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Given a triangulated category $\mathcal{D}$ with a t-structure $(\mathcal{D}_{\leq 0}, \mathcal{D}_{\geq 0})$ the cohomology functor can be defined as \begin{equation} H^k := \tau_{\geq 0}\circ \tau_{\leq 0} \circ [k] : \mathcal{D} \rightarrow \mathcal{A} \end{equation} where $\mathcal{A} := \mathcal{D}_{\leq 0} \cap \mathcal{D}_{\geq 0}$ is the heart of the t-structure. It is known that (e.g. https://en.wikipedia.org/wiki/T-structure#Cohomology_functors), given an exact triangle $A \rightarrow B \rightarrow C \rightarrow A[1]$ we get the long-exact-sequence \begin{equation} \cdots \rightarrow H^k(A) \rightarrow H^k(B) \rightarrow H^k(C) \rightarrow H^{k+1}(A) \rightarrow \cdots \end{equation} just like in the usual cohomological algebra. In the case of derived categories, I could see why this should be true (mainly because the cohomology functor simply computes the usual cohomology).

But is it easy to see that we get a long-exact-sequence from an exact triangle using only the definition of t-structures and the axioms of the general triangulated categories? Could someone suggest to me the easy way to do this?

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This should probably be a comment, but I am not able to comment yet. This is proved as theorem 1.3.6 in Faisceaux Pervers by Beilinson, Bernstein and Deligne. It also appears in English in Topological invariants of stratified spaces by Banagl as Proposition 7.1.12.

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