An introduction to the elementary theory of Heller triangulated categories

Matthias Künzer

January 24, 2013

Contents

1 Why derived categories?  2
1.1 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2
1.2 Derived functors, renovated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2
2 Why triangulated categories?  3
2.1 Verdier triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3
2.2 Exact functors between Verdier triangulated categories . . . . . . . . . . . . . . . . . . . . . . . .  5
2.3 Stability properties of the Verdier formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5
3 Heller triangulated categories  6
3.1 Heller’s original theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6
3.2 Extending Heller’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6
3.3 Heller triangulated categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8
3.4 n-triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9
3.5 Retrieving the Verdier context in the Heller context . . . . . . . . . . . . . . . . . . . . . . . . .  9
3.6 Exact functors between Heller triangulated categories . . . . . . . . . . . . . . . . . . . . . . . .  9
3.7 Stability properties of the Heller formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.8 Advantages of ϑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.9 An amusing observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Remarks on spectral sequences  12
4.1 Four indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.1 Grothendieck spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.2 First comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.3 Second comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1 Why derived categories?

1.1 Derived functors

Around 1960, Grothendieck struggled with increasingly complicated spectral sequence comparisons (^{1}). These spectral sequences arose as follows.

Given left exact functors A ^{\overline{}^{F}} B ^{\overline{}^{G}} C between abelian categories (^{2}), under normal circumstances (^{3}), the derived functor R^{n}(F G) of the composite F G can be approximated by the composites of the derived functors (R^{i}F)(R^{j}G), where i + j = n. In other words, we have the Grothendieck spectral sequence, which has E_{2}-terms (R^{i}F)(R^{j}G) and converges to R^{n}(F G).

For instance, roughly put, the first derivative

R^{1}(F G)

consists of

a part of (R^{1}F)(R^{0}G) plus a part of (R^{0}F)(R^{1}G) ;

which can be read as an approximative “product rule for the first derivative” (^{4}).

In practice, this is troublesome since it only yields an approximative relationship between

the derived functors of the composite  and  the composites of the derived functors ,

and since, moreover, this approximation is laborious.

Finally, if we want to compose three or more functors and relate their various derivatives, we are stuck.

1.2 Derived functors, renovated

The construction of such a derived functor R^{i}F proceeds in three steps.

(1) Resolve injectively.

(2) Apply the functor F.

(3) Take cohomology H^{i}.

Grothendieck saw that the troubles were caused by the third step and that dropping the third step, one should get a smooth formalism, in which the spectral sequence approximation mentioned above is turned into the simple and precise rule

(∗)  R(F G) ≃ (RF)(RG) .

1An example of an assertion of this kind may be seen in [7, 6.6.2]. According to Illusie, Grothendieck said : “The second part of EGA III is a mess, so, please, clean this up by introducing derived categories, write the Künneth formula in the general framework of derived categories.” [11, p. 1108].

2An abelian category has direct sums, kernels, cokernels, and the homomorphism theorem holds.

3The categories A and B are supposed to have enough injectives, and F to map injectives to G-acyclics.

4More precisely, R^{1}(F G) has a two-step filtration, one subfactor of which being a subfactor of (R^{1}F)(R^{0}G), the other being a subfactor of (R^{0}F)(R^{1}G) .

The price to pay was the development of this formalism, undertaken by Verdier around 1963 in [23].

Since we have dropped taking cohomology H^{i}, the renovated derived functor RF now takes values in complexes (over B). So in order to be able to compose, RF should also take as arguments complexes (over A).

Moreover, in order to ensure the validity of the composition rule (∗), one has to formally invert morphisms of complexes that induce isomorphisms in cohomology, called quasi-isomorphisms. This process yields the derived category

D^{+}(A) ,

having as objects complexes (^{5}) over A, and as morphisms fractions

f/s ,

where the numerator f is a morphism of complexes and where the denominator s is a quasiisomorphism of complexes.

So in full, the formula (∗) reads

(∗′)  (D^{+}(A) ^{^{R(F G)} \overline{}}D^{+}(C)) ≃ (D^{+}(A) ^{\overline{}^{RF}} D^{+}(B) ^{\overline{}^{RG}} D^{+}(C)) .

2 Why triangulated categories?

2.1 Verdier triangulated categories

The category A is abelian.

The derived category D^{+}(A) is not abelian (^{6}).

There exist hardly any short exact sequences in D^{+}(A), only split ones.

As substitute, the image in D^{+}(A) of a short exact sequence of complexes X′ [figure] X \overline{}X′′ fits into a diagram

X′ \overline{}X \overline{}X′′ \overline{}X^{′+1} ,

called a distinguished triangle, where X^{′+1} denotes the complex X′, shifted one step to the left (^{7}).

\xymatrix{ & X′′ \ar@{.>}[dl] \\ X′ \ar[rr] &  & X \ar@{-->}[ul]}

5Bounded to the left.

6Except if A is semisimple.

7And all differentials negated.

Now any morphism X_{1/0} \overline{}X_{2/0} in D^{+}(A) fits into such a distinguished triangle X_{1/0} \overline{}X_{2/0} \overline{}X_{2/1} \overline{}X_{1/0}^{+1} , and this completion is unique up to isomorphism (^{8}). We call X_{2/1} the cone of the morphism X_{1/0} \overline{}X_{2/0} (^{9}).

The compatibility of taking cones with composition is expressed by the following Verdier octahedron (^{10}), in which X_{0^{+1}/i} = X_{i/0}^{+1} for 1 ⩽ i ⩽ 3.

\xymatrix{ &  &  &  & 0 \\  &  &  & 0 \ar[r] & X_{0^{+1}/3} \ar[u] \\  &  & 0 \ar[r] & X_{3/2} \ar[r] \ar[u] & X_{0^{+1}/2} \ar[u] \\  & 0 \ar[r] & X_{2/1} \ar[r] \ar[u] & X_{3/1} \ar[r] \ar[u] & X_{0^{+1}/1} \ar[u] \\ 0 \ar[r] & X_{1/0} \ar[r] \ar[u] & X_{2/0} \ar[r] \ar[u] & X_{3/0} \ar[r] \ar[u] & 0 \ar[u]}

Here X_{j/i} is the cone on X_{i/0} \overline{}X_{j/0} for 1 ⩽ i < j ⩽ 3. Moreover, X_{3/2} is the cone on X_{2/1} \overline{}X_{3/1} .

[figure: X_{3/0}; X_{3/1}; X_{3/2}; X_{1/0}; X_{2/0}; X_{2/1}]

A theory of Verdier triangulated categories was developed by Verdier [23], which plays the same role for D^{+}(A) as the theory of abelian categories plays for A.

Here, a Verdier triangulated category is a triple (D, T, Ξ), consisting of an additive category D, an automorphism T = (−)^{+1} of D, called shift, and a set Ξ of distinguished triangles, satisfying a list of axioms, including the existence of a Verdier octahedron on each pair of composable morphisms. Then, Ξ is called a Verdier triangulation on (D, T).

8In contrast to what we are used to from kernels and cokernels in abelian categories, this isomorphism is not uniquely determined in general.

9This notion is motivated by the homotopy category of CW-complexes, which becomes a Verdier triangulated category after Spanier-Whitehead stabilisation, where this cone is an actual geometrically constructed cone.

10This alternative, non-octahedral form of this diagram was observed in [3, 1.1.14].

For example, the derived category D^{+}(A) is Verdier triangulated. Also already the homotopy category K^{+}(A), obtained as the category of complexes (^{11}) modulo split acyclic complexes, is Verdier triangulated. More generally, the stable category of a Frobenius category (^{12}) is Verdier triangulated; cf. [9, Th. 2.6].

One of the axioms records a curious phenomenon, without parallel in the context of abelian categories. For every distinguished triangle

X_{1/0} \overline{}X_{2/0} \overline{}X_{2/1} \overline{}X_{1/0}^{+1} ,

we get the rotated distinguished triangle

X_{2/1}^{−1} \overline{}X_{1/0} \overline{}X_{2/0} \overline{}X_{2/1} (^{13}) .

In a Verdier triangulated category, the cone of a morphism is at the same time a substitute for its kernel and its cokernel, but only in a weak form (^{14}). Therefore, a Verdier triangulated category is weakly abelian, i.e. it is an additive category in which each morphism is and has a weak kernel and a weak cokernel.

A set Ξ of distinguished triangles that satisfies the list of axioms except possibly for the existence of a Verdier octahedron on each diagram X_{1/0} \overline{}X_{2/0} \overline{}X_{3/0} , is called a Puppe triangulation on (D, T) [21]. Cf. §3.1 below.

2.2 Exact functors between Verdier triangulated categories

A strictly exact functor between Verdier triangulated categories is a shiftcompatible additive functor that maps distinguished triangles to distinguished triangles; i.e. that preserves cones.

Derived functors, such as the functor D^{+}(A) ^{\overline{}^{RF}} D^{+}(B) from §1.2, are strictly exact.

An exact functor between Verdier triangulated categories (D, T, Ξ) and (D′, T′, Ξ′) is a pair (V, a) consisting of an additive functor V : D \overline{}D′ and an isotransformation a : T V \overline{}V T′ such that each distinghuished triangle in D, mapped via V and isomorphically replaced via a, yields a distinguished triangle of D′.

So V : D \overline{}D′ is strictly exact if and only if (V, 1) is exact.

2.3 Stability properties of the Verdier formalism

Adjoints of exact functors are exact [20, App. 2, Prop. 11] [12, 1.6]. Already Grothendieck and Deligne observed in Algebraic Geometry the appearance of exact functors that are not derived functors, but adjoints to derived functors (^{15}).

11Bounded to the left.

12A Frobenius category is an exact category with a sufficient supply of relatively bijective objects.

13With a sign inserted.

14In the notation above, X_{2/0} \overline{}X_{2/1} is a weak cokernel of X_{1/0} \overline{}X_{2/0} , i.e. it satisfies the universal property of a cokernel, except for uniqueness of the induced morphism. Moreover, X_{2/1}^{−1} \overline{}X_{1/0} is a weak kernel of X_{1/0} \overline{}X_{2/0} , i.e. satisfies the universal property of a kernel, except for uniqueness of the induced morphism.

The Karoubi hull of an additive category is the universal additive category whose idempotents split [13, III.II]. We may form Karoubi hulls within the context of Verdier triangulated categories and exact functors, as shown by Balmer and Schlichting [1].

The localisation of a category at a subset of its morphisms is the universal category such that morphisms of this subset become invertible. For instance, the derived category D^{+}(A) is the localisation of the homotopy category K^{+}(A) at the subset of quasiisomorphisms, i.e. at the subset of morphisms with acyclic cone. We may form the localisation of a Verdier triangulated category at the subset of morphisms with cone in a given thick subcategory (^{16}) within the realm of Verdier triangulated categories and exact functors [23][22, Prop. 1.3].

3 Heller triangulated categories

3.1 Heller’s original theorem

Let D be a weakly abelian category; cf. §2.1. The Freyd category D̂ is the universal abelian category containing D [6]. Reducing modulo the full additive subcategory of projective objects, we obtain the stable category \underline{D̂}, which is Verdier triangulated (^{17}).

Now suppose D to carry a shift functor T. Then \underline{D̂} carries two shift functors, a first one induced by T, a second one given by the Verdier triangulated structure on \underline{D̂}.

Heller discovered a bijection between the set of Puppe triangulations Ξ on (D, T) and the set of isomorphisms from the first shift functor to the third power of the second shift functor satisfying an extra condition (^{18}) [10, Th. 16.4].

So such an isomorphism between these shift functors can be made responsable for a Puppe triangulation, as the extra datum needed to upgrade a weakly abelian category with shift (D, T) to a Puppe triangulated category (D, T, Ξ).

So we could just as well include this isomorphism instead of Ξ in our data.

3.2 Extending Heller’s theorem

Let D be a weakly abelian category. Let T be an automorphism of D.

In order to extend Heller’s result from Puppe triangulations to Verdier triangulations and beyond, all we need is a suitable replacement for \underline{D̂}.

15The functor Rf^{!}, constructed for certain morphisms f of schemes, is only abusively written with a “R”; cf.

[8, Exp. XVIII, Th. 3.1.4].

16A thick subcategory is a full subcategory closed under shift, forming cones and taking summands.

17The reason being that D is a big enough full subcategory in D̂ consisting of bijective objects, so that D̂ is a Frobenius abelian category. Cf. [9, §2.1].

18Such an isomorphism can be pre- and postcomposed with the Verdier shift on \underline{D̂}; the condition is that the result of precomposition is the negative of the result of postcomposition.

A weak square in D is a commutative quadrangle that is at the same time a weak pullback and a weak pushout (^{19}). A weak square is marked as

[figure: +; .]

Alternatively, a commutative quadrangle is a weak square if and only if its diagonal sequence is exact in the middle when viewed in D̂.

Let D^{+}(Δ̄_{2}^{#}) be the category of diagrams in D of the form

[figure: 0; ⋯; 0; X_{0^{+1}/2}; +; ⋯; 0; X_{2/1}; +; X_{0^{+1}/1}; +; ⋯; 0; X_{1/0}; +; X_{2/0}; +; 0; 0; X_{0/2^{−1}}; +; X_{1/2^{−1}}; +; 0; ⋮; +; ⋮; +; ⋮; ,]

where we do not require any relation between X_{0^{+1}/2} and X_{2/0}^{+1} etc. Viewed in the abelian category D̂, this is just the category of acyclic complexes consisting of objects in D.

Let D^{+}(Δ̄_{3}^{#}) be the category of diagrams in D of the form

[figure: 0; ⋯; 0; X_{0^{+1}/3}; +; ⋯; 0; X_{3/2}; +; X_{0^{+1}/2}; +; ⋯; 0; X_{2/1}; +; X_{3/1}; +; X_{0^{+1}/1}; +; ⋯; 0; X_{1/0}; +; X_{2/0}; +; X_{3/0}; +; 0; 0; X_{0/3^{−1}}; +; X_{1/3^{−1}}; +; X_{2/3^{−1}}; +; 0; ⋮; +; ⋮; +; ⋮; +; ⋮; ,]

19The respective universal property is supposed to hold, except for the uniqueness of the induced morphism.

where we do not require any relation between X_{0^{+1}/3} and X_{3/0}^{+1} etc.

Etc.

For n ⩾ 0, we let

\underline{D^{+}(Δ̄_{n}^{#})}

be the reduction of D^{+}(Δ̄_{n}^{#}) modulo the full additive subcategory of diagrams all of whose morphisms split. This category carries two shift functors, the outer shift [−]^{+1} and the inner shift [−^{+1}], characterised by, respectively,

\begin{aligned}([X]^{+1})_{β/α} & = & X_{α^{+1}/β} \\ ([X^{+1}])_{β/α} & = & (X_{β/α})^{+1} \end{aligned}

for X ∈ Ob \underline{D^{+}(Δ̄_{n}^{#})} = Ob D^{+}(Δ̄_{n}^{#}). In other words, the outer shift pulls the whole diagram down left, the inner shift applies T pointwise.

Then

\underline{D^{+}(Δ̄_{2}^{#})}  ≃  \underline{D̂} ,

where

the outer shift  corresponds to  the third power of the Verdier shift ,
the inner shift  corresponds to  the functor induced by T .

So we can transport an isomorphism as in Heller’s theorem from §3.1 to an isomorphism

[−]^{+1} \overline{}^{ϑ_{2}}_{∼} [−^{+1}]

from the outer to the inner shift functor on \underline{D^{+}(Δ̄_{2}^{#})}.

Using \underline{D^{+}(Δ̄_{2}^{#})} as a replacement for \underline{D̂} will enable us, in §3.3 below, to extend from \underline{D^{+}(Δ̄_{2}^{#})} to \underline{D^{+}(Δ̄_{n}^{#})} for n ⩾ 0, so as to include octahedra and bigger diagrams [3, 1.1.14], and to drop the extra condition on the isomorphism mentioned in §3.1.

3.3 Heller triangulated categories

Let D be a weakly abelian category. Let T be an automorphism of D.

Let a Heller triangulation on (D, T) be a tuple ϑ = (ϑ_{n})_{n⩾0} of isomorphisms ϑ_{n} : [−]^{+1} \overline{}[−^{+1}] from the outer shift [−]^{+1} to the inner shift [−^{+1}] on \underline{D^{+}(Δ̄_{n}^{#})} satisfying compatibilities with quasicyclic operations (^{20}) and with folding (^{21}).

20Deleting and doubling rows and columns in a periodic manner yield functors \underline{D^{+}(Δ̄_{n}^{#})} ^{\overline{}^{\underline{p}^{#}}} \underline{D^{+}(Δ̄_{m}^{#})}. We require that Xϑ_{n}\underline{p}^{#} = X\underline{p}^{#}ϑ_{m} for X ∈ Ob \underline{D^{+}(Δ̄_{n}^{#})} = Ob D^{+}(Δ̄_{n}^{#}).

21Suppose given n ⩾ 0 and X ∈ Ob \underline{D^{+}(Δ̄_{2n+1}^{#})} = Ob D^{+}(Δ̄_{2n+1}^{#}). We can canonically (^{22}) construct an object X\underline{𝔣}_{n} ∈ Ob \underline{D^{+}(Δ̄_{n+1}^{#})} = Ob D^{+}(Δ̄_{n+1}^{#}) that has (X\underline{𝔣}_{n})_{i/0} = X_{n+i/i−1} for 1 ⩽ i ⩽ n + 1 ; the diagram X\underline{𝔣}_{n} involves direct sums of objects occurring in X. The operation \underline{𝔣}_{n} can be turned into a functor from \underline{D^{+}(Δ̄_{2n+1}^{#})} to \underline{D^{+}(Δ̄_{n+1}^{#})}. We require that Xϑ_{2n+1}\underline{𝔣}_{n} = X\underline{𝔣}_{n}ϑ_{n+1} . Cf. [3, 1.1.13].

22Up to sign.

A Heller triangulated category then is a triple (D, T, ϑ) consisting of a weakly abelian category D, an automorphism T of D and a Heller triangulation ϑ on (D, T) [15, Def. 1.5.(ii.1)]. Often, we write just D := (D, T, ϑ).

For example, the derived category D^{+}(A) is Heller triangulated. Also the homotopy category K^{+}(A) is Heller triangulated. More generally, the stable category of a Frobenius category is Heller triangulated. Cf. [15, Cor. 4.7][18, Prop. 36] (^{23}).

3.4 n-triangles

Suppose given a Heller triangulated category (D, T, ϑ). Suppose given n ⩾ 0.

The base of a diagram X ∈ Ob D^{+}(Δ̄_{n}^{#}) is its subdiagram

\xymatrix{(X_{1/0} \ar[r] & X_{2/0} \ar[r] & ⋯ \ar[r] & X_{n−1/0} \ar[r] & X_{n/0}) & ∈ Ob D(Δ̇_{n})}

on the linearly ordered set Δ̇_{n} := {1, 2, …, n}.

A diagram X ∈ Ob D^{+}(Δ̄_{n}^{#}) is called an n-triangle if Xϑ_{n} = 1. A morphism X ^{\overline{}^{f}} Y between n-triangles X and Y is called periodic if [f]^{+1} = [f^{+1}].

The restriction functor to the base, mapping from the category of n-triangles and periodic morphisms to D(Δ̇_{n}), is full [15, Lem. 3.2]. If all idempotents split in D, then it is also surjective on objects [15, Lem. 3.1] (^{24}).

Such triangles are stable under quasicyclic operations and under folding [15, Lem. 3.4.(1, 2)].

3.5 Retrieving the Verdier context in the Heller context

Suppose given a Heller triangulated category (D, T, ϑ) in which all idempotents split.

Let Ξ be the set of 2-triangles in D. Then the triple (D, T, Ξ) is a Verdier triangulated category [15, Prop. 3.6] (^{25}).

Each 3-triangle is a Verdier octahedron; cf. §2.1. However, not every Verdier octahedron is a 3-triangle [17, Lem. 6] (^{26}).

{distinguished triangles} = {2-triangles}
{Verdier octahedra} ⊇ {3-triangles}

3.6 Exact functors between Heller triangulated categories

Suppose given Heller triangulated categories (D, T, ϑ) and (D′, T′, ϑ′).

23Cf. also [17, Prop. 22.(1).]

24More generally, this holds if D is a closed Heller triangulated category [18, Lem. 20]. Cf. also [15, Rem. 3.3].

25More generally, this holds if D is a closed Heller triangulated category [18, Rem. 18].

26Not even when requiring that it contains the triangles described in [3, 1.1.13]; cf. [17, Rem. 7].

A strictly exact functor from D to D′ is a shiftcompatible additive functor V : D \overline{}D′ that respects weak squares, and that satisfies

Xϑ_{n}\underline{V^{+}(Δ̄_{n}^{#})} = X\underline{V^{+}(Δ̄_{n}^{#})}ϑ′_{n}

for all n ⩾ 0 and all X ∈ Ob \underline{D^{+}(Δ̄_{n}^{#})} = Ob D^{+}(Δ̄_{n}^{#}), where \underline{V^{+}(Δ̄_{n}^{#})} acts by pointwise application of V.

An exact functor from D to D′ is a pair (V, a) consisting of an additive functor V : D \overline{}D′ respecting weak squares, and an isotransformation a : T V \overline{}V T′ such that

Xϑ_{n}\underline{V^{+}(Δ̄_{n}^{#})} · X\underline{a^{+}(Δ̄_{n}^{#})} = X\underline{V^{+}(Δ̄_{n}^{#})}ϑ′_{n}

for all n ⩾ 0 and all X ∈ Ob \underline{D^{+}(Δ̄_{n}^{#})} = Ob D^{+}(Δ̄_{n}^{#}).

So V : D \overline{}D′ is strictly exact if and only if (V, 1) is exact.

3.7 Stability properties of the Heller formalism

Adjoints of exact functors are exact [18, Prop. 28].

We may form the Karoubi hull within the context of Heller triangulated categories and exact functors [18, Prop. 12].

We may form the localisation at the subset of morphisms with cone in a given thick subcategory within the context of Heller triangulated categories and exact functors [18, Prop. 38].

The derived functor D^{+}(A) ^{\overline{}^{RF}} D^{+}(B) from §1.2 is exact, using that A is supposed to have enough injectives (^{27}).

It is also possible to characterise exactness of a functor, in a manner similar to §2.2, by preservation of n-triangles [18, Prop. 25]. The reason behind that possibility is that closed (^{28}) Heller triangulated categories can, alternatively, be defined via sets of n-triangles for n ⩾ 0 with suitable preservation properties with respect to quasicyclic operations and folding, as Thomas informed me.

3.8 Advantages of ϑ

Having n-triangles at our disposal allows constructions that have not been possible within the Verdier context. For instance, given two 3-triangles, a morphism between the bases can be prolonged to a morphism between the 3-triangles. This is no longer true, in general, once we replace “3-triangles” by “Verdier octahedra” [17, Lem. 6].

But why should we work primarily with ϑ, and only secondarily with n-triangles? A possible answer is that usage of ϑ allows low-effort proofs of the stability properties of the Heller formalism explained in §3.7; cf. [18, §2.2, §6, §5.2].

27Somewhat provisionally still, we may use [18, Prop. 28, Prop. 36], [17, Cor. 21], [15, Cor. 4.9] to arrive there. It would be preferable to use the derived functor construction via ind-categories along the lines of [8, Exp.

XVII, §1.2].

28A Heller triangulated category is called closed if it is closed under taking cones in its Karoubi hull.

Of course, the price to pay is to get accustomed to the administration of the n-triangles being done by a tuple of isomorphisms ϑ.

3.9 An amusing observation

Suppose given Heller triangulated category (D, T, ϑ) in which all idempotents split (^{29}).

A commutative quadrangle

\xymatrix{X′ \ar[r]^{f′} & Y′ \\ X \ar[r]_{f} \ar[u]^{x} & Y \ar[u]_{y}}

in D is called a dweak square (^{30}) if its diagonal sequence

\xymatrix{X \ar[r]^{^{(^{f x})}} & Y ⊕ X′ \ar[r]^{^{(\begin{matrix}y \\ −f′ \end{matrix} )}} & Y′}

appears as part of a 2-triangle. So a dweak square is in particular a weak square; cf. §3.2.

Alternatively, a commutative quadrangle is a dweak square if and only if it appears in some n-triangle for some n ⩾ 0.

Any corner [figure]can be completed to a dweak square. This completion is unique up to non-unique isomorphism. Accordingly in the dual situation.

Suppose given n ⩾ 1. Consider the set Chain_{n} of isoclasses of diagrams of the form X_{1} \overline{}X_{2} \overline{}⋯ \overline{}X_{n−1} \overline{}X_{n} in D, i.e. the set of isoclasses in D(Δ̇_{n}). We obtain two bijections

σ, τ : Chain_{n} ^{\overline{}^{∼}} Chain_{n}

as follows.

Let τ map the isoclass of X_{1} \overline{}X_{2} \overline{}⋯ \overline{}X_{n−1} \overline{}X_{n} to the isoclass of X_{1}^{+1} \overline{}X_{2}^{+1} \overline{}⋯ \overline{}X_{n−1}^{+1} \overline{}X_{n}^{+1} .

Let σ be defined as follows. Suppose given X_{1} \overline{}X_{2} \overline{}⋯ \overline{}X_{n−1} \overline{}X_{n}. Prolong this diagram by X_{n} \overline{}0. Complete to dweak squares along X_{1} \overline{}0, yielding a new row 0 \overline{}X′_{2} \overline{}⋯ \overline{}X′_{n−1} \overline{}X′_{n} \overline{}W_{1} . Complete to dweak squares along X′_{2} \overline{}0, yielding a new row 0 \overline{}X′′_{3} \overline{}⋯ \overline{}X′′_{n−1} \overline{}X′′_{n} \overline{}W_{2} . Etc. Then let σ map the isoclass of X_{1} \overline{}X_{2} \overline{}⋯ \overline{}X_{n−1} \overline{}X_{n} to the isoclass of W_{1} \overline{}W_{2} \overline{}⋯ \overline{}W_{n−1} \overline{}W_{n}.

Elementary properties of n-triangles force σ = τ.

If we only require D to be Verdier triangulated, both σ and τ are still definable, but it is unclear to me whether they coincide (^{31}).

29More generally, the following holds if D is a closed Heller triangulated category.

30An abbreviation for “distinguished weak square”. Also known as homotopy cartesian square, as homotopy bicartesian square, or as Mayer-Vietoris square.

31Suppose that [3, 1.1.13] holds in our Verdier triangulated category. Then σ and τ coincide if n ∈ {1, 2, 3}.

4 Remarks on spectral sequences

4.1 Four indices

Suppose given an abelian category A. Suppose given a filtered complex M with values in A, i.e. a chain of monomorphisms

M(−∞) [figure] ⋯ [figure] M(i) [figure] M(i + 1) [figure] ⋯ [figure] M(+∞),

indexed by {−∞} ⊔ Z ⊔ {+∞}, that satisfies certain technical conditions (^{32}).

We shall use the linearly ordered set

Z̄_{∞} := { i^{+k} : i ∈ {−∞} ⊔ Z ⊔ {+∞}, k ∈ Z } ,

where formally i^{+k} is defined as the pair (i, k), and where i^{+k} ⩽ j^{+ℓ} if k < ℓ or (k = ℓ and i ⩽ j).

Taking M as a base, we can form a diagram that is, morally, an ∞-triangle. It consists of shifted subfactor complexes M(β/α) for β^{−1} ⩽ α ⩽ β ⩽ α^{+1} in Z̄_{∞} and is called spectral object Sp(M) of M (^{33}). For γ/α ⩽ δ/β, i.e. γ ⩽ δ and α ⩽ β, the induced morphism

M(γ/α) \overline{}M(δ/β)

appears in this diagram Sp(M).

Let ME(δ/β//γ/α) ∈ Ob A be defined as the image of H^{0} of this morphism, i.e.

M(γ/α)H^{0} \overline{}ME(δ/β//γ/α) [figure] M(δ/β)H^{0} (^{34}) .

These objects ME(δ/β//γ/α) assemble to a big diagram with values in A, the spectral sequence

ME

of M (^{35}).

Suppose given ε^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ ε ⩽ α^{+1} in Z̄_{∞}. We obtain the short exact sequence

ME(ε/β//γ/α) [figure] ME(ε/β//δ/α) \overline{}ME(ε/γ//δ/α) ,

which can be made responsible for all exact sequences in general spectral sequences known to me. Cf. [15, Lem. 3.9], generalising a particular case of [24, §II.4.2.6].

Dropping certain “initial terms” (^{36}) from the spectral sequence ME, we obtain the proper spectral sequence

MĖ

of M.

32Viz. M(−∞) = 0, M(i) \overline{}M(i + 1) being pointwise split and the whole filtration being pointwise almost everywhere constant. Cf. [16, §3.1].

33This term has been coined by Verdier; cf. [24, §II.4].

34This definition slightly generalises the definition given in [5, App.]. The original definition in [24, §II.4.2.3] was closer to classical terminology, as found in [4, §XV.1].

35The classical spectral sequence terms are amongst the terms ME(δ/β//γ/α) ; cf. [16, §3.5].

36E_{1}-terms and similar ones; cf. [16, §3.6].

4.2 Comparisons

4.2.1 Grothendieck spectral sequences

Maintain the situation of §1.1. So A ^{\overline{}^{F}} B ^{\overline{}^{G}} C.

Suppose given X ∈ Ob A. Resolve X injectively. This yields a complex with values in A. Apply F pointwise. This yields a complex with values in B. Resolve this complex injectively, via the method of Cartan-Eilenberg [4, §XVII.1]. This yields a double complex with values in B. Apply G pointwise. This yields a double complex with values in C.

The total complex of this resulting double complex is obtained by forming direct sums over its diagonals. Replacing an increasing number of rows in this double complex by zero rows, and then taking the total complex, we obtain a descending chain of subcomplexes filtering our original total complex.

This filtered complex gives rise to the Grothendieck spectral sequence XE_{F,G}^{Gr} via the method of §4.1. This yields a functor E_{F,G}^{Gr} on A, mapping to the category of spectral sequences with values in C.

So we had to “resolve X twice”, with an intermediate application of F, and a final application of G, to carry out this construction.

4.2.2 First comparison

Suppose given abelian categories A, A′, B, C (^{37}). Suppose given objects X ∈ Ob A and X′ ∈ Ob A′. Let A × A′ ^{\overline{}^{F}} B be a biadditive functor such that (X, −)F and (−, X′)F are left exact. Let B ^{\overline{}^{G}} C be a left exact functor. Suppose further conditions to hold; see [16, §5.1].

[figure: X; X′; A × A′; F; B; G; C]

We have Grothendieck spectral sequence functors,

E_{(X,−)F, G}^{Gr} for A′ ^{^{(X,−)F}\overline{}}B ^{\overline{}^{G}} C ,
E_{(−,X′)F, G}^{Gr} for A ^{\overline{}^{(−,X′)F}} B ^{\overline{}^{G}} C .

We evaluate the former at X′ and the latter at X. Then the proper Grothendieck spectral sequences are isomorphic, i.e.

X′ Ė_{(X,−)F, G}^{Gr} ≃ X Ė_{(−,X′)F, G}^{Gr} ;

cf. [16, Th. 31]. So instead of “resolving X′ twice”, we may just as well “resolve X twice”.

37Of which A, A′ and B are supposed to have enough injectives.

4.2.3 Second comparison

Suppose given abelian categories A, B, B′, C (^{38}). Suppose given objects X ∈ Ob A and Y ∈ Ob B. Let A ^{\overline{}^{F}} B′ be a left exact functor. Let B × B′ ^{\overline{}^{G}} C be a biadditive functor such that (Y, −)G is left exact.

Let B ∈ Ob C^{[0}(B) be a resolution of Y such that (B^{k}, −)G is exact for all k ⩾ 0. Let A ∈ Ob C^{[0}(A) be an injective resolution of X. Suppose further conditions to hold; see [16, §6.1].

[figure: X; A; F; Y; B × B′; G; C]

We have the Grothendieck spectral sequence functor

E_{F, (Y,−)G}^{Gr} for A ^{^{F}\overline{}}B′  ^{^{(Y,−)G}\overline{}}C ,

which we evaluate at X.

On the other hand, we can consider the double complex (B, AF)G, where the indices of B count rows and the indices of A count columns. As described in §4.2.1, we can associate a spectral sequence to a double complex, in this case named E_{I}((B, AF)G).

Then the proper spectral sequences are isomorphic,

XĖ_{F,(Y,−)G}^{Gr} ≃ Ė_{I}((B, AF)G) .

So instead of “resolving X twice”, we may just as well “resolve X once and Y once”.

4.2.4 Applications

The comparisons in §4.2.2 and §4.2.3 may be used to reprove the following two theorems of Beyl.

The first theorem allows acyclic objects to be alternatively used to calculate Grothendieck spectral sequences [2, Th. 3.4]; cf. [16, Th. 40].

The second theorem allows the Hochschild-Serre-Hopf spectral sequence to be calculated with injective or, equivalently, with projective resolutions; the former fitting in the context of Grothendieck spectral sequences, the second being apt for manipulating concrete representing cocycles of cohomology classes; cf. [2, Th. 3.5], [16, Th. 52, 53].

Further applications can be found in [16, §8] (^{39}).

38Of which A and B′ are supposed to have enough injectives.

39If we were to reduce complexity in the assertions of §4.2, then, in the spirit of §1.2, we should directly work with suitably defined derived categories of double complexes; I do not know how to do that. We would probably get an additional shift functor.

References

[1] Balmer, P., Schlichting, M., Idempotent completion of triangulated categories, J. Alg. 236 (2), p. 819– 834, 2001.

[2] Beyl, R., The spectral sequence of a group extension, Bull. Sc. math., 2^{e} série, 105, p. 417–434, 1981.

[3] Bernstein, J.; Beilinson, A.A.; Deligne, P., Faisceaux pervers, Astérisque 100, 1982.

[4] Cartan, H., Eilenberg, S., Homological Algebra, Princeton, 1956.

[5] Deligne, P., Décompositions dans la catégorie dérivée, Proc. Symp. Pure Math. 55 (1), p. 115–127, 1994.

[6] Freyd, P., Stable Homotopy, Proc. Conf. Categorical Algebra, La Jolla, p. 121–172, Springer, 1965.

[7] Grothendieck, A. (redigé en coll. avec Dieudonné, J.), Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Pub. Math. IHÉS 17, 1963.

[8] Grothendieck, A. et al., Théorie des topos et cohomologie étale des schémas, SGA 4, SLN 269+270+305, 1972.

[9] Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, LMS LN 119, 1988.

[10] Heller, A., Stable homotopy categories, Bull. Am. Math. Soc. 74, p. 28–63, 1968.

[11] Illusie, L., Reminiscences of Grothendieck and His School, Notices AMS, Vol. 57 (9), October 2010.

[12] Keller, B., Vossieck, D., Sous les catégories dérivées, C. R. Acad. Sci. Paris, Série I, p. 225–228, 1987.

[13] Karoubi, M., K-théorie, Conférences au séminaire d’été de Montréal No. 36, Presses de Montr2éal, 1969.

[14] Künzer, M., On derived categories, diploma thesis, Stuttgart, 1996.

[15] Künzer, M., Heller triangulated categories, Homol. Homot. Appl. 9 (2), p. 233–320, 2007.

[16] Künzer, M., Comparison of spectral sequences involving bifunctors, Documenta Math. 13, p. 677–737, 2008.

[17] Künzer, M., Nonisomorphic Verdier octahedra on the same base, J. Homotopy and Related Structures 4 (1), p. 7–38, 2009.

[18] Künzer, M., On exact functors for Heller triangulated categories, preprint, arxiv 1301.0031, 2012.

[19] Maltsiniotis, G., Catégories triangulées supérieures, Pré-preprint, www.math.jussieu.fr/∼maltsin/textes.html, 2005.

[20] Margolis, H.R., Spectra and the Steenrod algebra, North-Holland, 1983.

[21] Puppe, D., On the formal structure of stable homotopy, Coll. on Algebraic Topology, Aarhus, p. 65–71, 1962.

[22] Rickard, J., Derived categories and stable equivalence, J. Pure Appl. Alg. 61, p. 303–317, 1989.

[23] Verdier, J.L., Catégories Derivées, published in SGA 4 1/2, SLN 569, p. 262–311, 1977 (written 1963).

[24] Verdier, J.L., Des catégories derivées des catégories abéliennes, éd. G. Maltsiniotis, Astérisque 239, 1996 (written 1967).

Matthias Künzer University of Stuttgart Fachbereich Mathematik

Pfaffenwaldring 57 D-70569 Stuttgart

kuenzer@mathematik.uni-stuttgart.de

w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/