Nonisomorphic Verdier octahedra on the same base

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 ∆3 ✛ d ∆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.

Let n 3. Let X be a periodic n-pretriangle. Suppose that Xd # is an (n -1)-triangle for all injective periodic monotone maps ∆n ✛ d ∆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.

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 X out of this subset, X| ∆n ≃ X| ∆n implies that there is a periodic isomorphism X ≃ X. We shall say that prolongation holds for this subset, if for X and X out of this subset and a morphism X| ∆n ✲ X| ∆n , there exists a periodic morphism X ✲ X that restricts on ∆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 ∆n ✛ d ∆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| ∆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 ∆n

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.

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 E, defined as the quotient of E modulo B. Second, there is the stable category E, defined as the quotient of the category

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