# Global Spec functor and base change

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.

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.

• There is a disagreement between the question and this answer here: the OP writes $f^*A$, the pullback, while you write $f_*A$, the pushforward. As such, this is not currently an answer. Jul 1, 2021 at 21:36