What does an endomorphism in a triangulated category give rise to?

Question

Let $D\xrightarrow[]\varphi D\xrightarrow[]kE\xrightarrow[]j\Sigma D$ be an exact triangle in a triangulated category. I am trying to figure out what structure emerges from this on the base of the axioms. For example, the octahedron axiom gives an exact triangle $E\to E^{(2)}\to E\xrightarrow[]{\Sigma k\circ j}\Sigma E$, where $E^{(2)}$ is from the exact triangle $D\xrightarrow[]{\varphi\circ\varphi}D\xrightarrow[]{}E^{(2)}\to\Sigma D$. All this resembles a beginning of a spectral sequence obtained from an exact couple (e. g. obviously the composite $d:E\xrightarrow[]j\Sigma D\xrightarrow[]{\Sigma k}\Sigma E$ satisfies $\Sigma d\circ d=0$) but it quickly becomes too entangled for me.

Can anyone give me a reference for analyzing such situations? At least in simpler situations when e. g. $\varphi\circ\varphi$ is $\varphi$ or zero or invertible? Or on the other hand in more general situations when $\varphi$ has nonzero degree (i. e. $\varphi:\Sigma^nD\to D$ for any $n\in\mathbb Z$)?

I have vague feeling that something like this takes place in connection with the Devinatz-Hopkins-Smith theorem, and that Bondal-Kapranov's twisted complexes might be relevant, but cannot come up with anything definite...

Answer 1

Here's an expanded version of my comment, addressing the spectral sequence part of the question. A filtered object in a triangulated category $T$ is simply a sequence

$$ \dots \to X_{n-1} \to X_n \to X_{n+1} \to \dots $$

When you apply a homological functor $H : T\to A$, where $A$ is an abelian category, filtered objects in $T$ become bigraded spectral sequences in $A$. The construction of these spectral sequences uses nothing more than the triangulated structure (but you need more structure to make them functorial in the filtered object). In fact, the spectral sequence arises from the exact couple

$$ H_*(X_*) \to H_*(X_*) \to H_*(C_*) \to H_*(X_*),$$

where $C_n=\operatorname{cofib}(X_{n-1}\to X_n)$ and $H_*=H\circ \Sigma^{-*}$. All this is described in much detail here, where some references are given.

The usual convergence result is the following. Suppose that $T$ and $A$ have enough sequential (homotopy) colimits and that:

• for every $r$, $H$ preserves the colimit of $n\mapsto \operatorname{cofib}(X_r\to X_{r+n})$,

• for every $r$, $H_r(X_n)=0$ for $n\ll 0$.

Then the spectral sequence converges strongly to $H_*(\operatorname{hocolim}_n X_n)$. Note that this applies to $H^{op}: T^{op}\to A^{op}$ as well, which gives a dual statement with limits.

Now, given an endomorphism $\phi\colon D\to\Sigma^d D$, I can think of two ways to get a filtered object:

(1) Just repeat the map $\phi$, with $X_n= \Sigma^{nd}D$,

(2) Let $X_0=0$, $X_{1}=\operatorname{fib}(\phi)$, $X_2=\operatorname{fib}(\phi^2)$, etc.

The potential targets are:

(1) $H_*$ of the colim-inversion of $\phi$,

(2) $H_*$ of the "$\phi$-primary torsion" in $D$.

In the dual story, we'll have potential targets:

(1') $H_*$ of the lim-inversion of $\phi$,

(2') $H_*$ of the "$\phi$-completion" of $D$.

I don't know if the spectral sequence (1) is useful. It converges strongly if $H$ preserves sequential colimits, $d<0$, and $H_*(D)$ is bounded below. It also converges trivially if $\phi$ has order $n$, the $(n+1)$st page being zero. But if $\phi$ is idempotent, the second page is zero and convergence fails completely.

The spectral sequence (2) is a Bockstein spectral sequence.