In the theory of triangulated categories there is a hefty literature on localisation -- the most common example in algebra being (variants of) localising the homotopy category of chain complexes over a ring $A$ with respect to the quasiisomorphisms, giving the derived category $\mathbf{D}(A)$, and similarly for $\mathcal{O}_X$-modules on a scheme $X$.

Although not all localisations are with respect to a multiplicative system, the above is (with respect to $H^0$). There are other examples that aren't: one curious and interesting one in the paper of Buan and Marsh, where they provide an example of triangulated category that is localised with regard to a system of morphisms that is not multiplicative, and out comes an abelian category.

Unlike localisation in commutative rings (or even noncommutative rings), I feel like there are many fewer examples of localisation that could be helpful, if only to practice computing in triangulated categories.

My question is: **what are the nice, and perhaps "crucial" examples if any, of localisation of triangulated categories, other than the above mentioned ones?**

I would be particularly interested in "toy examples" or curiosities that aren't necessarily deep but yet still slightly tricky and yet elucidate some curious ways to use localisation. Ideally, any examples should pertain to applications of triangulated categories and localisation to algebra (representation theory, homological algebra, rings), or algebraic geometry. **For example, papers such as the one mentioned above would be very welcome.** For instance one could localise with respect to the multiplicative system given by morphisms that are converted to isomorphisms via any $\mathrm{Hom}(A,-)$, because it is a cohomological functor---but I haven't yet figured out if this leads to anywhere interesting...