Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a finite number of steps from $T$ using the following operations : shift, direct sums, taking direct summands and cones.

Let $F,G : D^b(X) \rightarrow D^b(X)$ be two exact auto-equivalences and let $\rho : F \rightarrow G$ be a natural transformation. I assume that $\rho(T) : F(T) \rightarrow G(T)$ is zero. Are there some conditions on $T$, $F$,$G$ and $\rho$ (the weakest, the best) which would allow me to prove that $\rho$ vanishes as a natural transformation?

Thanks a lot!

Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a given finite number of steps from $T$ using the following operations : shift, direct sums, taking direct summands and cones.

Let $F,G : D^b(X) \rightarrow D^b(X)$ be two exact auto-equivalences and let $\rho : F \rightarrow G$ be a natural transformation. I assume that $\rho(T) : F(T) \rightarrow G(T)$ is zero. Are there some conditions on $T$, $F$,$G$ and $\rho$ (the weakest, the best) which would allow me to prove that $\rho$ vanishes as a natural transformation?

Thanks a lot!