Let $f:Y\to X$ be a map of affine schemes, where $X=\text{Spec}A$ and $Y=\text{Spec}B$. Let $M,N$ be modules over $A$ and $B$, respectively. I know the following three facts:
- The functors $f^{*}$ and $f_{*}$ on $\mathcal{O}_{X}$-modules and $\mathcal{O}_{Y}$-modules, respectively are adjoint to each other.
- $f_{*}\widetilde{N}=\widetilde{_{A}N}$, where $_{A}N$ denotes $N$ considered as an $A$-module.
- $f^{*}\widetilde{M}=\widetilde{M\otimes_{A}B}$.
Here, $\widetilde{M}$ denotes the $\mathcal{O}_{X}$-module associated to the $A$-module $M$, etc.
Starting with fact 3, the adjunction tells me that $\widetilde{M}=f_{*}\widetilde{M\otimes_{A}B}$, which, by fact 2, is $\widetilde{_{A}(M\otimes_{A}B)}$. However, I don't believe that $_{A}(M\otimes_{A}B)=M$ - for example, $A=M=\mathbb{Z}$ seems to make the left hand side equal to $B$ and the right hand side equal to $\mathbb{Z}$.
Am I making a mistake somewhere?
Adjunctions need not send isomorphisms to isomorphisms. For any scheme $X$, the identity ring map on global sections $\Gamma(X, \mathcal{O}_X) \to \Gamma(X, \mathcal{O}_X)$ induces, via adjunction, a natural morphism of schemes $X \to \text{Spec}(\Gamma(X, \mathcal{O}_X))$, which is not an isomorphism if $X$ is not affine.