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I have a question from Liu's book on Algebraic Geometry, doing Chapter 5 Question 1.16:

Let $f:X\rightarrow S$ be a morphism of schemes, with the following base change.

$$\require{AMScd} \begin{CD} X_{T}:=X\times_{S}T @>{f_{T}}>> T;\\ @V{p}VV @VV{\pi}V \\ X @>{f}>>S; \end{CD}$$

1) If $\mathcal{F}$ is an arbitrary $\mathcal{O}_{X}$-module, show that we have a canonical homomorphism: $\pi^{*}f_{*}\mathcal{F}\rightarrow (f_{T})_{*}p^{*}\mathcal{F}.$

[My attempt] I used the adjoint property to get the following (suppressing some notations for convenience)

$$\mbox{Hom}(\pi^{*}f_{*}\mathcal{F},(f_{T})_{*}p^{*}\mathcal{F})\\ \cong \mbox{Hom}(\mathcal{F},f^{*}f_{*}p_{*}p^{*}\mathcal{F})$$

from which we can obtain a canonical homomorphism. Question: Is this okay?

2) Suppose that $T\rightarrow S$ is flat, and $\mathcal{F}$ quasi-coherent. If $X$ is Noetherian (or $f$ is separated and quasi-compact), show that $\pi^{*}f_{*}\mathcal{F}\rightarrow (f_{T})_{*}p^{*}\mathcal{F}$ is an isomorphism.

Remark: I am stuck on this. Why do we need the condition that $X$ is Noetherian or $f$ is quasi-compact and separated??

Remark 2: I have no idea as well why we need flat condition.

3) Let $X$ and $f$ as in the previous question (I am not sure what he means. I think he means we need $X$ Noetherian and $f$ quasi-compact and separated). Let $\mathcal{G}$ be a locally free sheaf on $S$. Show that we have the canonical isomorphism: $$ f_{*}\mathcal{F}\otimes_{\mathcal{O}_{S}}\mathcal{G}\cong f_{*}(\mathcal{F}\otimes_{O_{X}}f^{*}\mathcal{G})$$

[My attempt] I did some preliminary calculations by cheating a bit: restricting to an open set where $\mathcal{G}$ is free, we may assume that $\mathcal{G}\cong\mathcal{O}_{S}^{(I)}$ for some indexing set $I$.

On one hand we have

$$\begin{eqnarray} &&f_{*}\mathcal{F}\otimes_{\mathcal{O}_{S}}\mathcal{G}\\ &\cong& f_{*}\mathcal{F}\otimes_{\mathcal{O}_{S}}\mathcal{O}_{S}^{(I)}\\ &\cong& (f_{*}\mathcal{F})^{(I)}\end{eqnarray}$$

On the other

$$\begin{eqnarray} && f_{*}(\mathcal{F}\otimes_{O_{X}}f^{*}\mathcal{G})\\ &\cong& f_{*}(\mathcal{F}\otimes_{O_{X}}f^{*}O_{S}^{(I)})\\ &\cong& f_{*}(\mathcal{F}\otimes_{O_{X}}(f^{*}O_{S})^{(I)})\\ &\cong& f_{*}(\mathcal{F}\otimes_{O_{X}}\mathcal{O}_{X}^{(I)})\\ &\cong& f_{*}(\mathcal{F}^{(I)})\cong (f_{*}\mathcal{F})^{(I)}. \end{eqnarray}$$

Of course I made a lot of logical leaps, because the problem is I still have not used the special condition! ($X$ Noetherian and $f$ quasi-compact, separated). Where is the problem and where should I use the condition.

Appreciate your help! Thanks!

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The assumption that $X$ is Noetherian or $f$ quasi-compact and separated is used to guarantee that $f_\ast \mathcal{F}$ is quasi-coherent. Without it, it is not true that the pushforward of a quasi-coherent sheaf is quasi-coherent. –  user38268 Jan 4 at 3:05
    
You can find the proof of Benja's assertion on Harshtorne's Chapter II Section V. –  ADR Jan 18 at 23:46

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