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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!

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  • $\begingroup$ There are some exercises in Hartshorne you may want to see..it says that if the inclusion $A\subset B$ has the property that the $A_{+}B=B_{+}$ then we have a morphism...if the graded inclusion is isomorphism for all d>>0, then the morphism is actually isomorphism..etc. $\endgroup$
    – user100841
    Commented Nov 6, 2017 at 20:36

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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.

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