VIRTUAL TRIANGLES

AVISHAY VAKNIN

Abstract. One of the difficulties in Neeman’s K-theory for triangulated categories is that distinguished triangles are not closed under the formation of mapping cones. Other classes of triangles that are closed under mapping cones are not closed under other operations in this K-theory. We introduce here a new type of triangles, namely the virtual triangles. Virtual triangles are closed under mapping cones and all the other operations in Neeman’s K-theory, including the t-structure truncation operation. We study the structure of virtual triangles, and describe their relation with other classes of triangles.

Contents

0. Introduction 1 0.1. Triangles and operations on them 1 0.2. Closure properties 4 0.3. Application to K-theory for triangulated categories 6 1. From Distinguished Triangles to Virtual Triangles 6 1.1. Triangulated categories 6 1.2. Exact, Virtual and Pseudo triangles 8 1.3. The structure of virtual triangles 12 2. The t-structure Truncation Operation 17 2.1. Basics 17 2.2. Simple triangles 19 2.3. The truncation operation on triangles 19 2.4. Truncating virtual triangles 23 3. Fuzzy Triangles and Non-exact Virtual Triangles 25 Acknowledgments 28 References 28

0. Introduction

0.1. Triangles and operations on them.

Definition 0.1. Let τ be an additive category with an invertible additive functor Σ.

1. A triangle is a sequence of objects and maps of the form

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} . (0.1)

A morphism between two triangles is a commutative diagram of the form

\xymatrix{X \ar[r]^{u} \ar[d]^{f} & Y \ar[r]^{v} \ar[d]^{g} & Z \ar[r]^{w} \ar[d]^{h} & ΣX \ar[d]^{Σf} \\ X′ \ar[r]^{u′} & Y′ \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX′} .

This is a morphism from the top triangle to the bottom triangle.

2. The mapping cone of this morphism is the triangle

\xymatrix{X′ ⊕ Y \ar[r]^{^{( \begin{matrix}u′ & g \\ 0 & −v \end{matrix} )}} & Y′ ⊕ Z \ar[r]^{^{( \begin{matrix}v′ & h \\ 0 & −w \end{matrix} )}} & Z′ ⊕ ΣX \ar[r]^{^{( \begin{matrix}w′ & Σf \\ 0 & −Σu \end{matrix} )}} & ΣX′ ⊕ ΣY} .

3. The right rotation of the triangle (0.1) is the triangle

\xymatrix{Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX \ar[r]^{−Σu} & ΣY}

and the left rotation is the triangle

\xymatrix{Σ^{−1}Z \ar[r]^{−Σ^{−1}w} & X \ar[r]^{u} & Y \ar[r]^{v} & Z}

4. The triangle

\xymatrix{0 \ar[r]^{0} & X \ar[r]^{1_{X}} & X \ar[r]^{0} & 0}

and its rotations are called trivial triangles. We say that a triangle is splitting, if it is isomorphic to a direct sum of trivial triangles.

5. The direct sum of two triangles

[figure: X; u; Y; v; Z; w; ΣX; X′; u′; Y′; v′; Z′; w′; ΣX′]

is the triangle

\xymatrix{X ⊕ X′ \ar[r]^{^{( \begin{matrix}u & 0 \\ 0 & u′ \end{matrix} )}} & Y ⊕ Y′ \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & v′ \end{matrix} )}} & Z ⊕ Z′ \ar[r]^{^{( \begin{matrix}w & 0 \\ 0 & w′ \end{matrix} )}} & ΣX ⊕ ΣX′}

Remark 0.2. 1. One can continue any triangle to the left by iterating the left rotation operation, and to the right by the right rotation operation. So there is a long sequence of objects and maps in τ corresponding to each triangle.

2. The right rotation of a triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} .

is isomorphic to the triangle

\xymatrix{Y \ar[r]^{−v} & Z \ar[r]^{−w} & ΣX \ar[r]^{−Σu} & ΣY} ,

the left rotation is isomorphic to

\xymatrix{Σ^{−1}Z \ar[r]^{−Σ^{−1}w} & X \ar[r]^{−u} & Y \ar[r]^{−v} & Z} ,

and any triangle the left rotation of its right rotation. We will feel free to use any proved statement in its “rotated versions”.

A triangulated category is an additive category with an additive endofunctor endowed with a class of distinguished triangles satisfying some axioms. For the explicit definition see Definition 1.1. The axioms imply the inclusion

{spliting} ⊆ {distinguished}.

Definition 0.3. Let τ be a triangulated category, and let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} (0.2)

be a triangle.

1. We say the map u is replaceable with a replacing map u′, if

\xymatrix{X \ar[r]^{u′} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

is a distinguished triangle. In the same way we define the replaceability of v or w. When u (v or w) is replaceable we will say the triangle 0.2 has a replaceable map in the left (middle or right).

2. We call (0.2) an exact triangle, if all of the maps u, v and w are replaceable.

3. When T_{1}, T_{2} and S are three triangles such that T_{1} ≅ T_{2}⊕S and S is splitting, we say, that T_{2} is an easy direct summand of T_{1}.

4. A virtual triangle is an easy direct summand of an exact triangle.

Remark 0.4. It is obvious that

{distinguished} ⊆ {exact} ⊆ {virtual}.

We want to define yet another class of triangles, the pseudo triangles. The pseudo triangles will contain the virtual triangles. First, we remind the reader of homology functors.

Definition 0.5. 1. A homology functor is a covariant additive functor from a triangulated category to an abelian category, which takes any long sequence, corresponding to a distinguished triangle, to a long exact sequence.

2. The contravariant version of this definition is the definition of cohomology functor.

3. A triangle is called a pseudo-triangle, if any homology or cohomology functor takes it to a long exact sequence.

There is yet another class of triangles, between the class of exact triangles and the class of virtual triangles.

Definition 0.6. Let τ be a triangulated category. A fuzzy triangle is a pseudo triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} ,

such that two of its maps are replaceable. In this case, we will say it is fuzzy in the remaining map. For example, if the maps u and w are replaceable, then we say that the triangle is fuzzy in v or just fuzzy in the middle.

Remark 0.7. By Lemma 3.2

{fuzzy} ⊆ {virtual}.

It is known, that the third edge of a distinguished triangle (the Z object in a distinguished triangles with fixed left map u : X → Y) is unique up to a non-canonical isomorphism. The same is true for exact triangles, and a similar phenomenon occurs with virtual triangles.

Definition 0.8. We will say that two objects Z, Z′ are quasi-isomorphic, if there is an object W such that Z ⊕ W ≅ Z′ ⊕ W

Proposition 0.9. The third edge in a virtual triangle is unique up to a quasiisomorphism.

Proof. See Proposition 1.11

There are triangulated categories where all the classes of triangles above coincide. This is the case with the derived category of a field. In general, it is far from the truth.

Theorem 0.10.

{splitting} ⫋ {distinguished} ⫋ {exact} ⫋ {fuzzy} ⫋ {virtual} ⫅ {pseudo}

The inclusion on the right is an equality if and only if the third edge of a pseudo triangle is unique up to a quasi-isomorphism. More precisely, there is a triangulated category with

1. a non-splitting distinguished triangle,

2. a non-distinguished exact triangle,

3. a non-exact fuzzy triangle and

4. a virtual triangle with no replaceable maps. There is an equality

{virtual} = {pseudo}

if and only if the third edge of a pseudo triangle is unique up to a quasi-isomorphism.

Proof. The abelian category of finitely generated abelian groups has a non-splitting short exact sequence, and hence, its derived category has a non-splitting distinguished triangle. In Lemma 1.8, we will see that exact triangles are closed under mapping cones. In particular, the mapping cone of a morphism between two distinguished triangles is exact. In [7] Neeman shows that the mapping cone of a morphism between two distinguished triangles is not distinguished in general. Hence, such a mapping cone is a non-distinguished exact triangle. In Section 3 we construct a non-exact fuzzy triangle and a virtual triangle with no replaceable map. The last statement is Corollary 1.18

0.2. Closure properties. In section 1, we study the closure properties of the various classes with respect to easy summands, mapping cones, and truncations. In section 2 we recall the definition of a t-structure on a triangulated category. A t-structure defines, under certain conditions, a truncation operation on triangles.

Definition 0.11. A triangle

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

is an n-bounded triangle, if X, Y, Z ∈ τ^{≤n}. A n-truncation octahedron on such a triangle is an octahedron on the composeable maps

Y → Z and Z → Z^{≥n}

i.e., an octahedron of the form

[figure: 0; Z^{≥n}; Z^{≥n}; 0; X; u; Y; v; jv; Z; w; j; ΣX; X; u^{t}; Y^{t}; v^{t}; a; Z^{<n}; wi; i; ΣX; 0; Σ^{−1}Z^{≥n}; b; Σ^{−1}Z^{≥n}; −Σ^{−1}k; 0]

where i, j and k are the natural maps, and all the columns are distinguished triangles. The third line T^{t} = (u^{t}, v^{t}, wi) in this octahedron is called an n-truncation of the triangle T = (u, v, w).

Lemma 0.12. The n-truncation octahedron on an n-bounded virtual triangle is uniquely defined up to a canonical isomorphism, and in this case, the truncation triangle is a virtual triangle.

Proof. This is the subject of Section 2.

The following theorem summarize the closure properties of the various classes of triangles.

Theorem 0.13. Let τ be a triangulated category.

1. Distinguished triangles are closed under isomorphisms, direct sums and direct summands. In general, they are not closed under mapping cones. If the category τ admits a t-structure , then they are also closed under the truncation operation.

2. Exact triangles are closed under isomorphisms, direct sums and mapping cones. If the category τ admits a t-structure , then they are also closed under the truncation operation. In general, exact triangles are not closed under easy direct summands. There is a non-exact fuzzy triangle and a virtual triangle with no replaceable map.

3. Virtual triangles are closed under isomorphisms, direct sums, mapping cones, and easy direct summands. If the category τ admits a t-structure , then they are also closed under the truncation operation.

4. Pseudo triangles are closed under isomorphisms, direct sums, mapping cones, and direct summands. A pseudo triangle is a virtual triangle, if and only if its third edge is quasi-isomorphic to the third edge of a distinguished triangle on the same left map.

5. The pseudo triangles are the virtual triangles, if and only if the third edge of a pseudo triangle is unique up to quasi isomorphism.

The last condition is satisfied by any triangulated category with countable direct sums. The reason is that any two objects, in such a category, are quasi-isomorphic. We should note that we do not know an example for a pseudo triangle which is not virtual, and we do not know if pseudo triangles are closed under the truncation operation or not.

0.3. Application to K-theory for triangulated categories. The motivation of this study is the K-theory for triangulated categories introduced by Neeman in [8, 9, 10, 5]. Using the triangulated structure, he defined a simplicial set, where the K-theory groups are the shifted homotopy groups of the realization of this simplicial set. This definition extends Quillen K-theory [11]; this follows from the theorem of the heart [5]. This theorem asserts, that when a triangulated category has a t-structure , then its K-theory is the same as Quillen K-theory of the heart of the t-structure .

The definition of this simplicial set uses the definition of bicartesian squares by distinguished triangles, generalizing a similar construction for abelian categories. Therefore, the proof is divided into two steps. The first one has to prove, that the simplicial set for the heart has the same homotopy type, as Quillen K-theory space. This is essentially due to Jardine [3]. The second step is the theorem of the heart, which shows, that the simplicial set for the heart has the same homotopy type, as the simplicial set for the whole triangulated category. The proof of this step in the main theme of the above list of Neeman’s papers. Even if it seems to be a long proof, it is based on understanding only few types of simplicial homotopies. The homotopies in these types are repeated procedures of creating new cartesian squares out of given diagram made of cartesian squares. The construction involves a sequence of operations on triangles, and in the proof, there is a careful verification, that each operation yields at the end a distinguished triangle. This sequence of operation includes: direct sums, t-truncation, the formation of mapping cones, and taking easy direct summands. In [7] Neeman shows, that in general, the set of distinguished triangles is not closed under the formation of mapping cones. To avoid this difficulty, Neeman uses Waldhausen models [12, 13]. In this way, the diagrams involved are highly correlated, so that the related mapping cones are distinguished triangles. Aside from taking us out of the world of triangulated categories, this usage causes the K-theory to lose its functorial property. Following Neeman’s suggestion at the end of [9], we define the virtual K-theory for triangulated categories. It is a modification of Neeman’s theory, where we replace distinguished triangles by virtual triangles, cartesian squares by virtual cartesian squares, and remove any condition which involves models. This K-theory is functorial by definition, and it has its own version of the theorem of the heart. This means, that two triangulated categories with isomorphic hearts have the same virtual K-theory. Unfortunately, now the simplicial set for the heart, is no longer the same as Jardine’s, so the compatibility with Quillen K-theory is open. Anyway, we know, that Neeman’s K-theory is a retract of the virtual K-theory.

1. From Distinguished Triangles to Virtual Triangles

In this section we study the structure of virtual triangles. We recall the definition of a triangulated category, and note some basic facts. Then we define the new types of triangles, namely, exact triangles, splitting triangles, pseudo triangles and virtual triangles, and we describe the hierarchy of inclusions among these. For more details on the basics of triangulated categories we refer to Chapter 1 of [6], Chapter 10 Section 2 in [14] and Appendix 2 in [4].

1.1. Triangulated categories.

Definition 1.1. A triangulated category is an additive category τ with invertible additive functor Σ, and a class of triangles, namely the distinguished triangles, subject to the following axioms.

1. Any map \xymatrix{X \ar[r]^{u} & Y} belongs to a distinguished triangle \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} , any trivial triangle is distinguished, and any triangle which is isomorphic to a distinguished triangle, is distinguished too.

2. Any rotation of a distinguished triangle is a distinguished triangle.

3. Consider a commutative diagram, where the dotted arrow is missing.

\xymatrix{X \ar[r]^{u} \ar[d]^{f} & Y \ar[r]^{v} \ar[d]^{g} & Z \ar[r]^{w} \ar@{.>}[d]^{h} & ΣX \ar[d]^{Σf} \\ X′ \ar[r]^{u′} & Y′ \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX′}

If the rows are distinguished triangles, then there is an arrow h, which complete the diagram to a morphism between triangles.

4. The last axiom is the octahedral axiom, and we give here two equivalent formulations.

(a) In the situation in axiom 3, one can choose the map h, so the mapping cone is a distinguished triangle.

(b) An octahedron in τ is an 4 × 4 commutative diagram of the form

\xymatrix{0 \ar[r] & ΣY \ar[r] & ΣZ \ar[r] & 0 \\ X′′ \ar[r] \ar[u] & Y′′ \ar[r] \ar[u] & Z′′ \ar[r] \ar[u] & ΣX′′ \ar[u] \\ X′ \ar[r] \ar[u] & Y′ \ar[r] \ar[u] & Z′ \ar[r] \ar[u] & ΣX′ \ar[u] \\ 0 \ar[r] \ar[u] & Y \ar[r] \ar[u] & Z \ar[r] \ar[u] & 0 \ar[u]}

where all the rows and all the columns are distinguished triangles. The reason to call such a diagram an octahedron is the following. If we identify the isomorphic objects

X′ ≅ X, Y ≅ Z

there are 6 non–zero objects in the diagram. There are 12 morphisms joining them, and these assemble into the 6 vertices and 12 edges of an octahedron. There are also 8 faces, all triangles, 4 of which are commutative and 4 distinguished.

The octahedral axiom says, that given an incomplete octahedron

\xymatrix{0 \ar[r] & ΣY \ar[r] & ΣZ \ar[r] & 0 \\ X′′ \ar[r] \ar[u] & Y′′ \ar[r] \ar[u] & Z′′ \ar[r] \ar[u] & ΣX′′ \ar[u] \\ X′ \ar@{.>}[r] \ar[u] & Y′ \ar@{.>}[r] \ar[u] & Z′ \ar[r] \ar[u] & ΣX′ \ar[u] \\ 0 \ar[r] \ar[u] & Y \ar[r] \ar[u] & Z \ar[r] \ar[u] & 0 \ar[u]}

where the dotted arrows are missing, then the dotted arrows may be filled in, so that the completed diagram is an octahedron. Another way to say it, is that any composeable maps

\xymatrix{ & ΣZ \\ Y′′ \ar[r] & Z′′ \ar[u]}

are contained in an octahedron.

Remark 1.2. 1. The two formulations of the octahedral axiom are indeed equivalent. Formulation 4b is much better known. Theorem 1.8 in [7] shows that it implies 4a. Proposition 1.4.6 in [6] shows that 4a implies 4b.

2. The axioms are self dual, so τ is triangulated, if and only if its opposite category is triangulated, with the same class of distinguished triangles, and Σ^{−1} as the suspension functor.

3. By definition, distinguished triangles are closed under isomorphism and rotations. It turns out that they are also closed under direct sums (Proposition 1.2.1 and Remark 1.2.2 in [6] and Proposition 10 in [4, Appendix 2]) and direct summands (Proposition 1.2.3 in [6]) .

4. The fill in maps in axiom 3 above are almost never unique, and in general, the mapping cone on a morphism between two distinguished triangles, is not distinguished. For more details see [7].

Definition 1.3. A homology functor is a covariant additive functor from a triangulated category to an abelian category, which takes any long sequence, corresponding to a distinguished triangle, to a long exact sequence. The contravariant version of this definition is the definition of cohomology functor.

Remark 1.4. A covariant representable functor is a homology functor, and a contravariant representable functor is a cohomology functor (to the abelian category of abelian groups). In particular, the composition of two sequential maps in a distinguished triangle vanishes (Lemma 1.1.10 in [6], Example 10.2.8 in [14] and Proposition 5 in [4, Appendix 2]) .

1.2. Exact, Virtual and Pseudo triangles.

Definition 1.5. Let τ be a triangulated category, and let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} (1.1)

be a triangle.

1. We say the map u is replaceable with a replacing map u′, if

\xymatrix{X \ar[r]^{u′} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

is a distinguished triangle. In the same way we define the replaceability of v or w. When u (v or w) is replaceable we will say the triangle 1.1 has a replaceable map in the left (middle or right).

2. We call (1.1) an exact triangle, if all of the maps u, v and w are replaceable.

3. We say that (1.1) is a pseudo-triangle, if any homology or cohomology functor takes it to a long exact sequence.

4. When T_{1}, T_{2} and S are three triangles such that T_{1} ≅ T_{2}⊕S and S is splitting, we say, that T_{2} is an easy direct summand of T_{1}.

5. A virtual triangle is an easy direct summand of an exact triangle.

Lemma 1.6 (5–lemma). Consider a morphism between two pseudo triangles.

\xymatrix{X \ar[r]^{u} \ar[d]^{f} & Y \ar[r]^{v} \ar[d]^{g} & Z \ar[r]^{w} \ar[d]^{h} & ΣX \ar[d]^{Σf} \\ X′ \ar[r]^{u′} & Y′ \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX′}

If two of the three maps f, g and h are isomorphisms, then the third is also an isomorphism.

Proof. By rotation we may assume that f and g are isomorphisms. When we apply a representable functor H on the diagram we get a morphism between two long exact sequences.

[figure: H(X); H(u); H(f); H(Y); H(v); H(g); H(Z); H(w); H(h); H(ΣX); H(−Σu); H(Σf); H(ΣY); H(Σg); H(X′); H(u′); H(Y′); H(v′); H(Z′); H(w′); H(ΣX′); H(−Σu′); H(ΣX′)]

All the vertical maps except the middle one H(h) are isomorphisms by the assumption. By the 5-lemma for abelian categories, H(h) is an isomorphism, and by Yoneda Lemma, h itself must be an isomorphism.

Lemma 1.7. Any pseudo triangle with one zero map splits, and hence, such a triangle is a distinguished triangle.

Proof. Let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be a pseudo triangle with one zero map. By rotation we may assume that w = 0. The sequence

\xymatrix{τ (Z, Y) \ar[r] & τ (Z, Z) \ar[r]^{0} & τ (Z, ΣX)}

is exact, so the left hand map is onto and there is a map t : Z → Y such that vt = 1. Now we have the triangle morphism

[figure: X; ^{( \begin{matrix}1 \\ 0 \end{matrix} )}; X ⊕ Z; ^{(^{0 1} )}; _{(_{u t})}; Z; 0; ΣX; X; u; Y; v; Z; w; ΣX′] .

By the 5-lemma 1.6, the map (^{u} ^{t}) is an isomorphism, and we have a triangle isomorphism between a splitting triangle and our triangle.

In Lemma 1.8, we will see that exact triangles are closed under mapping cones. In section 3, we construct a virtual triangle with no replaceable map. Hence, exact triangles are not closed under easy direct summands.

By definition, virtual triangles are closed under easy direct summands, and we will prove, that they are also closed under mapping cones. In section 2 we will see that virtual triangles are closed under the t-structure truncation operation, as well.

Clearly, pseudo triangles are closed under direct sums, direct summands and mapping cones, but it is not clear if they are closed under the t-structure truncation operation. In this section, we will formulate a necessary and sufficient condition, for the equality of the terms pseudo triangle and virtual triangles. We do not know any example of a triangulated category which does not satisfy the condition.

Lemma 1.8. 1. Exact triangles are closed under mapping cones 2. Virtual triangles are closed under mapping cones

Proof. 1. Consider a map between two exact triangles

\xymatrix{X \ar[r]^{u} \ar[d]^{f} & Y \ar[r]^{v} \ar[d]^{g} & Z \ar[r]^{w} \ar[d]^{h} & ΣX \ar[d]^{Σf} \\ X′ \ar[r]^{u′} & Y′ \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX′}

We will show that the left map on the mapping cone

\xymatrix{X′ ⊕ Y \ar[r]^{^{( \begin{matrix}u′ & g \\ 0 & −v \end{matrix} )}} & Y′ ⊕ Z \ar[r]^{^{( \begin{matrix}v′ & h \\ 0 & −w \end{matrix} )}} & Z′ ⊕ ΣX \ar[r]^{^{( \begin{matrix}w′ & Σf \\ 0 & −Σu \end{matrix} )}} & ΣX′ ⊕ ΣY} (1.2)

is replaceable, and the replaceability of the other maps is by rotation. We know that there are maps v_{1} and u′_{1} so the rows in the commutative diagram below are distinguished triangles.

\xymatrix{X \ar[r]^{u} \ar[d]^{f} & Y \ar[r]^{v_{1}} \ar@{.>}[d]^{g_{1}} & Z \ar[r]^{w} \ar[d]^{h} & ΣX \ar[d]^{Σf} \\ X′ \ar[r]^{u′_{1}} & Y′ \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX′}

By the octahedral axiom, there is a fill in map g_{1} for the dotted arrow, which makes the above diagram commutative, and the mapping cone

\xymatrix{X′ ⊕ Y \ar[r]^{^{( \begin{matrix}u′1 & g1 \\ 0 & −v1 \end{matrix} )}} & Y′ ⊕ Z \ar[r]^{^{( \begin{matrix}v′ & h \\ 0 & −w \end{matrix} )}} & Z′ ⊕ ΣX \ar[r]^{^{( \begin{matrix}w′ & Σf \\ 0 & −Σu \end{matrix} )}} & ΣX′ ⊕ ΣY}

a distinguished triangle. Hence, the map

( \begin{matrix}u′_{1} & g_{1} \\ 0 & −v_{1} \end{matrix} )

is a replacing map for the left map of (1.2).

2. Let T_{1} ⟶^{f} T_{2} be a map between two virtual triangles. We can choose a splitting triangle S, such that T_{1} ⊕ S and T_{2} ⊕ S are exact. The mapping cone of the map

T_{1} ⊕ S \underrightarrow{( \begin{matrix}f & 0 \\ 0 & 0 \end{matrix} )} T_{2} ⊕ S

which is exact, is isomorphic to the direct sum C ⊕ S ⊕ S′, where C is the mapping cone on f and S′ is the right rotation of S. The triangle S ⊕ S′ is a direct sum of trivial triangles, so C is an easy direct summand of an exact triangle.

To begin with the study of the structure of virtual triangles, we would like to prove some basic facts about pseudo triangles with some replaceable maps.

Lemma 1.9. Let

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be a pseudo triangle.

1. Let f : Y → W be a map. Then fu = 0, if and only if f factors through v.

2. Let g : W → Z be a map. Then wg = 0, if and only if g factors through v.

3. If T has a replacing map v′ for v, then v′ = av = vb for suitable a : Z → Z and b : Y → Y.

4. If v, u have replacing maps v′, u′, then there is an automorphism ϕ of Y, so that v′ = vϕ and u′ = ϕu.

Proof. 1. The statement is equivalent to exactness in the middle of the sequence

τ (X, W) ⟵^{τ (u,W)} τ (Y, W) ⟵^{τ (v,W)} τ (Z, W).

This sequence is exact since τ (−, W) is a cohomology functor and (u, v, w) is a pseudo triangle.

2. This is the dual statement of (1).

3. Apply (1) on the equality v′u = 0, and (2) on the equality wv′ = 0.

4. Given replacing maps u′, v′, complete the diagram below to a triangle morphism

[figure: X; u; Y; v′; ϕ; Z; w; ΣX; X; u′; Y; v; Z; w; ΣX]

and ϕ must be an automorphism of Y.

1.3. The structure of virtual triangles. It is known, that the third edge of a distinguished triangle (the Z objects in distinguished triangles with fixed left map u : X → Y) is unique up to a non-canonical isomorphism. The same is true for exact triangles, and a similar phenomenon occurs with virtual triangle.

Definition 1.10. We will say that two objects Z, Z′ are quasi-isomorphic, if there is an object W such that Z ⊕ W ≅ Z′ ⊕ W

Proposition 1.11. The third edge in a virtual triangle is unique up to a quasiisomorphism.

Proof. Let

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be a virtual triangle, and let

T′ : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX}

be a distinguished triangle on the same left map. We wish to show, that their third edges, Z and Z′, are quasi-isomorphic. There is a splitting triangle

S : \xymatrix{A \ar[r] & B \ar[r] & C \ar[r] & ΣA}

so that the triangle T ⊕ S is exact. Since the objects Z ⊕ C and Z′ ⊕ C are the third edges on exact triangles T ⊕ S and T′ ⊕ S, with the same map on the left ( \begin{matrix}u & 0 \\ 0 & 1 \end{matrix} ) : X ⊕ A ⟶ Y ⊕ B, they are isomorphic to each other. Hence, Z and Z′ are quasi-isomorphic.

Lemma 1.12. Let

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ A \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} )}} & Z ⊕ A \ar[r]^{^{(^{w 0} )}} & ΣX}

be a pseudo triangle with a replaceable map on the right. Then the pseudo triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

has also a replaceable map on the right. The same is true for the left side.

Proof. Let (_{w′} _{h}) be a replacing map. The composite

(_{w′} _{h}) ( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} ) = (^{0} ^{h})

vanishes, so h = 0 and the replacing map has the form (_{w′} _{0} ). The triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w′} & ΣX}

is a direct summand of the distinguished triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ A \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} )}} & Z ⊕ A \ar[r]^{^{(^{w′ 0} )}} & ΣX}

so it is distinguished too, and (u, v, w) has a replaceable map on the right. The claim for the left side is the dual statement.

Corollary 1.13. Let

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be a virtual triangle. Then there are three trivial triangles

M : \xymatrix{0 \ar[r] & A \ar[r] & A \ar[r] & Σ0}
L : \xymatrix{B \ar[r] & B \ar[r] & 0 \ar[r] & ΣB}
R : \xymatrix{C \ar[r] & 0 \ar[r] & ΣC \ar[r] & ΣC}

so that T ⊕ M has a replaceable map in the middle, T ⊕ L has a replaceable map in the left, and T ⊕ R has a replaceable map in the right.

Proof. There is a splitting triangle S such that T ⊕S is exact. The splitting triangle S is a direct sum of three trivial triangles

S = M ⊕ L ⊕ R.

By the left rotation of Lemma 1.12, T ⊕M ⊕L has a replaceable map in the middle. By the right rotation of Lemma 1.12 on the triangle T ⊕ M ⊕ L, the triangle T ⊕ M also has a replaceable map in the middle. The rest follows by rotation.

The next lemma and proposition are the key fact about pseudo triangles. In the lemma we describe the structure of pseudo triangles, and give an equivalent condition for replaceability of a map.

Definition 1.14. A commutative square

\xymatrix{Y \ar[r]^{v} \ar[d]^{b} & Z′ \ar[d]^{a} \\ Y′′ \ar[r]^{v′′} & Z}

is called cartesian, if there is a distinguished triangle

\xymatrix{Y \ar[r]^{^{( \begin{matrix}b \\ −v′ \end{matrix} )}} & Y′′ ⊕ Z′ \ar[r]^{^{(^{v′′ a})}} & Z \ar[r]^{d} & ΣY}

In this case the map d is called a differential of the square. If the triangle above is only virtual, we call the square a virtual cartesian square. In the same way we define pseudo cartesian squares, and splitting cartesian squares.

Lemma 1.15. Let τ be a triangulated category and let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} (1.3)

be a pseudo triangle.

1. One can draw the commutative diagram below, where the rows are distinguished triangles, and hence, the middle square is a splitting cartesian square.

[figure: X; u; Y; v′; b; Z′; w′; a; ΣX; X; u′′; Y′′; v′′; Z; w; ΣX] (1.4)

2. For any commutative diagram with distinguished triangles in the rows as above, the pseudo triangle (u, v, w) has middle replaceable map, if and only if there exists a map λ : Z′ → Y′′, such that b+λv is an isomorphism Y ⟶ Y′′.

Proof. 1. The rows in (1.4) are just completions of the maps u and w to distinguished triangles. The triangle (u, v′, w′) is distinguished, so Σuw′ = 0. Since (u, v, w) is a pseudo triangle, w′ factors through w, by a rotation of Lemma 1.9 (2). So there exist a map a : Z′ → Z such that w′ = wa. Now it remains to complete the diagram bellow to a morphism between distinguished triangles.

[figure: X; u; Y; v′; b; Z′; w′; a; ΣX; X; u′′; Y′′; v′′; Z; w; ΣX]

As an easy direct summand of the mapping cone, the sequence

\xymatrix{Y \ar[r]^{^{( \begin{matrix}b \\ −v′ \end{matrix} )}} & Y′′ ⊕ Z′ \ar[r]^{^{(^{v′′ a})}} & Z \ar[r]^{Σuw} & ΣY}

is a pseudo triangle. The differential Σuw vanishes, and hence, it is a splitting triangle. Hence, the middle square in the commutative diagram above, is a splitting cartesian square.

2. If there exists a map v_{1} such that (u, v_{1}, w) is a distinguished triangle, then complete the diagram below to a morphism between distinguished triangles.

[figure: X; u; Y; v_{1}; ψ; Z; w; ΣX; X; u′′; Y′′; v′′; Z; w; ΣX]

The map ψ must be an isomorphism Y ⟶ Y′′, and

ψu = u′′ = bu ⟹ (ψ − b)u = 0 ⟹ ∃λ : Z → Y′′, ψ = b + λv.

For the converse, assume that there exists a map λ : Z → Y′′, such that ψ = b + λv is an isomorphism Y ⟶ Y′′. Then ψu = bu = u′′, and the diagram below is an isomorphism of the distinguished triangle (u′′, v′′, w) with the triangle (u, v_{1}, w), where v_{1} = v′′ψ.

[figure: X; u; Y; v_{1}; ψ; Z; w; ΣX; X; u′′; Y′′; v′′; Z; w; ΣX]

Proposition 1.16. Let τ be a triangulated category, let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be a pseudo triangle in τ . Consider the following commutative diagram with distinguished triangles in the rows.

[figure: X; u; Y; v′; b; Z′; w′; a; ΣX; X; u′′; Y′′; v′′; Z; w; ΣX] (1.5)

Such a diagram exists, as we saw in Lemma 1.15. If the objects Z and Z′ are quasi-isomorphic, i.e., there exists an object W, such that

A = Z ⊕ W ≅ Z′ ⊕ W,

then the triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ A \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} )}} & Z ⊕ A \ar[r]^{^{(^{w 0} )}} & ΣX}

is a pseudo triangle with a replaceable map in the middle.

Proof. Let W be an object with an isomorphism

ϕ = ( \begin{matrix}k & l \\ m & n \end{matrix}) : Z′ ⊕ W ⟶ Z ⊕ W

We claim that the pseudo triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \\ 0 \end{matrix} )}} & Y ⊕ Z ⊕ W \ar[r]^{^{( \begin{matrix}v & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )}} & Z ⊕ Z ⊕ W \ar[r]^{^{(^{w 0 0} )}} & ΣX} (1.6)

has a replaceable map in the middle. By adding trivial triangles to the rows of (1.5), we get a commutative diagram in the shape of (1.4), which corresponds to the pseudo triangle (1.6).

[figure: X; ^{( \begin{matrix}u \\ 0 \\ 0 \end{matrix} )}; Y ⊕ Z ⊕ W; ^{( \begin{matrix}v′ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )}; _{( \begin{matrix}b & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )}; Z′ ⊕ Z ⊕ W; ^{(^{w′ 0 0} )}; _{( \begin{matrix}a & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )}; ΣX; X; _{( \begin{matrix}u′′ \\ 0 \\ 0 \end{matrix} )}; Y′′ ⊕ Z ⊕ W; _{( \begin{matrix}v′′ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )}; Z ⊕ Z ⊕ W; _{(_{w 0 0} )}; ΣX]

Clearly, in this commutative diagram, the rows are distinguished triangles. By 1.15, in order to prove that the middle map in (1.6) is replaceable, we need to show that

there is a map λ : Z′ ⊕ Z ⊕ W ⟶ Y′′ ⊕ Z ⊕ W , such that ( \begin{matrix}b & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) + λ ( \begin{matrix}v′ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

is an isomorphism Y ⊕ Z ⊕ W ⟶ Y′′ ⊕ Z ⊕ W. To this end, we will use the fact that the middle square in (1.5) is splitting. Complete the diagram below to a map between distinguished triangles.

[figure: Y; ^{( \begin{matrix}b \\ −v′ \end{matrix} )}; Y′′ ⊕ Z′; ^{(^{v′′ a})}; Z; 0; ΣY; Y; _{( \begin{matrix}1 \\ 0 \end{matrix} )}; Y ⊕ Z; _{(_{0 1} )}; ψ; Z; 0; ΣY]

The map ψ must be an isomorphism Y ⊕ Z ⟶ Y′′ ⊕ Z′, and since

ψ ( \begin{matrix}1 \\ 0 \end{matrix} ) = ( \begin{matrix}b \\ −v′ \end{matrix} ),

the map ψ must be in the form

ψ = ( \begin{matrix}b & t_{1} \\ −v′ & t_{2} \end{matrix} ).

Now we have a sequence of isomorphisms.

\xymatrix{Y ⊕ Z ⊕ W \ar[r]^{^{( \begin{matrix}b & t1 & 0 \\ −v′ & t2 & 0 \\ 0 & 0 & 1 \end{matrix} )}} & Y′′ ⊕ Z′ ⊕ W \ar[r]^{^{( \begin{matrix}1 & 0 & 0 \\ 0 & k & l \\ 0 & m & n \end{matrix} )}} & Y′′ ⊕ Z ⊕ W}

The composition is the isomorphism

( \begin{matrix}1 & 0 & 0 \\ 0 & k & l \\ 0 & m & n \end{matrix} ) ( \begin{matrix}b & t_{1} & 0 \\ −v′ & t_{2} & 0 \\ 0 & 0 & 1 \end{matrix} ) = ( \begin{matrix}b & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) + λ ( \begin{matrix}v′ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

where

λ = ( \begin{matrix}1 & 0 & 0 \\ 0 & k & l \\ 0 & m & n \end{matrix} ) ( \begin{matrix}0 & t_{1} & 0 \\ −1 & t_{2} & 0 \\ 0 & 0 & 1 \end{matrix} )

Corollary 1.17. 1. Let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be a pseudo triangle so that u belongs to a distinguished triangle on the same objects, i.e., we assume that there exists a distinguished triangle of the form

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v′} & Z \ar[r]^{w′} & ΣX} .

[One interesting special case is if the map w is replaceable; that is, the distinguished triangle is even more special, and we have v′ = v.] Then the pseudo triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ Z \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} )}} & Z ⊕ Z \ar[r]^{^{(^{w 0} )}} & ΣX}

has a replaceable map in the middle.

2. A pseudo triangle is virtual, if (and only if) its third edge is quasi-isomorphic to the third edge of a distinguished triangle with the same left map. Explicitly, let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be a pseudo triangle so that the third edge Z is quasi-isomorphic to the third edge in a distinguished triangle with the left map u

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX} .

Then the pseudo triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

is a virtual triangle.

3. A pseudo triangle with at least one replaceable map is virtual.

Proof. 1. It is just the case when W = 0 and Z = Z′ in proposition 1.16. 2. We assume that

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

is a pseudo triangle and that the triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX} .

is distinguished. We assume also that there exists an object W, such that Z′ ⊕ W ≅ Z ⊕ W. By the proposition, such a triangle is an easy direct summand of a pseudo triangle with a replaceable map in the middle. Now, by 1 above, (and rotation) this new triangle is an easy direct summand of a pseudo triangle with replaceable maps in the left and in the middle. Rotating again, this new pseudo triangle is an easy direct summand of a triangle, where all the maps are replaceable, i.e., an exact triangle. Therefore, the triangle we start with is an easy direct summand of an exact triangle, and by definition, it is a virtual triangle.

3. This a special case of (1) above and its rotations.

Corollary 1.18. Let τ be a triangulated category. Pseudo triangles are virtual triangles, if and only if, the third edge on a pseudo triangle is unique up to a quasiisomorphism.

2. The t-structure Truncation Operation

The basic fact is, that under certain conditions, a non-degenerate t-structure on a triangulated category ([1, Section 1.3], [2, Chapter 5 Section 3]) defines a truncation operation on triangles. This operation takes distinguished triangles to distinguished triangles. We will see that exact triangles are closed under the truncation operation, and we conclude, that virtual triangles are also closed under this operation. This is a conclusion of the observation, that a bounded virtual triangle is an easy direct summand of a bounded exact triangle, with the same bound. We do not know if the truncation of a pseudo triangle is a pseudo triangle.

2.1. Basics.

Definition 2.1. 1. A t-structure on a triangulated category τ is a pair of two full additive subcategories

τ^{≤0}  and τ^{≥0}

satisfying the following axioms.

(a) Στ^{≤0} ⊆ τ^{≤0} and τ^{≥0} ⊆ Στ^{≥0}. We define τ^{≤n} = Σ^{−n}τ^{≤0} and τ^{≥n} = Σ^{−n}τ^{≥0}, and we will write τ^{<n} for τ^{≤n−1} and τ^{>n} for τ^{≥n+1}.

(b) There is no non-zero map from an object of τ^{<0} to an object of τ^{≥0}.

(c) Any object X lies in a distinguished triangle

\xymatrix{A \ar[r] & X \ar[r] & B \ar[r] & ΣA}

where A ∈ τ^{<0} and B ∈ τ^{≥0}

2. We call the intersection τ^{≤0} ⋂τ^{≥0} the heart of τ with respect to this t-structure .

3. We say that the t-structure is non-degenerate if

⋂_{n}τ^{≤n} = ⋂_{n}τ^{≥n} = 0

As an immediate consequence we have

Lemma 2.2. For all n 1. τ^{<n} ⊆ τ^{≤n} and τ^{>n} ⊆ τ^{≥n}.

2. There is no non-zero map from an object of τ^{<n} to an object of τ^{≥n}.

The fundamental fact is that a t-structure determines two functors

(−)^{≥n} : τ → τ^{≥n}
(−)^{<n} : τ → τ^{<n}

called the truncation operations on objects and maps. These two functor come with the three natural transformations

i : (−)^{<n} → 1  j : 1 → (−)^{≥n}  k : (−)^{≥n} → Σ((−)^{<n}).

They are determined by the following lemma.

Lemma 2.3. Let A ∈ τ^{<n}, B ∈ τ^{≥n} and X, Y ∈ τ. Let

T : \xymatrix{A \ar[r] & Y \ar[r] & B \ar[r] & ΣA}

be a distinguished triangle.

1. The triangle

C : \xymatrix{X^{<n} \ar[r]^{i} & X \ar[r]^{j} & X^{≥n} \ar[r]^{k} & ΣX^{<n}}

is distinguished, and is called the canonical truncation triangle on the object X.

2. A map X → Y uniquely determines a triangle morphism C → T.

3. A map Y → X uniquely determines a triangle morphism T → C.

4. The functor (−)^{<n} is the right adjoint to the inclusion τ^{<n} ⊂ τ with i as a counit and the identity as a unit.

5. The functor (−)^{≥n} is the left adjoint to the inclusion τ^{≥n} ⊂ τ with j as a unit and the identity as a counit.

6. The heart is an abelian category, and the functor

H = (−)^{≥0} ∘ (−)^{≤0}

is a homology functor from τ to the heart of τ . Let H^{n} be the functor

H ∘ Σ^{n} = Σ^{n} ∘ (−)^{≥n} ∘ (−)^{≤n}.

7. If the t-structure is non-degenerate, then (a)

τ^{<n} = {X ∈ τ |H^{k}(X) = 0 ∀k ≥ n}
τ^{>n} = {X ∈ τ |H^{k}(X) = 0 ∀k ≤ n}

(b)

∀X ∈ τ X = 0 ⟺ (H^{n}(X) = 0 ∀k)

(c) A map f : X → Y is an isomorphism if and only if H^{n}(f) is an isomorphism for all n.

Proof. 1–5: Proposition 1.3.3 in [1] or Lemma 3.4.1 in [2].

6: Theorem 1.3.6 in [1] or Theorem 3.4 and Theorem 3.5.1a in [2]. 7a: Proposition 1.3.7 in [1] or Theorem 3.5.1c in [2]. 7b: Follows from 7a and the definition of non-degenerate t-structure . 7c: Theorem 3.5.1b in [2].

2.2. Simple triangles.

Definition 2.4. Let τ be a triangulated category with a t-structure . A triangle

T : \xymatrix{X \ar[r] & Y \ar[r] & Z \ar[r] & ΣX}

is called n-simple, if X ∈ τ^{≤n} and Z ∈ τ^{≥n}.

For example, any canonical triangle as in Lemma 2.3, is an n-simple triangle.

Lemma 2.5. Let \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} be an n-simple pseudo triangle.

1. Let f : A → Y be a map where A is an object in τ^{≤n}. If vf = 0, then f factors uniquely trough u.

2. Let g : Y → B be a map where B is an object in τ^{≥n}. If gu = 0, then g factors uniquely trough v.

Proof. 1. The existence is due to Lemma 1.9(2). Assume that a, b : A → X are two maps such that f = ua = ub. In this case, u(a − b) = 0 implies that a − b factors through Σ^{−1}Z. The object Σ^{−1}Z is in τ^{>n}, so any map from A vanishes and a = b.

2. This statement is dual to the previous one.

As an immediate corollary we have

Corollary 2.6. Let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

and

\xymatrix{X′ \ar[r]^{u′} & Y′ \ar[r]^{v′} & Z′ \ar[r]^{w′} & ΣX′}

be two n-simple pseudo triangles. Consider the commutative diagram where the dotted arrows are missed.

\xymatrix{X \ar[r]^{u} \ar@{.>}[d]^{f} & Y \ar[r]^{v} \ar[d]^{g} & Z \ar@{.>}[d]^{h} \\ X′ \ar[r]^{u′} & Y′ \ar[r]^{v′} & Z′}

If v′gu = 0, then there is a unique completions for the maps f, h.

It follows that the object X (if it exists) is uniquely determined by the map v up to a canonical isomorphism, and a choice for the object X strictly determines the map u. In the same way u determines the object Z up to a canonical isomorphism, and a choice for Z strictly determines the map w. As a result we can define the truncation operation and the truncation octahedron on a bounded triangle.

2.3. The truncation operation on triangles.

Definition 2.7. A triangle

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

is an n-bounded triangle, if X, Y, Z ∈ τ^{≤n}. An n-truncation octahedron on such a triangle is an 4×4 diagram which looks like an octahedron on the composeable maps

Y → Z and Z → Z^{≥n}

i.e., a diagram of the form

[figure: 0; Z^{≥n}; Z^{≥n}; 0; X; u; Y; v; jv; Z; w; j; ΣX; X; u^{t}; Y^{t}; v^{t}; a; Z^{<n}; wi; i; ΣX; 0; Σ^{−1}Z^{≥n}; b; Σ^{−1}Z^{≥n}; −Σ^{−1}k; 0]

where i, j and k are the natural maps, and all the columns are distinguished triangles. We do NOT require that the rows are distinguished. The third row T^{t} = (u^{t}, v^{t}, wi) in this diagram is called an n-truncation of the triangle T = (u, v, w).

Lemma 2.8. Let

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be an n-bounded triangle.

1. For a general triangle, we do not know if the truncation octahedron exists. But if u or w are replaceable maps, or more generally, if the objects X, Y, Z in T and the middle map v : Y → Z belong to a distinguished triangle,

T′ : \xymatrix{X \ar[r]^{u′} & Y \ar[r]^{v} & Z \ar[r]^{w′} & ΣX} ,

then there is a unique (up to a canonical isomorphism) truncation octahedron on T. The objects and the maps are uniquely defined by the objects and the maps in the triangle T. More precisely, the object Y^{t} and the maps a and b depend only on jv, the map v^{t} depends only on v, and the map u^{t} depends only on jv and u. As a result, the n-truncated triangle T^{t} is uniquely determined by T.

2. If T is distinguished, then the truncation octahedron is really an octahedron, which means that the third line T^{t} is a distinguished triangle.

3. Let S and T be two n-bounded triangles satisfying the condition in (1). The truncation octahedra on S and on T are well defined. Then the triangle T ⊕S satisfied the condition in (1), and the its truncation octahedron is the direct sum of the above two octahedra. In particular we have

(S ⊕ T)^{t} = S^{t} ⊕ T^{t}.

4. If T splits, then T^{t} splits too.

Proof. 1. The problem is to show the existence and uniqueness of the dotted maps in the third line in the following 4 × 4 diagram.

[figure: 0; Z^{≥n}; Z^{≥n}; 0; X; u; Y; v; jv; Z; w; j; ΣX; X; u^{t}; Y^{t}; v^{t}; a; Z^{<n}; wi; i; ΣX; 0; Σ^{−1}Z^{≥n}; b; Σ^{−1}Z^{≥n}; −Σ^{−1}k; 0]

For the existance ee can not apply the octahedral axiom, since we are not assuming that T is distinguished, but it is enough to show that Y^{t} ∈ τ^{≤n}. The reason is that in this case we have n-simple distinguished triangles (rotated to the left) in the columns, and the lemma follows from Corollary 2.6.

To show that H^{k}(Y^{t}) = 0 for all k > n, apply the octahedral axiom on the distinguished triangle T′ and the composeable maps

v : Y → Z and j : Z → Z^{≥n}.

Now we have a new truncation octahedron which is a real octahedron. All the triangles are distinguished. In this new diagram, the object Y^{t} remains the same, since it is determined by the map jv. The homology functor H^{k} on the third line of the new diagram yields the exact sequence

0 → H^{k}(Y^{t}) → 0.

for all k > n. Hence, H^{k}(Y^{t}) vanishes for all k > n.

2. By the octahedral axiom, there exists a choice for the third row on the truncation octahedron, which is a distinguished triangle. Since T^{t} is the unique choice, T^{t} is distinguished.

3. By definition, the direct sum of the truncation octahedra of S and 0f T is a truncation octahedron on S ⊕ T. Since the truncation octahedron is unique, the direct sum of these two truncation octahedra is the truncation octahedron on S ⊕ T. By definition, (S ⊕ T)^{t} is the third row of this octahedron, which is the direct sum S^{t} ⊕ T^{t}.

4. The truncation of a trivial triangle is trivial, so the truncation of a direct sum of trivial triangles is a direct sum of trivial triangles too.

Lemma 2.8 shows, that the truncation operation is well defined on bounded triangles with u or w replaceable. Hence truncation is well defined on exact bounded triangles.

Lemma 2.9. The truncation of a bounded exact triangle is exact.

Proof. Let

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be an n-bounded exact triangle with replacing maps u′, v′ and w′. Consider the truncation (u^{t}, v^{t}, w′i) of (u, v, w′) and the truncation (u′^{t}, v^{t}, wi) of (u′, v, w).

Lemma 2.8 says that the objects of these truncated triangles are the same, and maps which have the same symbol are really the same map. Therefore, the truncated triangle (u^{t}, v^{t}, wi) of (u, v, w) is well defined, and it has replaceable maps on the left and on the right. A priori, the truncation of the distinguished triangle (u, v′, w) takes the form (u_{1}^{t}, v′^{t}, wi) where u_{1}^{t} depends on u and on v′, and it might be different from u^{t}. We will prove that in our case, when we assume that w is replaceable, the triangle (u^{t}, v′^{t}, wi) is distinguished, and hence, it is the truncation triangle of (u, v′, w).

Let us draw the truncation octahedron on the distinguished triangle (u, v, w′).

[figure: 0; Z^{≥n}; Z^{≥n}; 0; X; u; Y; v; jv; Z; w′; j; ΣX; X; u^{t}; Y^{t}; v^{t}; a; Z^{<n}; w′i; i; ΣX; 0; Σ^{−1}Z^{≥n}; b; Σ^{−1}Z^{≥n}; −Σ^{−1}k; 0]

By the rotation to the left of Lemma 1.9(4), there is an automorphism ϕ of Z such that v′ = ϕv, w′ = wϕ. Now, (a, jv, −Σb) is a distinguished n-simple triangle, and hence so is the triangle

(a, ϕ^{≥n}jv, −Σb(ϕ^{≥n})^{−1}).

Since

jv′ = jϕv = ϕ^{≥n}jv,

the triangle (a, jv′, −Σb′) is also a distinguished n-simple triangle, where b′ = bΣ^{−1}(ϕ^{≥n})^{−1}. Therefore, when we replace in the truncation octahedron of (u, v, w),

[figure: 0; Z^{≥n}; Z^{≥n}; 0; X; u; Y; v; jv; Z; w; j; ΣX; X; u^{t}; Y^{t}; v^{t}; a; Z^{<n}; wi; i; ΣX; 0; Σ^{−1}Z^{≥n}; b; Σ^{−1}Z^{≥n}; −Σ^{−1}k; 0]

the maps v, v^{t}, b by the maps v′, v′^{t}, b′ respectively, we get a truncation octahedron on the distinguished triangle (u, v′, w).

[figure: 0; Z^{≥n}; Z^{≥n}; 0; X; u; Y; v′; jv′; Z; w; j; ΣX; X; u^{t}; Y^{t}; _{v′_{t}}; a; Z^{<n}; wi; i; ΣX; 0; Σ^{−1}Z^{≥n}; b′; Σ^{−1}Z^{≥n}; −Σ^{−1}k; 0]

The third row in this octahedron is the distinguished triangle (u^{t}, v′^{t}, wi), and v′^{t} is a middle replacing map for the triangle (u^{t}, v^{t}, wi).

2.4. Truncating virtual triangles. We have shown, that the truncation of an exact triangle is exact. Given an n-bounded virtual triangle T, by definition we know that there exits a splitting triangle S, such that T ⊕ S is exact. If T ⊕ S is n-bounded too, then its n-truncation is exact, and this truncated triangle contains the truncation of T as an easy direct summand. Hence, in this case, the truncation of T is also a virtual triangle. Therefore, in order to prove that virtual triangles are closed under truncations, we will show, that any n-bounded virtual triangle is an easy direct summand of an n-bounded exact triangle. The first step is the next lemma.

Lemma 2.10. Let τ be a triangulated category with a t-structure , and let

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} (2.1)

be a virtual triangle. By corollary 1.13, we can add to (2.1) a trivial triangle

\xymatrix{0 \ar[r] & W \ar[r] & W \ar[r] & Σ0}

to get a semi-triangle with a replaceable map in the middle. If X, Y, Z ∈ τ^{<n}, then one can choose the object W in τ^{<n}.

Proof. By corollary 1.13, there is an object W such that the virtual triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ W \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} )}} & Z ⊕ W \ar[r]^{^{(^{w 0} )}} & ΣX} (2.2)

has a replaceable map in the middle, with a replacing map

V = ( \begin{matrix}a & b \\ c & d \end{matrix} ).

That is, the triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ W \ar[r]^{V} & Z ⊕ W \ar[r]^{^{(^{w 0} )}} & ΣX} (2.3)

is distinguished. Applying to this triangle the homology functor H^{m} yields the exact sequence

\xymatrix{0 \ar[r] & H^{m}(W) \ar[r]^{H^{m}(V)=H^{m}(d)} & H^{m}(W) \ar[r] & 0}

for all m ≥ n. Therefore, the map V^{≥n} = d^{≥n} : W^{≥n} → W^{≥n} is an isomorphism.

In the 4 × 4 commutative diagram below, the top and the bottom rows are isomorphic to trivial triangles, the second row is the distinguished triangle (2.3), the middle columns are direct sums of a trivial triangles with canonical distinguished triangles, and the other two columns are trivial triangles.

[figure: 0; W^{≥n}; V^{≥n}; W^{≥n}; 0; X; _{( \begin{matrix}u \\ 0 \end{matrix} )}; Y ⊕ W; V; Z ⊕ W; _{(_{w 0} )}; ΣX; X; Y ⊕ W^{<n}; Z ⊕ W^{<n}; _{(_{w 0} )}; ΣX; 0; Σ^{−1}W^{≥n}; Σ^{−1}V^{≥n}; Σ^{−1}W^{≥n}; 0]

By the octahedral axiom, there are fill in maps so that the completed diagram is an octahedron. This octahedron is isomorphic to the truncation octahedron on the distinguished triangle (2.3). All the columns are n-simple distinguished triangles, so the the fill in maps for the dotted arrows are unique, and the third line is the triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ W^{<n} \ar[r]^{V^{<n}} & Z ⊕ W^{<n} \ar[r]^{^{(^{w 0} )}} & ΣX}

Therefore, in the virtual triangle

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ W^{<n} \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} )}} & Z ⊕ W^{<n} \ar[r]^{^{(^{w 0} )}} & ΣX}

the middle map has V^{<n} as a replacing map.

Corollary 2.11. An n-bounded virtual triangle is an easy direct summand of an n-bounded exact triangle.

Proof. Let

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

be an n-bounded virtual triangle. By Lemma 2.10, we may assume, without loss of generality, that this triangle T has a replaceable map in the middle. By rotation to the left of Corollary 1.17(1) the direct sum of T with

\xymatrix{Y \ar[r] & Y \ar[r] & 0 \ar[r] & ΣY}

has replaceable maps on the left and on the middle. By rotation to right of Corollary 1.17(1) on the last triangle, the direct sum with

\xymatrix{(X ⊕ Y) \ar[r] & 0 \ar[r] & Σ(X ⊕ Y) \ar[r] & Σ(X ⊕ Y)} .

is exact. Since the additional two trivial triangles are n-bounded, the triangle T is an easy direct summand of an n-bounded exact triangle.

Theorem 2.12. Let τ be a triangulated category. Then virtual triangles are stable under isomorphisms, direct sums, mapping cones and easy direct summands. If τ admits a non-degenerate t-structure , then virtual triangles are also closed under truncations.

Proof. The only operation which require a proof is the truncation. Consider an n-bounded virtual triangle.

T : \xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

By corollary 2.11, there is an n-bounded splitting triangle

S : \xymatrix{A \ar[r] & B \ar[r] & C \ar[r] & ΣA} ,

such that the direct sum T ⊕ S is an n-bounded exact triangle. By Lemma 2.8(3),

(T ⊕ S)^{t} = T^{t} ⊕ S^{t}

and (T ⊕ S)^{t} is exact (by Lemma 2.9). By Lemma 2.8(4), S^{t} splits. Therefore, T^{t} is an easy direct summand of an exact triangle, and by definition, it is a virtual triangle.

3. Fuzzy Triangles and Non-exact Virtual Triangles

In this section we introduce a new type of triangles, namely the fuzzy triangles. These triangles are virtual triangles which are very close to exact triangles. We describe their structure and the general way of constructing them. Using this description we construct a fuzzy triangle, which is not exact. This leads to a construction of a virtual triangle with no replaceable map.

Definition 3.1. Let τ be a triangulated category. A fuzzy triangle is a pseudo triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX} ,

such that two of its maps are replaceable. In this case, we will say it is fuzzy in the remaining map. For example, if the maps u and w are replaceable, then we say the triangle is fuzzy in v or just fuzzy in the middle.

Lemma 3.2. Let τ be a triangulated category.

1. Let

[figure: X; u; Y; v; b; Z; w′; a; ΣX; X; u′; Y; v; Z; w; ΣX] (3.1)

be a commutative diagram with distinguished triangles in the rows, and a splitting cartesian square in the middle (splitting means with zero differential). Then the triangle (u, v, w) is fuzzy in the middle. Moreover, any fuzzy in the middle triangle is obtained in this a way.

2. If the triangle

\xymatrix{X \ar[r]^{u} & Y \ar[r]^{v} & Z \ar[r]^{w} & ΣX}

is fuzzy in the middle, then

\xymatrix{X \ar[r]^{^{( \begin{matrix}u \\ 0 \end{matrix} )}} & Y ⊕ Z \ar[r]^{^{( \begin{matrix}v & 0 \\ 0 & 1 \end{matrix} )}} & Z ⊕ Z \ar[r]^{^{(^{w 0} )}} & ΣX}

is exact. By duality, one can add Y ’s instead of Z’s.

Proof. 1. Consider a commutative diagram

[figure: X; u; Y; v; b; Z; w′; a; ΣX; X; u′; Y; v; Z; w; ΣX]

where the rows are distinguished triangles, and the middle square is splitting cartesian square. We only need to show exactness at X. The middle square splits, so its differential Σuw vanishes. Apply any homology functor H on the triangle, and assume that x ∈ H(ΣX) and H(Σu)(x) = 0. Then H(Σu′)(x) = H(Σb)H(Σu)(x) = 0 ⟹ ∃z ∈ H(Z), x = H(w)(z). The proof for cohomology functor is dual.

The converse is essentially 1.15, where here we choose the distinguished triangles in the rows so that Z′ = Z and Y = Y′′.

2. This is a special case of corollary 1.17.1.

Lemma 3.3. Let n and m be relatively prime integers. Then the square

\xymatrix{Z \ar[r]^{n} \ar[d]^{m} & Z \ar[d]^{m} \\ Z \ar[r]^{n} & Z} (3.2)

is a splitting bicartesian square in the abelian category of abelian groups. This means, that the sequence

\xymatrix{0 \ar[r] & Z \ar[r]^{^{( \begin{matrix}m \\ −n \end{matrix} )}} & Z ⊕ Z \ar[r]^{^{(^{n m})}} & Z \ar[r] & 0}

is exact.

Proof. The numbers n and m are relatively prime, so there exist two integers l, k such that ln + km = 1, and we have a splitting, i.e., in the sequence below, the left map has a left inverse and the right map has a right inverse.

\xymatrix{Z \ar[r]^{^{( \begin{matrix}m \\ −n \end{matrix} )}} & Z ⊕ Z \ar[r]^{^{(^{n m})}} & Z}

It remains to show the exactness in the middle. Let ( \begin{matrix}x \\ y \end{matrix}) ∈ Z ⊕ Z be an element in the kernel of (^{n m}), which means that nx + my = 0. Then

lnx + lmy = 0 ⇒ (1 − km)x + lmy = 0 ⇒ x = m(kx − ly),

and in the same way y = n(ly − kx). So we have

( \begin{matrix}x \\ y \end{matrix}) = ( \begin{matrix}m \\ −n \end{matrix})z,

where z = kx − ly.

Corollary 3.4. In the triangulated category D^{c}(Ab), there exist a fuzzy triangle which is not exact.

Proof. Let n and m be two relatively prime integers, and choose two integers k, l satisfying

ln + km = 1.

The commutative diagram below is a morphism between two short exact sequences

[figure: 0; Z; n; m; Z; π; m; Z/(n); 0; 0; Z; n; Z; kπ; Z/(n); 0]

where π is the quotient map. Note that k is an automorphism of Z/(n). In the derived category this diagram represents a triangle morphism

[figure: X; u; Z; n; m; Z; w′; m; ΣX; X; u′; Z; n; Z; w; ΣX]

where X = Σ^{−1}(Z/(n)). The square (3.2) is a splitting bicartesian square, so it is a splitting cartesian square in D^{c}(Ab). By 3.2, the sequence (u, n, w) is a fuzzy in the middle triangle, and by 1.16, it is exact, if an only if, there is an integer λ such that m + λn is an automorphism of Z. But if m ≠ ±1(mod n) (for example n = 5 and m = 8) then the third edge in any distinguished triangle starting with m + λn : Z → Z is a non zero object. So the map m + λn is not an automorphism for any λ, and the triangle (u, n, w) is a non-exact fuzzy triangle.

Corollary 3.5. Let τ be the derived category of the abelian category of Z/(3)- graded abelian groups with degree preserving maps. There exists a virtual triangle in τ with no replaceable map.

Proof. The category τ is just the category

D(Ab) × D(Ab) × D(Ab).

The objects are sequences of three objects of D(Ab),

A
B
C

morphisms are sequences of three morphisms in D(Ab)

a : A → A′
b : B → B′
c : C → C′

and distinguished triangles are sequences of three distinguished triangles in D(Ab).

T_{0} : \xymatrix{A′ \ar[r] & A \ar[r] & A′′ \ar[r] & ΣA′}
T_{1} : \xymatrix{B′ \ar[r] & B \ar[r] & B′′ \ar[r] & ΣB′}
T_{2} : \xymatrix{C′ \ar[r] & C \ar[r] & C′′ \ar[r] & ΣC′}

This means that a triangle in τ is distinguished, if and only if, it has a distinguished triangle in each degree. Therefore, a triangle with one replaceable map in τ is an sequence of three triangles in D(Ab): T_{0}, T_{1}, T_{2} where each triangle has a replaceable map in the same place. Let

R : \xymatrix{A′ \ar[r] & A \ar[r] & A′′ \ar[r] & ΣA′}

be a non-exact fuzzy in the middle triangle in D(Ab). We view this triangle as a triangle in τ concentrated in degree 0.

R_{0} : \xymatrix{A′ \ar[r] & A \ar[r] & A′′ \ar[r] & ΣA′}
R_{1} : \xymatrix{0 \ar[r] & 0 \ar[r] & 0 \ar[r] & Σ0}
R_{2} : \xymatrix{0 \ar[r] & 0 \ar[r] & 0 \ar[r] & Σ0}

Let S be the triangle

S_{0} : \xymatrix{0 \ar[r] & 0 \ar[r] & 0 \ar[r] & Σ0}
S_{1} : \xymatrix{B′ \ar[r] & B \ar[r] & B′′ \ar[r] & ΣB′}
S_{2} : \xymatrix{0 \ar[r] & 0 \ar[r] & 0 \ar[r] & Σ0}

and let T be the triangle

R_{0} : \xymatrix{0 \ar[r] & 0 \ar[r] & 0 \ar[r] & Σ0}
R_{1} : \xymatrix{0 \ar[r] & 0 \ar[r] & 0 \ar[r] & Σ0}
R_{2} : \xymatrix{C′ \ar[r] & C \ar[r] & C′′ \ar[r] & ΣC′}

where S_{1} is the rotation to the left of R_{0} and T_{2} is the rotation of R_{0} to the right. The triangle

R_{0} : \xymatrix{A′ \ar[r] & A \ar[r] & A′′ \ar[r] & ΣA′}
S_{1} : \xymatrix{B′ \ar[r] & B \ar[r] & B′′ \ar[r] & ΣB′}
T_{2} : \xymatrix{C′ \ar[r] & C \ar[r] & C′′ \ar[r] & ΣC′}

is just the direct sum R⊕S ⊕T, and so it is a virtual triangle. Replacing one of the maps in this triangle leaves the triangle non-distinguished in at least one degree. Therefore, the triangle R ⊕ S ⊕ T has no replaceable map.

Acknowledgments. This work has taken place while the author was visiting the Australian National University, Canberra. The author would like to express his gratitude to Amnon Neeman for his interest, guidance, and many helpful discussions.

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  E-mail address: vaaknin@math.huji.ac.il