If $\pi : Proj(T_*) \to Proj(S_*)$ is induced by some $S_* \to T_*$, does some tensor product describe $\pi^*(\widetilde{M_*})$?

$\pi^*(\widetilde{M_*}) =_? T_* [\otimes]_{S_*} M_*$?, where $[\otimes]$ represents some appropriate tensor product? (Just naive analogy with the affine case.)

For example a definition of graded tensor product is given as an answer to my earlier question here: If $M_*$ and $N_*$ are graded modules over the *graded* ring $R_*$, what is the definition of $M_* \otimes_{R_*} N_*$?

But I am not sure if it is the tensor product that describes pullback of quasi coherent sheaves.

I think that under the additional equivalence relationship on described in that construction: $((S_* \otimes_{R_*} M_*)[r^{-1}])_0 = (S_*[r^{-1}])_0 \otimes_{R_*[r{-1}]} ((M_*)_r)_0$ is true, since the fractions of terms of the form $S_{-p} \otimes M_p$ can be balanced. So this seems to imply $\pi^*(\widetilde{M_*}) = T_* \otimes_{S_*} M_*$?, were the tensor product on the right is the graded tensor product given in the linked answer above (the usual tensor product of rings, then given the grading $m \otimes n$ has degree $deg(m) + deg(n)$.)

However I am not too confident in this kind of computation - I would appreciate a reference (especially if the source sets up all this symbolic nonsense cleanly!)


The proof of 5.12c in Hartshorne contains a cryptic answer (yes).


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