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If $\mathcal{H}$ is the heart of a bounded t-structure in a triangulated category $\mathcal{T}$, then for every object $E$ in $\mathcal{T}$ there exists a finite sequence of integers $k_1>k_2>\dots >k_n$ and a collection of triangles in $\mathcal{T}$ $$ E_{i-1}\stackrel{\phi_i}{\to}E_i\to A_i \to E_{i-1}[1] $$ for $i=0,\dots,n$, with $E_0=0$, $E_n=E$ and $A_i\in \mathcal{H}[k_i]$ for any $i=1,\dots,n$. This can be seen as a Postnikov tower for the initial morphism $0 \to E$, so by analogy with what happens in algebraic topology one is led to wonder whether more generally one has Postnikov towers for arbitrary morphisms in a bounded t-structure, i.e., the same as above, but this time with $E_0=X$, $E_n=Y$ and $\phi_n\circ\phi_{n-1}\circ\cdots\circ \phi_1$ equal to some given morphism $f:X\to Y$.

I have so far been unable to rigorously prove (or disprove) such a statement nor to locate it in the literature. As usual any suggestion in either direction will be appreciated.

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up vote 5 down vote accepted

Let $F$ be the fiber of $f:X \to Y$ (i.e. the cone shifted by $[-1]$) so that we have a distinguished triangle $$ F \to X \to Y. $$ Let $0 = F_0 \to F_1 \to \dots \to F_n = F$ be its Postnikov tower. Let $E_i$ be the cone of the map $F_i \to X$ (the composition of $F_i \to F$ and $F \to X$). Then $E_0 = Cone(0 \to X) = X$ and $E_n = Cone(F \to X) \cong Y$. On the other hand, by octahedron axiom $$ Cone(E_{i-1} \to E_i) = Cone (F_{i-1} \to F_i)[1], $$ so I think this is what you want.

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Thanks! Now that I see it, I see I had proved this a couple of months ago and then I had completely forgotten the reasoning (the Kubla Khan effect :) ). Thanks a lot! –  domenico fiorenza May 11 at 16:16

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