Let $k$ be a field and $A$ and $B$ be two graded $k$-algebras satisfying $A\subset B$. The $\operatorname{Proj}$ construction is not functorial but is there nothing to say about $\operatorname{Proj}(A)$ and $\operatorname{Proj}(B)$ ? (even under some additional assumptions you are free to make for exemples)


Many thanks for your enlightenments! 



If $A\subset B$, what to say about their $\operatorname{Proj}$?
Let $A\_+=\oplus\_{n\ge1}A\_n\subset A$ be the irrelevant ideal. Then the homogeneous ideal $A\_+B\subseteq B$ defines a subscheme $\mathcal N\subseteq{\rm Proj}(B)$ (the ``nullcone'' in case $A$ is a ring of invariants). Then the inclusion $A\subseteq B$ induces a projection $\pi:{\rm Proj}(B)\setminus\mathcal N\to{\rm Proj}(A)$. I don't think that one can say much more about this morphism other than that it is affine and, in case $B$ is a domain, dominant.


