Nonisomorphic Verdier octahedra on the same base

Matthias Künzer March 16, 2009

Abstract

We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle.

Contents

0 Introduction  2
0.1 Is being a 3-triangle characterised by 2-triangles? . . . . . . . . . . . . . . . . . . . . . . . . . . .  2
0.2 Is being an n-triangle characterised by (n − 1)-triangles? . . . . . . . . . . . . . . . . . . . . . . .  3
0.2.1 The example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3
0.2.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3
0.3 An appendix on transport of structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4
0.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4
0.5 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4
1 The classical stable category of (Z/p^{m})-mod  5
1.1 The category (Z/p^{m})-mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5
1.2 The shift on \underline{(Z/p^{m})-mod} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5
1.3 A Heller triangulation on \underline{(Z/p^{m})-mod} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6
1.4 A Verdier triangulation on \underline{(Z/p^{m})-mod} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7
2 Nonisomorphic periodic n-pretriangles  7
2.1 A (2n − 1)-triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7
2.2 An n-triangle and a periodic n-pretriangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8
3 Nonisomorphic Verdier octahedra  13
A Transport of structure  17
A.1 Transport of a Heller triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A.2 Detecting n-triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
A.3 Some lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A.3.1 Cleaning the diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A.3.2 Horseshoe lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A.3.3 Applying F̂ to a standard pure short exact sequence . . . . . . . . . . . . . . . . . . . . . 25
A.4 Stable vs. classically stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A.4.1 n-triangles in the stable category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A.4.2 The classical stable category under an additional hypothesis . . . . . . . . . . . . . . . . . 27

MSC2000: 18E30.

A.4.2.1 The hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.4.2.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.4.2.3 Standardisation by substitution of the rightmost column . . . . . . . . . . . . . 29
A.4.2.4 n-triangles in the classical stable category . . . . . . . . . . . . . . . . . . . . . . 30

0 Introduction

0.1 Is being a 3-triangle characterised by 2-triangles?

Verdier (implicitly) defined a Verdier octahedron to be a diagram in a triangulated category in the shape of an octahedron, four of whose triangles are distinguished, the four others commutative [5, Def. 1-1]; cf. also [1, 1.1.6]. It arises as follows.

To a morphism in a triangulated category, we can attach an object, called its cone. The morphism we start with and its cone are contained in a distinguished triangle. To the morphism we started with, we refer as the base of this distinguished triangle.

Now given a commutative triangle, we can form the cone on the first morphism, on the second morphism and on their composite, yielding three distinguished triangles. These three cones in turn are contained in a fourth distinguished triangle. The whole diagram obtained by this construction is a Verdier octahedron. We shall refer to the commutative triangle we started with as the base of this Verdier octahedron.

A distinguished triangle has the property of being determined up to isomorphism by its base. Moreover, any morphism between the bases of two distinguished triangles can be extended to a morphism between the whole distinguished triangles.

We shall show that the analogous assertion is not true for Verdier octahedra. In §3, we give an example of two nonisomorphic Verdier octahedra on the same base. In particular, the identity morphism between the bases cannot be prolonged to a morphism between the whole Verdier octahedra.

The reader particularly interested in Verdier octahedra can read §1.1, §1.2, §1.4 and §3.

In the terminology of Heller triangulated categories, a Verdier octahedron is a periodic 3-pretriangle X such that Xd^{#} is a 2-triangle (i.e. a distinguished triangle) for all injective periodic monotone maps Δ̄_{3} ^{\overline{}^{d}} Δ̄_{2}.

One of the two Verdier octahedra in our example will be a 3-triangle in the sense of [4, Def. 1.5], i.e. a “distinguished octahedron”, whereas the other will not.

Note that unlike a Verdier octahedron, a 3-triangle is uniquely determined up to isomorphism by its base in the Heller triangulated context; cf. [4, Lem. 3.4.(6)].

0.2 Is being an n-triangle characterised by (n − 1)-triangles?

The situation of §0.1 can be generalised in the following manner.

Suppose given a closed Heller triangulated category (C, T, ϑ); cf. [4, Def. 1.5], Definition 13.

The Heller triangulation ϑ = (ϑ_{n})_{n⩾0} on (C, T) can be viewed as a means to distinguish certain periodic n-pretriangles as n-triangles. Namely, a periodic n-pretriangle X is, by definition, an n-triangle if Xϑ_{n} = 1; cf. [4, Def. 1.5.(ii.2)]. For instance, 2-triangles are distinguished triangles in the sense of Verdier; 3-triangles are particular, “distinguished” Verdier octahedra.

0.2.1 The example

Let n ⩾ 3. Let X be a periodic n-pretriangle. Suppose that Xd^{#} is an (n − 1)-triangle for all injective periodic monotone maps Δ̄_{n} ^{\overline{}^{d}} Δ̄_{n−1}. One might ask whether X is an n-triangle.

We shall show in §2 by an example that this is, in general, not the case.

0.2.2 Consequences

Suppose given n ⩾ 3 and a subset of the set of periodic n-pretriangles. We shall say for the moment that determination holds for this subset if for X and X̃ out of this subset, X|Δ̇_{n} ≃ X̃|Δ̇_{n} implies that there is a periodic isomorphism X ≃ X̃. We shall say that prolongation holds for this subset, if for X and X̃ out of this subset and a morphism X|Δ̇_{n} \overline{}X̃|Δ̇_{n}, there exists a periodic morphism X \overline{}X̃ that restricts on Δ̇_{n} to that given morphism. If prolongation holds, then determination holds.

• Consider the subset of periodic n-pretriangles X such that Xd^{#} is an (n − 1)-triangle for all injective periodic monotone maps Δ̄_{n} ^{\overline{}^{d}} Δ̄_{n−1}. Our example shows that in general, determination and prolongation do not hold for this subset. In fact, if X is such an n-pretriangle, but not an n-triangle, then the n-triangle on the base X|Δ̇_{n} is not isomorphic to X; cf. [4, Lem. 3.4.(1, 4)].

• Bernstein, Beilinson and Deligne considered the subset of periodic n-pretriangles X such that Xd^{#} is a 2-triangle (i.e. a distinguished triangle) for all injective periodic monotone maps Δ̄_{n} ^{\overline{}^{d}} Δ̄_{2} [1, 1.1.14]. Our example shows that in general, determination and prolongation do not hold for this subset. In fact, this subset contains the previously described subset.

In both of the cases above, if n = 3, then the condition singles out the subset of Verdier octahedra.

• By [4, Lem. 3.4.(6); Lem. 3.2], determination and prolongation hold for the set of n-triangles.

So morally, our example shows that it makes sense to let the Heller triangulation ϑ distinguish n-triangles for all n ⩾ 0. There is no “sufficiently large” n we could be content with.

0.3 An appendix on transport of structure

Suppose given a Frobenius category E; that is, an exact category with enough bijective objects (relative to pure short exact sequences). Let B ⊆ E denote the full subcategory of bijective objects.

There are two variants of the stable category of E. First, there is the classical stable category \underline{E}, defined as the quotient of E modulo B. Second, there is the stable category \underline{\underline{E}}, defined as the quotient of the category of purely acyclic complexes with entries in B modulo the category of split acyclic complexes with entries in B. The categories \underline{E} and \underline{\underline{E}} are equivalent. The advantage of the variant \underline{\underline{E}} is that it carries a shift automorphism, whereas \underline{E} carries a shift autoequivalence.

In [4, Cor. 4.7], we have endowed \underline{\underline{E}} with a Heller triangulation. Now in our particular situation, also \underline{E} carries a shift automorphism. Since \underline{E} is better suited for calculations within that category, the question arises whether the equivalence \underline{E} ≃ \underline{\underline{E}} can be used to transport the structure of a Heller triangulated category from \underline{\underline{E}} to \underline{E}. This is indeed the case; cf. Proposition 22.(1). Moreover, we give recipes how to detect and how to construct n-triangles in \underline{E}; cf. Propositions 22.(2, 3), 25.

Roughly put, the variant \underline{\underline{E}} is rather suited for theoretical purposes, the variant \underline{E} is rather suited for practical purposes, and we had to pass a result from \underline{\underline{E}} to \underline{E}. Not surprisingly, to do so, we had to grapple with the various equivalences and isomorphisms involved.

0.4 Acknowledgements

I thank Amnon Neeman for pointing out, years ago, why a counterexample as in §2 should exist, contrary to what I had believed.

This example has been found using the computer algebra system Magma [2]. I thank Markus Kirschmer for help with a Magma program.

I thank the referee for helpful comments.

0.5 Notations and conventions

We use the conventions listed in [4, §0.6]. In addition, we use the following conventions.

(i) If x and y are elements of a set, we let ∂_{x,y} := 1 if x = y, and we let ∂_{x,y} := 0 if x ≠ y.

(ii) Given a ∈ Z, we write Z/a := Z/aZ. (iii) Given a ring R and R-modules X and Y, we write, by choice, _{R}(X, Y) = _{R -Mod}(X, Y) = Hom_{R}(X, Y).

Moreover, given k ⩾ 0, we write X^{⊕k} := ⨁_{i∈[1,k]} X.

(iv) An automorphism T of a category C is an endofunctor on C for which there exists an endofunctor S such that ST = 1_{C} and T S = 1_{C}. An autoequivalence T of a category C is an endofunctor on C for which there exists an endofunctor S such that ST ≃ 1_{C} and T S ≃ 1_{C}.

(v) Let n ⩾ 0. Recall that Δ̄_{n}^{△▽} = {β/α ∈ Δ̄_{n}^{#} : 0 ⩽ α ⩽ β ⩽ 0^{+1}} ⊆ Δ̄_{n}^{#}. We will often display an n-triangle or a periodic n-pretriangle in a Heller triangulated category C by showing its restriction to Δ̄_{n}^{△▽} ∖ ({α/α : 0 ⩽ α ⩽ 0^{+1}} ∪ {0^{+1}/0}). This is possible without loss of information, for we can reconstruct the whole diagram by adding zeroes on α/α for 0 ⩽ α ⩽ 0^{+1} and on 0^{+1}/0, and then by periodic prolongation.

(vi) Suppose given a Heller triangulated category C. A Verdier octahedron in C is a periodic 3-pretriangle X ∈ Ob C^{+, periodic}(Δ̄_{3}^{#}) such that Xd^{#} ∈ Ob C^{+, periodic}(Δ̄_{2}^{#}) is a 2-triangle for all injective periodic monotone maps Δ̄_{3} ^{\overline{}^{d}} Δ̄_{2}.

Henceforth, let p ⩾ 2 be a prime.

1 The classical stable category of (Z/p^{m}) -mod

1.1 The category (Z/p^{m}) -mod

Let m ⩾ 0. By E := (Z/p^{m}) -mod we understand the following category.

The objects are indexed by tuples (a_{i})_{i∈[0,m]} with a_{i} ∈ Z_{⩾0}. To such an index, we attach the object

⨁_{i∈[0,m]}(Z/p^{i})^{⊕a_{i}} .

As morphisms, we take Z/p^{m}-linear maps.

Note that we have not chosen a skeleton. The trick here is to pick several zero objects.

The duality contrafunctor _{Z/p^{m}}(−, Z/p^{m}) on E, which sends Z/p^{i} to Z/p^{i} for i ∈ [1, m], shows that an object in this category is injective if and only if it is projective. An object of E is bijective if and only if it is isomorphic to a finite direct sum of copies of Z/p^{m}. The category E is an abelian Frobenius category, with all short exact sequences stipulated to be pure; cf. e.g. [4, Def. A.5.(2)].

1.2 The shift on \underline{(Z/p^{m}) -mod}

To define a shift automorphism on the classical stable category \underline{E} = \underline{(Z/p^{m}) -mod}, we shall distinguish certain (pure) short exact sequences in E; cf. §A.4.2.1, [4, Def. A.7].

Let E_{k} := (\begin{matrix}1 \\ & . & . & . \\ & & & & 1 \end{matrix}) denote the unit matrix of size k × k; let E′_{k} := (\begin{matrix}& & & & 1 \\ & . & . & . \\ 1 \end{matrix}) denote the reversed unit matrix of size k × k.

As distinguished (pure) short exact sequences we take those of the form

[figure: ⨁_{i∈[0,m]}(Z/p^{i})^{⊕a_{i}}; •; ^{(\begin{matrix}p^{m}E_{a_{0}} \\ & p^{m−1}E_{a_{1}} \\ & & . & . & . \\ & & & & & p^{0}E_{am} \end{matrix} )}; (Z/p^{m})^{⊕ ∑_{i∈[0,m]} a_{i}}; ^{(\begin{matrix}& & & & & E′_{a_{0}} \\ & & & & E′_{a_{1}} \\ & . & . & . \\ E′_{am} \end{matrix} )}; ⨁_{i∈[0,m]}(Z/p^{i})^{⊕a_{m−i}}]

So roughly speaking, distinguished short exact sequences are direct sums of those of the form

Z/p^{i} [figure: p^{m−i}] Z/p^{m} ^{\overline{}^{1}} Z/p^{m−i} ,

where i ∈ [0, m]; we reorder the summands the cokernel term consists of.

With this choice, conditions (i, ii, iii) of §A.4.2.1 are satisfied.

On indecomposable objects and morphisms between them, the shift automorphism induced on \underline{E} by our set of distinguished short exact sequences is given by

(Z/p^{i} ^{\overline{}^{a}} Z/p^{j})^{+1} = (Z/p^{m−i} ^{\overline{}^{p^{i−j}a}} Z/p^{m−j}) ,

where i, j ∈ [0, m], and where a is a representative in Z. Note that if i < j, then a is divisible by p^{j−i}.

Note that Z/p^{i} ^{\overline{}^{a}} Z/p^{j} represents zero in \underline{E} if and only if a is divisible by p^{min(m−i, j)}.

1.3 A Heller triangulation on \underline{(Z/p^{m}) -mod}

Concerning the notation E^{□}(Δ̄_{n}^{△▽}), cf. §A.3. Given n ⩾ 0 and X ∈ Ob E^{□}(Δ̄_{n}^{△▽}), we form X^{τ} ∈ Ob \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}) with respect to the set of distinguished short exact sequences of §1.2 as described in §A.4.2.3. That is, we replace the rightmost column of X by the column obtained using distinguished short exact sequences, so that (X^{τ})_{0^{+1}/∗} = ((X^{τ})_{∗/0})^{+1} = (X_{∗/0})^{+1}; cf. §A.4.2.3.

Remark 1 If the short exact sequences

X_{α/0} [figure: ^{(^{x x})}] X_{α/α} ⊕ X_{0^{+1}/0} ^{\overline{}^{(\begin{matrix}x \\ −x \end{matrix})}} X_{0^{+1}/α}

appearing in the diagram X for 1 ⩽ α ⩽ n already are distinguished, then the image of X in Ob \underline{E}^{+}(Δ̄_{n}^{△▽}) equals X^{τ}.

Concerning the notion of a closed Heller triangulated category, cf. Definition 13 in §A.2.

Remark 2 The classical stable category \underline{E} = \underline{(Z/p^{m}) -mod} carries a closed Heller triangulation such that given n ⩾ 0 and X ∈ Ob E^{□}(Δ̄_{n}^{△▽}), the periodic prolongation of X^{τ} to an object of \underline{E}^{+, periodic}(Δ̄_{n}^{#}) is an n-triangle.

Proof. The assertion follows by Proposition 22.(1) in §A.4.2.1 and Proposition 25 in §A.4.2.4.

1.4 A Verdier triangulation on \underline{(Z/p^{m}) -mod}

By [3, Th. 2.6], \underline{E} = \underline{(Z/p^{m}) -mod} is a Verdier triangulated category, i.e. a triangulated category in the sense of Verdier [5, Def. 1-1].

This also follows by Remark 2 and by [4, Prop. 3.6], which says that any Heller triangulated category in which idempotents split is also Verdier triangulated. The 2-triangles in the Heller context are the distinguished triangles in the Verdier context.

Given a morphism X ^{\overline{}^{f}} Y in E, using the distinguished short exact sequence X [figure] B \overline{}X^{+1}, where B is bijective, we can form the morphism

[figure: Y; •; Z; X^{+1}; X; •; f; B; X^{+1}]

of short exact sequences, from which the sequence

[figure: X; f; Y; •; Z; X^{+1}]

represents a distinguished triangle in the Verdier triangulated category \underline{E}.

2 Nonisomorphic periodic n-pretriangles

Nonisomorphic periodic n-pretriangles whose periodic (n − 1)-pretriangles are all (n − 1)-triangles, to be specific.

Let n ⩾ 3. Let C := \underline{(Z/p^{2n}) -mod}, and let it be endowed with a shift automorphism as in §1.2 and a Heller triangulation as in §1.3.

2.1 A (2n − 1)-triangle

Let Y be the following (2n − 1)-triangle in C.

\xymatrix{ &  &  &  &  &  & Z/p^{1} \\  &  &  &  &  & Z/p^{1} \ar[r]^{−p} & Z/p^{2} \ar[u]^{1} \\  &  &  &  & Z/p^{1} \ar[r]^{p} & Z/p^{2} \ar[r]^{−p} \ar[u]^{1} & Z/p^{3} \ar[u]^{1} \\  &  &  & _{...} & ⋮ \ar[u]^{1} & ⋮ \ar[u]^{1} & ⋮ \ar[u]^{1} \\  &  & Z/p^{1} \ar[r]^{p} & ⋯ \ar[r]^{p} & Z/p^{2n−4} \ar[r]^{p} \ar[u]^{1} & Z/p^{2n−3} \ar[r]^{−p} \ar[u]^{1} & Z/p^{2n−2} \ar[u]^{1} \\  & Z/p^{1} \ar[r]^{p} & Z/p^{2} \ar[r]^{p} \ar[u]^{1} & ⋯ \ar[r]^{p} & Z/p^{2n−3} \ar[r]^{p} \ar[u]^{1} & Z/p^{2n−2} \ar[r]^{−p} \ar[u]^{1} & Z/p^{2n−1} \ar[u]^{1} \\ Z/p^{1} \ar[r]^{p} & Z/p^{2} \ar[r]^{p} \ar[u]^{1} & Z/p^{3} \ar[r]^{p} \ar[u]^{1} & ⋯ \ar[r]^{p} & Z/p^{2n−2} \ar[r]^{p} \ar[u]^{1} & Z/p^{2n−1} \ar[u]^{1}}

Here we have made use of the convention from §0.5 that we display of Y only its restriction to the subposet {β/α ∈ Δ̄_{2n−1}^{#} : 0 ⩽ α < β ⩽ 0^{+1}, β/α ≠ 0^{+1}/0}, which is possible without loss of information. Similarly below.

It arises from a diagram on Δ̄_{n}^{△▽} with values in (Z/p^{2n}) -mod that consists of squares, has entry Z/p^{2n} at position 0^{+1}/0, and has the quadrangle

\xymatrix{Z/p^{2n−2} \ar[r]^{−p} & Z/p^{2n−1} \\ Z/p^{2n−1} \ar[r]^{p} \ar[u]^{1} & Z/p^{2n} \ar[u]_{−1}}

in its lower right corner. This diagram contains the necessary distinguished short exact sequences with the necessary signs inserted for Y to be in fact a (2n − 1)-triangle; cf. Remarks 1, 2.

2.2 An n-triangle and a periodic n-pretriangle

We apply the folding operator 𝔣_{n−1} to the (2n − 1)-triangle Y obtained in §2.1, yielding the n-triangle Y 𝔣_{n−1}, which we shall display now; cf. [4, Lem. 3.4.(2), §1.2.2.3].

[figure: Z/p^{n}; Z/p^{1}⊕Z/p^{2n−1}; _{(\begin{matrix}−p^{n−1} \\ −1 \end{matrix})}; Z/p^{n}; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{2}⊕Z/p^{2n−2}; _{(\begin{matrix}−p^{n−2} \\ −1 \end{matrix})}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; _{...}; ⋮; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋮; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋮; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; ⋯; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{n−3}⊕Z/p^{n+3}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n−2}⊕Z/p^{n+2}; _{(\begin{matrix}−p^{2} \\ −1 \end{matrix})}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{2}⊕Z/p^{2n−2}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋯; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{n−2}⊕Z/p^{n+2}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n−1}⊕Z/p^{n+1}; _{( \begin{matrix}−p \\ −1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; Z/p^{n}; p; Z/p^{n}; p; _{(_{1 −p^{n−1}} )}; Z/p^{n}; p; _{(_{1 −p^{n−2}} )}; ⋯; p; Z/p^{n}; p; _{(_{1 −p^{2}} )}; Z/p^{n}; _{(_{1 −p})}]

Let X be the n-triangle obtained from Y 𝔣_{n−1} by isomorphic substitution along (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) on all terms consisting of two summands; cf. [4, Lem. 3.4.(4)]. So X can be displayed as follows.

[figure: Z/p^{n}; Z/p^{1}⊕Z/p^{2n−1}; _{(\begin{matrix}−p^{n−1} \\ 1 \end{matrix})}; Z/p^{n}; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{2}⊕Z/p^{2n−2}; _{(\begin{matrix}−p^{n−2} \\ 1 \end{matrix})}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; _{...}; ⋮; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋮; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋮; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; ⋯; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{n−3}⊕Z/p^{n+3}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n−2}⊕Z/p^{n+2}; _{(\begin{matrix}−p^{2} \\ 1 \end{matrix})}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{2}⊕Z/p^{2n−2}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋯; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{n−2}⊕Z/p^{n+2}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n−1}⊕Z/p^{n+1}; _{( \begin{matrix}−p \\ 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; Z/p^{n}; p; Z/p^{n}; p; _{(_{1 p^{n−1}} )}; Z/p^{n}; p; _{(_{1 p^{n−2}} )}; ⋯; p; Z/p^{n}; p; _{(_{1 p^{2}} )}; Z/p^{n}; _{(_{1 p})}]

Let X̃ be the following periodic n-pretriangle.

[figure: Z/p^{n}; Z/p^{1}⊕Z/p^{2n−1}; _{(\begin{matrix}−p^{n−1} \\ 1 \end{matrix})}; Z/p^{n}; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{2}⊕Z/p^{2n−2}; _{(\begin{matrix}−p^{n−2} \\ 1 \end{matrix})}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; _{...}; ⋮; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋮; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋮; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; ⋯; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{n−3}⊕Z/p^{n+3}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n−2}⊕Z/p^{n+2}; _{(\begin{matrix}−p^{2} \\ 1 \end{matrix})}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n}; p; Z/p^{1}⊕Z/p^{2n−1}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{2}⊕Z/p^{2n−2}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; ⋯; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; Z/p^{n−2}⊕Z/p^{n+2}; _{( \begin{matrix}p & 0 \\ p^{n−3} & 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )}; Z/p^{n−1}⊕Z/p^{n+1}; _{( \begin{matrix}−p \\ 1 \end{matrix} )}; _{( \begin{matrix}1 & 0 \\ −p^{n−3} & p \end{matrix} )}; Z/p^{n}; p; Z/p^{n}; p; Z/p^{n}; p; _{(_{1 p^{n−1}} )}; Z/p^{n}; p; _{(_{1 p^{n−2}} )}; ⋯; p; Z/p^{n}; p; _{(_{1 p^{2}} )}; Z/p^{n}; _{(_{1 p})}]

To verify that X̃ actually is an n-pretriangle, a comparison with X reduces us to show that the three quadrangles depicted in full in the lower right corner of X̃ are weak squares. Of these three, the middle quadrangle arises from the corresponding one of X

by an isomorphic substitution along Z/p^{n−1} ⊕ Z/p^{n+1}  \overline{}^{(\begin{matrix}1 & 0 \\ p^{n−3} & 1 \end{matrix})}_{∼}  Z/p^{n−1} ⊕ Z/p^{n+1}, and

thus is a weak square. For the lower one, we may apply [4, Lem. A.17] to the diagram (X̃_{1/0}, X̃_{n−1/0}, X̃_{n/0}, X̃_{0^{+1}/0}, X̃_{1/1}, X̃_{n−1/1}, X̃_{n/1}, X̃_{0^{+1}/1}) and compare with X to show that it is a weak square. For the right hand side one, we may apply [4, Lem. A.17] to the diagram (X̃_{n/0}, X̃_{n/1}, X̃_{n/2}, X̃_{n/n}, X̃_{0^{+1}/0}, X̃_{0^{+1}/1}, X̃_{0^{+1}/2}, X̃_{0^{+1}/n}) and compare with X to show that it is a weak square.

Given k ∈ [0, n], we let Δ̄_{n} ^{\overline{}^{d_{k}}} Δ̄_{n−1} be the periodic monotone map determined by [0, n − 1]d_{k} = [0, n] ∖ {k}.

Lemma 3 Suppose given k ∈ [0, n].

(1) The diagram X̃d_{k}^{#} is an (n − 1)-triangle.

(2) We have Xd_{k}^{#} ≃ X̃d_{k}^{#} in C^{+, periodic}(Δ̄_{n−1}^{#}).

Proof. Since X is an n-triangle, Xd_{k}^{#} is an (n − 1)-triangle; cf. [4, Lem. 3.4.(1)]. Since Xd_{k}^{#}|Δ̇_{n−1} = X̃d_{k}^{#}|Δ̇_{n−1}, the diagram X̃d_{k}^{#} is an (n − 1)-triangle if and only if it is isomorphic to Xd_{k}^{#} in C^{+, periodic}(Δ̄_{n−1}^{#}); cf. [4, Lem. 3.4.(4, 6)]. So assertions (1) and (2) are equivalent. We will prove (2).

When referring to an object on a certain position in the diagram Xd_{k}^{#} resp. X̃d_{k}^{#}, we shall also mention in parentheses its position as an object in the diagram X resp. X̃ for ease of orientation.

When constructing a morphism in C^{+, periodic}(Δ̄_{n−1}^{#}), we will give its components on {j/i : 0 ⩽ i ⩽ j ⩽ n − 1} ⊆ Δ̄_{n−1}^{#} ; the remaining components result thereof by periodic repetition.

Case k ∈ {1, n}. We have Xd_{1}^{#} = X̃d_{1}^{#} and Xd_{n}^{#} = X̃d_{n}^{#}.

Case k = 0. We claim that Xd_{0}^{#} is isomorphic to X̃d_{0}^{#} in C^{+, periodic}(Δ̄_{n−1}^{#}). In fact, an isomorphism Xd_{0}^{#} ^{\overline{}^{∼}} X̃d_{0}^{#} is given by

Z/p^{n−1} ⊕ Z/p^{n+1}  \overline{}^{(\begin{matrix}1 & 0 \\ p^{n−3} & 1 \end{matrix})}_{∼}  Z/p^{n−1} ⊕ Z/p^{n+1}

at position (n − 1)/0 (position n/1 in X resp. X̃), and by the identity elsewhere. This proves the claim.

Case k ∈ [2, n − 1]. We claim that Xd_{k}^{#} is isomorphic to X̃d_{k}^{#} in C^{+, periodic}(Δ̄_{n−1}^{#}). In fact, an isomorphism Xd_{k}^{#} ^{\overline{}^{∼}} X̃d_{k}^{#} is given as follows.

At position j/0 for j ∈ [1, n − 1] (position j/0 if j ⩽ k − 1 and (j + 1)/0 if j ⩾ k in X resp. X̃), it is given by the identity on Z/p^{n}.

At position j/i for i, j ∈ [1, k − 1] such that i < j (position j/i in X resp. X̃), it is given by the identity on Z/p^{j−i} ⊕ Z/p^{2n−j+i}.

At position j/i for i, j ∈ [k, n − 1] such that i < j (position (j + 1)/(i + 1) in X resp. X̃), it is given by the identity on Z/p^{j−i} ⊕ Z/p^{2n−j+i}.

At position j/i for i ∈ [1, k − 1] and j ∈ [k, n − 1] such that j/i ≠ (n − 1)/1 (position (j + 1)/i in X resp. X̃), it is given by

Z/p^{j+1−i} ⊕ Z/p^{2n−j−1+i}  \overline{}^{(\begin{matrix}1 & 0 \\ −p^{j−1−i} & 1 \end{matrix})}_{∼}  Z/p^{j+1−i} ⊕ Z/p^{2n−j−1+i}

At position (n−1)/1 (position n/1 in X resp. X̃), it is given by the identity on Z/p^{n−1} ⊕Z/p^{n+1}.

This proves the claim.

Lemma 4 X is not isomorphic to X̃ in C^{+, periodic}(Δ̄_{n}^{#}).

In particular, X̃ is not an n-triangle; cf. [4, Lem. 3.4.(6)].

Proof. We assume the contrary. By [4, 3.4.(4)], X and X̃ are n-triangles. Thus, by [4, 3.4.(6)], there is an isomorphism X ^{\overline{}^{∼}} X̃ that is identical at i/0 and at 0^{+1}/i for i ∈ [1, n]. Let

Z/p^{ℓ−k}⊕Z/p^{2n−ℓ+k} ^{\overline{}^{( \begin{matrix}a_{ℓ/k} & p^{2n−2ℓ+2k}b_{ℓ/k} \\ c_{ℓ/k} & d_{ℓ/k} \end{matrix} )}} Z/p^{ℓ−k}⊕Z/p^{2n−ℓ+k}

denote the entry of this isomorphism at ℓ/k, where 1 ⩽ k < ℓ ⩽ n.

If ℓ − k ⩾ 2, we have the following commutative quadrangle in C on ℓ/k \overline{}ℓ/(k + 1).

\xymatrix{Z/p^{ℓ−k}⊕Z/p^{2n−ℓ+k} \ar[d]_{_{( \begin{matrix}a_{ℓ/k} & p^{2n−2ℓ+2k}b_{ℓ/k} \\ c_{ℓ/k} & d_{ℓ/k} \end{matrix} )}} \ar[r]^{^{(\begin{matrix}1 & 0 \\ 0 & p \end{matrix})}} & Z/p^{ℓ−k−1}⊕Z/p^{2n−ℓ+k+1} \ar[d]^{_{( \begin{matrix}a_{ℓ/(k+1)} & p^{2n−2ℓ+2k+2}b_{ℓ/(k+1)} \\ c_{ℓ/(k+1)} & d_{ℓ/(k+1)} \end{matrix} )}} \\ Z/p^{ℓ−k}⊕Z/p^{2n−ℓ+k} \ar[r]_{(\begin{matrix}1 & 0 \\ 0 & p \end{matrix}) − ∂_{ℓ/k, n/1}(\begin{matrix}0 & 0 \\ p^{n−3} & 0 \end{matrix})} & Z/p^{ℓ−k−1}⊕Z/p^{2n−ℓ+k+1}}

We read off the congruences

(i)  c_{ℓ/k} − ∂_{ℓ/k, n/1} p^{n−3}d_{ℓ/k} ≡_{p^{ℓ−k−1}} pc_{ℓ/(k+1)}
(ii)  b_{ℓ/k} ≡_{p^{ℓ−k−1}} pb_{ℓ/(k+1)} .

From (i) we infer

(iii) c_{n/1} − p^{n−3}d_{n/1} ≡_{p^{n−2}} p^{1}c_{n/2} ≡_{p^{n−2}} p^{2}c_{n/3} ≡_{p^{n−2}} … ≡_{p^{n−2}} p^{n−2}c_{n/(n−1)} ≡_{p^{n−2}} 0 .

From (ii) we infer

(iv)  b_{n/1} ≡_{p^{n−2}} pb_{n/2} ≡_{p^{n−2}} p^{2}b_{n/3} ≡_{p^{n−2}} … ≡_{p^{n−2}} p^{n−2}b_{n/(n−1)} ≡_{p^{n−2}} 0 .

On n/1 \overline{}0^{+1}/1, we have the following commutative quadrangle in C.

[figure: Z/p^{n−1}⊕Z/p^{n+1}; _{(\begin{matrix}a_{n/1} & p^{2}b_{n/1} \\ c_{n/1} & d_{n/1} \end{matrix} )}; ^{(\begin{matrix}−p \\ 1 \end{matrix})}; Z/p^{n}; Z/p^{n−1}⊕Z/p^{n+1}; _{(\begin{matrix}−p \\ 1 \end{matrix})}; Z/p^{n}]

We read off the congruence

(v)  −pc_{n/1} + d_{n/1} ≡_{p^{n−1}} 1 .

On n/0 \overline{}n/1, we have the following commutative quadrangle in C.

[figure: Z/p^{n}; ^{(^{1 p})}; Z/p^{n−1}⊕Z/p^{n+1}; _{(\begin{matrix}a_{n/1} & p^{2}b_{n/1} \\ c_{n/1} & d_{n/1} \end{matrix} )}; Z/p^{n}; _{(_{1 p})}; Z/p^{n−1}⊕Z/p^{n+1}]

We read off the congruence

(vi)  pb_{n/1} + d_{n/1} ≡_{p^{n−1}} 1 .

By (iii) resp. (iv) we conclude from (v) resp. (vi) that

(v′)  (1 − p^{n−2})d_{n/1} ≡_{p^{n−1}} 1
(vi′)  d_{n/1} ≡_{p^{n−1}} 1 .

Substituting (vi′) into (v′), we obtain

1 − p^{n−2} ≡_{p^{n−1}} 1 ,

which is absurd.

3 Nonisomorphic Verdier octahedra

Since in §2, the category C is also a Verdier triangulated category, specialising to n = 3 yields two nonisomorphic Verdier octahedra on the same base. In this particular case, we shall now give a somewhat longer argument alternative to that given in §2 that is independent of [4], whose techniques might not be familiar to all readers. Nonetheless, §3 is a particular case of §2.

Let C := \underline{(Z/p^{6}) -mod}, and let it be endowed with a shift automorphism as in §1.2 and a Verdier triangulation as in §1.4.

Let the diagram X be given by

[figure: Z/p^{3}; Z/p^{1}⊕Z/p^{5}; _{(\begin{matrix}−p^{2} \\ 1 \end{matrix} )}; Z/p^{3}; p; Z/p^{1}⊕Z/p^{5}; _{(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})}; Z/p^{2}⊕Z/p^{4}; _{(\begin{matrix}−p \\ 1 \end{matrix})}; _{(\begin{matrix}1 & 0 \\ 0 & p \end{matrix})}; Z/p^{3}; p; Z/p^{3}; p; Z/p^{3}; p; _{(_{1 p^{2}})}; Z/p^{3}; _{(_{1 p})}; .]

Let the diagram X̃ be given by

[figure: Z/p^{3}; Z/p^{1}⊕Z/p^{5}; _{(\begin{matrix}−p^{2} \\ 1 \end{matrix} )}; Z/p^{3}; p; Z/p^{1}⊕Z/p^{5}; _{(\begin{matrix}p & 0 \\ 1 & 1 \end{matrix})}; Z/p^{2}⊕Z/p^{4}; _{(\begin{matrix}−p \\ 1 \end{matrix})}; _{(\begin{matrix}1 & 0 \\ −1 & p \end{matrix})}; Z/p^{3}; p; Z/p^{3}; p; Z/p^{3}; p; _{(_{1 p^{2}})}; Z/p^{3}; _{(_{1 p})}; .]

Lemma 5 Both X and X̃ are Verdier octahedra.

In contrast to the procedure in §2, to prove this, we will not make use of the folding operation.

Proof. For the periodic monotone map Δ̄_{3} ^{\overline{}^{d_{3}}} Δ̄_{2} that maps 0 \overline{}0, 1 \overline{}1 and 2 \overline{}2, we obtain Xd_{3}^{#} = X̃d_{3}^{#}, horizontally displayed as

[figure: Z/p^{3}; p; Z/p^{3}; ^{(^{1 p^{2}})}; Z/p⊕Z/p^{5}; ^{(\begin{matrix}−p^{2} \\ 1 \end{matrix} )}; Z/p^{3} .]

The following morphism of short exact sequences in (Z/p^{6}) -mod shows Xd_{3}^{#} to be a distinguished triangle.

[figure: Z/p^{3}; •; ^{(^{1 p^{2}})}; Z/p⊕Z/p^{5}; ^{(\begin{matrix}−p^{2} \\ 1 \end{matrix} )}; Z/p^{3}; Z/p^{3}; •; p^{3}; p; Z/p^{6}; 1; _{(_{0 1})}; Z/p^{3}]

For the periodic monotone map Δ̄_{3} ^{\overline{}^{d_{1}}} Δ̄_{2} that maps 0 \overline{}0, 2 \overline{}1 and 3 \overline{}2, we obtain the distinguished triangle Xd_{1}^{#} = X̃d_{1}^{#} = Xd_{3}^{#} again.

For the periodic monotone map Δ̄_{3} ^{\overline{}^{d_{2}}} Δ̄_{2} that maps 0 \overline{}0, 1 \overline{}1 and 3 \overline{}2, we obtain the diagram Xd_{2}^{#} = X̃d_{2}^{#}, horizontally displayed as

\xymatrix{Z/p^{3} \ar[r]^{p^{2}} & Z/p^{3} \ar[r]^{^{(^{1 p})}} & Z/p^{2}⊕Z/p^{4} \ar[r]^{^{(\begin{matrix}−p \\ 1 \end{matrix})}} & Z/p^{3} .}

The following morphism of short exact sequences in (Z/p^{6}) -mod shows Xd_{2}^{#} to be a distinguished triangle.

[figure: Z/p^{3}; •; ^{(^{1 p})}; Z/p^{2}⊕Z/p^{4}; ^{(\begin{matrix}−p \\ 1 \end{matrix})}; Z/p^{3}; Z/p^{3}; •; p^{3}; p^{2}; Z/p^{6}; 1; _{(_{0 1})}; Z/p^{3}]

For the periodic monotone map Δ̄_{3} ^{\overline{}^{d_{0}}} Δ̄_{2} that maps 1 \overline{}0, 2 \overline{}1 and 3 \overline{}2, we obtain the periodic isomorphism Xd_{0}^{#} ^{\overline{}^{∼}} X̃d_{0}^{#}, horizontally displayed as

[figure: Z/p⊕Z/p^{5}; ^{(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix})}; Z/p^{2}⊕Z/p^{4}; ^{(\begin{matrix}1 & 0 \\ 0 & p \end{matrix})}; _{(\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix})}; ≀; Z/p⊕Z/p^{5}; ^{(\begin{matrix}0 & −p^{4} \\ 1 & 0 \end{matrix} )}; Z/p⊕Z/p^{5}; Z/p⊕Z/p^{5}; _{(\begin{matrix}p & 0 \\ 1 & 1 \end{matrix})}; Z/p^{2}⊕Z/p^{4}; _{(\begin{matrix}1 & 0 \\ −1 & p \end{matrix})}; Z/p⊕Z/p^{5}; _{(\begin{matrix}0 & −p^{4} \\ 1 & 0 \end{matrix} )}; Z/p⊕Z/p^{5} .]

So we are reduced to show that Xd_{0}^{#} is a distinguished triangle, which it is as a direct sum of two distinguished triangles, as the following morphisms of short exact sequences in (Z/p^{6}) -mod show.

[figure: Z/p; •; p; Z/p^{2}; 1; Z/p; Z/p^{4}; •; p; Z/p^{5}; 1; Z/p; Z/p^{5}; •; p; 1; Z/p^{6}; 1; 1; Z/p; Z/p^{5}; •; p; 1; Z/p^{6}; 1; 1; Z/p]

Lemma 6 The Verdier octahedra X and X̃ are not isomorphic in C^{+, periodic}(Δ̄_{3}^{#}).

That is, there is no isomorphism between the displayed parts of X resp. of X̃ such that its entries on the rightmost vertical column arise by an application of the shift functor of C to its entries on the lower row.

We will not use the fact that X is a 3-triangle, which in conjunction with [4, 3.4.(4, 6)] would permit us to restrict ourselves to consider isomorphisms that are identical on the lower row and the rightmost vertical column, as we did in Lemma 4.

Proof. We assume the contrary and depict an isomorphism X ^{\overline{}^{∼}} X̃ as follows.

[figure: Z/p^{3}; w; Z/p⊕Z/p^{5}; _{( \begin{matrix}−p^{2} \\ 1 \end{matrix} )}; _{( \begin{matrix}a′′ & p^{4}b′′ \\ c′′ & d′′ \end{matrix} )}; Z/p^{3}; p; v; Z/p⊕Z/p^{5}; _{( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}a′ & p^{4}b′ \\ c′ & d′ \end{matrix} )}; Z/p^{2}⊕Z/p^{4}; _{(\begin{matrix}1 & 0 \\ 0 & p \end{matrix})}; _{( \begin{matrix}−p \\ 1 \end{matrix} )}; _{( \begin{matrix}a & p^{2}b \\ c & d \end{matrix} )}; Z/p^{3}; p; u; Z/p^{3}; p; u; Z/p^{3}; p; _{(_{1 p^{2}} )}; v; Z/p^{3}; _{(_{1 p})}; w; Z/p^{3}; Z/p⊕Z/p^{5}; _{( \begin{matrix}−p^{2} \\ 1 \end{matrix} )}; Z/p^{3}; p; Z/p⊕Z/p^{5}; _{( \begin{matrix}p & 0 \\ 1 & 1 \end{matrix} )}; Z/p^{2}⊕Z/p^{4}; _{(\begin{matrix}1 & 0 \\ −1 & p \end{matrix})}; _{( \begin{matrix}−p \\ 1 \end{matrix} )}; Z/p^{3}; p; Z/p^{3}; p; Z/p^{3}; p; _{(_{1 p^{2}} )}; Z/p^{3}; _{(_{1 p})}]

Note that all vertical quadrangles commute in C.

The commutative quadrangles on 1/0 \overline{}2/0 \overline{}3/0 yield u ≡_{p^{2}} v ≡_{p^{2}} w.

The commutative quadrangle on 3/0 \overline{}3/1 yields pb + d ≡_{p^{2}} w.

The commutative quadrangle on 3/1 \overline{}0^{+1}/1 yields −pc + d ≡_{p^{2}} u.

The commutative quadrangle on 3/1 \overline{}3/2 yields b ≡_{p} 0 and c ≡_{p} d.

Altogether, we have

u ≡_{p^{2}} w ≡_{p^{2}} pb + d ≡_{p^{2}} d ≡_{p^{2}} u + pc ≡_{p^{2}} u + pd ,

whence

0 ≡_{p} d ≡_{p} w .

Since Z/p^{3} ^{\overline{}^{w}} Z/p^{3} is an isomorphism in C, we have w ≢_{p} 0. This is absurd.

In [1, 1.1.13], it is described how an octahedron gives rise to two “extra” triangles. As cone of the diagonal of a quadrangle appearing in that octahedron, we take the direct sum of the nondiagonal terms of the subsequent quadrangle, the morphisms being taken from the octahedron, with one minus sign inserted to ensure that the composition of two morphisms in the constructed triangle vanishes.

Remark 7 The triangles arising from X and from X̃ as described in [1, 1.1.13] are distinguished.

Proof. (^{1}) The morphism of short exact sequences in (Z/p^{6}) -mod

[figure: Z/p^{3}; •; ^{(^{1 p^{2} −p})}; Z/p ⊕ Z/p^{5} ⊕ Z/p^{3}; ^{(\begin{matrix}p & 0 \\ 0 & 1 \\ 1 & p \end{matrix})}; Z/p^{2} ⊕ Z/p^{4}; Z/p^{4} ⊕ Z/p^{2}; •; _{(\begin{matrix}p^{2} & 0 \\ 0 & p^{4} \end{matrix} )}; _{(\begin{matrix}−p \\ p^{2} \end{matrix})}; Z/p^{6} ⊕ Z/p^{6}; _{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})}; _{(\begin{matrix}0 & −p & 1 \\ 0 & 1 & 0 \end{matrix})}; Z/p^{2} ⊕ Z/p^{4}]

and the isomorphism of diagrams with coefficients in C

[figure: Z/p^{3}; ^{(^{1 p^{2} −p})}; Z/p⊕Z/p^{5}⊕Z/p^{3}; ^{(\begin{matrix}p & 0 \\ 0 & 1 \\ 1 & p \end{matrix})}; Z/p^{2}⊕Z/p^{4}; ^{(\begin{matrix}−p^{2} \\ p \end{matrix} )}; _{(\begin{matrix}1−p & 0 \\ 1 & 1 \end{matrix})}; ≀; Z/p^{3}; Z/p^{3}; _{(_{1 p^{2} −p})}; (Z/p⊕Z/p^{5})⊕Z/p^{3}; _{(\begin{matrix}p & 0 \\ 1 & 1 \\ 1 & p \end{matrix})}; Z/p^{2}⊕Z/p^{4}; _{(\begin{matrix}−p^{2} \\ p \end{matrix} )}; Z/p^{3}]

show one of the triangles mentioned in loc. cit. to be distinguished in X and in X̃.

The morphism of short exact sequences in (Z/p^{6}) -mod

[figure: Z/p^{2} ⊕ Z/p^{4}; •; ^{(\begin{matrix}1 & 0 & p \\ 0 & p & −1 \end{matrix})}; Z/p ⊕ Z/p^{5} ⊕ Z/p^{3}; ^{(\begin{matrix}−p^{2} \\ 1 \\ p \end{matrix} )}; Z/p^{3}; Z/p^{3}; •; p^{3}; _{(_{p p^{2}})}; Z/p^{6}; 1; _{(_{0 1 0})}; Z/p^{3}]

1Strictly speaking, we should reorder summands in the diagrams that follow; cf. §1.1. But then the proof would be more difficult to read.

and the isomorphism of diagrams with coefficients in C

[figure: Z/p^{3}; ^{(^{p p^{2}})}; Z/p^{2}⊕Z/p^{4}; ^{(\begin{matrix}1 & 0 & p \\ 0 & p & −1 \end{matrix})}; (Z/p⊕Z/p^{5})⊕Z/p^{3}; ^{(\begin{matrix}−p^{2} \\ 1 \\ p \end{matrix} )}; _{(\begin{matrix}1 & 0 & −p^{2} \\ 0 & 1 & 1 \\ 1 & 0 & 1+p \end{matrix})}; ≀; Z/p^{3}; Z/p^{3}; _{(_{p p^{2}})}; Z/p^{2}⊕Z/p^{4}; _{(\begin{matrix}1 & 0 & p \\ −1 & p & −1 \end{matrix})}; (Z/p⊕Z/p^{5})⊕Z/p^{3}; _{(\begin{matrix}−p^{2} \\ 1 \\ p \end{matrix} )}; Z/p^{3}]

show the other of the triangles mentioned in loc. cit. to be distinguished in X and in X̃.

A Transport of structure

We use the notation of [4, §1, §2].

A.1 Transport of a Heller triangulation

Concerning weakly abelian categories, see e.g. [4, §A.6.3]. Recall that an additive functor between weakly abelian categories is called subexact if it induces an exact functor on the Freyd categories; cf. [4, §1.2.1.3]. For instance, an equivalence is subexact.

Setup 8

Suppose given a Heller triangulated category (C, T, ϑ); cf. [4, Def. 1.5]. Suppose given a weakly abelian category C′ and an automorphism T′ on C′, called shift ; cf. [4, Def. A.26].

Assume given subexact functors C ^{\overline{}^{F}} C′ and C′ ^{\overline{}^{G}} C, and isotransformations 1_{C′} \overline{}^{ε}_{∼} GF and T′ G \overline{}^{σ}_{∼} G T.

Suppose given n ⩾ 0. By abuse of notation, we write F := \underline{F^{+}(Δ̄_{n}^{#})} : \underline{C^{+}(Δ̄_{n}^{#})} \overline{}\underline{C^{′+}(Δ̄_{n}^{#})} for the functor obtained by pointwise application of F.

Similarly, we write ε := \underline{ε^{+}(Δ̄_{n}^{#})} : \underline{(1_{C′})^{+}(Δ̄_{n}^{#})} ^{\overline{}^{∼}} \underline{(GF)^{+}(Δ̄_{n}^{#})} for the isotransformation obtained by pointwise application of ε.

More generally speaking, for notational convenience, induced functors of type \underline{A^{+}(Δ̄_{n}^{#})} will often be abbreviated by A, and induced transformations of type \underline{α^{+}(Δ̄_{n}^{#})} will often be abbreviated by α. For instance, given X ∈ Ob \underline{C^{+}(Δ̄_{n}^{#})}, we will allow ourselves to write X T = X\underline{T^{+}(Δ̄_{n}^{#})} ( = [X^{+1}]).

Given X′ ∈ Ob \underline{C^{′+}(Δ̄_{n}^{#})} = Ob C^{′+}(Δ̄_{n}^{#}), we define the isomorphism [X′]^{+1} \overline{}^{X′ϑ′_{n}}_{∼} [X^{′+1}] in \underline{C^{′+}(Δ̄_{n}^{#})} by the following commutative diagram.

[figure: [X′]^{+1}; X′ϑ′_{n}; ∼; [X′ε]^{+1}; ≀; [X^{′+1}]; [X^{′+1}]ε; ≀; [X′GF]^{+1}; [X^{′+1}]GF; ≀; X′σF; [X′G]^{+1}F; X′Gϑ_{n}F; ∼; [(X′G)^{+1}]F]

In other words, we let

X′ϑ′_{n} := ([X′ε]^{+1})(X′Gϑ_{n}F)(X′σ^{−}F)([X^{′+1}]ε^{−}) .

As a composite of isotransformations, (X′ϑ′_{n})_{X′ ∈ Ob \underline{C^{′+}(Δ̄_{n}^{#})}} is an isotransformation. Let ϑ′ := (ϑ′_{n})_{n⩾0}.

Lemma 9 The triple (C′, T′, ϑ′) is a Heller triangulated category.

Cf. [4, Def. 1.5]. We will say that ϑ′ is transported from (C, T, ϑ) via F and G. Strictly speaking, we should mention ε and σ here as well.

Proof. Suppose given m, n ⩾ 0, a periodic monotone map Δ̄_{n} ^{\overline{}^{q}} Δ̄_{m} and X′ ∈ Ob \underline{C^{′+}(Δ̄_{n}^{#})}. We claim that X′\underline{q}^{#}ϑ′_{m} = X′ϑ′_{n}\underline{q}^{#}. We have

\begin{aligned}X′\underline{q}^{#}ϑ′_{m} & = & ([X′\underline{q}^{#}ε]^{+1})(X′\underline{q}^{#}Gϑ_{m}F)(X′\underline{q}^{#}σ^{−}F)([X′\underline{q}^{#}^{+1}]ε^{−}) , \\ X′ϑ′_{n}\underline{q}^{#} & = & ([X′ε]^{+1}\underline{q}^{#})(X′Gϑ_{n}F \underline{q}^{#})(X′σ^{−}F \underline{q}^{#})([X^{′+1}]ε^{−}\underline{q}^{#}) . \end{aligned}

By respective pointwise definition, we have [X′\underline{q}^{#}ε]^{+1} = [X′ε]^{+1}\underline{q}^{#} (using that q is periodic), X′\underline{q}^{#}σ^{−}F = X′σ^{−}F \underline{q}^{#} and [X′\underline{q}^{#}^{+1}]ε^{−} = [X^{′+1}]ε^{−}\underline{q}^{#}. Moreover, since (C, T, ϑ) is Heller triangulated, we get

X′\underline{q}^{#}Gϑ_{m}F = X′G\underline{q}^{#}ϑ_{m}F = X′Gϑ_{n}\underline{q}^{#}F = X′Gϑ_{n}F \underline{q}^{#} .

This proves the claim.

Suppose given n ⩾ 0 and X′ ∈ Ob \underline{C^{′+}(Δ̄_{2n+1}^{#})}. We claim that X′\underline{𝔣}_{n}ϑ′_{n+1} = X′ϑ′_{2n+1}\underline{𝔣}_{n}. We have

\begin{aligned}X′\underline{𝔣}_{n}ϑ′_{n+1} & = & ([X′\underline{𝔣}_{n}ε]^{+1})(X′\underline{𝔣}_{n}Gϑ_{n+1}F)(X′\underline{𝔣}_{n}σ^{−}F)([X′\underline{𝔣}_{n}^{+1}]ε^{−}) , \\ X′ϑ′_{2n+1}\underline{𝔣}_{n} & = & ([X′ε]^{+1}\underline{𝔣}_{n})(X′Gϑ_{2n+1}F \underline{𝔣}_{n})(X′σ^{−}F \underline{𝔣}_{n})([X^{′+1}]ε^{−}\underline{𝔣}_{n}) . \end{aligned}

By additivity of F, G and T′ and by respective pointwise definition, we have [X′\underline{𝔣}_{n}ε]^{+1} = [X′ε]^{+1}\underline{𝔣}_{n} (using shiftcompatibility of \underline{𝔣}_{n}), X′\underline{𝔣}_{n}σ^{−}F = X′σ^{−}F \underline{𝔣}_{n} and [X′\underline{𝔣}_{n}^{+1}]ε^{−} = [X^{′+1}]ε^{−}\underline{𝔣}_{n}. Moreover, since (C, T, ϑ) is Heller triangulated, we get

X′\underline{𝔣}_{n}Gϑ_{n+1}F = X′G\underline{𝔣}_{n}ϑ_{n+1}F = X′Gϑ_{2n+1}\underline{𝔣}_{n}F = X′Gϑ_{2n+1}F \underline{𝔣}_{n} .

This proves the claim.

A.2 Detecting n-triangles

Setup 10

Suppose given a Heller triangulated category (C, T, ϑ); cf. [4, Def. 1.5]. Suppose given an additive category C′ and an automorphism T′ on C′, called shift.

Suppose given mutually inverse equivalences C ^{\overline{}^{F}} C′ and C′ ^{\overline{}^{G}} C. Note that G ⊣ F, whence there exist isotransformations 1_{C′} \overline{}^{ε}_{∼} GF and F G \overline{}^{η}_{∼} 1_{C} such that both (F ε)(ηF) = 1_{F} and (εG)(Gη) = 1_{G} hold. We fix such ε and η.

Suppose given an isotransformation T′ G \overline{}^{σ}_{∼} G T.

Note that C′ is weakly abelian, being equivalent to the weakly abelian category C.

Let ϑ′ be transported from (C, T, ϑ) via F and G.

That is, we let X′ϑ′_{n} := ([X′ε]^{+1})(X′Gϑ_{n}F)(X′σ^{−}F)([X^{′+1}]ε^{−}) for n ⩾ 0 and X′ ∈ Ob \underline{C^{′+}(Δ̄_{n}^{#})}, defining ϑ′ := (ϑ′_{n})_{n⩾0}.

By Lemma 9, the triple (C′, T′, ϑ′) is a Heller triangulated category.

Moreover, let

(T F \overline{}^{ρ}_{∼} F T′) = (T F \overline{}^{η^{−} T F}_{∼} F G T F \overline{}^{F σ^{−}F}_{∼} F T′ GF \overline{}^{F T′ ε^{−}}_{∼} F T′) .

Notation 11 Suppose given n ⩾ 0. Concerning the full subposet

Δ̄_{n}^{△▽} = {β/α ∈ Δ_{n}^{#} : 0 ⩽ α ⩽ β ⩽ 0^{+1}} ⊆ Δ̄_{n}^{#} ,

cf. [4, §2.5.1].

(1) Suppose given X′ ∈ Ob C^{′+, periodic}(Δ̄_{n}^{#}), where periodic means [X′]^{+1} = [X^{′+1}]; cf. [4, §2.5.3]. Consider the diagram X′G|Δ̄_{n}^{△▽}. Denote by X′G|σ ; Δ̄_{n}^{△▽} ∈ Ob C^{′+}(Δ̄_{n}^{#}) the diagram X′G|Δ̄_{n}^{△▽} with (X′G)_{0^{+1}/i} = X′_{0^{+1}/i}G = X′_{i/0} T′ G isomorphically replaced via X′_{i/0} σ by X′_{i/0}G T. Denote by X′G^{σ} ∈ Ob C^{+, periodic}(Δ̄_{n}^{#}) its periodic prolongation, characterised by X′G^{σ}|Δ̄_{n}^{△▽} = X′G|σ ; Δ̄_{n}^{△▽}; cf. [4, §2.5.3]. Using, for k ⩾ 0,

([X′G]^{+k} ^{\overline{}^{∼}} [X′G^{+1}]^{+(k−1)} ^{\overline{}^{∼}} ⋯ ^{\overline{}^{∼}} [X′G^{+k}])|Δ̄_{n}^{△▽} ,

given by

X′_{(j/i)^{+k}}G = X′_{j/i} T^{′k} G T^{0}  \overline{}^{T^{′k−1} σ T^{0}}_{∼} X′_{j/i} T^{′k−1} G T^{1}  \overline{}^{T^{′k−2} σ T^{1}}_{∼}  ⋯  \overline{}^{T^{′0} σ T^{k−1}}_{∼} X′_{j/i} T^{′0} G T^{k}

at j/i for 0 ⩽ i ⩽ j ⩽ 0^{+1}, and similarly for k ⩽ 0, using T^{′−} G \overline{}^{T^{′−} σ^{−} T^{−}}_{∼} G T^{−}, we obtain an isomorphism

X′G \overline{}^{φ}_{∼} X′G^{σ} in C^{+}(Δ̄_{n}^{#}) such that φ_{i/0} = 1_{X′_{i/0}G} and φ_{0^{+1}/i} = X′_{i/0} σ for i ∈ [1, n].

(2) Suppose given X ∈ Ob C^{+, periodic}(Δ̄_{n}^{#}). Consider the diagram XF |Δ̄_{n}^{△▽}. Denote by XF |ρ ; Δ̄_{n}^{△▽} ∈ Ob C^{+}(Δ̄_{n}^{#}) the diagram XF |Δ̄_{n}^{△▽} with (XF)_{0^{+1}/i} = X_{0^{+1}/i}F = X_{i/0} T F isomorphically replaced via X_{i/0} ρ by X_{i/0}F T′. Denote by XF^{ρ} ∈ Ob C^{′+, periodic}(Δ̄_{n}^{#}) its periodic prolongation, characterised by XF^{ρ}|Δ̄_{n}^{△▽} =

XF |ρ ; Δ̄_{n}^{△▽}; cf. [4, §2.5.3].

Similarly as in (1), we have an isomorphism XF \overline{}^{ψ}_{∼} XF^{ρ} in C^{′+}(Δ̄_{n}^{#}) such that ψ_{i/0} = 1_{X_{i/0}F} and ψ_{0^{+1}/i} = X_{i/0} ρ for i ∈ [1, n].

Lemma 12 Suppose given n ⩾ 0.

(1) Suppose given X′ ∈ Ob C^{′+, periodic}(Δ̄_{n}^{#}). Then X′ is an n-triangle if and only if X′G^{σ} is an n-triangle.

(2) Suppose given X ∈ Ob C^{+, periodic}(Δ̄_{n}^{#}). Then X is an n-triangle if and only if XF^{ρ} is an n-triangle.

Cf. [4, Def. 1.5.(ii.2)].

Proof. Ad (1). Since ϑ_{n} is a transformation, there exists a commutative quadrangle

\xymatrix{[X′G^{σ}]^{+1} \ar[r]^{X′G^{σ}ϑ_{n}}_{∼} & [(X′G^{σ})^{+1}] \\ [X′G]^{+1} \ar[r]^{X′Gϑ_{n}}_{∼} \ar[u]_{[φ]^{+1}}^{≀} & [(X′G)^{+1}] \ar[u]_{[φ^{+1}]}^{≀}}

in \underline{C^{+}(Δ̄_{n}^{#})}. Therefore, X′G^{σ} is an n-triangle if and only if [φ]^{+1} = (X′Gϑ_{n})[φ^{+1}]. By [4, Prop. 2.6], this equation is equivalent to [φ]^{+1}|Δ̇_{n} = (X′Gϑ_{n})|Δ̇_{n}[φ^{+1}]|Δ̇_{n}; cf. [4, §2.1.1]; in other words, to

X′σ|Δ̇_{n} = X′Gϑ_{n}|Δ̇_{n}

as morphisms from X′ T′ G|Δ̇_{n} to X′G T |Δ̇_{n} in \underline{C(Δ̇_{n})}. This, in turn, is equivalent to X′σ = X′Gϑ_{n} as morphisms from X′ T′ G to X′G T in \underline{C^{+}(Δ̄_{n}^{#})} by [4, Prop. 2.6].

Now X′ being an n-triangle is equivalent to X′ϑ′_{n} = 1; i.e. to

([X′ε]^{+1})(X′Gϑ_{n}F)(X′σ^{−}F)([X^{′+1}]ε^{−}) = 1 .

Since [X′]^{+1} = [X^{′+1}], we have [X′ε]^{+1} = [X′]^{+1}ε = [X^{′+1}]ε, whence this equation is equivalent to (X′Gϑ_{n}F)(X′σ^{−}F) = 1. Since F is an equivalence, this amounts to (X′Gϑ_{n})(X′σ^{−}) = 1, as was to be shown.

Ad (2). Since ϑ′_{n} is a transformation, there exists a commutative quadrangle

\xymatrix{[XF^{ρ}]^{+1} \ar[r]^{XF^{ρ}ϑ′_{n}}_{∼} & [(XF^{ρ})^{+1}] \\ [XF]^{+1} \ar[r]^{XF ϑ′_{n}}_{∼} \ar[u]_{[ψ]^{+1}}^{≀} & [(XF)^{+1}] \ar[u]_{[ψ^{+1}]}^{≀}}

in \underline{C^{′+}(Δ̄_{n}^{#})}. Therefore, XF^{ρ} is an n-triangle if and only if [ψ]^{+1} = (XF ϑ′_{n})[ψ^{+1}]. By [4, Prop. 2.6], this equation is equivalent to [ψ]^{+1}|Δ̇_{n} = (XF ϑ′_{n})|Δ̇_{n}[ψ^{+1}]|Δ̇_{n}; in other words, to

Xρ|Δ̇_{n} = XF ϑ′_{n}|Δ̇_{n}

as morphisms from X T F |Δ̇_{n} to XF T′ |Δ̇_{n} in \underline{C′(Δ̇_{n})}. This, in turn, is equivalent to Xρ = XF ϑ′_{n} as morphisms from X T F to XF T′ in \underline{C^{′+}(Δ̄_{n}^{#})} by [4, Prop. 2.6]. Which amounts to

(Xη^{−} T F)(XF σ^{−}F)(XF T′ ε^{−}) = ([XF ε]^{+1})(XF Gϑ_{n}F)(XF σ^{−}F)([XF^{+1}]ε^{−}) ;

i.e. to

Xη^{−} T F = ([Xη^{−}F]^{+1})(XF Gϑ_{n}F) .

Since [Xη^{−}F]^{+1} = [Xη^{−}]^{+1}F and since ϑ_{n} is a transformation, the right hand side equals (Xϑ_{n}F)([(Xη^{−})^{+1}]F), and therefore we can continue the string of equivalent assertions with

Xη^{−} T F = (Xϑ_{n}F)([(Xη^{−})^{+1}]F) ;

i.e. with Xϑ_{n}F = 1; i.e. with Xϑ_{n} = 1; i.e. with X being an n-triangle.

Definition 13 A Heller triangulated category (C, T, ϑ) is said to be closed if every morphism X ^{\overline{}^{f}} Y therein can be completed to a 2-triangle; i.e. if for all morphisms X ^{\overline{}^{f}} Y in C, there exists U ∈ Ob C^{+, ϑ=1}(Δ̄_{2}^{#}) with (X ^{\overline{}^{f}} Y) = (U_{1/0} ^{\overline{}^{u}} U_{2/0}). If this is the case, then also the Heller triangulation ϑ is called closed.

For instance, a Heller triangulated category whose idempotents split is closed; cf. [4, Prop. 3.6].

Recall that ϑ′ is transported from (C, T, ϑ) via F and G.

Lemma 14 If (C, T, ϑ) is a closed Heller triangulated category, then (C′, T′, ϑ′) is a closed Heller triangulated category.

Proof. By Lemma 9, it remains to prove closedness of (C′, T′, ϑ′). Suppose given X′ ^{\overline{}^{u′}} Y′ in C′. We have to prove that it can be prolonged to a 2-triangle. Using closedness of (C, T, ϑ), we find a 2-triangle

X′G ^{\overline{}^{u′G}} Y′G ^{\overline{}^{v}} Z ^{\overline{}^{w}} X′G T .

We claim that

M′ := (X′ ^{\overline{}^{u′}} Y′ ^{\overline{}^{(Y′ε)(vF)}} ZF ^{\overline{}^{(wF)(X′σ^{−}F)(X′ T′ ε^{−})}} X′ T′)

is a 2-triangle in C′. By Lemma 12.(1), it suffices to show that M′G^{σ} is a 2-triangle in C. Consider the periodic isomorphism with upper row M′G^{σ} and lower row a 2-triangle

[figure: X′G; u′G; Y′G; (Y′εG)(vF G); ZF G; (wF G)(X′σ^{−}F G)(X′ T′ ε^{−}G)(X′σ); X′G T; X′G; u′G; Y′G; v; Z; w; ≀; Zη^{−}; X′G T .]

In fact, we have (Y′εG)(vF G) = (Y′Gη^{−})(vF G) = v(Zη^{−}) and

(Zη^{−})(wF G)(X′σ^{−}F G)(X′ T′ ε^{−}G)(X′σ) = w(X′G T η^{−})(X′σ^{−}F G)(X′ T′ ε^{−}G)(X′σ)
= w(X′σ^{−})(X′ T′ Gη^{−})(X′ T′ ε^{−}G)(X′σ)
= w(X′σ^{−})(X′σ)
= w .

This shows that M′G^{σ} is a 2-triangle; cf. [4, Lem. 3.4.(4)]. This proves the claim. Remark 15 Suppose given n ⩾ 0.

(1) Given X ∈ Ob C^{+}(Δ̄_{n}^{#}), we have (Xϑ_{n}F)(Xρ) = XF ϑ′_{n} in \underline{C^{′+}(Δ̄_{n}^{#})}.

(2) Given X′ ∈ Ob C^{′+}(Δ̄_{n}^{#}), we have (X′ϑ′_{n}G)(X′σ) = X′Gϑ_{n} in \underline{C^{+}(Δ̄_{n}^{#})}. Proof. Ad (1). We have

(Xϑ_{n}F)(Xρ) = (Xϑ_{n}F)(Xη^{−} T F)(XF σ^{−}F)(XF T′ ε^{−})
= (Xϑ_{n}F)([(Xη^{−})^{+1}]F)(XF σ^{−}F)([XF^{+1}]ε^{−})
= ([Xη^{−}]^{+1}F)(XF Gϑ_{n}F)(XF σ^{−}F)([XF^{+1}]ε^{−})
= ([Xη^{−}F]^{+1})(XF Gϑ_{n}F)(XF σ^{−}F)([XF^{+1}]ε^{−})
= ([XF ε]^{+1})(XF Gϑ_{n}F)(XF σ^{−}F)([XF^{+1}]ε^{−})
= XF ϑ′_{n} .

Ad (2). We have

(X′ϑ′_{n}G)(X′σ) = ([X′ε]^{+1}G)(X′Gϑ_{n}F G)(X′σ^{−}F G)([X^{′+1}]ε^{−}G)(X′σ)
= ([X′ε]^{+1}G)(X′Gϑ_{n}F G)(X′σ^{−}F G)([X^{′+1}]Gη)(X′σ)
= ([X′ε]^{+1}G)(X′Gϑ_{n}F G)([X′G^{+1}]η)(X′σ^{−})(X′σ)
= ([X′εG]^{+1})(X′Gϑ_{n}F G)([X′G^{+1}]η)
= ([X′Gη^{−}]^{+1})(X′Gϑ_{n}F G)([X′G^{+1}]η)
= ([X′G]^{+1}η^{−})(X′Gϑ_{n}F G)([X′G^{+1}]η)
= (X′Gϑ_{n})([X′G^{+1}]η^{−})([X′G^{+1}]η)
= X′Gϑ_{n} .

A.3 Some lemmata

Let E be a Frobenius category, let B ⊆ E be its full subcategory of bijective objects; cf. e.g. [4, Def. A.5]. We use the notations and conventions of [4, §A.2.3], in particular those of [4, Ex. A.6.(2)]. Let n ⩾ 0. Let E ⊆ Δ̄_{n}^{#} be a convex full subposet, i.e. whenever ξ, ζ ∈ E and λ ∈ Δ̄_{n}^{#} such that ξ ⩽ λ ⩽ ζ, then λ ∈ E; cf. [4, §2.2.2.1]. For instance, Δ_{n}^{△▽} ⊆ Δ̄_{n}^{#} is such a convex full subposet; cf. Notation 11. A pure square in E is a commutative quadrangle (A, B, C, D) with pure short exact diagonal sequence (A, B ⊕ C, D); cf. [4, §A.4]. Denote by E^{□}(E) ⊆ E(E) the full subcategory determined by

Ob E^{□}(E) := \{ X ∈ Ob E(E) : \begin{matrix}1) & X_{α/α} is in Ob B for all α ∈ Δ̄_{n} such that α/α ∈ E, and \\ & X_{α^{+1}/α} is in Ob B for all α ∈ Δ̄_{n} such that α^{+1}/α ∈ E. \\ 2) & For all δ^{−1} ⩽ α ⩽ β ⩽ γ ⩽ δ ⩽ α^{+1} in Δ̄_{n} \\ & such that γ/α, γ/β, δ/α and δ/β are in E, \\ & the quadrangle \\ X_{γ/β} & x & [figure] & X_{δ/β} & X_{γ/α} & x & [figure] & x & [figure] & □ & X_{δ/α} & x & [figure] \\ & is a pure square. \end{matrix} \} .

A particular case of this definition has been considered in [4, §4.1].

A.3.1 Cleaning the diagonal

Lemma 16 Suppose given X ∈ Ob E^{□}(Δ̄_{n}^{△▽}). Suppose given β ∈ Δ̄_{n} such that 0 ⩽ β ⩽ 0^{+1}. There exists X̃ ∈ Ob E^{□}(Δ̄_{n}^{△▽}) such that the following conditions (1a, 1b, 2) hold.

(1a) We have X̃_{α/α} = X_{α/α} for 0 ⩽ α ⩽ 0^{+1} such that α ≠ β. (1b) We have X̃_{β/β} = 0.

(2) There exists an isomorphism X̃ ^{\overline{}^{∼}} X in \underline{E}(Δ̄_{n}^{△▽}).

Proof. Pars pro toto, we consider the case n = 4 and β = 2. We display X as follows.

[figure: X_{0^{+1}/0^{+1}}; X_{4/4}; x; X_{0^{+1}/4}; x; X_{3/3}; x; X_{4/3}; x; x; □; X_{0^{+1}/3}; x; X_{2/2}; x; X_{3/2}; x; x; □; X_{4/2}; x; x; □; X_{0^{+1}/2}; x; X_{1/1}; x; X_{2/1}; x; x; □; X_{3/1}; x; x; □; X_{4/1}; x; x; □; X_{0^{+1}/1}; x; X_{0/0}; x; X_{1/0}; x; x; □; X_{2/0}; x; x; □; X_{3/0}; x; x; □; X_{4/0}; x; x; □; X_{0^{+1}/0}; x]

Set X̃ to be the following diagram.

[figure: X_{0^{+1}/0^{+1}}; X_{4/4}; x; X_{0^{+1}/4}; x; X_{3/3}; x; X_{4/3}; x; x; □; X_{0^{+1}/3}; x; 0; X_{3/2}; x; x; □; X_{4/2}; x; x; □; X_{0^{+1}/2}; x; X_{1/1}; x; X_{2/1}; _{(_{x x})}; □; X_{3/1}⊕X_{2/2}; _{( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}x \\ −x \end{matrix} )}; □; X_{4/1}⊕X_{2/2}; _{( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}x \\ −x \end{matrix} )}; □; X_{0^{+1}/1}⊕X_{2/2}; _{( \begin{matrix}x \\ −x \end{matrix} )}; X_{0/0}; x; X_{1/0}; x; x; □; X_{2/0}; _{(_{x x})}; x; □; X_{3/0}⊕X_{2/2}; _{( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} )}; □; X_{4/0} ⊕ X_{2/2}; _{( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} )}; _{( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} )}; □; X_{0^{+1}/0} ⊕ X_{2/2}; _{( \begin{matrix}x & 0 \\ 0 & 1 \end{matrix} )}]

Using the Gabriel-Quillen-Laumon embedding theorem, we see that X̃ is actually an object of E^{□}(Δ̄_{n}^{△▽}); cf. [4, §A.2.2; Lem. A.11].

Since X_{2/2} is bijective, inserting the zero morphism on all copies of X_{2/2} and the identity on all other summands yields an isomorphism X̃ ^{\overline{}^{∼}} X in \underline{E}(Δ̄_{n}^{△▽}).

Lemma 17 Suppose given X ∈ Ob E^{□}(Δ̄_{n}^{△▽}).

There exists X′ ∈ Ob E^{□}(Δ̄_{n}^{△▽}) such that the following conditions (1, 2) hold.

(1) We have X′_{α/α} = 0 for all α ∈ Δ̄_{n} such that 0 ⩽ α ⩽ 0^{+1}.

(2) There exists an isomorphism X′ ^{\overline{}^{∼}} X in \underline{E}(Δ̄_{n}^{△▽}).

Proof. This follows by application of Lemma 16 consecutively for β = 0, β = 1, …, β = 0^{+1}.

A.3.2 Horseshoe lemma

Recall that B^{ac} denotes the category of purely acyclic complexes with entries in B, i.e. of complexes with entries in B that decompose into pure short exact sequences in E; cf. [4, §A.2.3].

Suppose given Y ∈ Ob E. An object B of B^{ac} is called a (both-sided) bijective resolution of Y if Y is isomorphic to Im(B^{0} \overline{}B^{1}). Note that a bijective resolution of a bijective object is split acyclic.

We have a full and dense functor (^{2})

\begin{matrix}B^{ac} & ^{\overline{}^{F̂}} & E \\ B & \overline{} & Im(B^{0} \overline{}B^{1}) \end{matrix}

We make the additional convention that if the image factorisation of a pure morphism d in E is chosen to be d = d̄ḋ, then we choose the image factorisation −d = d̄(−ḋ) over the same image object.

2A functor induced by F̂ will play the role of F of Setup 10; cf. §A.4.2.2 below.

Pointwise application yields a functor B^{ac}(Δ̄_{n}^{△▽}) ^{\overline{}^{F̂}} E(Δ̄_{n}^{△▽}), which is an abuse of notation. Suppose given X ∈ Ob E^{□}(Δ̄_{n}^{△▽}) such that X_{α/α} = 0 for all 0 ⩽ α ⩽ 0^{+1}. In particular, X_{β/α} \overline{}X_{γ/α} \overline{}X_{γ/β} is a pure short exact sequence for 0 ⩽ α ⩽ β ⩽ γ ⩽ 0^{+1}. Recall that for n ∈ Δ̄_{n}, we have n + 1 = 0^{+1}; cf. [4, §1.1].

Lemma 18 Suppose given a bijective resolution C_{α+1/α} of X_{α+1/α} for all α ∈ Δ̄_{n} such that 0 ⩽ α ⩽ n. Then there exists B ∈ Ob(B^{ac})^{□}(Δ̄_{n}^{△▽}) such that (1, 2, 3) hold.

(1) We have BF̂ ≃ X in E^{□}(Δ̄_{n}^{#}).

(2) We have B_{α/α} = 0 for all 0 ⩽ α ⩽ 0^{+1}.

(3) We have B_{α+1/α} = C_{α+1/α} for all α ∈ Δ̄_{n} such that 0 ⩽ α ⩽ n.

If n = 2, and if we restrict to {1/0, 2/0, 2/1} ⊆ Δ̄_{2}^{△▽}, we recover the classical horseshoe lemma in its bothsided Frobenius category variant.

Proof. For 0 ⩽ α ⩽ n, we denote (C_{α+1/α}^{0} ^{\overline{}^{d̄}} X_{α+1/α}) := (C_{α+1/α}^{0} \overline{}C_{α+1/α}F̂ ^{\overline{}^{∼}} X_{α+1/α}).

By duality and by induction, it suffices to find a morphism Y \overline{}X in E^{□}(Δ̄_{n}^{△▽}) such that (i, ii, iii) hold.

(i) We have Y_{β/α} ∈ Ob B for all 0 ⩽ α ⩽ β ⩽ 0^{+1}.

(ii) We have Y_{α/α} = 0 for all 0 ⩽ α ⩽ 0^{+1}. (iii) We have (Y_{α+1/α} \overline{}X_{α+1/α}) = (C_{α+1/α}^{0} ^{\overline{}^{d̄}} X_{α+1/α}) for all α ∈ Δ̄_{n} such that 0 ⩽ α ⩽ n.

Note that any morphism Y \overline{}X fulfilling (i, ii, iii) consists pointwise of pure epimorphisms, and that the kernel of such a morphism Y \overline{}X taken in E(Δ̄_{n}^{△▽}) is in Ob E^{□}(Δ̄_{n}^{△▽}). To construct Y \overline{}X, we let

Y_{γ/α} := ⨁_{β ∈ Δ̄_{n}, α⩽β<γ} C_{β+1/β}^{0}

for 0 ⩽ α ⩽ γ ⩽ 0^{+1}. For γ/α ⩽ γ′/α′, the diagram morphism Y_{γ/α} \overline{}Y_{γ′/α′} is stipulated to be identical on the summands C_{β+1/β}^{0} with α′ ⩽ β < γ and zero elsewhere. This yields Y ∈ Ob E^{□}(Δ̄_{n}^{△▽}). Given 0 ⩽ α ⩽ γ ⩽ 0^{+1}, we let Y_{γ/α} \overline{}X_{γ/α} be defined as follows. For 0 ⩽ β ⩽ n, we choose Y_{β+1/β} ^{\overline{}^{e}} X_{β+1/0} such that

(Y_{β+1/β} ^{\overline{}^{e}} X_{β+1/0} ^{\overline{}^{x}} X_{β+1/β}) = (Y_{β+1/β} ^{\overline{}^{d̄}} X_{β+1/β}) .

The component of the morphism

(Y_{γ/α} \overline{}X_{γ/α}) = ( ⨁_{β ∈ Δ̄_{n}, α⩽β<γ} C_{β+1/β}^{0} \overline{}X_{γ/α})

at β is defined to be the composite

(C_{β+1/β}^{0} \overline{}X_{γ/α}) := (C_{β+1/β}^{0} ^{\overline{}^{e}} X_{β+1/0} ^{\overline{}^{x}} X_{γ/α}) .

A.3.3 Applying F̂ to a standard pure short exact sequence

Recall that for X ∈ Ob B^{ac}, we have chosen, in a functorial manner, a pure short exact sequence X [figure] X I \overline{}X T with a bijective middle term X I, where the letter I stands for “injective”; cf. [4, §A.2.3].

Lemma 19 Suppose given X ∈ Ob B^{ac}. There exists an isomorphism of pure short exact sequences in E as follows.

[figure: XF̂; •; X I F̂; ≀; X T F̂; XF̂; •; X^{1}; X T F̂]

Therein, the upper sequence results from an application of F̂ to the pure short exact sequence X [figure] X I \overline{}X T in B^{ac}. The lower sequence is taken from the purely acyclic complex X.

Proof. Consider the following part of the pure short exact sequence X [figure] X I \overline{}X T in B^{ac}; cf. [4, Ex. A.6].

[figure: X^{2}; •; ^{(^{1 d})}; X^{2} ⊕ X^{3}; ^{( \begin{matrix}−d \\ 1 \end{matrix} )}; X^{3}; X T F̂; •; ḋ; X^{1}; •; _{(_{1 d})}; d; d̄; X^{1} ⊕ X^{2}; _{( \begin{matrix}−d \\ 1 \end{matrix} )}; _{( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )}; X^{2}; −d; X I F̂; •; δ̇; κ^{−}d̄; XF̂; •; ḋ; •; ḋ; •; ḋκ; X^{1}; •; _{(_{1 0} )}; d̄; κ; ∼; X T F̂; •; −ḋ; X^{0}; •; _{(_{1 d})}; d; d̄; X^{0} ⊕ X^{1}; _{( \begin{matrix}−d \\ 1 \end{matrix} )}; _{(\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})}; _{( \begin{matrix}0 \\ 1 \end{matrix} )}; δ̄; X^{1}; −d; d̄]

We have added the image factorisations X^{0} ^{\overline{}^{d̄}} XF̂ [figure: ḋ] X^{1} and X^{0}⊕X^{1} ^{\overline{}^{δ̄}} X I F̂ [figure: δ̇] X^{1}⊕X^{2} of the respective differentials, resulting from an application of F̂. Factoring the differential of X T as

(X^{1} ^{\overline{}^{−d}} X^{2}) = (X^{1} ^{\overline{}^{d̄}} X T F̂ [figure: −ḋ] X^{2})

follows the additional convention made above.

Moreover, we have added the image factorisation X^{0}⊕X^{1} ^{\overline{}^{( \begin{matrix}0 \\ 1 \end{matrix} )}} X^{1} [figure: ^{(^{1 0} )}] X^{1}⊕X^{2} and, accordingly, the isomorphism X^{1} \overline{}^{κ}_{∼} X I F̂ that satisfies κδ̇ = (^{1 0}) and (\begin{matrix}0 \\ 1 \end{matrix}) κ = δ̄.

The horizontal pure short exact sequence XF̂ [figure: ḋ] X^{1} ^{\overline{}^{d̄}} X T F̂ lets all four arising parallelograms commute.

Now the sequence X [figure] X I \overline{}X T maps to XF̂ ^{\overline{}^{ḋκ}} X I F̂ ^{\overline{}^{κ^{−}d̄}} X T F̂, for the commutativities

(ḋκ)δ̇ = ḋ(^{1 d}) and δ̄(κ^{−}d̄) = (\begin{matrix}−d \\ 1 \end{matrix}) d̄ hold.

In particular, the sequence (X [figure] X I \overline{}X T)F̂ actually is purely short exact.

A.4 Stable vs. classically stable

Let E be a Frobenius category, let B ⊆ E be its full subcategory of bijective objects.

A.4.1 n-triangles in the stable category

Recall that B^{ac} denotes the category of purely acyclic complexes with entries in B; cf. §A.3.2. Let B^{sp ac} ⊆ B^{ac} denote the subcategory of split acyclic complexes. Let \underline{\underline{E}} = B^{ac}/B^{sp ac} denote the stable category of E; cf. [4, Def. A.7]. Let T be the automorphism on \underline{\underline{E}} that shifts a complex to the left by one position, inserting signs; cf. [4, Ex. A.6.(1)]. Then (\underline{\underline{E}}, T) carries a Heller triangulation ϑ; cf. [4, Cor. 4.7]. In fact, we may, and will, choose the tuple of isotransformations ϑ = (ϑ_{n})_{n⩾0} constructed in the proof of [4, Th. 4.6].

Suppose given n ⩾ 0 and X ∈ Ob(B^{ac})^{□}(Δ̄_{n}^{#}); cf. [4, §4.1] or §A.3. Now X maps to an object X ∈ Ob \underline{\underline{\underline{E}}^{+}(Δ̄_{n}^{#})};

cf. [4, Lem. A.29]. Thus we have an isomorphism [X]^{+1} ^{\overline{}^{Xϑ_{n}}} [X^{+1}] in \underline{\underline{\underline{E}}^{+}(Δ̄_{n}^{#})}. By the construction in the

proof of [4, Th. 4.6], there is a representative [X]^{+1} ^{\overline{}^{Xθ}} [X^{+1}] in \underline{\underline{E}}^{+}(Δ̄_{n}^{#}) of Xϑ_{n} such that in particular, there exists a morphism of pure short exact sequences

[figure: X_{i/0}; •; ^{(^{x x})}; X_{i/i}⊕X_{0^{+1}/0}; ^{( \begin{matrix}x \\ −x \end{matrix} )}; X_{0^{+1}/i}; Xθ̂_{i/0}; X_{i/0}; •; X_{i/0} I; X_{i/0}^{+1}]

for each i ∈ [1, n]; where Xθ̂_{i/0} is a representative in B^{ac} for the morphism Xθ_{i/0} in \underline{\underline{E}}; where the upper pure short exact sequence stems from the diagram X; and where the lower pure short exact sequence is the standard one as in [4, Ex. A.6.(1)]. In particular, Xθ_{i/0} is an isomorphism in \underline{\underline{E}}.

Let X^{ϑ} ∈ Ob \underline{\underline{E}}^{+, periodic}(Δ̄_{n}^{#}) be defined as periodic prolongation of the image of the diagram X|Δ̄_{n}^{△▽} in Ob \underline{\underline{E}}^{+}(Δ̄_{n}^{△▽}) with X_{0^{+1}/i} isomorphically replaced via Xθ_{i/0} by X_{i/0}^{+1} for all i ∈ [1, n]. For short, the rightmost column of the image of X|Δ̄_{n}^{△▽} becomes standardised; cf. [4, §2.1.3]. Using

([X]^{+k} ^{\overline{}^{∼}} [X^{+1}]^{+(k−1)} ^{\overline{}^{∼}} ⋯ ^{\overline{}^{∼}} [X^{+k}])|Δ̄_{n}^{△▽}

for k ⩾ 0, and similarly for k ⩽ 0, we obtain an isomorphism X ^{\overline{}^{ω}} X^{ϑ} in \underline{\underline{E}}^{+}(Δ̄_{n}^{#}) such that ω_{i/0} = 1_{X_{i/0}} and ω_{0^{+1}/i} = Xθ_{i/0} for i ∈ [1, n]; cf. Notation 11.(1).

Lemma 20 Given n ⩾ 0 and X ∈ Ob(B^{ac})^{□}(Δ̄_{n}^{#}), the periodic n-pretriangle X^{ϑ} is an n-triangle.

The following proof is similar to the proof of Lemma 12.

Proof. We have to show that X^{ϑ}ϑ_{n} = 1; cf. [4, Def. 1.5.(ii.2)]. Since ϑ_{n} is a transformation, we have a commutative quadrangle

\xymatrix{[X^{ϑ}]^{+1} \ar[r]^{X^{ϑ}ϑ_{n}}_{∼} & [(X^{ϑ})^{+1}] \\ [X]^{+1} \ar[r]^{Xϑ_{n}}_{∼} \ar[u]_{[ω]^{+1}}^{≀} & [X^{+1}] \ar[u]_{[ω^{+1}]}^{≀}}

in \underline{\underline{\underline{E}}^{+}(Δ̄_{n}^{#})}. So we have to show that (Xϑ_{n})[ω^{+1}] = [ω]^{+1}. By [4, Prop. 2.6], it suffices to show that (Xϑ_{n})|Δ̇_{n}[ω^{+1}]|Δ̇_{n} = [ω]^{+1}|_{Δ̇_{n}}. Now [ω^{+1}]|Δ̇_{n} = 1_{[X^{+1}]|Δ̇n} and, by construction, (Xϑ_{n})|Δ̇_{n} = [ω]^{+1}|Δ̇_{n}.

Corollary 21 The Heller triangulated category (\underline{\underline{E}}, T, ϑ) is closed.

Cf. Definition 13. Proof. We can extend any morphism X_{1/0} \overline{}X_{2/0} of B^{ac} to an object of (B^{ac})^{□}(Δ̄_{2}^{#}) by choosing X_{1/0} [figure] X_{1/1} with X_{1/1} bijective and by choosing X_{0^{+1}/0} = 0, then forming pushouts, then choosing X_{2/1} [figure] X_{2/2} with X_{2/2} bijective, etc. Dually in the other direction. Then we apply Lemma 20.

A.4.2 The classical stable category under an additional hypothesis

A.4.2.1 The hypothesis

Let \underline{E} := E/B denote the classical stable category of E. Suppose given a set D of distinguished pure short exact sequences in E such that the following conditions hold.

(i) The middle term of each distinguished pure short exact sequence is bijective.

(ii) For all X ∈ Ob E, there exists a unique distinguished pure short exact sequence with kernel term X. (iii) For all X ∈ Ob E, there exists a unique distinguished pure short exact sequence with cokernel term X.

A.4.2.2 Consequences

We shall define an endofunctor T′ of \underline{E}. On objects. Given X ∈ Ob \underline{E} = Ob E, there exists a unique distinguished pure short exact sequence with kernel term X. Let X T′ be the cokernel term of this sequence. On morphisms. The image under T′ of the residue class in \underline{E} of a morphism X ^{\overline{}^{f}} Y in E is represented by the morphism X T′ ^{\overline{}^{g}} Y T′ in E if there exists a morphism of distinguished pure short exact sequences as follows.

[figure: X; •; f; B; X T′; g; Y; •; C; Y T′]

Then T′ is an automorphism of \underline{E}; i.e. there exists an inverse T^{′−}, constructed dually, such that T′ T^{′−} = 1\underline{E} and T^{′−} T′ = 1\underline{E}. As usual, we shall write X^{+1} := X T′ for X ∈ Ob \underline{E}; etc. The functor

\begin{matrix}\underline{\underline{E}} & ^{\overline{}^{F}} & \underline{E} \\ B & \overline{} & Im(B^{0} \overline{}B^{1}) \end{matrix}

induced by F̂ is an equivalence; cf. §A.3.2, [4, Lem. A.1]. Splicing purely acyclic complexes from distinguished

pure short exact sequences, we obtain an inverse equivalence \underline{\underline{E}} ^{\overline{}^{G}} \underline{E}. Define T′ G ^{\overline{}^{σ}} G T at Y ∈ Ob \underline{E} = Ob E by letting

(Y σ^{i}) := (−1_{(Y G)^{i+1}})^{i} .

Note that Y σF = 1_{Y T′ GF} = 1_{Y G T F}; cf. §A.3.2.

Suppose given Y ∈ Ob E. We have a commutative diagram

[figure: (Y G)^{1}; Y; ∼; •; Y GF; •; (Y G)^{0}; ,]

consisting of two image factorisations of the differential (Y G)^{0} \overline{}(Y G)^{1} and the induced isomorphism between the images Y ^{\overline{}^{∼}} Y GF that makes the upper and the lower triangle in this diagram commute.

The residue class in \underline{E} of this induced morphism Y ^{\overline{}^{∼}} Y GF shall be denoted by Y \overline{}^{Y ε}_{∼} Y GF. Letting Y

vary, this gives rise to an isotransformation 1\underline{E} \overline{}^{ε}_{∼} GF. Since ε is a transformation, we have (Y ε)(Y GF ε) =

(Y ε)(Y εGF), whence GF ε = εGF. Thus there is an isotransformation F G \overline{}^{η}_{∼} 1\underline{\underline{E}} such that (εG)(Gη) = 1_{G} and (F ε)(ηF) = 1_{F}. Namely, for η we may take the inverse image under F of F ε^{−}.

So we are in the situation of Setup 10 of §A.2. Define (T F ^{\overline{}^{ρ}} F T′) as in §A.2.

Proposition 22

(1) By transport from (\underline{\underline{E}}, T, ϑ) via F and G, we obtain a closed Heller triangulation ϑ′ on (\underline{E}, T′).

(2) Suppose given X′ ∈ Ob \underline{E}^{+, periodic}(Δ̄_{n}^{#}). Then X′ is an n-triangle if and only if X′G^{σ} is an n-triangle.

(3) Suppose given X ∈ Ob \underline{\underline{E}}^{+, periodic}(Δ̄_{n}^{#}). Then X is an n-triangle if and only if XF^{ρ} is an n-triangle.

Proof. Assertion (1) follows by Lemmata 9 and 14; cf. Corollary 21. Assertions (2, 3) follow by Lemma 12.

Recall that XF = XF̂ in Ob \underline{E} = Ob E for X ∈ Ob \underline{\underline{E}} = Ob B^{ac}.

Lemma 23 Suppose given X ∈ Ob B^{ac}. We have a morphism of pure short exact sequences

[figure: XF; •; X^{1}; X T F; XF; •; (XF G)^{1}; XF T′]

in E such that its morphism X T F \overline{}XF T′ represents Xρ in \underline{E}. Here, the upper pure short exact sequence is taken from the purely acyclic complex X; the lower pure short exact sequence is distinguished.

Proof. Given X ∈ Ob B^{ac}, we can form a commutative diagram in E as follows.

[figure: X^{2}; (XF G)^{2}; X T F; •; XF T′; •; XF T′ ε; XF T′ GF = XF G T F; •; X^{1}; (XF G)^{1}; XF; •; XF; •; XF ε; XF GF; •; X^{0}; (XF G)^{0}]

The morphisms (XF G)^{0} \overline{}XF, XF [figure] (XF G)^{1}, (XF G)^{1} \overline{}XF T′ and XF T′ [figure] (XF G)^{2} appear in distinguished pure short exact sequences. Moreover, by abuse of notation, we have written XF ε resp. XF T′ ε for representatives in E of the respective morphisms in \underline{E}.

The partially displayed morphism of complexes X \overline{}XF G represents X ^{\overline{}^{Xη^{−}}} XF G in \underline{\underline{E}}, for F maps the morphism represented by X \overline{}XF G to XF ε = Xη^{−}F.

Therefore, the composite morphism

(X T F \overline{}XF T′ ^{^{XF T′ ε}\overline{}}XF T′ GF = XF G T F)

from this diagram represents

(X ^{\overline{}^{Xη^{−}}} XF G) T F ;

note that there are no signs to be inserted at the respective pure epimorphisms of the image factorisation chosen by F; cf. §A.3.2. Thus the morphism X T F \overline{}XF T′ from this diagram represents

(X T F ^{\overline{}^{Xη^{−} T F}} XF G T F = XF T′ GF ^{\overline{}^{XF T′ ε^{−}}} XF T′ ) = (X T F ^{\overline{}^{Xρ}} XF T′ ) .

A.4.2.3 Standardisation by substitution of the rightmost column

We mimic the construction X \overline{}X^{ϑ} made in §A.4.1, now for E instead of B^{ac}.

Denote by E^{□}(Δ̄_{n}^{△▽}) ^{\overline{}^{M′}} \underline{E}^{+}(Δ̄_{n}^{△▽}) the residue class functor; cf. §A.3, [4, §2.1.3, §4.1]. Denote by \underline{E}^{+, (□)}(Δ̄_{n}^{△▽}) the full subcategory of \underline{E}^{+}(Δ̄_{n}^{△▽}) whose set of objects is given by

Ob \underline{E}^{+, (□)}(Δ̄_{n}^{△▽}) := ( Ob E^{□}(Δ̄_{n}^{△▽}))M′ .

So \underline{E}^{+, (□)}(Δ̄_{n}^{△▽}) is defined to be the “full image” in \underline{E}^{+}(Δ̄_{n}^{△▽}) of the residue class functor M′.

Suppose given n ⩾ 0 and X ∈ Ob E^{□}(Δ̄_{n}^{△▽}). Recall that Δ̇_{n} = [1, n] is identified with {i/0 : i ∈ [1, n]} ⊆ Δ̄_{n}^{△▽}. Write X_{∗/0} := X|Δ̇_{n} = (i \overline{}X_{i/0}) ∈ Ob E(Δ̇_{n}) and X_{0^{+1}/∗} := (i \overline{}X_{0^{+1}/i}) ∈ Ob E(Δ̇_{n}), analogously for morphisms; analogously for objects in \underline{E}^{+}(Δ̄_{n}^{△▽}) and their morphisms.

Let the isomorphism X_{0^{+1}/∗} \overline{}^{Xτ}_{∼} X_{∗/0}^{+1} in \underline{E}(Δ̇_{n}) be defined by morphisms of pure short exact sequences

[figure: X_{i/0}; •; ^{(^{x x})}; X_{i/i}⊕X_{0^{+1}/0}; ^{( \begin{matrix}x \\ −x \end{matrix} )}; X_{0^{+1}/i}; Xτ̂_{i}; X_{i/0}; •; B_{i/0}; X_{i/0}^{+1}]

for i ∈ [1, n]; where Xτ̂_{i} is a representative in E for the morphism Xτ_{i} in \underline{E}; where the upper pure short exact sequence stems from the diagram X; and where the lower pure short exact sequence is distinguished.

In this way, we get an isotransformation τ between the functors (−)_{0^{+1}/∗} and (−)_{∗/0}^{+1} from \underline{E}^{+, (□)}(Δ̄_{n}^{△▽}) to \underline{E}(Δ̇_{n}).

Let \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}) be the (in general not full) subcategory of \underline{E}^{+}(Δ̄_{n}^{△▽}) given by the set of objects

Ob \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}) := \{Y ∈ Ob \underline{E}^{+}(Δ̄_{n}^{△▽}) : Y_{0^{+1}/∗} = Y_{∗/0}^{+1} in Ob \underline{E}(Δ̇_{n})\} ,

and by the set of morphisms

\underline{E}^{+, periodic}(Δ̄_{n}^{△▽})(Y, Y′) = \{f ∈ _{\underline{E}^{+}(Δ̄_{n}^{△▽})}(Y, Y′) : f_{0^{+1}/∗} = f_{∗/0}^{+1} in \underline{E}(Δ̇_{n})\} ,

for Y, Y′ ∈ Ob \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}).

Given X ∈ Ob E^{□}(Δ̄_{n}^{△▽}), we let X^{τ} ∈ Ob \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}) be defined as the diagram X with X_{0^{+1}/∗} isomorphically replaced via Xτ by X_{∗/0}^{+1}. For short, the rightmost column of X becomes standardised to obtain X^{τ}.

Given X, X′ ∈ Ob E^{□}(Δ̄_{n}^{△▽}), a morphism X ^{\overline{}^{f}} X′ in \underline{E}^{+}(Δ̄_{n}^{△▽}) induces a morphism X^{τ} ^{\overline{}^{f^{τ}}} X^{′τ} in \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}). Namely, we let f_{β/α}^{τ} := f_{β/α} for 0 ⩽ α ⩽ β ⩽ n, and we let f_{0^{+1}/∗}^{τ} be characterised by the commutative quadrangle

\xymatrix{X_{0^{+1}/∗} \ar[r]^{Xτ}_{∼} \ar[d]_{f_{0^{+1}/∗}} & X_{∗/0}^{+1} \ar[d]^{f_{0^{+1}/∗}^{τ}} \\ X′_{0^{+1}/∗} \ar[r]^{X′τ}_{∼} & _{X′{+1 \atop ∗/0}}}

in \underline{E}(Δ̇_{n}). In particular, since τ is an isotransformation, we have f_{0^{+1}/∗}^{τ} = (f_{∗/0}^{τ})^{+1}.

Remark 24 The constructions made above define a functor

\underline{E}^{+, (□)}(Δ̄_{n}^{△▽}) ^{\overline{}^{(−)^{τ}}}  \underline{E}^{+, periodic}(Δ̄_{n}^{△▽})
X  \overline{}X^{τ} .

A.4.2.4 n-triangles in the classical stable category

Proposition 25 Suppose given n ⩾ 0 and X ∈ Ob E^{□}(Δ̄_{n}^{△▽}).

The periodic prolongation of X^{τ} ∈ Ob \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}) to an object of \underline{E}^{+, periodic}(Δ̄_{n}^{#}) is an n-triangle with respect to the triangulation ϑ′ on (\underline{E}, T′) obtained as in Proposition 22.

Proof. By Lemma 17, there exists X′ ∈ Ob E^{□}(Δ̄_{n}^{△▽}) such that X′_{α/α} = 0 for all 0 ⩽ α ⩽ 0^{+1} and such that X is isomorphic to X′ in \underline{E}^{+}(Δ̄_{n}^{△▽}). By Remark 24, the object X^{τ} is isomorphic to X^{′τ} in \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}). Thus the periodic prolongation of X^{τ} is an n-triangle if and only if that of X^{′τ} is; cf. [4, Lem. 3.4.(4)].

Therefore, we may assume that X_{α/α} ≃ 0 for all 0 ⩽ α ⩽ 0^{+1}.

Let X̃ ∈ Ob(B^{ac})^{□}(Δ̄_{n}^{△▽}) be such that there exists an isomorphism X \overline{}^{â}_{∼} X̃F̂ in E^{□}(Δ̄_{n}^{△▽}) and such that X̃_{α/α} = 0 for all 0 ⩽ α ⩽ 0^{+1}; cf. Lemma 18. Denote by X \overline{}^{a}_{∼} X̃F the isomorphism in \underline{E}^{+}(Δ̄_{n}^{△▽}) represented by X \overline{}^{â}_{∼} X̃F̂.

Let ˜ X̃ ∈ Ob(B^{ac})^{□}(Δ̄_{n}^{#}) be such that ˜ X̃|Δ̄_{n}^{△▽} = X̃. By Lemma 20, the periodic n-pretriangle ˜ X̃^{ϑ} ∈

Ob \underline{\underline{E}}^{+, periodic}(Δ̄_{n}^{#}) is an n-triangle. Note that ˜ X̃^{ϑ} depends only on X̃, not on the choice of ˜ X̃.

Thus, by Proposition 22.(3), ˜ X̃^{ϑ}F^{ρ} ∈ Ob \underline{E}^{+, periodic}(Δ̄_{n}^{#}) is an n-triangle. Therefore, it suffices to show that X^{τ} and ˜ X̃^{ϑ}F^{ρ}|Δ̄_{n}^{△▽} are isomorphic in \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}), for then their periodic prolongations are isomorphic in \underline{E}^{+, periodic}(Δ̄_{n}^{#}), which in turn shows the periodic prolongation of X^{τ} to be an n-triangle; cf. [4, Lem. 3.4.(4)].

We have a composite isomorphism

X^{τ} ^{\overline{}^{∼}} X \overline{}^{a}_{∼} X̃F = ˜ X̃F |Δ̄_{n}^{△▽}  ^{\overline{}^{∼}} ˜ X̃^{ϑ}F |Δ̄_{n}^{△▽}  ^{\overline{}^{∼}} ˜ X̃^{ϑ}F^{ρ}|Δ̄_{n}^{△▽}

in \underline{E}^{+}(Δ̄_{n}^{△▽}). We claim that it lies in \underline{E}^{+, periodic}(Δ̄_{n}^{△▽}).

Suppose given i ∈ [1, n]. On i/0, this composite equals a_{i/0}. Thus we have to show that on 0^{+1}/i, this composite equals a_{i/0}^{+1} = a_{i/0} T′. Consider, to this end, the following morphisms of pure short exact sequences in E.

[figure: X_{i/0}; •; (X_{i/0}G)^{1}; X_{i/0} T′; X_{i/0}; •; x; â_{i/0}; ≀; X_{0^{+1}/0}; −x; â_{0^{+1}/0}; ≀; X_{0^{+1}/i}; Xτ̂_{i}; â_{0^{+1}/i}; ≀; _{X̃_{i/0}F̂}; •; _{x̃F̂}; _{X̃_{0^{+1}/0}F̂}; −x̃F̂; _{X̃_{0^{+1}/i}F̂}; Xθ̂_{i/0}F̂; _{X̃_{i/0}F̂}; •; _{X̃_{i/0} I F̂}; _{X̃_{i/0} T F̂}; _{X̃_{i/0}F̂}; •; (X̃_{i/0})^{1}; _{X̃_{i/0} T F̂}; _{X̃_{i/0}F̂}; •; (X̃_{i/0}F̂G)^{1}; _{X̃_{i/0}F̂ T_{′}}]

The fourth sequence is purely short exact by Lemma 19.

The first morphism from above arises by definition of τ̂_{i}; cf. §A.4.2.3. The second morphism is taken from â. The third morphism arises by definition of θ̂_{i/0} and an application of F̂; cf. §A.4.1. The fourth morphism is given by Lemma 19. The fifth morphism is given by Lemma 23.

The first and the sixth pure short exact sequence are distinguished, and so the claim and hence the proposition follow.

References

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

[2] Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (3-4), p. 235–265, 1997 (cf. magma.maths.usyd.edu.au).

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

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

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

Matthias Künzer

Lehrstuhl D für Mathematik

RWTH Aachen Templergraben 64 D-52062 Aachen

kuenzer@math.rwth-aachen.de www.math.rwth-aachen.de/∼kuenzer