Let $* \to \mathbb A^1$ be the embedding of the point defined by $t=0$, where $t$ is the coordinate for $\mathbb A^1$. Let $X$ be a smooth projective variety (say $\mathbb P^1$), and $i\colon X \to X \times \mathbb A^1$ be the embedding of the central fibre. Let ${\mathrm D}(-)$ denote the bounded derived category of coherent sheaves. My question is:
Assume $F\in{\mathrm D}(X \times \mathbb A^1)$ such that $F \xrightarrow{t} F$ is zero map (in ${\mathrm D}(X \times \mathbb A^1)$), where $t$ denotes the morphism which is acted by $t$ in each degree. Then, is it true that there exists a $E\in {\mathrm D}(*)$, such that $F= i_*E$?
In the non-derived setting (i.e. replace $\mathrm {D}$ by $\mathrm{Coh}$) this is true by definition. I would like to know if this is true also in the derived setting.
My attempt. Basically, what I want is to find a representative of $F$ such that $F \xrightarrow{t} F$ is homotopic to zero (then we can define such a $E$). However, I guess the following is not true in general (unless $Y$ is affine):
Let $f\colon F\to F$ be a chain map of bounded complex of locally free sheaves on $Y$. Assume $f$ is zero in the derived category. Then, is $f$ homotopic to zero?
(Edit: Indeed, for example, see this counter-example.) However, one may take advantage of the form $F \xrightarrow{t} F$ for $Y=X \times \mathbb A^1$, so I am not sure if my first question is true or false.
Thanks in advance!
