By an admissable subcategory $A$ in a triangulated category $B$, I mean a triangulated subcategory that has $A \oplus B \in A$, then $A$, $B \in A$, and so that there is either a right or left adjoint to the inclusion of $A$ into $B$.


Then, if $X$ is a projective variety (possibly with mild assumptions on the singularities), is $D^b\_{perf}(X)$ admissible in $D^b(X)$? It clearly has the first condition, so I am curious about if an adjoint can be constructed.


I would like to know what is the best perfect approximation to a given complex.


Given a module, we can write $0 \to Tor(M) \to M \to M / Tor(M) \to 0$. For some rings (e.g. $\mathbb{Z}$), this is good enough, I think. But I don't know if there is or isn't something smarter one can do that works on any variety.


Follow up question: If so, then what is the (left or right) orthogonal to $D^b\_{perf}(X)$ in $D^b(X)$?


I am trying to understand if $D^b(X)$ can be understood in general as a decomposition of some category of well behaved objects (perfect complexes), and some other category. What is that other category? Something torsion something?



Let $X$ be a projective variety. Is the bounded derived category of perfect complexes admissable in $D^b(X)$?
$D^b\_{perf}(X)$ is not admissible in $D^b(X)$, unless $X$ is smooth (in which ase $D^b\_{perf}(X) = D^b(X)$). Still one can consider the quotient category $D^b(X)/D^b\_{perf}(X)$. It is called triangulated category of singularities, and was much studied.


