Conditions for the map of sheaves $F\to F\otimes_{\mathcal O_Y}f_\ast \mathcal O_X$ to be an isomorphism

Question

Let $f:X\to Y$ be an immersion of locally noetherian separated schemes, for instance (not necessarily reduced) varieties over a field $k$. We can us assume $Y$ is affine.

Let $F$ be a quasi-coherent sheaf on $Y$. There is a canonical map $$ F\overset{u}{\longrightarrow}F\otimes_{\mathcal O_Y}f_\ast \mathcal O_X. $$ I wonder under what conditions on $f$ or on the sheaf $F$ the map $u$ is an isomorphism.

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1. If $f$ is a closed immersion, then $u$ is surjective. Even in simple cases, like the thickening of a point $f:X=\textrm{Spec }\mathbb C\to Y=\textrm{Spec }\mathbb C[\epsilon]$, I do not seem to get beyond surjectivity.
2. If $f$ is an open immersion, I have the following example in mind. Suppose $f$ is the immersion of a distinguished open set, namely $f:X=\textrm{Spec }B_h\to Y=\textrm{Spec }B$ for some $h\in B$. Then for a $B$-module $F$, we are asking whether the $B$-linear localization map $F\to F\otimes_BB_h=F_h$ is an isomorphism. I would not expect this to happen very often. (For instance if $F=B$ then I do not think one can aim at anything better than injectivity.)

Even with such easy examples I tried to build, the condition I am after seems to fail pretty often for fixed $F$. So, keeping for simplicity the condition that $f$ be an immersion, I would like to ask:

What conditions on $F$ (possibly combined with additional conditions on $f$) make the map $u$ an isomorphism?

Example. Take case 2 ($f$ open immersion). Since $\mathcal O_Y\to f_\ast \mathcal O_X$ is an isomorphism when restricted to $X$, I was thinking to fix the previous bug by including the condition that the (scheme-theoretic) support of $F$ is contained in the image of $f$. Would that make sense?

Thanks!

Answer 1

Here is the answer in the case of a closed immersion of schemes:

1) The affine case
Let $Y=\operatorname {Spec}(A)$ be an affine scheme and let $ F=\widetilde M$ be a quasi-coherent sheaf, associated to the $A$-module $M$.
Suppose $X=V(I)=\operatorname {Spec}(A/I)\hookrightarrow Y$ is the closed subscheme associated to the ideal $I\subset A$.
Then the map $F\overset{u}{\longrightarrow}F\otimes_{\mathcal O_Y}f_\ast \mathcal O_X$ corresponds to the $A$-module morphism $$M\overset{v}{\longrightarrow}M\otimes_AA/I=M/IM$$ That map is always surjective and will be injective iff $IM=0$, which means exactly that $M$ is obtained by restricting scalars to $A$ from an $A/I$-module $N$ i.e. $M=N_{[A]}$.

2) Globalization: the general case
Let $Y$ be a scheme, $f: X\hookrightarrow Y$ a closed subscheme and $ F$ a quasi coherent sheaf on $Y$.
Proposition
The canonical map $F\longrightarrow F\otimes_{\mathcal O_Y}f_\ast \mathcal O_X $ of quasi-coherent shaves on $Y$ is always surjective.
It is bijective if and only if there exists a quasi-coherent sheaf $E$ on $X$ such that $F=f_\ast(E)$.