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For a Noetherian scheme $X$, let $D^b(X)$ denote the bounded derived category of coherent sheaves on $X$.

Let $X$ be a Noetherian scheme, $i:Y \hookrightarrow X$ a closed subscheme and $F$ an object of $D^b(Y)$. Is any object $G$ of $D^b(X)$ which is a direct summand of $i_*(F)$ isomorphic to $i_*(F')$ for some object $F'$ of $D^b(Y)$?

It is easy to find examples where $F'$ cannot be a direct summand of $F$, but I have not been able to find a counterexample to the above statement. I am mostly interested in the case that $X$ is a variety over some field and $Y$ is reduced, but this might not be the natural setting.

If the question has a positive answer (perhaps with some extra conditions on $X$ and $Y$) then I would also be interested in knowing the answer to the question with $D^b(X)$ replaced by other flavours of the derived category, e.g., $D^-(X)$, the derived category of quasicoherent sheaves,...

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    $\begingroup$ Note that if you replace $i$ by more general proper morphism, e.g. resolution $\pi: Y \to X$ of rational singularities this is presumably unknown: according to Kawamata Lemma 7.4 in arxiv.org/pdf/1903.00801.pdf, the image of $\pi_*$ generates the $D^b(X)$ up to direct summands, but without direct summands this is not known (at least not in the literature and I don't know how to prove it). $\endgroup$
    – Evgeny Shinder
    Commented Jun 8, 2021 at 9:34
  • $\begingroup$ @EvgenyShinder: Thanks, I had not thought about this setting. I will take a look at Kawamata's paper. $\endgroup$
    – naf
    Commented Jun 8, 2021 at 9:40

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