Let $f:X \rightarrow Y$ be any morphism of schemes, $A$ be a quasi-coherent sheaf of $\mathcal{O}_X$ algebra (Or Noetherian schemes, coherent sheaf if you want), S be a scheme over $Y$ defined by $A$ using the global Spec. Now can we conclude that $X \times_Y S$ is isomorphic with the scheme defined by $f^*A$ ¿ I think that will be true to check locally but not that definite with that.
Global Spec functor and base change
algebraic-geometryschemessheaf-theory
Best Answer
Question: "Now can we conclude that $X×_Y S$ is isomorphic with the scheme defined by $f_∗A$? I think that will be true to check locally but not that definite with that."
Answer: If $f: X:=Spec(B) \rightarrow S:=Spec(A)$ and if $A_S$ is any quasi coherent sheaf of $\mathcal{O}_S$-algebras it follows $R:=\Gamma(S,A_S)$ is an $A$-algebra with $A_X \cong \tilde{R}$ the sheafification of $R$. Let $T:=Spec(A_S)$. It follows
$$X\times_S T\cong Spec(B\otimes_A R).$$
And $f^*(A_S) \cong \tilde{B\otimes_A R}$ is the sheafification of $B\otimes_A R$, hence
$$Spec(f^*A_S) \cong Spec(B\otimes_A R) \cong X\times_S T.$$
This construction globalize.
If $A_X$ is a quasi coherent sheaf of $\mathcal{O}_X$-algebras, there is a map
$$f^{\#}: \mathcal{O}_S \rightarrow f_*\mathcal{O}_X$$
hence $f_*A_X$ is a quasi coherent sheaf of $\mathcal{O}_S$-algebras and you may construct $\pi: Spec(f_*A_X)\rightarrow S$ which is a scheme over $S$
Note: The push forward $g_*F$ of a quasi coherent sheaf $F$, where $g: Y \rightarrow Y'$ is a map of schemes, is not always quasi coherent. It is quasi coherent when $Y$ is Noetherian.