}}(P)

is an equivalence.

From Yoneda’s lemma it follows that the representable functors Γ^{d,V} with V ∈ P_{k} form a family of small projective generators of Rep Γ_{k}^{d}. Now assume n ≥ d. The decomposition (2.8) then implies that the functors Γ^{λ} with λ ∈ Λ(n, d) form a family of small projective generators. Thus P = Γ^{d,k^{n}} is a small projective generator since each Γ^{λ} occurs as a direct summand. It remains to observe that End_{Γ_{k}^{d}}(P) = S_{k}(n, d) and that Hom_{Γ_{k}^{d}}(P, −) equals evaluation at k^{n}, by Yoneda’s lemma. □

Remark 2.11. Let n ≥ d. The evaluation functor Rep Γ_{k}^{d} ⟶^{∼} Mod S_{k}(n, d) restricts to an equivalence rep Γ_{k}^{d} ⟶^{∼} mod S_{k}(n, d), where mod S_{k}(n, d) denotes the full subcategory of modules that are finitely generated projective over k.

Remark 2.12. The tensor product for the category Rep Γ_{k}^{d} from Proposition 2.4 induces via transport of structure a tensor product for the category Mod S_{k}(n, d) when n ≥ d. It seems that this tensor product has not been noticed before, despite the fact that polynomial representations of the general linear groups have been studied for more than a hundred years.

Representations of symmetric groups. Schur–Weyl duality yields a relation between representations of the general linear groups and representations of the symmetric groups. In our context this takes the following form. Let ω = (1, …, 1) be a sequence of length d. Then Γ^{ω} is the functor taking V to V^{⊗d} and

End_{Γ_{k}^{d}}(Γ^{ω}) ≅ k𝔖_{d},

where k𝔖_{d} denotes the group algebra of the symmetric group 𝔖_{d}. This observation gives rise to the functor Hom_{Γ_{k}^{d}}(Γ^{ω}, −): Rep Γ_{k}^{d} → Mod k𝔖_{d}, which appears in Schur’s thesis [27, Sections III, IV]; see also [16, §6]. The functor is not relevant for the rest of this work, but we mention that it has a fully faithful left adjoint Mod k𝔖_{d} → Rep Γ_{k}^{d} taking X ∈ Mod k𝔖_{d} to the functor

V ↦ X ⊗_{k𝔖_{d}} Hom_{Γ_{k}^{d}}(Γ^{d,V} , Γ^{ω})

and a fully faithful right adjoint Mod k𝔖_{d} → Rep Γ_{k}^{d} taking X ∈ Mod k𝔖_{d} to the functor

V ↦ Hom_{k𝔖_{d}}(Hom_{Γ_{k}^{d}}(Γ^{ω}, Γ^{d,V}), X).

3. Exterior powers and Koszul duality

Koszul duality expresses the intimate homological relation between symmetric and exterior powers. For strict polynomial functors, Koszul duality has been introduced by Chałupnik [8, §2] and Touzé [30, §3] as the derived functor of ℋom_{Γ_{k}^{d}}(Λ^{d}, −). In this section we treat the non-derived version. A crucial ingredient is the compatibility of the internal and external tensor products; for this we follow closely Touzé [29, §2]. For some earlier work relating symmetric and exterior powers for Schur algebras, we refer to [1, 2, 3].

Exterior powers. Given V ∈ P_{k}, let ΛV = ⨁_{d≥0} Λ^{d}V denote the exterior algebra, which is obtained from the tensor algebra T V = ⨁_{d≥0} V^{⊗d} by taking the quotient with respect to the ideal generated by the elements v ⊗ v, v ∈ V. Note that multiplication in ΛV is graded commutative and that

ΛV ⊗ ΛW ≅ Λ(V ⊕ W).

For each d ≥ 0, the k-module Λ^{d}V is free provided that V is free; see [5, III.7.8]. Thus Λ^{d}V belongs to P_{k} for all V ∈ P_{k}, and this gives a functor Γ^{d}P_{k} → P_{k}, since the ideal generated by the elements v ⊗ v is invariant under the action of 𝔖_{d} on V^{⊗d}. There is a natural isomorphism

Λ^{d}(V^{∗}) ≅ (Λ^{d}V)^{∗}

induced by (f_{1} ∧ ⋯ ∧ f_{d})(v_{1} ∧ ⋯ ∧ v_{d}) = det(f_{i}(v_{j})), and therefore (Λ^{d})^{∘} ≅ Λ^{d}. For each V ∈ Γ^{d}P_{k}, we use the notation

Λ^{d,V} = Λ^{d} ∘ Hom(V, −) ≅ Λ^{d} ⊗_{Γ_{k}^{d}} Γ^{d,V} .

Internal versus external tensor product. We investigate the compatibility of internal and external tensor product.

Lemma 3.1. Let X_{1}, X_{2} ∈ Rep Γ_{k}^{d} and Y_{1}, Y_{2} ∈ Rep Γ_{k}^{e}. Then there is a natural morphism

τ: (X_{1} ⊗_{Γ_{k}^{d}} X_{2}) ⊗ (Y_{1} ⊗_{Γ_{k}^{e}} Y_{2}) ⟶ (X_{1} ⊗ Y_{1}) ⊗_{Γ_{k}^{d+e}} (X_{2} ⊗ Y_{2}).

Proof. Consider first the case that X_{2} = Γ^{d,V} and Y_{2} = Γ^{e,W} for some V, W ∈ P_{k}. From the decomposition (2.8) it follows that Γ^{d,V} ⊗ Γ^{e,W} is canonically a direct summand of Γ^{d+e,V ⊕W}. Also, we use the description of Γ^{d,V} ⊗_{Γ_{k}^{d}} − from Lemma 2.5. The projections V ⊕ W → V and V ⊕ W → W induce a morphism

X_{1} ∘ Hom(V, −) ⊗ X_{2} ∘ Hom(W, −) ⟶ (X_{1} ⊗ Y_{1}) ∘ Hom(V ⊕ W, −)

and composition with the canonical projection

(X_{1} ⊗ Y_{1}) ∘ Hom(V ⊕ W, −) ⟶ (X_{1} ⊗ Y_{1}) ⊗_{Γ_{k}^{d+e}} (Γ^{d,V} ⊗ Γ^{e,W})

gives τ which is natural in V ∈ Γ^{d}P_{k} and W ∈ Γ^{e}P_{k}. This last observation yields τ for general X_{2} and Y_{2}. □

Lemma 3.2. Let X ∈ Rep Γ_{k}^{d} and Y ∈ Rep Γ_{k}^{e}. Then the composite

(Λ^{d} ⊗_{Γ_{k}^{d}} X) ⊗ (Λ^{e} ⊗_{Γ_{k}^{e}} Y) ⟶^{τ} (Λ^{d} ⊗ Λ^{e}) ⊗_{Γ_{k}^{d+e}} (X ⊗ Y) → Λ^{d+e} ⊗_{Γ_{k}^{d+e}} (X ⊗ Y)

(where the second map is induced by the multiplication Λ^{d}⊗Λ^{e} → Λ^{d+e}) is a natural isomorphism.

Proof. Consider first the case that X = Γ^{d,V} and Y = Γ^{e,W} for some V, W ∈ P_{k}. We have an isomorphism of graded representations

(Λ ⊗_{Γ_{k}} Γ^{V}) ⊗ (Λ ⊗_{Γ_{k}} Γ^{W}) ≅ Λ^{V} ⊗ Λ^{W}
≅ Λ^{V ⊕W}
≅ Λ ⊗_{Γ_{k}} Γ^{V ⊕W}
≅ Λ ⊗_{Γ_{k}} (Γ^{V} ⊗ Γ^{W})

which restricts in degree d + e to an isomorphism

⨁_{i+j=d+e} Λ^{i,V} ⊗ Λ^{j,W} ⟶^{∼} Λ^{d+e} ⊗_{Γ_{k}^{d+e}} ( ⨁_{i′+j′=d+e} Γ^{i′,V} ⊗ Γ^{j′,W} ).

The (i, j) = (d, e) summand then maps onto the (i′, j′) = (d, e) summand (requires checking). This establishes the isomorphism for the case X = Γ^{d,V} and Y = Γ^{e,W}. Naturality in V, W yields the general case. □

Remark 3.3. The multiplication maps Λ^{d} ⊗ Λ^{e} → Λ^{d+e} and Λ^{e} ⊗ Λ^{d} → Λ^{d+e} are equal up to (−1)^{de}σ, where σ: Λ^{d} ⊗ Λ^{e} ⟶^{∼} Λ^{e} ⊗ Λ^{d} permutes the factors of the tensor product. This sign comes from the graded commutativity of the exterior algebra and appears as well in the isomorphism of Lemma 3.2 when the factors X and Y are permuted.

For a sequence λ = (λ_{1}, λ_{2}, …, λ_{n}) of non-negative integers we write

Λ^{λ} = Λ^{λ_{1}} ⊗ ⋯ ⊗ Λ^{λ_{n}}.

Proposition 3.4. For each partition λ ∈ Λ(n, d) there is an isomorphism

Λ^{λ} ≅ Λ^{d} ⊗_{Γ_{k}^{d}} Γ^{λ}.

Proof. Apply Lemma 3.2. □

Koszul duality. We compute Λ^{d} ⊗_{Γ_{k}^{d}} − and ℋom_{Γ_{k}^{d}}(Λ^{d}, −) using the relation between exterior and symmetric powers.

Lemma 3.5. Given d ≥ 1 and V ∈ P_{k}, there are exact sequences

⨁_{i=1}^{d−1} V^{⊗i−1} ⊗ Γ^{2}V ⊗ V^{⊗d−i−1} ⟶^{1⊗Δ⊗1} V^{⊗d} ⟶ Λ^{d}V ⟶ 0

⨁_{i=1}^{d−1} V^{⊗i−1} ⊗ Λ^{2}V ⊗ V^{⊗d−i−1} ⟶^{1⊗Δ⊗1} V^{⊗d} ⟶ S^{d}V ⟶ 0

where Δ: Γ^{2}V → V ⊗ V is the component of the comultiplication which is dual to the multiplication V^{∗} ⊗ V^{∗} → S^{2}(V^{∗}), and Δ: Λ^{2}V → V ⊗ V is the component of the comultiplication given by Δ(v ∧ w) = v ⊗ w − w ⊗ v.

Proof. See [3, p. 214–216] or [31, p. 6]. □

Proposition 3.6. Let d ≥ 0. Then

Λ^{d} ⊗_{Γ_{k}^{d}} Λ^{d} ≅ S^{d}.

Proof. Tensoring the projective presentation of Λ^{d} in Lemma 3.5 with Λ^{d} yields an exact sequence which identifies with the presentation of S^{d} via the isomorphism of Proposition 3.4. Here, we use the isomorphism (2.7). □

Corollary 3.7. For each partition λ ∈ Λ(n, d) there is an isomorphism

Λ^{d} ⊗_{Γ_{k}^{d}} Λ^{λ} ≅ S^{λ}.

Proof. Combine Lemma 3.2 and Proposition 3.6. To be explicit, we have

Λ^{d} ⊗_{Γ_{k}^{d}} (Λ^{λ_{1}} ⊗ ⋯ ⊗ Λ^{λ_{n}}) ≅ (Λ^{λ_{1}} ⊗_{Γ_{k}^{λ_{1}}} Λ^{λ_{1}}) ⊗ ⋯ ⊗ (Λ^{λ_{n}} ⊗_{Γ_{k}^{λn}} Λ^{λ_{n}}) ≅ S^{λ}. □

For a set C of objects of an additive category, we denote by add C the full subcategory consisting of all finite direct sums of objects in C and their direct summands.

Corollary 3.8. Let d, n ≥ 1. The functor Λ^{d} ⊗_{Γ_{k}^{d}} − induces equivalences

add{Γ^{λ} | λ ∈ Λ(n, d)} ⟶^{∼} add{Λ^{λ} | λ ∈ Λ(n, d)}
add{Λ^{λ} | λ ∈ Λ(n, d)} ⟶^{∼} add{S^{λ} | λ ∈ Λ(n, d)}

with quasi-inverses induced by ℋom_{Γ_{k}^{d}}(Λ^{d}, −).

Proof. From Lemma 2.7 we have for X ∈ Rep Γ_{k}^{d}

ℋom_{Γ_{k}^{d}}(Λ^{d}, X)^{∘} ≅ Λ^{d} ⊗_{Γ_{k}^{d}} X^{∘}.

The assertion then follows from Proposition 3.4 and Corollary 3.7, using that (S^{λ})^{∘} ≅ Γ^{λ} and (Λ^{λ})^{∘} ≅ Λ^{λ} for each partition λ. □

4. Derived Koszul duality

In this section we establish the Koszul duality for strict polynomial functors, working in the unbounded derived category and over an arbitrary ring of coefficients. We refer to [8, Corollary 2.3] and [30, Theorem 3.4] for the corresponding results of Chałupnik and Touzé.

The derived category of strict polynomial functors. Let D(Rep Γ_{k}^{d}) denote the unbounded derived category of the abelian category Rep Γ_{k}^{d}. The objects are ℤgraded complexes with differential of degree +1 and the morphisms are chain maps with all quasi-isomorphisms inverted. This is a triangulated category which admits set-indexed products and coproducts. The set of morphisms between two complexes X, Y in D(Rep Γ_{k}^{d}) is denoted by Hom_{D(Γ_{k}^{d})}(X, Y). An object in D(Rep Γ_{k}^{d}) is called perfect if it is isomorphic to a bounded complex of finitely generated projective objects.

Derived functors. We construct the derived functors of the internal tensor product and the internal Hom functor.

Proposition 4.1. The bifunctors − ⊗_{Γ_{k}^{d}} − and ℋom_{Γ_{k}^{d}}(−, −) for Rep Γ_{k}^{d} have derived functors

− ⊗_{Γ_{k}^{d}}^{L} −: D(Rep Γ_{k}^{d}) × D(Rep Γ_{k}^{d}) ⟶ D(Rep Γ_{k}^{d})
Rℋom_{Γ_{k}^{d}}(−, −): D(Rep Γ_{k}^{d})^{op} × D(Rep Γ_{k}^{d}) ⟶ D(Rep Γ_{k}^{d})

with a natural isomorphism

Hom_{D(Γ_{k}^{d})}(X ⊗_{Γ_{k}^{d}}^{L} Y, Z) ≅ Hom_{D(Γ_{k}^{d})}(X, Rℋom_{Γ_{k}^{d}}(Y, Z)).

To compute these functors, one chooses for a complex X in D(Rep Γ_{k}^{d}) a Kprojective resolution pX → X (in the sense of [28]) and gets for each Y in D(Rep Γ_{k}^{d})

X ⊗_{Γ_{k}^{d}}^{L} Y = pX ⊗_{Γ_{k}^{d}} Y
Rℋom_{Γ_{k}^{d}}(X, Y) = ℋom_{Γ_{k}^{d}}(pX, Y).

This involves total complexes which are defined in degree n by

(X ⊗_{Γ_{k}^{d}} Y)^{n} = ⨁_{p+q=n} X^{p} ⊗_{Γ_{k}^{d}} Y^{q}
ℋom_{Γ_{k}^{d}}(X, Y)^{n} = ∏_{p+q=n} ℋom_{Γ_{k}^{d}}(X^{−p}, Y^{q}).

Note that pX = X when X is a bounded above complex of projective objects.

Proof. Let K(Rep Γ_{k}^{d}) denote the homotopy category of Rep Γ_{k}^{d} and consider the quotient functor Q: K(Rep Γ_{k}^{d}) → D(Rep Γ_{k}^{d}) that inverts all quasi-isomorphisms. This functor has a left adjoint p (taking a complex to its K-projective resolution) and a right adjoint i (taking a complex to its K-injective resolution).

Given X in D(Rep Γ_{k}^{d}), one obtains the following pairs of adjoint functors.

\xymatrix{D(Rep Γ_{k}^{d}) \ar[r]^{p} & K(Rep Γ_{k}^{d}) \ar[l]^{Q} \ar[r]^{X⊗_{Γ^{d}_{k}} −} & K(Rep Γ_{k}^{d}) \ar[l]^{ℋom _{Γ^{d}_{k}} (X,−)} \ar[r]^{Q} & D(Rep Γ_{k}^{d}) \ar[l]^{i}}

The composite from left to right gives the derived functor X ⊗_{Γ_{k}^{d}}^{L} −, while the composite from right to left gives the derived functor Rℋom_{Γ_{k}^{d}}(X, −). □

Lemma 4.2. Let P ∈ D(Rep Γ_{k}^{d}) be perfect. Then P ⊗_{Γ_{k}^{d}}^{L} − and Rℋom_{Γ_{k}^{d}}(P, −) preserve set-indexed products and coproducts.

Proof. The objects X ∈ D(Rep Γ_{k}^{d}) such that X ⊗_{Γ_{k}^{d}}^{L} − and Rℋom_{Γ_{k}^{d}}(X, −) preserve

set-indexed products and coproducts form a triangulated subcategory of D(Rep Γ_{k}^{d}) which contains all finitely generated projective objects when viewed as complexes concentrated in degree zero. It follows that each perfect object belongs to this subcategory. □

Duality. The duality taking X ∈ Rep Γ_{k}^{d} to X^{∘} has a derived functor. For a complex X = (X^{n}, d_{X}^{n}) we define its dual X^{∘} by

(X^{∘})^{n} = (X^{−n})^{∘} and d_{X^{∘}}^{n} = (−1)^{n+1}(d_{X}^{−n−1})^{∘}

and its derived dual by

X^{⋄} = (pX)^{∘}.

Note that

ℋom_{Γ_{k}^{d}}(X, S^{d}) ≅ X^{⋄}.

Lemma 4.3. Given X, Y ∈ D(Rep Γ_{k}^{d}), there is a natural isomorphism

Hom_{D(Γ_{k}^{d})}(X, Y^{⋄}) ≅ Hom_{D(Γ_{k}^{d})}(Y, X^{⋄}).

Proof. The assertion means that (−)^{⋄} as a functor D(Rep Γ_{k}^{d})^{op} → D(Rep Γ_{k}^{d}) is self-adjoint. From Lemma 2.8 we know that (−)^{∘} as a functor (Rep Γ_{k}^{d})^{op} → Rep Γ_{k}^{d} is self-adjoint. Passing to complexes, one obtains the following pairs of adjoint functors.

\xymatrix{D(Rep Γ_{k}^{d})^{op} \ar[r]^{p} & K(Rep Γ_{k}^{d})^{op} \ar[l]^{Q} \ar[r]^{(−)^{∘}} & K(Rep Γ_{k}^{d}) \ar[l]^{(−)^{∘}} \ar[r]^{Q} & D(Rep Γ_{k}^{d}) \ar[l]^{p}}

The assertion now follows. □

Hom-tensor identities. We collect some basic identities for the internal products − ⊗_{Γ_{k}^{d}}^{L} − and Rℋom_{Γ_{k}^{d}}(−, −).

Lemma 4.4. Given X, Y, Z ∈ D(Rep Γ_{k}^{d}), the natural morphism

Rℋom_{Γ_{k}^{d}}(X, Y) ⊗_{Γ_{k}^{d}}^{L} Z ⟶ Rℋom_{Γ_{k}^{d}}(X, Y ⊗_{Γ_{k}^{d}}^{L} Z)

is an isomorphism provided that X or Z is perfect.

Proof. Adapt the proof of Lemma 2.6. □

Lemma 4.5. Let X, Y be in D(Rep Γ_{k}^{d}) and suppose that X is perfect. Then there are natural isomorphisms

X ⊗_{Γ_{k}^{d}}^{L} Y^{⋄} ≅ Rℋom_{Γ_{k}^{d}}(X, Y)^{⋄}
(X ⊗_{Γ_{k}^{d}}^{L} Y)^{⋄} ≅ Rℋom_{Γ_{k}^{d}}(X, Y^{⋄}).

Proof. Adapt the proof of Lemma 2.7. □

Lemma 4.6. Given X, Y ∈ D(Rep Γ_{k}^{d}), there is a natural isomorphism

Rℋom_{Γ_{k}^{d}}(X, Y^{⋄}) ≅ Rℋom_{Γ_{k}^{d}}(Y, X^{⋄}).

Proof. Adapt the proof of Lemma 2.8, using Lemma 4.3. □

Koszul duality. We compute Λ^{d} ⊗_{Γ_{k}^{d}}^{L} − and Rℋom_{Γ_{k}^{d}}(Λ^{d}, −) using the relation between exterior and symmetric powers.

Lemma 4.7. Given d ≥ 1 and V ∈ P_{k}, the normalised bar resolutions computing Ext_{S(V^{∗})}^{∗}(k, k) ≅ ΛV and Ext_{Λ(V^{∗})}^{∗}(k, k) ≅ SV yield exact sequences

0 → Γ^{d}V → ⨁_{i_{1}+i_{2}=d} Γ^{i_{1}}V ⊗ Γ^{i_{2}}V → ⋯

→ ⨁_{i_{1}+···+i_{d−1}=d} Γ^{i_{1}}V ⊗ ⋯ ⊗ Γ^{i_{d−1}}V → V^{⊗d} → Λ^{d}V → 0

0 → Λ^{d}V → ⨁_{i_{1}+i_{2}=d} Λ^{i_{1}}V ⊗ Λ^{i_{2}}V → ⋯

→ ⨁_{i_{1}+···+i_{d−1}=d} Λ^{i_{1}}V ⊗ ⋯ ⊗ Λ^{i_{d−1}}V → V^{⊗d} → S^{d}V → 0

where i_{1}, i_{2}, … run through all positive integers.

Proof. See [1, p. 359] or [31, p. 6]. □

The resolution of Λ^{d} shows that it is a perfect object in D(Rep Γ_{k}^{d}) when viewed as a complex concentrated in degree zero.

Proposition 4.8. Let d ≥ 0. Then

Λ^{d} ⊗_{Γ_{k}^{d}}^{L} Λ^{d} ≅ S^{d}.

Proof. Tensoring the projective resolution of Λ^{d} in Lemma 4.7 with Λ^{d} yields an exact sequence which identifies with the resolution of S^{d} via the isomorphism of Proposition 3.4. □

Theorem 4.9. The functors Λ^{d}⊗_{Γ_{k}^{d}}^{L}− and Rℋom_{Γ_{k}^{d}}(Λ^{d}, −) provide mutually quasiinverse equivalences

D(Rep Γ_{k}^{d}) ⟶^{∼} D(Rep Γ_{k}^{d}).

Proof. Using Hom-tensor identities, the isomorphism (Λ^{d})^{⋄} ≅ Λ^{d}, and Proposition 4.8, one computes for a perfect complex P

Rℋom_{Γ_{k}^{d}}(Λ^{d}, Λ^{d} ⊗_{Γ_{k}^{d}}^{L} P) ≅ Rℋom_{Γ_{k}^{d}}(Λ^{d}, Λ^{d}) ⊗_{Γ_{k}^{d}}^{L} P
≅ (Λ^{d} ⊗_{Γ_{k}^{d}}^{L} Λ^{d})^{⋄} ⊗_{Γ_{k}^{d}}^{L} P
≅ (S^{d})^{⋄} ⊗_{Γ_{k}^{d}}^{L} P
≅ Γ^{d} ⊗_{Γ_{k}^{d}}^{L} P
≅ P

and

Λ^{d} ⊗_{Γ_{k}^{d}}^{L} Rℋom_{Γ_{k}^{d}}(Λ^{d}, P^{⋄}) ≅ Λ^{d} ⊗_{Γ_{k}^{d}}^{L} Rℋom_{Γ_{k}^{d}}(P, Λ^{d})
≅ Rℋom_{Γ_{k}^{d}}(P, Λ^{d} ⊗_{Γ_{k}^{d}}^{L} Λ^{d})
≅ Rℋom_{Γ_{k}^{d}}(P, S^{d})
≅ Rℋom_{Γ_{k}^{d}}(Γ^{d}, P^{⋄})
≅ P^{⋄}.

Now one uses the following dévissage argument. The objects X in D(Rep Γ_{k}^{d}) such that the adjunction morphism

X ⟶ Rℋom_{Γ_{k}^{d}}(Λ^{d}, Λ^{d} ⊗_{Γ_{k}^{d}}^{L} X)

is an isomorphism form a triangulated subcategory which is closed under setindexed coproducts, since Rℋom_{Γ_{k}^{d}}(Λ^{d}, −) and Λ^{d} ⊗_{Γ_{k}^{d}}^{L} − are exact and preserve set-indexed coproducts by Lemma 4.2. This subcategory contains all perfect objects and coincides therefore with D(Rep Γ_{k}^{d}); see [18, §4.2]. The dual argument takes care of the adjunction morphism

Λ^{d} ⊗_{Γ_{k}^{d}}^{L} Rℋom_{Γ_{k}^{d}}(Λ^{d}, X) ⟶ X

since the objects of the form P^{⋄} cogenerate D(Rep Γ_{k}^{d}); see [19, §2]. □

The following consequence is the contravariant version of Koszul duality.

Corollary 4.10. Given X ∈ D(Rep Γ_{k}^{d}), there is a natural isomorphism

Rℋom_{Γ_{k}^{d}}(Rℋom_{Γ_{k}^{d}}(X, Λ^{d}), Λ^{d}) ≅ X^{⋄⋄}.

Proof. We combine Hom-tensor identities, the isomorphism (Λ^{d})^{⋄} ≅ Λ^{d}, and Theorem 4.9. This gives

Rℋom_{Γ_{k}^{d}}(Rℋom_{Γ_{k}^{d}}(X, Λ^{d}), Λ^{d}) ≅ Rℋom_{Γ_{k}^{d}}(Rℋom_{Γ_{k}^{d}}(Λ^{d}, X^{⋄}), Λ^{d})
≅ Rℋom_{Γ_{k}^{d}}(Λ^{d}, Rℋom_{Γ_{k}^{d}}(Λ^{d}, X^{⋄})^{⋄})
≅ Rℋom_{Γ_{k}^{d}}(Λ^{d}, Λ^{d} ⊗_{Γ_{k}^{d}}^{L} X^{⋄⋄})
≅ X^{⋄⋄}.  □

The bounded derived category of finite representations. Let D^{b}(rep Γ_{k}^{d}) denote the bounded derived category of the exact category rep Γ_{k}^{d}.

Lemma 4.11. The inclusion rep Γ_{k}^{d} → Rep Γ_{k}^{d} induces a fully faithful and exact functor

D^{b}(rep Γ_{k}^{d}) ⟶ D(Rep Γ_{k}^{d}).

Proof. The proof of Theorem 2.10 shows that Γ^{d,k^{d}} is a projective generator of Rep Γ_{k}^{d} which belongs to rep Γ_{k}^{d}. Thus each object X in rep Γ_{k}^{d} admits an epimorphism P → X such that P is a projective object and belongs to rep Γ_{k}^{d}. It follows that the inclusion rep Γ_{k}^{d} → Rep Γ_{k}^{d} induces a fully faithful functor D^{b}(rep Γ_{k}^{d}) → D(Rep Γ_{k}^{d}); see for instance [32, Proposition III.2.4.1]. □

The duality (rep Γ_{k}^{d})^{op} ⟶^{∼} rep Γ_{k}^{d} taking X to X^{∘} is an exact functor and induces therefore a duality

D^{b}(rep Γ_{k}^{d})^{op} ⟶^{∼} D^{b}(rep Γ_{k}^{d}).

Note that X^{⋄} ≅ X^{∘} for all X ∈ D^{b}(rep Γ_{k}^{d}).

Given X, Y ∈ D^{b}(rep Γ_{k}^{d}), the objects X ⊗_{Γ_{k}^{d}}^{L} Y and Rℋom_{Γ_{k}^{d}}(X, Y) belong

to D^{b}(rep Γ_{k}^{d}) provided that X is perfect. This observation yields the following immediate consequence of Theorem 4.9.

Corollary 4.12. The functors Λ^{d} ⊗_{Γ_{k}^{d}}^{L} − and Rℋom_{Γ_{k}^{d}}(Λ^{d}, −) induce mutually quasi-inverse equivalences

D^{b}(rep Γ_{k}^{d}) ⟶^{∼} D^{b}(rep Γ_{k}^{d}).  □

A consequence of Corollary 4.10 is the following.

Corollary 4.13. The functor Rℋom_{Γ_{k}^{d}}(−, Λ^{d}) induces an equivalence

D: D^{b}(rep Γ_{k}^{d})^{op} ⟶^{∼} D^{b}(rep Γ_{k}^{d})

satisfying D^{2} ≅ Id. □

Finite global dimension. The Schur algebra S_{k}(n, d) has finite global dimension provided that k is a field or k = ℤ. This is a classical fact [2, 10] and follows also from the more general fact that Schur algebras are quasi-hereditary; precise dimensions were computed by Totaro [31]. We extend this result as follows.

Proposition 4.14. Suppose that k is noetherian and let X, Y ∈ rep Γ_{k}^{d}. Then Ext_{Γ_{k}^{d}}^{i}(X, Y) = 0 for all i ≥ 2d.

The proof is based on the following elementary lemma which is taken from [9]; it also explains the noetherianess hypothesis. For a prime ideal 𝔭 ⊆ k, let k(𝔭) denote the residue field k_{𝔭}/𝔭_{𝔭}.

Lemma 4.15. Let A be a noetherian k-algebra and M a finitely generated Amodule. Suppose that A and M are k-projective. Then M is projective over A if and only if M ⊗_{k} k(𝔭) is projective over A ⊗_{k} k(𝔭) for all prime ideals 𝔭 ⊆ k.

Proof. One direction is clear. So suppose that M ⊗_{k}k(𝔭) is projective over A⊗_{k}k(𝔭) for all 𝔭. It suffices to prove the assertion when k is local with maximal ideal 𝔪, and we may assume that k is complete since k is noetherian. Thus A is semi-perfect and a projective cover P → M ⊗_{k} k(𝔪) lifts to a projective cover P → M, which is an isomorphism since P ⊗_{k} k(𝔪) → M ⊗_{k} k(𝔪) is one. It follows that M is projective over A. □

Proof of Proposition 4.14. We work in the category of modules over S_{k}(d, d) which is equivalent to Rep Γ_{k}^{d} by Theorem 2.10. Note that for all n ≥ 1 and each ring homomorphism k → k′

S_{k′}(n, d) ≅ S_{k}(n, d) ⊗_{k} k′.

This is a consequence of the base change formula

Γ_{k′}^{d}(V ⊗_{k} k′) ≅ (Γ_{k}^{d}V) ⊗_{k} k′

which holds for each V ∈ P_{k}. The global dimension of S_{k(𝔭)}(d, d) is bounded by 2d for all prime ideals 𝔭 ⊆ k, by [31, Theorem 2]. Thus the assertion follows from Lemma 4.15. □

Schur and Weyl functors. Fix a positive integer d. A partition of weight d is a sequence λ = (λ_{1}, λ_{2}, …) of non-negative integers satisfying λ_{1} ≥ λ_{2} ≥ … and ∑ λ_{i} = d. Its conjugate λ′ is the partition where λ′_{i} equals the number of terms of λ that are greater or equal than i.

Fix a partition λ of weight d. Each integer r ∈ {1, …, d} can be written uniquely as sum r = λ_{1} + … λ_{i−1} + j with 1 ≤ j ≤ λ_{i}. The pair (i, j) describes the position (ith row and jth column) of r in the Young diagram corresponding to λ. The partition λ determines a permutation σ_{λ} ∈ 𝔖_{d} by σ_{λ}(r) = λ′_{1} + … λ′_{j−1} + i, where 1 ≤ i ≤ λ_{j}. Note that σ_{λ′} = σ_{λ}^{−1}.

Fix a partition λ of weight d, and assume that λ_{1} + ⋯ + λ_{n} = d = λ′_{1} + ⋯ + λ′_{m}. Following [3, II.1], there is defined for V ∈ P_{k} the Schur module S_{λ}V as image of the map

Λ^{λ′_{1}}V ⊗ ⋯ ⊗ Λ^{λ′_{m}}V ⟶^{Δ⊗···⊗Δ} V^{⊗d} ⟶^{s_{λ′}} V^{⊗d} ⟶^{∇⊗···⊗∇} S^{λ_{1}}V ⊗ ⋯ ⊗ S^{λ_{n}}V.

Here, we denote for an integer r by Δ: Λ^{r}V → V^{⊗r} the component of the comultiplication given by

Δ(v_{1} ∧ ⋯ ∧ v_{r}) = ∑_{σ∈𝔖_{r}} sgn(σ)v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(r)},

∇: V^{⊗r} → S^{r}V is the multiplication, and s_{λ}: V^{⊗d} → V^{⊗d} is given by

s_{λ}(v_{1} ⊗ ⋯ ⊗ v_{d}) = v_{σ_{λ}(1)} ⊗ ⋯ ⊗ v_{σ_{λ}(d)}.

The corresponding Weyl module W_{λ} is by definition the image of the analogous map

Γ^{λ_{1}}V ⊗ ⋯ ⊗ Γ^{λ_{n}}V ⟶^{Δ⊗···⊗Δ} V^{⊗d} ⟶^{s_{λ}} V^{⊗d} ⟶^{∇⊗···⊗∇} Λ^{λ′_{1}}V ⊗ ⋯ ⊗ Λ^{λ′_{m}}V.

Note that (W_{λ}V)^{∗} ≅ S_{λ}(V^{∗}). Moreover, S_{λ}V is free when V is free, by [3, Theorem II.2.16]. Thus S_{λ}V and W_{λ}V belong to P_{k} for all V ∈ P_{k}.

The definition of Schur and Weyl modules gives rise to functors S_{λ} and W_{λ} in Rep Γ_{k}^{d} for each partition λ of weight d. In [2, §4], a finite resolution Γ(W_{λ}) of W_{λ} in terms of divided powers is constructed, and in [2, Theorem 6.1] it is shown that Λ^{d} ⊗_{Γ_{k}^{d}} Γ(W_{λ}) is a resolution of S_{λ′}, basically using an explicit description of the functor

add{Γ^{μ} | μ ∈ Λ(n, d)} ⟶^{∼} add{Λ^{μ} | μ ∈ Λ(n, d)}

from Corollary 3.8. Summarising this discussion, we have the following result.

Proposition 4.16. For each partition λ of weight d, there is an isomorphism

Λ^{d} ⊗_{Γ_{k}^{d}}^{L} W_{λ} ≅ S_{λ′}.  □

The functor Λ^{d} ⊗_{Γ_{k}^{d}}^{L} − gives an equivalence D(Rep Γ_{k}^{d}) ⟶^{∼} D(Rep Γ_{k}^{d}), by Theorem 4.9. Thus the classical formula

Ext_{Γ_{k}^{d}}^{∗}(W_{λ}, W_{μ}) ≅ Ext_{Γ_{k}^{d}}^{∗}(S_{λ′}, S_{μ′}).

due to Akin–Buchsbaum [2, Theorem 7.7] and Donkin [11, Corollary 3.9] follows.

An explicit example. Set k = 𝔽_{2}. We compute rep Γ_{k}^{2}. This is an abelian length category, that is, each object has finite composition length. The simple objects are indexed by the partitions α = (2) and ω = (1, 1). We describe the indecomposable objects by specifying the factors of a composition series; see also [14] for a description of the corresponding Schur algebra which is of finite representation type. The indecomposable projective objects are

Γ^{2} = Γ^{α} = [ \begin{matrix}α \\ ω \end{matrix}] and Γ^{ω} = [ \begin{matrix}ω \\ α \\ ω \end{matrix} ]

and the indecomposable injective objects are

S^{2} = S^{α} = [ \begin{matrix}ω \\ α \end{matrix}] and S^{ω} = [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] .

The resolutions from Lemma 4.7 have the form

0 ⟶ Γ^{α} ⟶ Γ^{ω} ⟶ Λ^{α} ⟶ 0
0 ⟶ Λ^{α} ⟶ Λ^{ω} ⟶ S^{α} ⟶ 0

and therefore

Λ^{2} = Λ^{α} = [^{ω}] and Λ^{ω} = [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] .

The Schur and Weyl functors are

S_{α} = [ \begin{matrix}ω \\ α \end{matrix}]  S_{ω} = [^{ω}]  W_{α} = [ \begin{matrix}α \\ ω \end{matrix}]  W_{ω} = [^{ω}] .

The decomposition (2.8) of Γ_{k}^{2,k^{2}} has the form

Γ_{k}^{2,k^{2}} = Γ^{α} ⊕ Γ^{ω} ⊕ Γ^{α}

and the endomorphism algebra of Γ_{k}^{2,k^{2}} is the Schur algebra S_{k}(2, 2). The functor Hom_{Γ_{k}^{2}}(Γ_{k}^{2,k^{2}}, −) induces an equivalence rep Γ_{k}^{2} ⟶^{∼} mod S_{k}(2, 2). The following table gives for each pair of indecomposable representations in rep Γ_{k}^{2} their tensor product.

\begin{matrix}⊗_{Γ_{k}^{2}} & [^{α}] & [^{ω}] & [ \begin{matrix}α \\ ω \end{matrix}] & [ \begin{matrix}ω \\ α \end{matrix}] & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] \\ [^{α}] & [^{α}] & 0 & [^{α}] & 0 & 0 \\ [^{ω}] & 0 & [ \begin{matrix}ω \\ α \end{matrix}] & [^{ω}] & [ \begin{matrix}ω \\ α \end{matrix}] & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] \\ [ \begin{matrix}α \\ ω \end{matrix}] & [^{α}] & [^{ω}] & [ \begin{matrix}α \\ ω \end{matrix}] & [ \begin{matrix}ω \\ α \end{matrix}] & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] \\ [ \begin{matrix}ω \\ α \end{matrix}] & 0 & [ \begin{matrix}ω \\ α \end{matrix}] & [ \begin{matrix}ω \\ α \end{matrix}] & [ \begin{matrix}ω \\ α \end{matrix}] & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] \\ [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] & 0 & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ] & [ \begin{matrix}ω \\ α \\ ω \end{matrix} ]⊕[ \begin{matrix}ω \\ α \\ ω \end{matrix} ] \end{matrix}

I am grateful to Dieter Vossieck for pointing out the following connection.

Remark 4.17 (Vossieck). Let k[ε] denote the k-algebra of dual numbers (ε^{2} = 0). The category rep Γ_{k}^{2} is equivalent to the category of finitely generated modules over the Auslander algebra [4] of k[ε]. The category of k[ε]-modules admits (at least) three different tensor products which induce tensor products on the category of modules over the Auslander algebra (via Day convolution): We have − ⊗_{k[ε]} − and − ⊗_{k} −, where in the second case the action of k[ε] is either given by ε ↦ ε ⊗ ε (the group scheme μ_{2}) or by ε ↦ ε ⊗ 1 + 1 ⊗ ε (the group scheme α_{2}). The last case yields a tensor product which is equivalent to − ⊗_{Γ_{k}^{2}} −.

5. Koszul versus Ringel and Serre duality

In this section we explain the connection between the Koszul duality from Theorem 4.9 and Ringel duality for Schur algebras, as developed in work of Ringel [25] and Donkin [11]. Then we show that Koszul duality applied twice yields a Serre functor in the sense of Reiten and Van den Bergh [24].

Ringel duality. Fix integers d, n ≥ 1 with n ≥ d and recall from Theorem 2.10 that evaluation at k^{n} gives an equivalence

Hom_{Γ_{k}^{d}}(Γ^{d,k^{n}}, −): Rep Γ_{k}^{d} ⟶^{∼} Mod S_{k}(n, d).

The module

T = Hom_{Γ_{k}^{d}}(Γ^{d,k^{n}}, Λ^{d,k^{n}})

is the characteristic tilting module^{3} for the Schur algebra S_{k}(n, d); see [11, 25]. In particular, it induces an equivalence

RHom_{S_{k}(n,d)}(T, −): D(Mod S_{k}(n, d)) ⟶^{∼} D(Mod End_{S_{k}(n,d)}(T)).

3A module T over any ring A is a tilting module iff RHom_{A}(T, −) induces an equivalence D(Mod A) ⟶^{∼} D(Mod End_{A}(T)). If A is quasi-hereditary, then this extra structure determines a characteristic tilting module. Its endomorphism ring A′ is called Ringel dual of A; it is again quasi-hereditary and A′′ is Morita equivalent to A; see [25]. Note that Schur algebras are quasihereditary.

The connection between this equivalence and the equivalence of Theorem 4.9 is explained as follows. First observe that

Λ^{d,k^{n}} = Λ^{d} ⊗_{Γ_{k}^{d}} Γ^{d,k^{n}}.

It follows that the composite of the equivalences Λ^{d} ⊗_{Γ_{k}^{d}}^{L} − and Hom_{Γ_{k}^{d}}(Γ^{d,k^{n}}, −) induces an isomorphism

ϕ: S_{k}(n, d) = End_{Γ_{k}^{d}}(Γ^{d,k^{n}}) ⟶^{∼} End_{Γ_{k}^{d}}(Λ^{d,k^{n}}) ⟶^{∼} End_{S_{k}(n,d)}(T).

Theorem 5.1. The following diagram commutes up to a natural isomorphism.

\xymatrix{D(Rep Γ_{k}^{d}) \ar[d]_{≀}^{RHom(Γ^{d,k^{n}} ,−)} \ar[rr]_{∼}^{Rℋom (Λ^{d},−)} &  & D(Rep Γ_{k}^{d}) \ar[d]^{≀}_{RHom(Γ^{d,k^{n}} ,−)} \\ D(Mod S_{k}(n, d)) \ar[r]_{∼}^{RHom(T,−)} & D(Mod End(T)) \ar[r]_{∼}^{ϕ_{∗}} & D(Mod S_{k}(n, d))}

Proof. We have

RHom_{Γ_{k}^{d}}(Γ^{d,k^{n}}, Rℋom_{Γ_{k}^{d}}(Λ^{d}, −)) ≅ RHom_{Γ_{k}^{d}}(Λ^{d,k^{n}}, −)
≅ ϕ_{∗} RHom_{S_{k}(n,d)}(T, RHom_{Γ_{k}^{d}}(Γ^{d,k^{n}}, −))

where the first isomorphism is adjunction and the second is induced by evaluation at k^{n}. □

Serre duality. Fix an integer d ≥ 0 and suppose that k is a field. Our aim is to describe a Serre duality functor for D^{b}(rep Γ_{k}^{d}).

Recall from work of Reiten and Van den Bergh [24] the following notion of Serre duality. Let C be a k-linear triangulated category and suppose that Hom_{C}(X, Y) is finite dimensional over k for all objects X, Y in C. A Serre functor is by definition an equivalence F: C ⟶^{∼} C together with a natural isomorphism

Hom_{C}(X, −)^{∗} ⟶^{∼} Hom_{C}(−, F X)

for each X ∈ C. When a Serre functor exists, it is unique up to isomorphism. Our description of a Serre functor is based on the following lemma.

Lemma 5.2. Let X, Y be in Rep Γ_{k}^{d} and suppose that X is finitely generated projective. Then there is a natural isomorphism

Hom_{Γ_{k}^{d}}(X, Y)^{∗} ≅ Hom_{Γ_{k}^{d}}(Y, S^{d} ⊗_{Γ_{k}^{d}} X).

Proof. Using Hom-tensor identities, we compute for X = Γ^{d,V}

Hom_{Γ_{k}^{d}}(Γ^{d,V} , Y)^{∗} ≅ Y (V)^{∗}
≅ Hom_{Γ_{k}^{d}}(Γ^{d,V^{∗}}, Y^{∘})
≅ Hom_{Γ_{k}^{d}}((S^{d} ⊗_{Γ_{k}^{d}} Γ^{d,V})^{∘}, Y^{∘})
≅ Hom_{Γ_{k}^{d}}(Y, S^{d} ⊗_{Γ_{k}^{d}} Γ^{d,V}).  □

Proposition 5.3. Let k be a field. Given X, Y ∈ D^{b}(rep Γ_{k}^{d}), there is a natural isomorphism

Hom_{D(Γ_{k}^{d})}(X, Y)^{∗} ≅ Hom_{D(Γ_{k}^{d})}(Y, S^{d} ⊗_{Γ_{k}^{d}}^{L} X).

Proof. One can assume that X and Y are bounded complexes of finitely generated projective objects, by Proposition 4.14. Form the total complex Hom_{Γ_{k}^{d}}(X, Y) defined in degree n by

Hom_{Γ_{k}^{d}}(X, Y)^{n} = ∏_{p+q=n} Hom_{Γ_{k}^{d}}(X^{−p}, Y^{q}).

Note that this involves in each degree a product having only finitely many non-zero factors. Taking cohomology in degree zero gives

H^{0} Hom_{Γ_{k}^{d}}(X, Y) ≅ Hom_{K(Γ_{k}^{d})}(X, Y) ≅ Hom_{D(Γ_{k}^{d})}(X, Y),

where the first isomorphism is a general fact while the second uses that X is perfect. Now the assertion follows from the isomorphism in Lemma 5.2, using that the duality with respect to k is exact. □

In Corollary 4.12 it has been shown that Λ^{d} ⊗_{Γ_{k}^{d}}^{L} − induces an equivalence

D^{b}(rep Γ_{k}^{d}) ⟶^{∼} D^{b}(rep Γ_{k}^{d})

and it would be interesting to have a description of its powers (Λ^{d} ⊗_{Γ_{k}^{d}}^{L} −)^{n}. The first step is the following case n = 2.

Corollary 5.4. Let k be a field. The functor (Λ^{d} ⊗_{Γ_{k}^{d}}^{L} −)^{2} ≅ S^{d} ⊗_{Γ_{k}^{d}}^{L} − induces a

Serre functor D^{b}(rep Γ_{k}^{d}) ⟶^{∼} D^{b}(rep Γ_{k}^{d}).

Proof. Combine Propositions 4.8 and 5.3. □

I am grateful to Bernhard Keller for pointing out the following connection.

Remark 5.5 (Keller). A perhaps related phenomenon was observed in section 10.3 of [18]: If A is an augmented dg algebra and A^{∗} its Koszul dual, there is a canonical triangle functor F_{A} from the derived category of A^{∗} to that of A and a canonical triangle functor can_{A} from the derived category of A to that of A^{∗∗}. With these notations, there is a canonical morphism F_{A}F_{A^{∗}}can_{A} → S, where S is the ‘Serre functor’ of A, i.e. the derived tensor product by the k-dual bimodule of A.

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Henning Krause, Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany.

   E-mail address: hkrause@math.uni-bielefeld.de