I am trying to understand the proof of proposition 4.17 in "Fourier-Mukai transforms in algebraic geometry" by D. Huybrechts about the structure of the group of autoequivalences of $D^b(X)$ in the case of ample (anti-)canonical bundle.
So here is a brief outline of the set-up for my question:
Suppose our autoequivalence $F$ maps $\mathcal{O}_X$ to itself and $w_X^{\otimes k}$ to itself too. Then we get a map $$\oplus H^0(X,w_X^{\otimes k})=\oplus Hom(\mathcal{O}_X, w_X^{\otimes k}) \xrightarrow{\phi_F} \oplus Hom(F(\mathcal{O}_X), F(w_X^{\otimes k}))=\oplus Hom(\mathcal{O}_X, w_X^{\otimes k})=\oplus H^0(X,w_X^{\otimes k})$$ $$s \mapsto F(s)$$
It is also shown that this in fact induces a graded ring automorphism.
Since we assume $w_X$ to be ample, we have $X \cong Proj (\oplus H^0(X, w_X^{\otimes k}))$ so that we get an automorphism $X \xrightarrow{\phi_F^{\flat}} X$. Then it is claimed that if we compose $F$ with the pullback $(\phi_F^{\flat})^{*}$ action on $D^b(X)$ we actually get an automorphism which acts as identity on $\oplus H^0(X, w_X^{\otimes k})$.
What is the best way to see it?
It is well-known that for $X=Proj (S_{\bullet})$ the category of quasi-coherent sheaves is equivalent to saturated graded $S_{\bullet}$-modules, but is there an analog for the graded ring itself? Something along the lines: if $X$ is $Proj$ of a saturated graded ring (whatever that might be), then automorphisms of $X$ naturally correspond to automorphisms of the graded ring (naturally being given by $\sharp$ and $\flat$, as in the affine case).
