Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism? A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less traditional examples.
Let $A$ be a commutative ring, and $I \subset A$ an ideal. Then tensoring by $A / I$ is idempotent and so we can look at the localisation functor on $D(A)$, the derived category. This is exact in the triangulated sense, as tensoring is right-exact. Am I correct in thinking the colocalisation should just be the kernel of this tensor map, i.e. multiplying by $I$? If so, I expect for every complex of modules $M^\bullet$ that we should get a quasi-isomorphism $$I^2 M^\bullet \to I M^\bullet$$ from the natural inclusion. Is there an elementary way of seeing this? I'm finding it difficult to follow the machinery that guarantees a natural isomorphism $\Gamma^2 \to \Gamma$ over $D(A)$ for whichever colocalisation functor $\Gamma$ is actually associated to the derived tensor $A / I \otimes \_ $ defined over $D(A)$.
 A: The derived tensor product with $A/I$ is typically not idempotent (because ${\rm Tor}_{*}^A(A/I, A/I)$ is nontrivial), so this won't give you a localization sequence.
One case where it is idempotent is when $I = eA$ for $e \in A$ an idempotent.  In this case, tensoring with $A/I$ is the same as localizing at $1-e$, and the localization sequence does take the form $eM \to M \to M/eM$.
A: An example that might help the OP (who has "tried to come up with less traditional examples") is spelled out in Bill Dwyer's book chapter Localizations (Examples 2.5-2.7). This example does not exactly match the question the OP posed, but it's related and I felt it would be useful to share. As with the OP's example, the tensor product must be derived.
Dwyer assumes $I$ is generated by a finite set of elements, and considers the localization at $I$, and the corresponding colocalization. Dwyer constructs the colocalization via the derived tensor product, much like the OP wanted. In this case, the localization sequence recovers the theory of local homology and cohomology.
