DERIVED FUNCTORS OF GRADED THEORIES

Hans-Joachim Baues

We introduce the notion of a graded theory T which corresponds to an algebraic theory as studied by Lawvere [13]; see (8.1) below. The models of a graded theory form a category model(T). Many graded algebraic objects in algebra and topology are examples of such models. We show that the category model(T) can be equivalently described by an iterated Grothendieck construction Γ = (Γ^{1}, Γ^{2}, …). Here Γ^{1} : A_{0} → A_{1} is a functor and Γ^{n+1} : Γ^{n}A_{n} → A_{n+1} is a functor defined on the Grothendieck construction of Γ^{n}, n ≥ 1. The categories A_{i}, i ≥ 0, are abelian categories. We study the derived functors L_{i}Γ^{n+1} and we obtain certain vanishing lines for such functors. In section § 10 we consider the derived functors of a functor

F : model(T) → B

where B is an abelian category. This yields in particular corollary (10.6) which is a generalized version of the main result 3.1 of Blanc [4]. We introduce new transparent methods to prove the result.

There are many fields of application like • the Bousfield-Kan spectral sequence [6], [7],

• the Quillen homology of algebras [16], and

• the Blanc homology spectral sequence [3], [4],

• the homotopy operation spectral sequence [18]. We refer the reader to [4] and [18] for the details on these examples. Our motivating example is the spherical graded theory (8.1”) obtained by the homotopy category of finite one point unions of spheres S^{n} with n ≥ 2. The homotopy groups {π_{n}X}_{n≥2} of a space describe a model of the spherical theory. See also [9], [10], [17] where such models are called π -algebras. Applications on π -algebras of the theory in this paper are given in [18]. In particular the associated iterated Grothendieckconstruction Γ = (Γ_{1}, Γ_{2}, …) of a graded theory T and the reduction functors of Γ below turn out to be of crucial importance for the computation of homotopy groups of a space; see [18]. Dual results are also available for the graded theory of Eilenberg-Mac Lane spaces; see (8.1′′′).

§ 1 Grothendieck construction

We study partial resolutions in a Grothendieck construction ΓA of a functor Γ : C → A. The results of this section will be used inductively in section § 2 where we consider iterated Grothendieck constructions. This leads to a vanishing line of derived functors in § 3.

Typeset by AMS-TEX

Let A be an abelian category, for example A = Ab the category of abelian groups. We say that an epimorphism in A is a surjective map in A and similarly we say that a monomorphism in A is an injective map in A.

(1.1) Definition. Let C be a category and let Γ : C → A be a functor. Then we obtain the category ΓA as follows. Objects are triple (X, A, η) where X is an object in C and where η : Γ(X) → A is a morphism in A. Morphisms (X, A, η) → (Y, B, λ) in ΓA are pairs (f, g) where f : X → Y is a morphism in C such that the diagram

Γ(X) ⟶^{Γ(f)} Γ(Y)
↓^{η}  ↓^{λ}
A ⟶^{g} B

commutes in A. We call ΓA the Grothendieck construction of Γ. An object (X, A, η) in ΓA is also denoted by η = (X, A, η).

Recall that a coequalizer of a pair f, g : X → Y of morphisms in a category C is a morphism u : Y → U such that uf = ug and for all α : Y → Z in C with αf = αg there is a unique ᾱ : U → Z with α = ᾱu. A functor Γ : C → C′ preserves a coequalizer diagram

(1.2)  X ⇉ Y → U

in C if the induced diagram

ΓX ⇉ ΓY → ΓU

is a coequalizer diagram in C′. One readily checks that coequalizers exist in an abelian category A and that an exact sequence

(1.3)  A_{1} ⟶^{i} A_{0} ⟶^{q} A → o

in A corresponds to a coequalizer diagram

(2)  A_{1} ⇉_{g}^{f} A_{0} ⟶^{q} A

where f = 0 and g = i.

We have canonical functors

(1.4)

C × A

↑Q̃
C ⟶^{I_{1}} ΓA ⟵^{I_{o}} C × A
↓^{Q}

A Here Q carries the object η in ΓA to the cokernel

(1)  Q(η) = cok(η : Γ(X) → A)

of η in the abelian category A and Q̃ carries (X, A, η) to the pair (X, Qη). Moreover I_{0} and I_{1} are full inclusions of categories. The functor I_{1} carries X to the object

(2)  I_{1}(X) = (X, ΓX, 1_{ΓX})

given by the identity 1_{ΓX} of ΓX. The functor I_{0} carries the pair (X, A) in C × A to the object

(3)  I_{0}(X, A) = (X, A, 0)

given by the zero homomorphism 0 : Γ(X) → A. Clearly Q̃I_{0} is the identical functor. We also write

I_{0}(X) = I_{0}(X, 0) = (X, 0, 0) for X ∈ C,
I_{0}(A) = I_{0}(∗, A) = (∗, A, 0) for A ∈ A.

For each object (X, A, η) in ΓA we have the commutative diagram

(4)  \begin{matrix}Γ(X) & & Γ(X) & & Γ(X) \\ ^{1}↓ & & ↓^{η} & & ↓^{0} \\ Γ(X) & ⟶^{η} & A & ⟶^{q} & Q(η) & ⟶ & 0 \end{matrix}

in which the bottom row is an exact sequence in A. This gives us the canonical morphisms

(1.5)  I_{1}(X) ⟶^{(1,η)} (X, A, η) ⟶^{(1,q)} I_{0}(X, Q(η))

in ΓA.

We now consider chain complexes and simplicial objects. Let chain (A) be the category of chain complexes A_{∗} in A with A_{i} = 0 for i < 0. Moreover for a category C let sC be the category of simplicial objects X_{·} in C. By the Dold-Kan theorem we have isomorphisms of categories

(1.6)  chain (A) ⇄_{N}^{K} sA

where N is the ‘normalization functor’ with NK ≅ id and KN ≅ id. Here id is the identical functor. Compare [8]. For a chain complex A_{∗} the homology H_{i}A_{∗} is defined. We also write for a simplicial object A_{·} in sA

(1.7)  π_{i}(A_{·}) = H_{i}(NA_{·})

with i ≥ 0. These are the homotopy groups of A_{·}. For a chain complex A_{∗} the corresponding object in degree n is denoted by A_{n}. For a simplicial object A_{·}, however, the object in degree n is denoted by A_{(n)}. We write

(1)  length(A_{·}) ≤ n

if the chain complex NA_{·} satisfies (NA_{·})_{k} = 0 for k > n. That is, NA_{·} is a chain complex concentrated in degree 0, …, n. Clearly for an exact sequence

A_{·} → B_{·} → C_{·} → 0

in sA we have

(2)  length(B_{·}) ≤ Max{length(A_{·}), length(C_{·})}

This is a consequence of the exactness of the functor N_{n}, see [8].

(1.8) Definition. A resolution of an object A in A is a chain complex A_{∗} in chain (A) with H_{0}A_{∗} = A and H_{i}(A_{∗}) = 0 for i > 0. This is a projective resolution if all A_{i} in A_{∗} are projective in A. We say that A has enough projectives if each object A in A has a projective resolution. Moreover we say that A has global dimension gd(A) ≤ n if each object A in A has a projective resolution A_{∗} with A_{i} = 0 for i > n. Hence gd(Ab) = 1. More generally for any subring R of the rationals ℚ the category of R-modules has global dimension ≤ 1. A category of vector spaces has global dimension = 0.

(1.9) Definition. Let ΓA be the Grothendieck construction of Γ : C → A as in (1.1). A simplicial object

η_{·} = (X_{·}, A_{·}, η_{·})

in sΓA is given by a simplicial map

η_{·} : Γ(X_{·}) → A_{·}

in sA with X_{·} ∈ sC and A_{·} ∈ sA. We say that η_{·} is a partial resolution if the map

Nη_{·} : NΓ(X_{·}) → NA_{·} ∈ chain (A)

induced by the normalization functor N has the following properties (a), (b) and (c):

(a) Nη_{·} is injective,

(b) the cokernel of Nη_{·} is projective in each degree,

(c) the chain complex NA_{·} is a resolution of A = H_{0}NA_{·}.

Now assume that the coequalizer diagram

X_{(1)} ⇉ X_{(0)} → X

exists in the category C where X_{(1)} ⇉ X_{(0)} is given by the simplicial object X_{·}. Then we write

X = π_{0}(X_{·})

We have a simplicial map X_{·} → X (where X is the constant simplicial object given by X) which induces the map

q : π_{0}(ΓX_{·}) → π_{0}ΓX = ΓX

We say that η_{·} above is an X_{·} -partial resolution of η : ΓX → A ∈ ΓA if the diagram

π_{0}ΓX_{·} ⟶^{(η_{·})_{∗}} π_{0}A_{·}
∥
q↓  H_{0}NA_{·}
∥
ΓX ⟶^{η} A

commutes in A.

(1.10) Proposition. Assume A has enough projectives. Then for each simplicial object X_{·} in sC and for each η : Γ(π_{0}X_{·}) → A in A there exists an X_{·} -partial resolution η_{·} : ΓX_{·} → A_{·} of η. Moreover the exact sequence in sA

0 → ΓX_{·} ⟶^{η_{·}} A_{·} ⟶^{q_{·}} Qη_{·} → 0

induces an isomorphism

∂ : π_{i+1}(Qη_{·}) ≅ π_{i}(ΓX_{·}) for i ≥ 1

and an exact sequence

0 → π_{1}(Qη_{·}) → π_{0}Γ(X_{·}) ⟶^{ηq} A → π_{0}(Qη_{·}) → 0

Proof. Since A has enough projectives we know by the arguments in (I.6.12) in [2] that chain(A) is a cofibration category. See also [15]. Weak equivalences are homology isomorphisms and cofibrations are injective maps X_{∗} → A_{∗} for which the cokernel is projective. Now consider the chain map

η̃: N(ΓX_{·}) → A

Here A is the chain complex concentrated in degree 0 given by A and η̃ is the chain map for which the following diagram commutes

H_{0}(NΓX_{·}) ⟶^{(η̃)_{∗}} H_{0}A
∥
π_{0}ΓX_{·}  ∥
q↓
Γπ_{0}X_{·}  ⟶^{η} A

Then the factorization axiom in a cofibration category shows that there exists a factorization

N(ΓX_{·}) ↣^{i} C_{∗} ⟶^{q} A

where i is a cofibration and q is a weak equivalence and qi = η̃_{·} Now let

η_{·} : ΓX_{·} ≅ KNΓX_{·} ⟶^{Ki} KC_{∗} = A_{·}

be given by the functor K in (1.6). q.e.d.

Addendum. We can choose the X_{·} -partial resolution η_{·} : ΓX_{·} → A_{·} of η in such a way that

length(Qη_{·}) ≤ length(ΓX_{·}) + gd(A) + 1

Each partial resolution η_{·} : ΓX_{·} → A_{·} ∈ sΓA yields a short exact sequence

(1.11)  0 → Γ(X_{·}) ⟶^{η_{·}} A_{·} ⟶^{q_{·}} Qη_{·} → 0

in sA defined as in (1.9) (4). Moreover this corresponds to a morphism

(*)  I_{1}(X_{·}) ⟶^{(1,η_{·})} (X_{·}, A_{·}, η_{·})

in sΓA as in (1.4) (5).

(1.12) Lemma. Let Λ : ΓA → A′ be a functor to the abelian category A′. Then a partial resolution η_{·} : IX_{·} → A_{·} ∈ sΓA yields by (*) an exact sequence in sA′:

0 → ΛI_{1}(X_{·}) → Λη_{·}

Proof. We show that the map (1, η_{(n)}) : I_{1}X_{(n)} → η_{(n)} = (X_{(n)}, A_{(n)}, η_{(n)}) is split injective in each degree n ≥ 0. In fact, since (Qη_{·})_{(n)} = Qη_{(n)} is projective we see by (1.11) that η_{(n)} : Γ(X_{(n)}) → A_{(n)} admits a retraction r. Such a retraction defines also a retraction (1, r) of (1, η_{(n)}) in (*). q.e.d.

§ 2 Polynomial and reducible functors

Using cross effects we define polynomial functors and the notion of a reducible functor Λ : ΓA → A where ΓA is the Grothendieck construction of a functor Γ. Such reducible functors are crucial for the results on vanishing lines of derived functors below.

A category C with sums and zero object is a category in which finite sums exist and for which the initial object ∗ (which is the empty sum) is also a final object. We therefore get for objects X, Y in C a zero map 0 : X → ∗ → Y and for a sum X ∨ Y in C we obtain retractions

\begin{matrix}r_{1} = (1, 0) : X ∨ Y → X \\ r_{2} = (0, 1) : X ∨ Y → Y \end{matrix}

For example an abelian category A is a category with sums A ⊕ B and zero object 0. If Γ : C → A is a functor with Γ(∗) = 0 then we define the cross effect

(2.1)  Γ(X | Y) = kernel {(r_{1})_{∗}, (r_{2})_{∗} : Γ(X ∨ Y) → Γ(X) ⊕ Γ(Y)}

Hence we have a natural direct sum decomposition

Γ(X ∨ Y) = Γ(X) ⊕ Γ(Y) ⊕ Γ(X | Y)

and we have the map

P : Γ(X | X) ⊂ Γ(X ∨ X) ⟶^{(1,1)_{∗}} Γ(X)

which is natural in X.

(2.2) Lemma. If C is a category with sums and zero object and if Γ : C → A satisfies Γ(∗) = 0 then also ΓC is a category with sums and zero object.

Proof. The sum of η : Γ(X) → A and λ : Γ(Y) → B in ΓA is the map

(1)  η ∨ λ : Γ(X ∨ Y) → A ⊕ Γ(X | Y) ⊕ B

which is (η(r_{1})_{∗}, r_{12}, λ(r_{2})_{∗}) where r_{12} is the projection given by (4.1). The zero object ∗ in ΓC is the identity Γ(∗) = 0 of the zero object in A. q.e.d.

We point out that the functor I_{1} : C → ΓA carries sums to sums, that is I_{1}(X ∨ Y) ≅ I_{1}(X) ∨ I_{1}(Y); see (1.4). The functor I_{0}, however, has not this property. The sum of (X, A), (Y, B) ∈ C × A is (X, A) ∨ (Y, B) = (X ∨ Y, A ⊕ B). We have the commutative diagram in A

(2)  \begin{matrix}Γ(X ∨ Y) & & Γ(X ∨ Y) & & Γ(X ∨ Y) \\ ^{r_{12}}↓ & & ↓^{(0,r_{12},0)} & & ↓^{(0,0)} \\ Γ(X | Y) & ⟶^{i} & A ⊕ Γ(X | Y) ⊕ B & ⟶ & A ⊕ B \end{matrix}

in which the bottom row is short exact. The inclusion i admits a retraction p which is the projection satisfying p(o, r_{12}, 0) = r_{12}. Hence we obtain a ”split short exact sequence” in ΓA r_{12} ↣ I_{0}(X, A) ∨ I_{0}(Y, B) ↠ I_{0}((X, A) ∨ (Y, B)).

(2.3) Definition. Let A be an abelian category and let C be a category with sums and zero object. Let Γ : C → A be a functor with Γ(∗) = 0. Then for X_{1}, …, X_{q} ∈ C the q − th cross effect Γ(X_{1} | … | X_{q}), q ≥ 1, is the kernel of the map

Γ(X_{1} ∨ … ∨ X_{q}) → ⨁_{i=1.}^{q} Γ(∂_{i})

for which the i − th coordinate is induced by

\{ \begin{matrix}1 ∨ 0 ∨ 1 : X_{1} ∨ … ∨ X_{q} → ∂_{i} with \\ ∂_{i} = X_{1} ∨ … ∨ X_{i−1} ∨ ∗ ∨ X_{i+1} ∨ … ∨ X_{q} \end{matrix}

We say that Γ has degree q if Γ(X_{1} | … | X_{q+1}) = 0 for all X_{1}, …, X_{q+1} ∈ C. The functor Γ is linear, resp. quadratic if Γ has degree 1, resp. 2. Moreover Γ is polynomial if there is 1 ≤ q < ∞ such that Γ has degree q. We also call q the polynomial degree of Γ. See [11].

(2.4) Definition. An epimorphism in ΓA is a morphism (1, f) : η → λ of the form

(1)  \begin{matrix}ΓX & & ΓX \\ ^{η}↓ & & ↓^{λ} \\ A & ⟶^{f} & B \end{matrix}

where f is an epimorphism in A. Let A′ be an abelian category. We say that a functor

(2)  Λ : ΓA → A′

preserves epimorphisms if Λ carries an epimorphism (1) to an epimorphism. Similarly we say that a functor λ : C × A → A′ preserves epimorphisms if Λ(1, f) is an epimorphism for each morphism (1, f) : (X, A) → (X, B) where f is an epimorphism. Moreover we say that Λ is I_{1} -reduced if ΛI_{1}(X) = 0 for X ∈ C.

Now let C be a category with sums and zero object ∗ and let Γ : C → A be a functor into the abelian category A satisfying Γ(∗) = 0. Moreover let

Λ : ΓA → A′

be a functor into the abelian category A′. Then we define a new functor

(2.3)  Λ′ : ΓA → A′

by the following natural exact sequence with η : ΓX → A ∈ ΓA,

(1)  Λ(I_{1}X | I_{0}ΓX) → Λ(η | I_{0}ΓX) → Λ′(η) → 0

Here the left hand side is induced by (1, η) : I_{1}X → η, see (1.5), and we use the notation on I_{0}A with A = ΓX in (1.4) (3). It is clear that Λ′ is a well defined functor. Moreover if Λ is I_{1} -reduced we obtain a natural map

(2)  Λ′(η) ⟶^{P′} Λ(η)

as follows. Consider the commutative diagram in which the top row is exact.

Λ(I_{1}X | I_{0}ΓX) ⟶ Λ(η | I_{0}ΓX) ⟶ Λ′(η) ⟶ 0
↓  ↓
Λ(I_{1}X | I_{1}X) ⟶ Λ(η | I_{1}ΓX)  ↓_{P′}
∥  ↓
Λ(I_{1}X | I_{1}X) ⟶ Λ(η | η) ⟶^{P} Λ(η)

Here we use the canonical maps (0, 1) : I_{0}ΓX → I_{1}ΓX and (1, η) : I_{1}X → η. Since Λ is I_{1} reduced we see that the composite of the maps of the bottom row is zero. Hence the factorization P′ is well defined.

(2.4) Proposition. Assume Λ : ΓA → A′ is I_{1} -reduced and preserves epimorphisms. Then there is a functor Λ′′ : C × A → A′ together with an exact sequence

Λ′(η) ⟶^{P′} Λ(π) ⟶^{π} Λ′′(X, Qη) → 0

which is natural in (η : ΓX → A) ∈ ΓA. Moreover Λ′ is I_{1} -reduced and Λ′ and Λ′′ preserve epimorphisms.

We call Λ′′ a reduction functor of Λ since the domain category of Λ′′ does not involve the functor Γ. We point out that the functor Λ′′ and the projection π is completely determined by P′ since (2.4) implies that we have for η = I_{0}(X, A) the exact sequence

(2.5)  Λ′(I_{0}(X, A)) ⟶^{P′} Λ(I_{0}(X, A)) ⟶^{π_{0}} Λ′′(X, A) → 0

Moreover for (1, q) : η → I_{0}(X, Qη) the diagram

Λ(η)  ⟶^{π} Λ′′(X, Qη)
(1,q)_{∗}↓  ∥
Λ(I_{0}(X, Qη)) ⟶^{π_{0}} Λ′′(X, Qη)

so that π in (2.4) is given by π_{0}(1, q)_{∗} = π.

Proof of (2.4). Consider the diagram

(1)  \begin{matrix}ΓX & & ΓX \\ ^{η}↓ & & ↓^{λ} \\ A & ⟶_{g}^{f} & B \\ ^{q}↓ & & ↓^{q} \\ Qη & ⟶^{α} & Qλ \end{matrix}

where (1, f) and (1, g) are both maps η → λ in ΓA which induce α = Q(1, f) = Q(1, g). Hence there exists

(2)  Qη ⟶^{δ} L ⊂^{i} B

with f + iδq = g. Here L is the image of λ which is the kernel of q. We obtain g as the composite of maps in ΓA

(3)  \begin{matrix}ΓX & & ΓX & & ΓX & ΓX \\ ^{η}↓ & & ↓^{(η,0)} & & ↓^{(λ,0)} & ↓^{λ} \\ A & ⟶^{(1,q)} & A ⊕ Qη & ⟶^{f⊕δ} & B ⊕ L & ⟶^{(1,i)} & B \end{matrix}

The morphism (η, 0), (λ, 0) in the diagram can be expressed as sums in ΓA of the form

(4)  \{ \begin{matrix}(η, 0) = η ∨ I_{0}(Qη) \\ (λ, 0) = λ ∨ I_{0}(L) \end{matrix}

and hence using cross effects of Λ we get by (3) the maps

(5)

Λ(η) ⟶^{H̄} Λ(η) ⊕ Λ(η | I_{0}Qη) ⊕ ΛI_{0}(Qη)
↓^{(1,g)_{∗}}  ↓^{Λ(f)⊕Λ(f|δ)⊕ΛI_{0}(δ)}
Λ(λ) ⟵_{P̄} Λ(λ) ⊕ Λ(λ | I_{0}L) ⊕ ΛI_{0}(L)

Here the maps H̄ and P̄ have the coordinates

(6)  \{ \begin{matrix}H̄ = (1, H, (1, q)_{∗}) \\ P̄ = (1, P′′, 0) \end{matrix}

The third coordinate of P̄ is actually trivial by the following argument: Let λ_{0} : ΓX → L be the map with iλ_{0} = λ. Then we have the composite

0 = ΛI_{1}X ↠^{(1,λ_{0})_{∗}} Λ(λ_{0}) ⟶^{(1,i)_{∗}} Λ(λ)

where (1, λ_{0})_{∗} is an epimorphism since Λ preserves epimorphisms and where (1, i)_{∗} = 0 since Λ is I_{1} -reduced. Hence we see that also

(7)  \begin{matrix}Γ(∗) & ⟶^{0} & ΓX & & ΓX \\ ↓^{0} & & ^{λ_{0}}↓ & & ^{λ}↓ \\ L & ⟶^{1} & L & ⟶^{i} & B \end{matrix}

induces the trivial map ΛI_{0}L → Λ(λ). But this map is the third coordinate of P̄ in (6). We now consider the composite

(8)  P′′′ : Λ(λ | I_{0}ΓX) ⟶^{(1|I_{0}λ_{0})_{∗}} Λ(λ | I_{0}L) ⟶^{P′′} Λ(λ)

Here (1 | I_{0}λ_{0})_{∗} is an epimorphism since Λ preserves epimorphisms. Hence we have

(9)  image(P′′) = image(P′′′) = image(P′)

where P′′ = P (1 | (0, i)) is given by (7) and where P′ is given by (2.3) (2). Hence if we devide out image(P′) we obtain a natural exact sequence

(10)  Λ(λ, I_{0}L) → Λ(λ) → Λ̃(λ) → 0

where Λ̃ : ΓA → A′ is a well defined functor. By (5) we see that (1, f) and (1, g) in (1) induce the same map

(11)  (1, f)_{∗} = (1, g)_{∗} : Λ̃(η) → Λ̃(λ)

Hence there is a unique functor Λ′′ as in (2.4) satisfying Λ̃(η) = Λ′′(X, Qη).

Now it is clear by (2.3) (1) that Λ′ preserves epimorphisms since λ has this property. Moreover by (2.3) (1) it is clear that λ′ is I_{1}-reduced. q.e.d.

(2.6) Definition. Let Λ : ΓA → A′ be a functor which preserves epimorphisms. Then we define inductively functors

Λ_{0} : C → A′
Λ_{i} : C × A → A′, i ≥ 1,
Λ_{(i)} : ΓA → A′

as follows. Let Λ_{0}(X) = Λ(I_{1}X) and let Λ_{(1)} be defined by the exact sequence

(1)  Λ(I_{1}X) → Λ(η) → Λ_{(1)}(η) → 0

which is natural in (η : ΓX → A) ∈ ΓA. Then Λ_{(1)} is I_{1} -reduced and preserves epimorphisms. Hence we obtain by (2.4) for i ≥ 1 the exact sequence

(2)  Λ_{(i+1)}(η) → Λ_{(i)}(η) → Λ_{i}(X, Qη) → 0

where Λ_{(i+1)} = Λ′_{(i)} and Λ_{i} = Λ′′_{(i)}. We call the sequence of functors Λ_{i}, i ≥ 0, the reduction functors of Λ. Moreover we say that Λ is reducible if there is N < ∞ such that for all split injective (η : ΓX → A) ∈ ΓA we have

(3)  Λ_{(i)}(η) = 0 for i > N

In this case we say that Λ has reduction degree ≤ N. By (1) and (2) and by the definition in (2.3) (1), (2.5) we obtain inductively the natural epimorphisms (k ≥ 0)

(4)  Λ(I_{0}(X, A) | I_{0}ΓX | … | I_{0}ΓX) ⟶^{q_{k}} Λ_{k+1}(X, A) → 0

where the left hand side is the (k + 1) -cross effect of Λ with k-times I_{0}ΓX.

(2.7) Theorem. Let C be a category with sums and zero object ∗ and let Γ : C → A be a functor with Γ(∗) = 0 which is polynomial of degree ≤ k. Moreover let Λ : ΓA → A′ be a polynomial functor which preserves epimorphisms. Then Λ is reducible of reduction degree ≤ k and all reduction functors Λ_{i}, i ≥ 0, are polynomial and preserve epimorphisms for i ≥ 1.

Proof. We have surjections

Λ(η) → Λ_{(1)}(η) → 0 (1)

Λ_{(1)}(η | I_{0}ΓX) → Λ_{(2)}(η) → 0

Λ_{(i)}(η | I_{0}ΓX) → Λ_{(i+1)}(η) → 0, i ≥ 1, (2)

by (2.3) (1). Hence since Λ is polynomial we see that also Λ_{(1)} is and therefore also Λ_{(2)}. In fact if Λ_{(i)} is polynomial then by (2) also Λ_{(i+1)} is polynomial, i ≥ 1. Also it is clear by (1) and (2) that all Λ_{(i)} preserve epimorphisms for i ≥ 1 and therefore by (2.4) also all Λ_{i}, i ≥ 1, preserve epimorphisms in A. In order to prove that Λ is reducible we have to consider Λ_{(i)}(η) where η is split injective. In this case we have an isomorphism η = I_{1}(X)∨I_{0}(A) where A = Qη and we consider the functor Ω_{(i)} : A → A′ given by

(3)  Ω_{(i)}(A) = Λ_{(i)}(I_{1}(X) ∨ I_{0}(A))

It is clear by (1) that Ω_{(1)} is polynomial of degree ≤ k since Λ has degree ≤ k. Moreover by (2) we see that degree (Ω_{(i+1}) ≤ degree(Ω_{(i)}) − 1. Hence Ω_{(k+1)} has degree 0 and hence Ω_{(k+1)}(A) = Ω_{(k+1)}(0). This shows that Λ_{(k+1)}(η) = 0 since Λ_{(k+1)} is I_{1}-reduced. q.e.d.

(2.8) Remark. The results and the notation above generalize in an obvious way for a functor

Λ : (ΓA) × B → A′

where B is an abelian category. In fact, we can consider the functor

Γ′ : C ⟶^{Γ} A ⟶^{i} A × B

where i is the inclusion which carries A to (A, 0). Then

Γ′(A × B) = (ΓA) × B

In the following we have B = A^{r} where A^{r} = A × … × A is the r-fold product category of A, r ≥ 0.

§ 3 Iterated Grothendieck constructions and derived functors

Let A be an abelian category. We consider a sequence

Γ = (Γ^{1}, Γ^{2}, …, Γ^{m})

of functors Γ^{i}, 1 ≤ i ≤ m. This is an infinite sequence (Γ^{1}, Γ^{2}, …) if m = ∞.

(3.1) Definition. The sequence Γ of functors is an iterated Grothendieck construction on A if the following property (1) holds:

(1)  \{ \begin{matrix}Γ^{1} : A → A and \\ Γ^{k} : Γ^{k−1}A → A for 2 ≤ k ≤ m \end{matrix}

Here Γ^{k−1}A is the Grothendieck construction of Γ^{k−1} in (1.1). There is a more general form of an iterated Grothendieck construction Γ on a sequence (A^{0}, A^{1}, …, A^{m}) of abelian categories A^{i} satisfying

(1’)  \{ \begin{matrix}Γ^{1} : A^{0} → A^{1} and \\ Γ^{k} : Γ^{k−1}A^{k−1} → A^{k} and 2 ≤ k ≤ m \end{matrix}

The notation and results considered below for the case (1) have an obvious generalization for the case (1’).

An object X = (A^{≤m}, η^{≤m}) ∈ Γ^{m}A is given by an (m + 1) -tuple

A^{≤m} = (A^{0}, A^{1}, …, A^{m})

of objects in A and by an m-tuple

η^{≤m} = (η^{1}, η^{2}, …, η^{m})

of morphisms in A. Here X defines the restriction X^{k} = (A^{≤k}, η^{≤k}) for k = 0, …, m where we set X^{0} = A^{0} and the morphisms η^{k} are morphisms

η^{k} : Γ^{k}(X^{k−1}) → A^{k} (1 ≤ k ≤ m)

We also write X^{k−1} = (η^{1}, …, η^{k−1}) for k ≥ 2 so that morphisms

(2)  \{ \begin{matrix}η^{1} : Γ^{1}(A^{0}) → A^{1} \\ η^{k} : Γ^{k}(η^{1}, …, η^{k−1}) → A^{k} (2 ≤ k ≤ m) \end{matrix}

describe the object X = (A^{≤m}, η^{≤m}). A morphism

(3)  f^{≤m} : (A^{≤m}, η^{≤m}) → (B^{≤m}, λ^{≤m}) in Γ^{m}A

is a tuple f^{≤m} = (f^{0}, …, f^{m}) of morphisms f^{i} : A^{i} → B^{i} with 0 ≤ i ≤ m such that all f^{i} are compatible with the η^{i}, that is, for 0 < i < m the diagram

Γ^{i}(A^{≤i−1}, η^{≤i−1}) ⟶^{(f^{≤i−1})_{∗}} Γ^{i}(B^{≤i−1}, λ^{≤i−1})
↓^{η^{i}}  _{λ^{i}}↓
A^{i}  ⟶^{f^{i}}  B^{i}

commutes. For 1 ≤ k < m we have full inclusions

(4)  A = Γ^{0}A ⟶^{I_{1}} Γ^{1}A ⟶^{I_{1}} Γ^{2}A ⟶^{I_{1}} … ⟶^{I_{1}} Γ^{k}A ⟶^{I_{1}} Γ^{k+1}A

given by I_{1} in (1.4). For m = ∞ let Γ^{∞}A be the limit category of the sequence (4). Objects in Γ^{∞}A are of the form (A^{∞}, η^{1}, η^{2}, …) where A^{∞} = (A^{0}, A^{1}, …) is a sequence of objects in A and where η^{1}, η^{2}, … is a sequence of morphisms η^{k} in A with the property in (2) for k ≥ 1.

We obtain the canonical functors

(3.2)  Q̄, K̄ : Γ^{m}A → A^{m} (m ≤ ∞)

Here Q̄ carries the object (A^{≤m}, η^{≤m}) to the object (Qη^{1}, …, Qη^{m}) ∈ A^{m} where Qη^{i} = cokernel (η^{i}) as in (1.4) (1). We call Q̄ the functor of indecomposables on Γ^{m}A. Moreover we define accordingly the functor K̄ by kernels; that is, K̄ carries (A^{≤m}, η^{≤m}) to the object (Kη^{1}, …, Kη^{m}) ∈ A^{m} where Kη^{i} = kernel (η^{i}). We call K̄ the kernel functor on Γ^{m}A.

Partial resolutions (1.9) lead to the following definition.

(3.3) Definition. Let X = (A^{≤n}, η^{≤n}) be an object in Γ^{n}A. A resolution of X in Γ^{n}A is a simplicial object

X_{·} = (A_{·}^{≤n}, η_{·}^{≤n}) ∈ sΓ^{n}A

with the properties (a) and (b).

(a) N(A_{·}^{0}) is a projective resolution of A^{0}.

(b) For 1 ≤ k < n the simplicial object η_{·}^{k+1} given by X_{·} is an X_{·}^{k} -partial resolution of η^{k+1} where X_{·}^{k} is the restriction of X_{·}.

(3.4) Lemma. Assume A has enough projectives. Then resolutions exist in Γ^{n}A for all n ≤ m.

This is an inductive application of (1.10).

(3.5) Definition. Let B be an abelian category, let Γ be an iterated Grothendieck construction and let

Λ : Γ^{m}A → B

be a functor. If A has enough projectives we define the derived functors

L_{i}Λ : Γ^{m}A → B for i ≥ 0

by the homotopy groups

(L_{i}Λ)(X) = π_{i}(ΛX_{·})

where X_{·} is a resolution of X ∈ Γ^{m}A as in (3.3). We have the natural map

(L_{0}Λ)(X) = π_{0}(ΛX_{·}) → Λ(π_{0}X_{·}) = Λ(X).

We write L_{0}Λ = Λ if this map is an isomorphism for all X. As usual, different resolutions yield canonically isomorphic derived functors. Hence the derived functors L_{i}Λ are well defined up to canonical isomorhism since resolutions exist by (3.4). Using the functor of indecomposables Q̄ in (3.2) we get the following result by the second part of (1.10).

(3.6) Proposition. The derived functors L_{i}Q̄ of Q̄ : Γ^{m}A → A^{m} satisfy in degree k with 1 ≤ k ≤ m the formula

(L_{i+1}Q̄)_{k} = L_{i}Γ^{k} for i ≥ 1

and if L_{0}Γ^{k} = Γ^{k} one has the exact sequence

0 → (L_{1}Q̄)_{k}(X) → Γ^{k}X^{k−1} ⟶^{η^{k}} A^{k} → (L_{0}Q̄)_{k}(X) → 0

Here X = (A^{≤m}, η^{≤m}) ∈ Γ^{m}A yields by restriction X^{k} = (A^{≤k}, η^{≤k}) ∈ Γ^{k}A. This implies L_{0}Q̄ = Q̄ and L_{1}Q̄ = K̄ is the kernel functor K̄ in (3.2).

§ 4 Vanishing line for derived functors

Let A be an abelian category which has enough projectives and let Γ = (Γ^{1}, …) be an iterated Grothendieck construction on A. Then the derived functors

(4.1)  L_{i}Γ^{n+1} : Γ^{n}A → A

are defined by (3.5). We say that these functors have a vanishing line if for all n there exists N_{n+1} < ∞ such that L_{i}Γ^{n+1} = 0 for i > N_{n+1}. We now describe conditions on Γ which imply that a vanishing line exists. We say that Γ preserves zero objects if Γ^{n}(∗) = 0 for all n ≥ 1. In this case (2.2) shows that all categories Γ^{n}A have sums and a zero object. Hence cross effects Γ^{n+1}(X | Y) are defined for X, Y ∈ Γ^{n}A. Moreover we say that Γ = (Γ^{1}, …) preserves multi-epimorphisms if for all n ≥ 0 the functor Γ^{n+1} carries a map f^{≤n} as in (3.1) (3) for which all f^{i}, 0 ≤ i ≤ n, are epimorphisms to an epimorphism. Hence in this case the reduction functors of Γ^{n+1}, n ≥ 0, are defined by (2.4). But in addition also all reduction functors of reduction functors are inductively defined by (2.4) since all these functors preserve again multi-epimorphisms.

(4.2) Definition. Assume Γ = (Γ^{1}, …) preserves zero objects and multi-epimorphisms. Then Γ is said to be reducible if all Γ^{n+1}, n ≥ 0, are reducible and if the same holds for all reduction functors. Moreover also the reduction functors of reduction functors (obtained inductively) again are reducible.

Let ℕ_{0} = {0, 1, 2, … } and ℕ = {1, 2, … } and let

(4.3)  ε = (ε^{1}, ε^{2}, …, ε^{r}) ∈ (ℕ_{0})^{r}

be a tuple of elements ε^{i} ∈ ℕ_{0} with 1 ≤ i ≤ r. Let ℓ(ε) = r be the length of ε and let

(4.4)  \{ \begin{matrix}ε^{−1}(ℕ) = \{i; 1 ≤ i ≤ r, ε^{i} ≥ 1\} \\ ε^{−1}(0) = \{i; 1 ≤ i ≤ r, ε^{i} = 0\} \end{matrix}

Clearly {1, …, r} = ε^{−1}(ℕ) ∪ ε^{−1}(0). Moreover let | ε | be the number of elements in ε^{−1}(ℕ). We also write for k ∈ ℕ_{0} and ε as above

(ε, k) = (ε^{1}, ε^{2}, …, ε^{r}, k)

so that ℓ(ε, k) = ℓ(ε) + 1. If the iterated Grothendieck construction Γ = (Γ_{1}, …) is reducible we obtain for n ≥ 0 functors

(4.5)  Γ_{ε}^{n+1} : (Γ^{n}A) × A^{|ε|} → A

where ε is an element as in (4.3) with ℓ(ε) ≥ 0. Here (4.5) coincides with Γ^{n+1} if ℓ(ε) = 0. If (4.5) is defined for ℓ(ε) ≤ r and n ≥ 0 then we define Γ_{(ε,k)}^{n+1} by the reduction functor

Γ_{(ε,k)}^{n+1} = (Γ_{ε}^{n+2})_{k} of Γ_{ε}^{n+2}

Hence the functors (4.5) applied to split injective objects are trivial if all ε^{i}, 1 ≤ i ≤ ℓ(ε), are sufficiently large. We call (4.5) the reduction functors associated to the reducible Grothendieck construction Γ.

(4.6) Definition. We say that a functor Λ : A^{k} → A is bounded by a bounding function λ : ℕ_{0}^{k} → ℕ_{0} if for all simplicial objects A_{·}^{i} ∈ sA (1 ≤ i ≤ k) with length (A_{·}^{i}) ≤ a^{i} < ∞ one has

lengthΛ(A_{·}^{1}, …, A_{·}^{k}) ≤ λ(a^{1}, …, a^{k}).

Accordingly we say that the reducible iterated Grothendieck construction Γ is bounded if all functors

Γ_{ε}^{1} : A × A^{|ε|} → A

associated to Γ are bounded by some bounding function γ_{ε} with γ_{ε} = 0 if Γ_{ε}^{1} = 0. In fact, since all objects of A × A^{|ε|} are split injective we see that Γ_{ε}^{1} = 0 if all ε^{i}, i ∈ {1, …, l(ε)}, are sufficiently large.

(4.7) Theorem. Let Γ be an iterated Grothendieck construction on A which preserves zero objects and multi-epimorphisms and let Γ be reducible and bounded. Moreover assume that the global dimension of A is finite. Then the derived functors L_{i}Γ^{n+1} in (4.1) have a vanishing line N_{n+1}, n ≥ 0.

Let d < ∞ be the global dimension of A. Then we define inductively numbers a^{n}, 0 ≤ n, as follows

a^{0} = d
a^{n+1} = 1 + d + N_{n+1} for n ≥ 0, where
N_{n+1} = Max{γ_{ε}(a^{0}, a^{i_{1}}, …, a^{i_{|ε|}}), ε ∈ (ℕ_{0})^{n}}

and ε^{−1}(ℕ) = {i_{1} < … < i_{|ε|}}; see (4.4).

(4.8) Addendum. L_{i}Γ^{n+1} = 0 for i > N_{n+1}.

Proof. Let X_{·} = (A_{·}^{≤m}, η_{·}^{≤m}) be a resolution of X ∈ Γ^{m}A. We can choose X_{·} inductively as in the addendum (A) of (1.10). Hence we have for 1 ≤ n < m

(1)  b^{n} ≤ length(A_{·}^{n}) ≤ c^{n} + d + 1

where

\{ \begin{matrix}b^{n} = length Q(η_{·}^{n}) and \\ c^{n} = length (Γ^{n}X_{·}^{n−1}) \end{matrix}

Moreover we have by the inductive application of (2.7) the following diagram in which the row and the columns are exact sequences in sA given by (2.4). In the diagram we assume for simplicity that the reduction degree is always ≤ 1. It is easy to extend the diagram to the general case.

Γ_{(0,0)}^{n−1}(X_{·}^{n−2})  Γ_{(1,0)}^{n−1}(X_{·}^{n−2}, Qη_{·}^{n})
↓  ↓
Γ_{(0)}^{n}(X^{n−1})_{·}  ⟶ Γ^{n+1}(X_{·}^{n}) ⟶  Γ_{(1)}^{n}(X_{·}^{n−1}, Qη_{·}^{n})
↓  ↓
Γ_{(1,0)}^{n−1}(X_{·}^{n−2}, Qη_{·}^{n−1}, Qη_{·}^{n−1})  Γ_{(1,1)}^{n−1}(X_{·}^{n−2}, Qη_{·}^{n−1}, Qη_{·}^{n})

The 4 corners of this diagram are given by the functors Γ_{ε}^{n−1} with ℓ(ε) = 2. We can extend the diagram in such a way that we obtain a large diagram in which the corners are given by the functors Γ_{ε} with ℓ(ε) = n. This shows by (1.7) (2) that

(2)  c^{n+1} = length Γ^{n+1}(X_{·}^{n}) ≤ Max{c_{ε}^{1}, ε ∈ (ℕ_{0})^{n}}

where

c_{ε}^{1} = length Γ_{ε}^{1}(A_{·}^{0}, Qη_{·}^{i_{1}}, …, Qη_{·}^{i_{|ε|}})

with ε^{−1}(ℕ) = {i_{1} < … < i_{|ε|}}. Since γ_{ε} is a bounding function for Γ_{ε}^{1} we get

(3)  c_{ε}^{1} ≤ γ_{ε}(length (A_{·}^{0}), b^{i_{1}}, …, b^{i_{|ε|}})

We have to show c^{n+1} ≤ N_{n+1}. By (1.8) we can choose lenght(A_{·}^{0}) = d = a^{0}. Now assume b^{k} ≤ a^{k} for k ≤ n. Then we get by (3)

(3)  c_{ε}^{1} ≤ γ_{ε}(a^{0}, a^{i_{1}}, …, a^{i_{|ε|}}), ε ∈ (ℕ_{0})^{n},

and hence by (2)

(5)  c^{n+1} ≤ N_{n+1} hence

(6) b^{n+1} ≤ 1 + d + c^{n+1} ≤ 1 + d + N_{n+1} = a^{n+1} by (1) for n + 1 < m.

This yields the result. q.e.d.

§ 5 Polynomial iterated Grothendieck constructions

We consider an iterated Grothendieck construction Γ = (Γ^{1}, …) which preserves zero ojects so that all categories Γ^{n}A have sums. We say that Γ is polynomial if all Γ^{n+1}, n ≥ 0, are polynomial functors; see (2.3).

(5.1) Theorem. Let Γ = (Γ^{1}, …) be an iterated Grothendieck construction which preserves zero obejcts and multi-epimorphisms and which is polynomial. Then Γ is reducible and bounded.

Proof. This is an inductive application of (2.7) which shows that all reduction functors exist and are polynomial. Hence also all functors Γ_{ε}^{1} are polynomial and thus they are bounded by the following lemma. q.e.d.

(5.2) Lemma. Let Λ : A → B be a functor between abelian categories. If Λ(0) = 0 and if Λ has degree k then one has for any simplicial object A_{·} ∈ sA the formula

length Λ(A_{·}) ≤ k · length (A_{·})

This is proved in 4.23 of [8]. See also [14]. Combining (5.1) and (3.7) we thus get:

(5.3) Theorem. Let Γ be an iterated Grothendieck construction on A which preserves zero objects and multi-epimorphisms. Assume that Γ is polynomial and assume that the global dimension of A is finite. Then the derived functors L_{i}Γ^{n+1} have a vanishing line N_{n+1}, n ≥ 0, as in (4.7).

As a special case of (5.3) we obtain the following results.

(5.4) Corollary. Let Γ = (Γ^{1}, …) be an iterated Grothendieck construction on A which preserves zero objects and multi-epimorphisms. If all Γ^{n+1} are linear, n ≥ 0, and if d is the global dimension of A with d < ∞ then

L_{i}Γ^{n+1} = 0 for i > d + n(d + 1) = N_{n+1}.

Proof. All Γ_{ε}^{1} are linear and hence by (5.2) we have

γ_{ε}(a^{0}, a^{i_{1}}, …, a^{i_{|ε|}}) = Max {a^{0}, a^{i_{1}}, …, a^{i_{|ε|}}}

. Hence (5.4) is a consequence of (4.7). q.e.d.

The next definition and corollary is motivated by a result of Blanc [4]; see (10.7) below.

(5.5) Definition. Let Γ = (Γ^{1}, …) be an iterated Grothendieck construction on A which preserves zero objects. We say that an object X = (A^{≤n}, η^{≤n}) ∈ Γ^{n}A, n ≥ 0, has connectivity degree | X | if A^{k} = 0 for k <| X | and A^{k} ≠ 0 for k =| X |. Given a natural number r ≥ 1 we say that Γ = (Γ^{1}, …) is r -polynomial if for n ≥ 0 and q ≥ 2 and any set of objects X_{1}, …, X_{q} ∈ Γ^{n}A with

∑_{i=1}^{q}(r+ | X_{i} |) > r + n + 1

we have Γ^{n+1}(X_{1} | … | X_{q}) = 0. Since r ≥ 1 this implies that Γ^{n+1} is polynomial.

(5.6) Corollary. Let Γ = (Γ^{1}, …) be an iterated Grothendieck construction on A which preserves zero objects and multi-epimorphisms. Moreover assume that Γ is r-polynomial with r > d ≥ 0. Here d is the global dimension of A. Then

L_{i}Γ^{n+1} = 0 for i > d + n(d + 1).

If r > n then all Γ^{i+1} with 0 ≤ i ≤ n in (5.5) are linear and in this case (5.6) corresponds to (5.4). Moreover if Γ is (r+1) -polynomial then Γ is also r-polynomial and hence it suffices to prove the corollary for r = d + 1.

Proof of (5.6). Using (5.3) and (4.7) the corollary is a consequence of the following fact which we shall prove below. Let A_{·}^{i} ∈ sA be a simplicial object for 0 ≤ i ≤ n and assume

(1)  length (A_{·}^{i}) ≤ d + i(d + 1).

Then all functors Γ_{ε}^{1} with ε ∈ (ℕ_{0})^{n} satisfy

(2)  lenght (Γ_{ε}^{1}(A_{·}^{0}, A_{·}^{i_{1}}, …, A_{·}^{i_{|ε|}}) ≤ d + n(d + 1)

where ε^{−1}(ℕ) = {i_{1} < … < i_{|ε|}}. For the proof of (2) we first observe that the functor I_{0} in (1.4) satisfies

(3)  I_{0}(X, A) = I_{0}(X) ∨ I_{0}(A)

For simplicity we assume now that the reduction degree is always ≤ 1. The general case with reduction degree > 1 is proved by a similar argument where the inductive application of the surjection (2.6) (4) is needed for the description of Γ_{ε}^{1} as a quotient of cross effects involving the functors Γ^{i}, i ≥ 1.

Since we assume reduction degree ≤ 1 the functor Γ_{ε}^{1} is only needed for ε^{i} ∈ {0, 1}. Hence we obtain by (2.6) (4) a surjection

Γ^{n+e(ε)+1}J_{ε}(B^{0}, B^{i_{1}}, …, B^{i_{r}}) → Γ_{ε}^{1}(B^{0}, B^{i_{1}}, …, B^{i_{r}})

where ε^{−1}(ℕ) = {i_{1} < … < i_{r}} and B^{i} ∈ A. Here we define the full inclusion

(4)  J_{ε} : A × A^{r} → Γ^{n}A

as follows. Let J_{τ} = I_{1} for τ = 0 and let J_{τ} = I_{0} for τ = 1. Then J_{ε} is the composite J_{ε} = J_{ε^{1}} ∘ … ∘ J_{ε^{r}}. Now (3) implies that the functor J_{ε} in (4) satisfies

(5)  J_{ε}(B^{0}, B^{1}, … B^{|ε|}) = ⋁_{j=0}^{|ε|} J_{ε}^{j}(B^{j})

Here J_{ε}^{j} : A → Γ^{n}A is a functor for which the connectivity degree | J_{ε}^{j}(B) | with B ∈ A (see (5.5)) satisfies

| J_{ε}^{j}(B) |= \{ \begin{matrix}0 if j = 0 \\ i_{j} if 1 ≤ j ≤| ε | \end{matrix}

with {i_{1} < … < i_{j}} = ε^{−1}(ℕ). Now (5) and the definition of Γ^{n+1}J_{ε} shows that we have the cross effect decomposition

(6) \{ \begin{matrix}Γ^{n+1}J_{ε}(B^{0}, B^{1}, …, B^{|ε|}) = ⨁_{t≥1} ⨁_{0≤a_{1}<...<a_{t}≤|ε|} Λ_{ε,a}(B^{a_{1}}, …, B^{a_{t}}) \\ Λ_{ε,a}(B^{a_{1}}, … B^{a_{t}}) = Γ^{n+1}(J_{ε}^{a_{1}}(B^{a_{1}}) | … | J_{ε}^{a_{t}}(B^{a_{t}})) \end{matrix}

Here the assumption that Γ^{n+1} is r-polynomial implies that Λ = Λ_{ε,a} is a functor as in the following lemma. Hence this lemma implies (2) by use of the decomposition (6). For this we set C^{τ} = B^{a_{τ}} = A^{j_{τ}} ∈ A with j_{τ} = i_{a_{τ}} for a_{τ} ≥ 1 and j_{τ} = 0 for a_{τ} = 0. Moreover we set r = d + 1. q.e.d.

(5.7) Lemma. Let n ≥ 0, r ≥ 1, t ≥ 1 and 0 ≤ j_{1} < … < j_{t} ≤ n be given. Let Λ : A^{t} → A be a functor with the property that

Λ(C_{1}^{1} | … | C_{s_{1}}^{1} , C_{1}^{2} | … | C_{s_{2}}^{2} , …, C_{1}^{t} | … | C_{s_{t}}^{t}) = 0

for all s_{1}, s_{2}, …, s_{t} ≥ 1 satisfying

∑_{τ=1}^{t} s_{τ} (j_{τ} + r) > r + n + 1.

Then for C_{·}^{τ} ∈ sA with

length (C_{·}^{τ}) ≤ r − 1 + j_{τ} r for τ = 1, …, t

one has

length Λ(C_{·}^{1}, …, C_{·}^{t}) ≤ r − 1 + nr.

For t = 1 the lemma is a consequence of (5.2). In fact, for t = 1 we have s_{1} ≥ deg(Λ) + 1 and hence we get

lenghtΛ(C_{·}^{1}) ≤ deg(Λ)(r − 1 + j_{1}r)

≤ (s_{1} − 1)(r − 1 + j_{1}r)

< ( \frac{r+n+1}{j_{1}+r} − 1 ) (r − 1 + j_{1}r) ≤^{(∗)} r + nr

Here the proof of (*) for 0 ≤ j_{1} ≤ n and n ≥ 0, r ≥ 1 is a nice exercise.

Proof of (5.7). The normalization of C_{·}^{τ} yields the objects

(1)  K_{i}^{τ} = (NC_{·}^{τ})_{i}

with K_{i}^{τ} = 0 for i < 0 and i > r − 1 + j_{τ} r. Moreover we have by the Dold-Kan theorem

(2)  C_{(m)}^{τ} = ⨁_{0≤λ^{τ} ≤m} ⨁_{I∈F_{λ,m}} K_{m−λ^{τ}}^{τ}

where F_{λ,m} is the set of all sequences I = {0 ≤ u_{1} < … < u_{λ} < m}. We allow λ = 0. We have to show that

(3)  Λ(C_{(m)}^{1}, …, C_{(m)}^{t}) is degenerate

for m > r − 1 + nr or equivalently for m ≥ r + nr where r ≥ 1. Using (2) and (3) we get a decomposition of (3) by terms of the form

(4)  Λ(K_{m−λ_{1}^{1}}^{1} | … | K_{m−λ_{s_{1}}^{1}}^{1} , …, K_{m−λ_{1}^{t}}^{t} | … | K_{m−λ_{s_{t}}^{t}}^{t})

with

(5)  \{ \begin{matrix}m − λ_{α}^{τ} ≤ r − 1 + j_{τ} r for 1 ≤ α ≤ s_{τ} \\ ∑_{τ=1}^{t} s_{τ} (j_{τ} + r) ≤ n + r + 1 \end{matrix}

Hence (5) implies

(6)  (m − λ_{α}^{τ})/r < j_{τ} for 1 ≤ α ≤ s_{τ}

and

(7)  ∑_{τ=1}^{t} ∑_{α=1}^{s_{τ}}(r + (m − λ_{α}^{τ})/r) < n + r + 1

Using (7) we get

(8) ∑_{τ=1}^{t} ∑_{α=1}^{s_{τ}}(m − λ_{α}^{τ}) < rn + r^{2} + r − ∑_{τ=1}^{t} s_{τ} r^{2} ≤ rn + r^{2} + r − tr^{2} ≤ rn + r

Hence for rn + r ≤ m the left hand side of (8) is smaller than m and this implies by Lemma 5.2.2 in Blanc [4] that (4) and hence (3) is degenerate. q.e.d.

§ 6 Models of theories

A theory T is a category with initial object ∗ and with finite sums denoted by X ∨ Y for X, Y ∈ T. We consider ∗ as the empty sum. A map between theories T and T′ is a functor F : T → T′ which preserves sums. Let Set be the category of sets and let T^{op} be the opposite category of T. A model of the theory T is a functor

(6.1)  M : T^{op} → Set

(i.e. a contravariant functor T → Set) such that M carries sums in T to products of sets in Set. This means that M carries the inclusions i_{X} : X → X ∨ Y, i_{Y} : Y → X ∨ Y to the projections p_{X} = i_{X}^{∗} : M(X ∨ Y) → M(X) and p_{Y} = i_{Y}^{∗} : M(X ∨ Y) → M(Y) of the product set

M(X ∨ Y) = M(X) × M(Y)

In particular M carries the empty sum ∗ in T to the final object ∗ in Set consisting of a single point. See Lawvere [13]. For example given an object X in T then we obtain the model

Mor_{X} = T(−, X) : T^{op} → Set

which carries Y to the set T(Y, X) of morphisms in T. Then Mor_{X} is called a free model. Let model(T) be the category of models; objects are models M, M′ as above and morphisms are natural transformations M → M′; for this we assume that the category T is small so that model(T) is a well defined category. We have the Yoneda embedding of categories

(6.2)  T ⊂ model(T)

which carries X to Mor_{X}. A map F : T′ → T between theories induces a functor

F^{∗} : model(T) → model(T′)

which carries M to M ∘ F. Here F^{∗} is also called an ”algebraic functor”. By 3.7.7, 3.7.5 of Borceux [5] we know that any algebraic functor F^{∗} has a left adjoint.

A cogroup X = (X, 0, μ, ν) in a theory T is an object X together with maps

(6.3)  0 : X → ∗, μ : X → X ∨ X, ν : X → X

satisfying the usual identities (see Baues [1]) such that (M(X), 0^{∗}, μ^{∗}, ν^{∗}) is a group. The cogroup X is abelian if the interchange map T : X ∨ X → X ∨ X with T i_{1} = i_{2}, T i_{2} = i_{1} satisfies T μ = μ. Then M(X) is an abelian group. A map f : X → Y between cogroups is linear if μf = (f ∨ f)μ. If X and Y are cogroups then X ∨ Y is in the canonical way also a cogroup.

We call T a theory of cogroups if each object X of T has the structure of a cogroup such that this structure is compatible with sums in C; compare Baues [1].

(6.4) Example. A ringoid (or a pre-additive category) is a category R with the property that all morphism sets R(A, B) with A; B ∈ R are abelian groups and composition in R is bilinear. A ringoid with only one object is a ring. An additive category A is a ringoid in which sums A ∨ B = A ⊕ B exist. Then each object in A is an abelian cogroup with the comultiplication

(1)  μ_{A} = i_{1} + i_{2} : A → A ∨ A

In fact, an additive category is the same as a theory A in which each object is an abelian cogroup and in which all morphisms are linear.

Each ringoid R yields in a canonical way the additive category Add(R) which is the additive completion of R. Objects in Add(R) are tuple (X_{0}, …, X_{r}), r ≥ 0, of objects in R and morphisms (X_{0}, …, X_{r}) → (Y_{0}, …, Y_{k}) are matrices of morphisms in R. An additive functor F : R → R′ between ringoids is a functor which induces for X, Y ∈ R a homomorphism F : R(X, Y) → R′(F X, F Y) between abelian groups. A (right) R -module M is an additive functor

(2)  M : R^{op} → Ab

Let mod(R) be the category of (right) R -modules; morphisms are natural transformations. Then one readily checks that

(3)  mod(R) = model(Add(R))

where the right hand side is the category of models of the theory Add(R). We point out that mod(R) is an abelian category.

(6.5) Lemma. Let C be a theory of cogroups and let

X ⟶^{f,g} Y ⟶^{s} X

be a diagram in model(C) with fs = gs = 1_{Y}. Then X ⇉ Y → N is a coequalizer in model(C) if and only if for all C ∈ C the induced sequence

kernel{f : X(C) → Y (C)} ⟶^{g′} Y (C) → N(C) → 0

is an exact sequence in the category of groups. Here g′ is the restriction of g.

Proof. Assume the sequence is exact for each C and let h : Y → U be given with hf = hg then h yields a unique quotient map h̄ : N(C) → U(C) since hg(x) = 0 for f(x) = 0. Here h̄ is natural in C. Hence X ⇉ Y → N is a coequalizer. On the other hand if Y → N is the coequalizer of (f, g) then we get a map α : cokernel(g′) → N and also a map β : N → cokernel(g′) where β is the inverse of α. q.e.d.

(6.6) Definition. Let C be a theory in which the initial object ∗ is a zero object. Then we have the inclusion

C → model(C)

which carries X to Mor_{X} = C(−, X). We call sums

X = ⋁_{i∈J}Mor_{X_{i}} in model(C)

with X_{i} ∈ C free objects in model(C). Here J is finitely generated if J is finite. Such sums exist for all index sets J. A free presentation g : Y → X of an object N ∈ model(C) is a coequalizer diagram

Y ∨ X ⇉_{(g,1)}^{(0,1)} X → N

in model(C) where X and Y are free objects. Here 0 : Y → ∗ → X is defined by the zero object ∗ in C. We say that two free presentations g : Y → X and g′ : Y′ → X′ of N are strongly equivalent if X = X′ and if there is a commutative diagram

Y ⟶^{g} X ⟶ N
∥  ∥
Y′ ⟶^{g′} X′ ⟶ N

(6.7) Lemma. Let C be a theory of cogroups. Then all objects N in model(C) have a free presentation

Proof. We can choose a free X together with a surjective map q : X → N. For example let

X = ⋁_{C∈C} ⋁_{α∈N(C)} C(−, C_{α})

where C_{α} = C for all α ∈ N(C). Then q is defined by q(g) = g^{∗}(α) for g ∈ C(C′, C_{α}), C′ ∈ C. This is a map between models for which

q : X(X′) → N(X′)

is surjective for all X′ ∈ C. Since N(∗) = 0 we see that kernel(q) is again a model of C and hence there exists a free object Y and a surjective map ḡ such that g is the composite

g : Y ⟶^{ḡ} kernel(q) ⊂ X.

Now it is easy to see that (0, 1) and (g, 1) form a coequalizer as in (6.6). q.e.d.

§ 7 Split extension of theories

We here describe certain split extensions of theories and we show that the category of models of such an extension is canonically the Grothendieck construction of a functor Γ.

(7.1) Definition. Let C be a theory and let R be an additive category and let

(1)  D : R^{op} × C → Ab

be a bifunctor which is additive in R and which satisfy D(U, ∗) = 0 for U ∈ R. Then we define a category T as follows. Objects are pairs (U, X) with U ∈ R and X ∈ C and morphisms

(2)  (f, g, δ) : (U, X) → (V, Y)

in T are given by morphisms f : U → V ∈ R, g : X → Y ∈ C and elements δ ∈ D(U, Y). Composition of such morphisms is defined by

(3)  (f, g, δ)(f_{1}, g_{1}, δ_{1}) = (ff_{1}, gg_{1}, f_{1}^{∗}δ + g_{∗}δ_{1})

We have the projection functor

q : T → R × C

which is the identity on objects and which carries (f, d, δ) to (f, g). We call T together with q the split linear extension of R × C by D; see Baues [2]. By the properties of D in (1) we see that sums in R × C denoted by

(4)  (U, X) ∨ (V, Y) = (U ⊕ V, X ∨ Y)

are also sums in T.

Given a split linear extension T of R × C we have full inclusions of theories

(7.2)  C ⟶^{λ} T ⟵^{ρ} R

with λ(X) = (0, X) and ρ(U) = (U, ∗). We clearly have the equation of abelian groups

T(ρU, (V, Y)) = R(U, V) ⊕ D(U, Y)

by use of (7.1) (2). Here ρU is an abelian cogroup in T.

Let Δ be the left adjoint of

λ^{∗} : model(T) → model(C)

with adjunction maps

\{ \begin{matrix}ε : N → λ^{∗}Δ(N) for N ∈ model(C) \\ τ : Δλ^{∗}(M) → M and M ∈ model(T) \end{matrix}

The functor Δ carries the free model C(−, X) with X ∈ C to the free model T(−, λX) ∈ model(T). In fact we have

model(T)(T(−, λX), M) = M(λX) = model(C)(C(−, λX), Mλ)

and this shows Δ(C(−, X)) = T(−, λX).

Using Δ and ρ we obtain the composite functor

(7.3)  ρ^{∗}Δ : model(C) ⟶^{Δ} model(T) ⟶^{ρ^{∗}} model(R) = A

Here model(R) = mod(R) = A is an abelian category since R is an additive category. For a set K let ℤ[K] be the free abelian group generated by K; we have the inclusion K ⊂ ℤ[K] denoted by x ↦ [x] ∈ ℤ[K] for x ∈ K. We also obtain the functor

(7.4)  Γ : model(C) → model(R) = A

which carries N ∈ model(C) to Γ(N) ∈ A with

Γ(N)(U) = D(U, −) ⊗_{C} ℤ[N(−)]

Here the right hand side is the abelian group obtained as a quotient of

⨁_{Y ∈C}D(U, Y) ⊗ ℤ[N(Y)]

by the relations

δ ⊗ [f^{∗}y] ∼ (f_{∗}δ) ⊗ [y]

for f : Y → Z ∈ C, δ ∈ D(U, Y), y ∈ N(Z). It is clear that Γ(N)(U) is a well defined contravariant functor in U ∈ R and that Γ(N) is natural in N so that the functor Γ in (7.4) is well defined.

(7.5) Lemma. The functors Γ and ρ^{∗}Δ in (7.4) and (7.3) respectively coincide, that is Γ = ρ^{∗}Δ.

Proof. We define a functor

Δ′ : model(C) → model(T)

by

(1)  Δ′(N)(U, X) = Γ(N)(U) × N(X)

for (U, X) ∈ T and by

(2)  (f, g, δ)^{∗} : Γ(N)(V) × N(Y) → Γ(N)(U) × N(X)

\{ \begin{matrix}(f, g, δ)^{∗}(w, z) = (f^{∗}w + δ ⊗ [z], g^{∗}z) \\ w ∈ Γ(N)(V), z ∈ N(Y) \\ f : U → V ∈ R \\ g : V → Y ∈ C \\ δ ∈ D(U, Y) \end{matrix}

One readily checks that Δ′ is a well defined functor; compare (7.1) (3). Now we claim that Δ = Δ′ is the left adjoint of λ^{∗}. In fact since λ^{∗}(Δ′)(N) = N we have a natural map Δ → Δ′ given by the identity of N. On the other hand we have for each M ∈ model(T) and α : N → Mλ a unique map

ᾱ : Δ′(N) → M ∈ model(T)

with ᾱλ^{∗} = α. In fact, since (U, X) = ρU ∨ λX is a sum we have for the inclusions i_{1}, i_{2} the commutative diagram

Γ(N)(U) ⟵^{(i_{1})^{∗}} Δ′(N)(U, X) ⟶^{(i_{2})^{∗}} N(X)
↓ᾱ  ↓ᾱ  ↓^{α}
M(ρU) ⟵^{(i_{1})^{∗}} M(U, X) ⟶^{(i_{2})^{∗}} M(λX)

Here ᾱ on the left hand side satisfies for δ ∈ D(U, Y), y ∈ N(Y) the equation

ᾱ(δ ⊗ [y]) = ᾱ(f, g, δ)^{∗}(0, y); see (2),
= (f, g, δ)ᾱ(0, y)
= (f, g, δ)^{∗}(0, αy)

with (f, g, δ) : (U, ∗) → (U, Y) given by f = 1 = identity of U, g = 0 : ∗ → Y and δ ∈ D(U, Y). Hence the extension ᾱ is unique and therefore we see that Δ′ → Δ → Δ′ is the identity. Clearly also Δ → Δ′ → Δ is the identity. q.e.d.

(7.6) Corollary. The adjunction map ε in (7.2) is an isomorphism. Moreover the adjunction map τ is given for N = Mλ by the commutative diagram

Δ(N)(U, X)=Γ(N)(U) × N(X)
τ↓  ↓_{τ′}
M(U, X) =M(ρU) × M(λX)

Here τ′ carries the pair (δ ⊗ [y], z) to ((1, 0, δ)^{∗}(0, y), z) with z ∈ N(X) and (0, y) ∈ M(U, Y) = M(ρU) × M(λY) and (1, 0, δ) : (U, ∗) → (U, Y) ∈ T.

(7.7) Proposition. Let T be the split linear extension of R×C by D. Then there is a canonical isomorphism of categories

ψ : model(T) = ΓA

where ΓA is the Grothendieck construction of Γ in (7.3).

Proof. Let M be a model of T and let N = Mλ be the induced model of C. We have the adjunction map

τ : Δ(N) = (Δλ^{∗})(M) → M in model(T)

and hence τ induces a map

(1)  τ_{∗} : Γ(N) = ρ^{∗}Δ(N) → ρ^{∗}M = A in A

Now ψ carries M to the object (N, A, τ_{∗}) in ΓA. It is clear that ψ is a well defined functor. Next we define a functor

(2)  χ : ΓA → model(T)

which is the inverse of ψ. Let (N, A, η) be an object in ΓA with N ∈ model(C) and η : ΓN → A ∈ A. We define M = χ(N, A, η) with Mλ = N as follows. The contravariant functor

M : T → Set

carries an object (U, X) to the product set

(3)  M(U, X) = A(U) × N(X)

A morphism (f, g, δ) : (U, X) → (V, Y) in T induces the function

(f, g, δ)^{∗} : M(V, Y) = A(V) × N(Y) → A(U) × N(X) = M(U, X)

defined for v ∈ A(V), y ∈ N(Y) by

(4)  (f, g, δ)^{∗}(v, y) = (f^{∗}v + ∂, g^{∗}(y))

Here ∂ = ηδ^{∗}ε(y) is the image of y under the composition

y ∈ N(Y) ⟶^{ε} (λ^{∗}Δ(N))(Y) = Δ(N)(λY) ⟶^{δ^{∗}}

Δ(N)(ρU) = (ρ^{∗}Δ(N))(U) = (Γ(N))(U) ⟶^{η} A(U)

Here δ : ρU = (U, ∗) → (0, Y) = λY is the morphism in T given by (0, 0, δ). Using (6.5) (3) we have to check

(f_{1},g_{1}, δ_{1})^{∗}(f, g, δ)^{∗}(v, g)
= (f_{1}, g_{1}, δ_{1})^{∗}(f^{∗}(v) + ηδ^{∗}ε(y), g^{∗}(y))
= (f_{1}^{∗}(f^{∗}(v) + ηδ^{∗}ε(y)) + ηδ_{1}^{∗}ε(g^{∗}(y)), g_{1}^{∗}g^{∗}(y))
= ((ff_{1})^{∗}(v) + η(α_{1}^{∗}δ + ξ_{∗}δ_{1})^{∗}ε(y), (gg_{1})^{∗}(y))

This follows from the equation

f_{1}^{∗}ηδ^{∗}ε(y) + ηδ_{1}^{∗}ε(g^{∗}y) = η(f_{1}^{∗}δ + g_{∗}δ_{1})^{∗}ε(y)

where η is a homomorphism. Hence M is a well defined object in model(T) and the functor χ which carries (N, A, η) to M is well defined. Now one can check that χ is the inverse of ψ. For example we obtain χψ(M) = M since

(5)  M(U, X) = MρU × MλX = (χψ(M))(U, X)

Here the left hand side holds since M is a model and since (U, X) = ρU ∨ λX is a sum in T. The right hand side is a consequence of the definitions of χ and ψ. Since (5) is compatible with induced maps we see that M = χψ(M). On the other hand we get for (N, A, η) ∈ ΓA above the natural equation

(6)  ψχ(N, A, η) = ψ(M) = (N, A, η)

where M is defined as in (3). For this we have to understand the adjunction map τ : δ(N) → M described in (7.6). Now (4) shows that (6) holds. q.e.d.

(7.8) Addendum. For the functors I_{0} and I_{1} in (1.4) the following diagrams commute.

\begin{matrix}model(C) & ⟶^{I_{1}} & ΓA \\ ∥ & ∥ & ψ \\ model(C) & ⟶^{Δ} & model(T) \end{matrix}

\begin{matrix}model(C) × A & ⟶^{I_{0}} & ΓA \\ ∥ & ∥ & ψ \\ model(R × C) & ⟶^{q^{∗}} & model(T) \end{matrix}

Proof. For N ∈ model(C) and M = ΔN the map τ in (7.7) (1) is the identity. Hence τ_{∗} is the identity and therefore ψΔ(N) = I_{1}(N). Next we have

model(R × C) = model(R) × model(C)

with model(R) = A so that the second diagram is well defined. For N and A ∈ A we consider χI_{0}(N, A) = M with χ defined in (7.7) (3). Since η = 0 in (7.7) (4) we see that (f, g, δ)^{∗}(v, y) = (f^{∗}v, g^{∗}y). But this shows that M = q^{∗}(N, A). q.e.d.

Now let T be the split linear extension of R × C by D : R^{op} × C → Ab. Let

free(C) ⊂ model(C)

for the full subcategory consisting of free objects. Then we have the canonical functor

(7.9)  D̄ : R^{op} × free(C) → Ab

which extends D such that the functor X ↦ D(U, X) preserves filtered colimits for X ∈ free(C). We also have the equation

D̄(U, X) = model(T)(ρ^{∗}Mor_{U}, λ^{∗}X)

Using D̄ we get the following useful description of the functor Γ = ρ^{∗}Δ.

(7.10) Lemma. Let C be a theory of cogroups with a zero object ∗. Moreover let a free presentation of N ∈ model(C) be given as in (6.6). Then the sequence

D̄(−, Y ∨ X)_{2} ⟶^{(g,1)_{∗}} D̄(−, X) → Γ(N) → 0

is an exact sequence in mod(R) where

D̄(U, X ∨ Y)_{2} = kernel{(0, 1)_{∗} : D̄(U, Y ∨ X) → D̄(U, X)}

for U ∈ R and where (g, 1)_{∗} is the restriction of the map (g, 1)_{∗} : D̄(U, Y ∨ X) → D̄(U, X).

Proof. We consider the functor

Δ′′ : model(C) → model(T)

defined as follows. For a finitely generated free object X in model(C) let

(1)  Δ′′(X) = T(−, λ^{∗}X)

Moreover if a finitely generated free presentation of N ∈ model(C) is given as in (6.6) then let Δ′′(N) be defined by the exact sequence of groups with Z ∈ T

(2)  T(Z, λ^{∗}(X ∨ Y))_{2} → T(Z, λ^{∗}X) → Δ′′(N)(Z) → 0

One can check that the functor Δ′′ does not depend on the choice of the free presentation and hence is well defined; if the presentation is not finitely generated we use compatibility with filtered colimits. We claim that Δ′′ = Δ is the left adjoint Δ of λ^{∗}. In fact Δ, as a left adjoint, preserves coequalizers so that the free presentation in (6.6) is mapped to the coequalizer

(3)  Δ(X ∨ Y) ⇉_{(g,1)_{∗}}^{(0,1)_{∗}} Δ(X) → Δ(N) → 0

For a free object X, however, it is clear that Δ(X) = Δ′′(X) by (1). Hence by (2) and (3) we obtain the canonical surjective map

(4)  Δ′′(N)(Z) → Δ(N)(Z)

which is natural in M and Z. On the other hand we know by (6.6) and (2) that

(5)  λ^{∗}Δ′′(N) = N

Hence we have also a canonical map Δ(N) → Δ′′(N) which is the inverse of (4). q.e.d.

(7.11) Corollary. Let C be a theory of cogroups with a zero object ∗. Then Γ carries a free presentation in model(C) to a coequalizer in A.

Proof. For a free object X we have by (7.10) D̄(−, X) = Γ(X). Hence the corollary is an immediate consequence of (7.10). q.e.d.

The next two lemmas are needed in the proof of theorem (7.14) below.

(7.12) Lemma. Let g be a free presentation of M in model(T). Then g′ = λ^{∗}g is a free presentation of λ^{∗}M in model (C).

Proof. For (V, Y) ∈ T and X ∈ C we have

(λ^{∗}T(−, (V, Y)))(X) = T(λX, (V, Y))
= T((0, X), (V, Y))
= C(X, Y)
= C(−, Y)(X)

Hence λ^{∗} carries a free object to a free object. Using (6.5) we see that λ^{∗}g is a free presentation of λ^{∗}M. q.e.d.

The next lemma uses the functor

Q̃ : ΓA → model(C) × A

given by (1.4).

(7.13) Lemma. Let g′ be a free presentation of N in model(C) and let g′′ be a free presentation of B in model(R) = A. If M ∈ model(C) is given with Q̃ψ(M) = (N, B) then there exists a free presentation g of M with Q̃ψ(g) = (g′, g′′′) where g′′′ is strongly equivalent to g′′.

Proof. Let η : Γ(N) → A be given by χ(M) so that the cokernel of η is Qη = B. Then g′ : Y → X and g′′ : G → F yield the following commutative diagram with exact rows and columns in A.

Γ(Y ∨ X)_{2} ⟶ Γ(X) ⟶ Γ(N) ⟶ 0
↓^{η_{1}}  ↓^{η_{0}}  ↓^{η}
A_{1}  ⟶ A_{0} ⟶ A ⟶ 0
↓  ↓  ↓
G ⊕ D ⟶^{g′′′} F ⟶ B ⟶ 0
↓  ↓  ↓
0  0  0

Here we set A_{0} = Γ(X) ⊕ F and A_{1} = Γ(Y ∨ X)_{2} ⊕ G ⊕ D where D is free in A. We choose for F → B a lift F → A which defines the surjective map A_{0} → A. We can choose g′′′ in such a way that g′′′(D) = 0 and g′′′ | G = g′′. Now η_{0} and η_{1} correspond to free objects in model(C) which yield via the diagram above the free presentation g of M with the properties in (7.13). q.e.d.

In the following result we describe crucial properties of the reduction functors Λ_{i} in (2.6).

(7.14) Theorem. Let C be a theory of cogroups with zero object ∗ and let T be a split linear extension of R × C as in (7.1). Moreover let

Λ : model(T) = ΓA → A′

be a functor which carries a free presentation to a coequalizer. Then Λ preserves epimorphisms and the reduction functors

\begin{matrix}Λ_{0} : model(C) → A′ \\ Λ_{i} : model(C) × A → A′, i ≥ 1, \end{matrix}

carry a free presentation to a coequalizer.

Here a free presentation in the product category model(C) × A is a pair of free presentations in model(C) and in A = model(R) respectively. In addition to (7.14) we get the following result. We say that a functor f : A → A′ between abelian categories is weakly right exact if each exact sequence A ⟶^{g} B → C → 0 in A yields an exact sequence

F (A ⊕ B)_{2} ⟶^{(g,1)_{∗}} F B → F C → 0

in A′.

(7.15) Addendum. Given A ∈ A the reduction functors

Λ_{0}, Λ_{i}(−, A) : model(C) → A′

with i ≥ 1 carry a free presentation in model(C) to a coequalizer in A′. Moreover for X ∈ model(C) the reduction functor

Λ_{i}(X, −) : A → A′

is weakly right exact.

Proof of (7.14). Let η : ΓN → A ∈ ΓA and let

(1)  \begin{matrix}ΓY & ⟶^{Γg} & ΓX & ⟶ & ΓN \\ ↓^{η_{Y}} & & ↓^{η_{X}} & ↓^{η} \\ A_{1} & ⟶^{g′} & A_{0} & ⟶ & A \end{matrix}

be morphisms in ΓA such that (g, g′) : η_{Y} → η_{X} is a free presentation of η. Here we identify model(T) = ΓA via ψ in (7.7). Let (1, f) : η → λ be an A -epimorphism. Then we can choose as in (7.13) a free presentation λ′′ → λ′ of λ for which the following diagram commutes; in particular we may choose λ′ = η_{X}.

η_{Y} ⟶ η_{X} ⟶^{p} η
∥  ↓^{(1,f)}
λ′′ ⟶ λ′ ⟶^{q} λ

Hence since Λ carries p and q to epimorphisms also Λ(1, f) is an epimorphism. Next we observe that the inclusion (see (7.8))

(2)  Δ : model(C) ⟶^{I_{1}} ΓA = model(T)

carries a free model to a free model; in fact for X ∈ C the functor Δ carries C(−, X) to T(−, λX). Moreover Δ preserves coequalizers. Hence we see that Λ_{0} = ΛI_{1} carries free presentations to coequalizers. Next we prove that the functors

(3)  model(T) = ΓA ⟶^{Λ_{(i)}} A′, i ≥ 1,

carry a free presentation to a coequalizer. In fact, since λ^{∗} in (7.12) carries a free presentation to a free presentation we see that the top row and the row in the middle of the next diagram are exact.

(4)

ΛI_{1}(Y ∨ X)_{2} ⟶ ΛI_{1}X ⟶ ΛI_{1}(N) ⟶ 0
↓  ↓  ↓
Λ(η_{Y} ∨ η_{X})_{2} ⟶ Λ(η_{X}) ⟶ Λ(η) ⟶ 0
↓  ↓  ↓
Λ_{(1)}(η_{Y} ∨ η_{X})_{2} ⟶ Λ_{(1)}(η_{X}) ⟶ Λ_{(1)}(η) ⟶ 0
↓  ↓  ↓
0  0  0

Moreover all columns are exact by (2.6) (1). This implies that also the bottom row is exact. Therefore Λ_{(1)} has the property in (3). Now assume Λ_{(i)} = Ω, i ≥ 1, has the property in (3). Then we have by (2.3) (1) the exact sequence

(5)  Ω(I_{1}N | I_{0}ΓN) → Ω(η | I_{0}ΓN) → Λ_{(i+1)}(η) → 0

for η ∈ ΓA.

Let C → B → A → 0 be an exact sequence in A which for example coincides with the exact sequence

(6)  Γ(Y ∨ X)_{2} → Γ(X) → Γ(N) → 0

given by (1) and (7.11). Then we can choose as in (7.13) a diagram in ΓA

γ′′
↓
γ′  ⟶  γ  ⟶ η ∨ I_{0}A
↓  ↓  ∥
η_{Y} ∨ I_{0}C ⟶ η_{X} ∨ I_{0}B ⟶ η ∨ I_{0}A

in which the column in the middle and the row in the middle are free presentations. Since Ω has property (3) we thus see that the induced sequence

Ω((η_{Y} ∨ I_{0}C) ∨ (η_{X} ∨ I_{0}B))_{2} → Ω(η_{X} ∨ I_{0}B) → Ω(η ∨ I_{0}A) → 0

is exact and therefore also the sequence

(7)  Ω(η_{Y} ∨ η_{X} | I_{0}C ∨ I_{0}B)_{2} → Ω(η_{X} | I_{0}B) → Ω(η | I_{0}A) → 0

is exact. Using (5) we get the following commutative diagram where A → B → C → 0 is given by (6).

Ω(I_{1}Y ∨ I_{1}X | I_{0}C ∨ I_{0}B)_{2} ⟶ Ω(I_{1}X | I_{0}B) ⟶ Ω(I_{1}N | I_{0}A) ⟶ 0
↓  ↓  ↓
Ω(η_{Y} ∨ η_{X} | I_{0}C ∨ I_{0}B)_{2} ⟶ Ω(η_{X} | I_{0}B) ⟶ Ω(η | I_{0}A) ⟶ 0
↓  ↓  ↓
Λ_{(i+1)}(η_{Y} ∨ η_{X})_{2}  ⟶ Λ_{(i+1)}(η_{X}) ⟶ Λ_{(i+1)}(η) ⟶ 0
↓  ↓  ↓
0  0  0

Here the columns are exact by (5) and the top row and the row in the middle are exact by (7). Hence the bottom row is exact. This proves (3) for Λ_{(i+1)}. Hence all Λ_{(i)}, i ≥ 1, carry a free presentation to a coequalizer.

Now we can use the definition of Λ_{i} by the exact sequence

(8)  Λ_{(i+1)}(I_{0}(X, A)) ⟶^{P′} Λ_{(i)}(I_{0}(X, A)) → Λ_{i}(X, A) → 0

in (2.5) and (2.6) (2). Using (7.13) we obtain for a free presentation (g′, g′′) of (X, A) in model(C) × A a free presentation g of I_{0}(X, A) in model(T) such that Q̃(g) = (g′, g′′′) where g′′′ is strongly equivalent to g′′. Since Λ_{(i)} and Λ_{(i+1)} satisfy (3) we thus see by (8) that Λ_{i} carries the free presentation (g′, g′′) to a coequalizer. q.e.d.

Proof of (7.15). We choose a commutative diagram in A

G′ ⟶ F′ ⟶ 0
↓  ↓  ↓
G ⟶ F ⟶ A ⟶ 0
↓  ↓  ∥
C ⟶ B ⟶ A ⟶ 0
↓  ↓  ↓
0  0  0

The rows and the columns are exact in A and F, F′, G, G′ are free in A. For η ∈ ΓA we thus get the diagram

(η_{Y} , G′) ⟶ (η_{Y} , F′) ⟶ (∗, 0)
↓  ↓  ↓
(η_{Y} , G) ⟶ (η_{X}, F) ⟶ (η, A)
↓  ↓  ∥
(∗, C) ⟶ (η, B) ⟶ (η, A)

Here the top row and the row in the middle and also the left hand column in the middle are free presentations in model(C)×A. Hence they yield by Λ_{i} coequalizers and this shows that

Λ_{i}((∗, C) ∨ (η, B))_{2} ⟶ Λ_{i}(η, B) ⟶ Λ_{i}(η, A) ⟶ 0
∥
Λ_{i}(η, C ⊕ B)_{2}

is exact. Hence the functor Λ_{i}(η, −) is weakly right exact for all η ∈ ΓA. Next we consider the diagram

(∗, G)
↓
(η_{Y} , G) ⟶ (η_{X}, F) ⟶ (η, A)
↓  ↓  ∥
(η_{Y} , 0) ⟶ (η_{X}, A) ⟶ (η, A)

here the row in the middle and also the column in the middle are free presentations in model(C) × A. Hence they are mapped to coequalizers by Λ_{i}. This shows that the sequence

Λ_{i}((η_{Y} , 0) ∨ (η_{X}, A))_{2} ⟶ Λ_{i}(η_{X}, A) ⟶ Λ_{i}(η, A) ⟶ 0
∥
Λ_{i}(η_{Y} ∨ η_{X}, A)_{2}

is exact. Hence for each A the functor Λ_{i}(−, A) carries a free presentation to a coequalizer. q.e.d.

§ 8 Graded theories

We introduce the notion of a graded theory and we show that the category of models of a graded theory is an iterated Grothendieck construction.

(8.1) Definition. A graded theory T is a theory T with a zero object ∗ together with a set S of objects in T and a function

| |: S → ℤ

which carries S ∈ S to the degree | S | with | S |≥ 0. Moreover the following properties (a), (b), (c) are satisfied.

(a) The objects X of T are exactly the finite sums of objects in S, that is

X = S_{1} ∨ … ∨ S_{r}

with S_{1}, …, S_{r} ∈ S and 0 ≤ r < ∞. Let | X |= Min{| S_{i} |; 1 ≤ i ≤ r} be the degree of X and let dim(X) = Max{| S_{i} |; 1 ≤ i ≤ r} be the (formal) dimension of X. For n ≥ 0 we define the n-skeleton of X by

X^{n} = ⋁_{\begin{matrix}i∈\{1,... ,r\} \\ |S_{i}|≤n \end{matrix}} S_{i}

We have the canonical inclusion X^{n} ⊂ X and dim(X) ≤ n if and only if X^{n} = X and | X |≥ k if and only if X^{k} = ∗.

(b) Each object S of S is an abelian cogroup and thus by (a) also each object X of T is an abelian cogroup. Moreover each map f : S → X in T with | X |≥| S | is linear and T(S, X) = 0 for | X |>| S |.

(c) Each map f : S → U ∨ Y in T where | U |≥| S | and dim(Y) <| S | splits as a sum f = i_{1}f_{1} + i_{2}f_{2} where f_{1} : S → U and f_{2} : S → Y.

As a special case of graded theories T we can consider such T for which all ringoids R_{n}, n ≥ 0, coincide so that the associated abelian categories A_{n}, n ≥ 0, satisfy A_{0} = A_{1} = . . .. In particular, one has the following special case of such a graded theory.

(8.1’) Definition. Let R be a ring. A graded R-theory is a graded theory T with the following properties. For n ≥ 0 there is at most one object S = S_{n} ∈ S with | S |= n and the endomorphism ring R_{n} of S_{n} coincides with the ring R.

(8.1”) Example. Let r ≥ 2 and let

S_{r} = {S^{r}, S^{r+1}, … }

be the set of all spheres S^{n} with n ≥ r. We define the degree function

\{ \begin{matrix}|  |: S_{r} ⟶ ℕ_{0} by \\ | S^{n+r} |= n for n ≥ 0 \end{matrix}

Hence the degree of the sphere does not coincide with the dimension of the sphere. Let Top^{∗}/ ≃ be the homotopy category of pointed spaces and basepoint preserving maps. We associate with S_{r} the full subcategory

T(S_{r}) ⊂ Top^{∗}/ ≃

consisting of finite one point unions S_{1} ∨ … ∨ S_{k} with S_{i} ∈ S_{r} for 1 ≤ i ≤ k, k ≥ 0. Then it is easily seen that T(S_{r}) is a graded ℤ -theory. Moreover T(S_{r}) is (r − 1) -polynomial in the sense of (9.4) below as follows from the Hilton-Milnor theorem [12]. Compare [17], [9], [10].

(8.1”’) Example. Let r ≥ 1 and let A be a set of abelian groups. Moreover let

K_{r}(A) = {K(A, n + r); n ≥ 0, A ∈ A}

be the corresponding set of Eilenberg-Mac Lane spaces K(A, n + r). We define the degree function

\begin{matrix}| |: K_{r}(A) → ℕ_{0} by \\ | K(A, n + r) |= n for n ≥ 0 \end{matrix}

We associate with K_{r}(A) the full subcategory

K_{r}(A) ⊂ Top^{∗}/ ≃

consisting of products K_{1} × … × K_{k} with K_{i} ∈ K_{r}(A) for 1 ≤ i ≤ k, k ≥ 0. Then it is easily seen that the opposite category K_{r}(A)^{op} is a graded theory. Moreover K_{r}(A)^{op} is r-polynomial in the sense of (9.4) below. The system of cohomology groups M = {H^{n+r}(X; A), n ≥ 0, A ∈ A} together with all cohomology operations is a model of K_{r}(A)^{op}.

Let R(A) be the full subcategory of Ab consisting of all abelian groups A ∈ A. Then R(A) is a ringoid and all abelian categories A_{i} associated with the theory K_{r}(A) coincide with the category of modules

A_{i} = mod (R(A)), i ≥ 0.

If Add (R(A)) is an abelian category then the global dimension of mod (R(A)) is at most 2. For example this holds if A = Cyc consists of all cyclic groups ℤ and ℤ/p^{i} where p is a prime and i ≥ 1. This also holds if A = Cyc_{p} consists of all ℤ/p^{i}, i ≥ 1, where p is a fixed prime. Hence for these cases we can apply the results on the vanishing line below. This enriches considerably the classical theory of modules are the Steenrod-algebra which corresponds to the case A = {ℤ/p}.

Remark. Blanc [4] defines for a ring R a category of regular graded algebras CRGA. It is easy to check that for each CRGA there exists a graded R^{op} -theory T such that

CRGA = model(T).

In fact, T is the category of finitely generated free RGA’s; see Blanc [4] 2.1.4. The category model(T) given by graded R-theories, however, are considerably more general than CRGA’s in the sense of Blanc [4]. Blanc [4] 2.4.2, 5.1.1 uses free simplicial resolutions with a CW-basis of an object in the category CRGA. Such resolutions are special cases of the resolutions in Γ^{∞}A defined in § 2. Hence derived functors in this paper yield the derived functors considered by Blanc [4] as a special case.

The definition of a graded theory T implies the following properties of T. Let T_{n}, n ≥ 0, be the full subcategory of T consisting of objects X with dim(X) ≤ n. Moreover let R_{n} be the full subcategory of T consisting of objects S ∈ S with | S |= n. Then (8.1) (b) shows that R_{n} is a ringoid with R_{0} = T_{0}. Let

(8.2)  A_{n} = mod(R_{n}) = model(Add(R_{n}))

be the abelian category of right R_{n} -modules; see (6.4). These are the abelian categories associated to the graded theory T. We have full inclusions of theories

(8.3)  T_{n} ⟶^{λ} T_{n+1} ⟵^{ρ} Add(R_{n+1})

Here λ is the inclusion of the subcategory and ρ carries U = (S_{1}, …, S_{r}) ∈ Add(R_{n+1}) with S_{i} ∈ S and | S_{i} |= n + 1 for 1 ≤ i ≤ r to the sum

(1)  ρ(U) = S_{1} ∨ … ∨ S_{r}

in T. Moreover we define a bifunctor

(2)  \begin{matrix}D_{n+1} : Add(R_{n+1})^{op} × T_{n} → Ab \\ D_{n+1}(U, Y) = T(ρU, Y) \end{matrix}

Here D_{n+1} has the properties of the bifunctor D in (7.1), that is D_{n+1} is additive in U and D_{n+1}(U, ∗) = 0.

(8.4) Lemma. Let T be a graded theory. Then the theory T_{n+1} is a split extension of Add(R_{n+1}) × T_{n} by D_{n+1}.

Proof. Each object X of T_{n+1} has the unique decomposition

X = ρU ∨ λY

where dim(Y) ≤ n and U ∈ Add(R_{n+1}). A morphism h in T_{n+1} is given by

h : X = ρU ∨ λY → ρU′ ∨ λY′ = X′

Using (8.1) (b), (c) we see that h carries λY to λY′, that is, there is g : Y → Y′ ∈ T_{n} with hi_{2} = i_{2}λ(g). Moreover hi_{1} splits by (8.1) (c), that is, there are f : U → U′ and δ ∈ D_{n+1}(U, Y′) such that hi_{1} = i_{1}f +i_{2}δ. This shows that T_{n+1} is canonically isomorphic to the split extension described in (7.1). q.e.d.

The lemma shows that we can apply (7.7) inductively. This leads to the following crucial result.

(8.5) Theorem. Let T be a graded theory and let A = (A_{0}, A_{1}, …) be the sequence of abelian categories associated to T by A_{n} = mod(R_{n}) for n ≥ 0. Then there exists a sequence of functors Γ = (Γ^{1}, Γ^{2}, …) with

\{ \begin{matrix}Γ^{1} : A_{0} → A_{1} \\ Γ^{n+1} : Γ^{n}A_{n} → A_{n+1}, n ≥ 0 \end{matrix}

such that the category of models of T is the iterated Grothendieck construction

model(T) = Γ^{∞}A.

If there is m ≥ 0 with | S |≤ m for all S ∈ S then we get

model(T) = Γ^{m}A_{m}.

Using (7.7) we obtain inductively the isomorphism of categories, n ≥ 0,

(8.6)  ψ : model(T_{n}) = Γ^{n}A_{n}

and the functors

Γ^{n+1} : Γ^{n}A_{n} = model(T_{n}) ⟶^{Δ} model(T_{n+1}) ⟶^{ρ^{∗}} mod(R_{n+1}) = A_{n+1}

Here Δ is the left adjoint of the functor λ^{∗} induced by λ : T_{n} → T_{n+1} and ρ^{∗} is induced by ρ : Add(R_{n+1}) → T_{n+1}; compare (8.3). Now (8.5) is a direct consequence of (7.7). It is clear that mod(R_{n}) has enough projectives for n ≥ 0. This shows by (2.7) that resolutions exist in Γ^{n}A_{n} for all n ≥ 0. Moreover one readily checks that all Γ^{n} preserve zero objects so that sums in Γ^{n}A can be described by cross effects as in (4.2). The multi-epimorphism in Γ^{n}A_{n} correspond via ψ in (8.6) exactly to surjective maps in model(T_{n}); here we say that a map f : M → N in model(T_{n}) is surjective if for all objects X ∈ T_{n} the induced map f : M(X) → N(X) is a surjective map between sets.

Using (7.14) inductively we get the following result; compare (4.2).

(8.7) Theorem. Let T be a graded theory and let Γ = (Γ^{1}, …) be the associated iterated Grothendieck construction. Then Γ^{n+1} carries a multi-epimorphism in Γ^{n}A_{n} to an epimorphism in A_{n+1}, n ≥ 0. Hence all reduction functors

Γ_{ε}^{n+1} : (Γ^{n}A_{n}) × A_{i_{1}} × … × A_{i_{|ε|}} → A_{n+ℓ(ε)+1}

are defined for all ε ∈ (ℕ_{0})^{ℓ(ε)} with ℓ(ε) ≥ 0 and n + ε^{−1}(ℕ) = {i_{1} < … < i_{|ε|}}. These reduction functors carry a multi-epimorphism to an epimorphism and they carry a free presentation to a coequalizer.

The theorem shows that the reduction functor Γ_{ε}^{n+1} is determined by its restriction to finitely generated free objects. This is very helpful for the computation of the reduction functors. As in (7.15) we get the following additional property of the reduction functors Γ_{ε}^{n+1} in (8.7).

(8.8) Addendum. For X ∈ Γ^{n}A_{n} and A_{i} ∈ A_{i} the functor Γ_{ε}^{n+1}(−, A_{i_{1}}, …, A_{i_{|ε|}}) carries a free presentation in Γ^{n}A_{n} to a coequalizer and the functor

Γ_{ε}^{n+1}(X, A_{i_{1}}, …, A_{i_{t}}, −, A_{i_{t+2}}, …, A_{i_{|ε|}})

with 0 ≤ t <| ε | is weakly right exact.

Proof of (8.7) and (8.8). It suffices to show that Γ^{n+1} carries a multi-epimorphism f^{≤n} : η^{≤n} → λ^{≤n} to an epimorphism. For this we observe that a free presentation Y → X → η^{≤n} can be used to obtain a free presentation Y′ → X → λ^{≤n} where X → λ^{≤n} is the composite of X → η^{≤n} and f^{≤n}. Hence since Γ^{n+1} carries a free presentation to an epimorphism we see that Γ^{n+1} carries f^{≤n} to an epimorphism. Now we can apply (7.14) and (7.15) inductively. This yields the results. q.e.d.

§ 9 Derived functors for graded theories

Let T be a graded theory. We have seen in (8.6) that resolutions exist in model(T_{n}). Hence we obtain for each object X ∈ model(T_{n}) the derived functors (n ≥ 0)

(9.1)  L_{i}Γ^{n+1}(X) = π_{i}(Γ^{n+1}X_{·})

where X_{·} is a resolution of X in s model(T_{n}).

(9.2) Proposition. L_{0}Γ^{n+1} = Γ^{n+1} for n ≥ 0.

Proof. Since free presentations exist we see that there exists a resolution X_{·} for which

X_{(1)} ⇉ X_{(0)} → X

is a free presentation of X as in (6.6). Now (7.11) shows that Γ^{n+1} carries a free presentation to a coequalizer. This implies L_{0}Γ^{n+1} = Γ^{n+1}. q.e.d.

The next result is an immediate corollary of (8.7) and (4.7).

(9.3) Theorem. Let T be a graded theory and let Γ = (Γ^{1}, …) be the associated iterated Grothendieck construction with Γ^{n+1} : Γ^{n}A_{n} → A_{n+1}. Assume the abelian category A_{n} has finite global dimension d_{n} < ∞ for n ≥ 0. Moreover assume that all reduction functors Γ_{ε}^{n+1} are reducible and that all reduction functors Γ_{ε}^{1} are bounded by γ_{ε} with γ_{ε} = 0 if Γ_{ε}^{1} = 0. Then the derived functors L_{i}Γ^{n+1} have a vanishing line N_{n+1} < ∞, that is

L_{i}Γ^{n+1} = 0 for i > N_{n+1}.

Here N_{n+1} is inductively defined by

\{ \begin{matrix}a^{0} = d_{0} \\ a^{n+1} = 1 + d_{n+1} + N_{n+1} for n ≥ 0 \\ N_{n+1} = Max\{γ_{ε}(a^{0}, a^{i_{1}}, …, a^{i_{|ε|}}), ε ∈ (ℕ_{0})^{n}\} \end{matrix}

The maximum is taken over all ε ∈ (ℕ_{0})^{n}. Such an ε defines i_{1}, …, i_{|ε|} by ε^{−1}(ℕ) = {i_{0} < … < i_{|ε|}}.

A graded theory T is polynomial if for all S ∈ S the functors

(9.4)  π^{S} : T → Ab with π^{S}(X) = T(S, X)

are polynomial. Moreover T is polynomial of degree ≤ k if all these functors are of degree ≤ k. In particular T is linear, resp. quadratic if T is polynomial of degree 1, resp. 2. One readily checks that T is linear if and only if all morphisms in T are linear; see (6.3). For r ≥ 1 we say that T is r-polynomial if for n ≥ 0 and q ≥ 2 and any set of objects X_{1}, …, X_{q} in T with

∑_{i=1}^{q}(r+ | X_{i} |) > r + n + 1

and S ∈ S with | S |= n + 1 we have

π^{S}(X_{1} | … | X_{q}) = 0.

Since r ≥ 1 this shows that T is polynomial.

(9.5) Lemma. Let T be a graded theory with associated iterated Grothendieck construction Γ = (Γ^{1}, …). Then T is polynomial (resp. polynomial of degree k, resp. r-polynomial) if and only if Γ is polynomial (resp. polynomial of degree k, resp. r-polynomial).

Proof. Since cross effects behave well with respect to coequalizers we see that (9.5) is a consequence of (7.10) and (8.4). q.e.d.

(9.6) Theorem. Let T be a graded theory which is polynomial and let Γ = (Γ^{1}, …) be the associated iterated Grothendieck construction. Assume that all associated abelian categories A_{n} have finite global dimension d_{n} < ∞ for n ≥ 0. Then the derived functors L_{i}Γ^{n+1}, n ≥ 0, i ≥ 0, have a vanishing line N_{n+1} < ∞ with

L_{i}Γ^{n+1} = 0 for i > N_{n+1}

where N_{n+1} can be computed as in (9.3).

Proof. This is a consequence of (9.5), (9.3) and (5.3). q.e.d.

(9.7) Corollary. Let T be a theory as in (9.6). If T is linear then

L_{i}Γ^{n+1} = 0 for i > d_{0} + ∑_{k=1}^{n}(d_{k} + 1)

(9.8) Corollary. Let T be a theory as in (9.6). If T is r-polynomial with r > d = Max{d_{0}, d_{1}, …, d_{n}}. Then

L_{i}Γ^{n+1} = 0 for i > d + n(d + 1).

Here (9.7) corresponds to (5.4) and (9.8) corresponds to (5.6).

§ 10 Functors on categories of models

Let T be a graded theory and let model(T) be the category of models of T. We here consider functors

(10.1)  F : model(T) → B

where B is an abelian category. The derived functors L_{i}F are defined for i ≥ 0 by

(L_{i}F)(M) = π_{i}(F M_{·})

where M_{·} is a resolution of M; such resolutions exist by (3.4) and (8.5).

(10.2) Definition. For each model M of T and k ≥ 0 we define a new model M^{k} and a surjective map

(1)  q : M → M^{k} in model(T)

by setting M^{k}(S) = M(S) for | S |≤ k and M^{k}(S) = 0 for | S |> k. Then q is the identity on M(S) for | S |≤ k and is trivial otherwise. For Y ∈ T we have

(2)  M^{k}(Y) = M(Y^{k})

where Y^{k} is the k-skeleton of Y. Hence M^{k} is also defined by the composite

(3)  M^{k} : T ⟶^{j} T_{k} ⟶^{i} T ⟶^{M} Set

Here j carries Y to Y^{k} and i is the inclusion functor. Hence M^{k} is a well defined model of T. We say that the functor F in (10.1) has height(F) ≤ k if for all M ∈ model(T) the induced map

(4)  q_{∗} : F (M) → F (M^{k})

is an isomorphism.

(10.3) Lemma. A functor F has height(F) ≤ k if and only if F is naturally isomorphic to a composite functor

model(T) ⟶^{i^{∗}} model(T_{k}) ⟶^{G} B

Proof. If F is given then we have G = Γj^{∗} with j defined as in (10.2) (3). q.e.d.

(10.4) Theorem. Let T be a graded theory which is polynomial and for which the associated abelian categories A_{i} have global dimension d_{i} < ∞ for i ≥ 0. Let F be a functor as in (10.1) which preserves zero objects and which carries surjective maps to epimorphisms. Moreover assume that F is polynomial and of finite height. Then there exists N < ∞ with L_{i}F = 0 for i > N.

(10.5) Corollary. Assume T and F have the properties in (10.4). If T is linear and if F is linear with height (F) = k. Then

L_{i}F = 0 for i > d_{0} + ∑_{i=1}^{k}(d_{i} + 1)

For M ∈ model(T) we write | M |≥ m if M(S) = 0 for S ∈ S with | S |< m. The functor F in (10.1) with height(F) = k < ∞ is called r-polynomial (r ≥ 1) if for q ≥ 2 and any set of objects M_{1}, …, M_{q} ∈ model(T) with

∑_{i=1}^{q}(r+ | M_{i} |) > r + k + 1

we have F (M_{1} | … | M_{q}) = 0. Since r ≥ 1 this implies that F is polynomial.

(10.6) Corollary. Assume T and F have the properties in (10.4). Assume T is r-polynomial and assume F has finite height(F) = k < ∞ and F is r-polynomial. If r > d = Max{d_{0}, …, d_{k}} then

L_{i}F = 0 for i > d + k(d + 1)

Proof of (10.4). Let Γ = (Γ^{1}, …) be the iterated Grothendieck construction associated to model(T). Then the assumptions in (10.4) imply that

Γ_{G} = (Γ^{1}, Γ^{2}, …, Γ^{k}, G)

is an iterated Grothendieck construction with G given by F as in (10.3). Here we use (8.6). Moreover Γ_{G} satisfies assumptions as in (5.3). This shows that (10.4) is a consequence of the generalized form of (5.3). Similarly (10.5) and (10.6) are consequences of the generalized form of (5.4) and (5.6) respectively. q.e.d.

(10.7) Remark. Theorem (10.4) and corollary (10.6) yield the correct list of assumptions needed for the result 3.1 of Blanc [4].

References

[1] H.-J. Baues, Combinatorial foundation of homology and homotopy, MPI für Math. Bonn (1997) 250 pages.

[2] H.-J. Baues, Algebraic homotopy, Cambridge studies in advanced math. 15, Cambridge University Press 1988, 466 pages.

[3] D. Blanc, A Hurewicz spectral sequence for homology, Ph.D. thesis, M.I.T., 1988, Trans. Amer. Math. Soc. 318 (1990) 335-354.

[4] D. Blanc, Derived functors of graded algebra, J. Pure and Appl. Alg. 64 (1990) 239-262.

[5] F. Borceaux, Handbook of categorical algebra 2, Cambridge University Press (1994) 443 pages.

[6] A.K. Bousfield and D.M. Kan, The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 (1972) 79-106.

[7] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics 304 (Springer, Berlin, 1972).

[8] A. Dold and D. Puppe, Homologie nicht-additiver Funktoren, Ann. Inst. Fourier 11 (1961) 201-312.

[9] W.G. Dwyer abd D.M. Kan, The enveloping ring of a π-algebra, Advances in Homotopy Theory. London Math. Soc. Lecture Note 139 (1989) 49-60.

[10] W.G. Dwyer and D.M. Kan, Homology and cohomology of π-algebras, Trans. Amer. Math. Soc. 342 (1994) 257-273.

[11] S. Eilenberg and S. Mac Lane, On the groups H(π, n), I, Ann. of Math. 58 (1953) 55-106.

[12] P.J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955) 154-172.

[13] F.W. Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. USA 50 (1963) 869-872.

[14] T. Pirashvili, Simplicial degrees of functors, Preprint (1997) SFB Bielefeld.

[15] D.G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 20, (Springer, Berlin, 1963).

[16] D.Q. Quillen, On the (co-)homology of commutative rings, Proc. Sympos. Pure Math. 17 (1970) 65-87.

[17] C.R. Stover, A van Kampen spectral sequence for higher homotopy groups, Ph.D. thesis, M.I.T., 1988 Topology 29 (1990) 9-26.

[18] H.J. Baues and P. Goerss, A homotopy operation spectral sequence for the computation of homotopy groups, Preprint (1997).