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Let $\mathcal{A}$ be an abelian category, and $D$ its bounded derived category. An object $M \in D$ may be described as a list of cohomology objects $H^i = H^i(M)$ together with some complicated glueing data.

I am interested only in the case when $\mathcal{A}$ has homological dimension two. For example, $\mathcal{A}$ can be a category of coherent sheaves on a smooth surface. In this case the glueing data amounts to a collection of classes $\xi_i \in \mathrm{Ext}^2(H^i, H^{i-1})$ between each pair of adjacent cohomology objects, with no restrictions on the choices.

By definition, an object $M \in D$ is quasiisomorphic to a direct sum of complexes concentrated in a single degree (i.e., shifts of objects from $\mathcal{A}$) if and only if each $\xi_i$ vanishes.

Similarly, some objects in $D$ are quasiisomorphic to direct sums of complexes concentrated in two adjacent degrees. Is it possible to characterize this property by vanishing of some obstructions built in terms of the presentation of an object as a collection $\{ (H_i, \xi_i) \}_{i \in \mathbb{Z}}$ above?

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As pointed out in nikola karabatic's answer, a decomposition of M as a direct sum of two-term complexes induces decompositions $H^i=H^i_a\oplus H^i_b$ for each $i$. These have the property that $\xi_i\in\text{Ext}^2(H^i_a,H^{i-1}_b)$, where $\text{Ext}^2(H^i_a,H^{i-1}_b)$ is regarded as a direct summand of $\text{Ext}^2(H^i,H^{i-1})$ in the obvious way. Or equivalently, there are vanishing obstructions in the other three summands of $\text{Ext}^2(H^i,H^{i-1})$.

Conversely, if there are such decompositions of the $H^i$, then the reconstruction of $M$ from the data $\{(H^i,\xi_i)\}_{i\in\mathbb{Z}}$ gives a direct sum of two term complexes with cohomology $H^i_a$ in degree $i$ and $H^{i-1}_b$ in degree $i-1$.

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This is not a complete answer, but too long for a comment.

A splitting into a sum of two chain complexes yields a splitting of every $H_i$ into two parts $H_{i}^1\oplus H_{i}^2$, coming from the complex in degrees $i-1,i$ and $i, i+1$ respectively. This splitting should be part of the data defining the obstruction (otherwise I see no chance to do so).

Given this data, consider some fixed index $i$. We consider the distinguished triangle $$\tau_{[i-1,i]}M \rightarrow \tau_{[i-1,i+1]}M\rightarrow H_{i+1}M[-i-1]\rightarrow \tau_{[i-1,i]}M[1].$$ Here $\tau_{[i-1,i]}M=\tau_{\ge i}\tau_{\le i} M$ and $\tau_{\le n}$ denotes the canonical truncation. The cohomology splits as $H_{i+1}M=H_{i+1}^1\oplus H_{i+1}^2$. Let $M'$ be the preimage of $H_{i+1}^1$ in $\tau_{[i-1,i+1]}M$, then we have a triangle $$\tau_{[i-1,i]}M \rightarrow M' \rightarrow H_{i+1}^1 M[-i-1]\rightarrow \tau_{[i-1,i]}M[1]$$ and want to know whether this splits. For this splitting, there is an obstruction $\eta_i$ in $\mathrm{Hom}_{D(\mathcal{A})}(H_{i+1}^1 M[-i-1], \tau_{[i-1,i]}M[1])$ given as the image of the identity in the long exact sequence $$ \ldots\rightarrow \mathrm{Hom}_{D(\mathcal{A})}(H_{i+1}^1 M[-i-1],M')\rightarrow \mathrm{Hom}_{D(\mathcal{A})}(H_{i+1}^1 M[-i-1], H_{i+1}^1 M[-i-1]) \rightarrow \mathrm{Hom}_{D(\mathcal{A})}(H_{i+1}^1 M[-i-1], \tau_{[i-1,i]}M[1])\rightarrow \ldots $$ The complex $\tau_{[i-1,i]}M[1]$ is given by your $\xi_i$. Similarly, you get an obstruction $\mu_i$ by considering $\tau_{[i-2,i]}M$.

Upshot: The obstructions $\eta_i$ and $\mu_i$ constructed above, depending on a splitting of the cohomology, certainly vanish if $M$ splits as a sum of two-term complexes. I don't know whether the converse is true.

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  • $\begingroup$ I'm a little confused by some of the indexing, but I think that the fact that $\mathcal{A}$ has homological dimension two significantly simplifies this, since then the map $H^{i+1}M[-i-1]\to\tau_{[i-1.i]}M[1]$ in your first distinguished triangle (I think you meant $H^{i+1}M$ to be shifted in degree?) is determined by a map $H^{i+1}M[-i-1]\to H^iM[-i-2]$ (which is exactly the element of $\text{Ext}^2(H^{i+1}M,H^iM)$ referred to in the question. $\endgroup$ – Jeremy Rickard Sep 20 at 9:36
  • $\begingroup$ Sorry, in the previous comment $[-i-2]$ should be $[-i+1]$. $\endgroup$ – Jeremy Rickard Sep 20 at 12:10
  • $\begingroup$ As Jeremy Rickard said, in the case of homological dimension 2 this description becomes almost tautological: given a splitting $H^i = H^i_a \oplus H^i_b$, it produces a splitting of the complex if and only if $\xi_{i+1}$ factors through $H^i_a$ and $\xi_i$ factors through $H^i_b$. And that is more or less "the complex splits into a direct sum of two-term complexes". I would be more interested in an obstruction that doesn't start with a decomposition of a cohomology object. Intuitively the condition should be some "orthogonality" between glueing maps, but I don't know in which sense. $\endgroup$ – Dmitry Pirozhkov Sep 20 at 15:24
  • $\begingroup$ fixed the indexing and @Jeremy Rickard: you are right that this was too complicated. I upvoted your answer. $\endgroup$ – nikola karabatic Sep 20 at 17:49
  • $\begingroup$ @Dmitry Pirozhkov: I still cannot see how it should be possible to define an obstruction without the splitting as input data. There might be many ways to split the cohomology, yielding different decompositions on homology, for example if all $\xi_i$ equal $0$. (Perhaps my fantasy is just too limited.) $\endgroup$ – nikola karabatic Sep 20 at 17:51

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