Hartshorne Corollary 9.4 precise meaning of $\mathcal{F}_y$ and $\mathcal{F} \otimes k(y)$

Question

Let $f: X \to Y= \operatorname{Spec A}$ be a separated morphism of finite type of noetherian schemes and let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. For a point $y \in Y$, let $X_y$ be the fiber over $y$ and let $\mathcal{F}_y$ be the induced sheaf.

Let $i: X_y \to X$. Is $\mathcal{F}_y$ the restriction of $\mathcal{F}$ onto the fiber $X_y$, i.e $\mathcal{F}_y = i^{-1} \mathcal{F}$, which is not necessarily locally free or even an $\mathcal{O}_X$-module, or is $\mathcal{F}_y$ the pullback, $i^* \mathcal{F}$ which is locally free and an $\mathcal{O}_x$-module?

I imagine $\mathcal{F}_y$ should be the latter but I am not exactly sure what Hartshorne has in mind here.

Furthermore, in the same corollary, he goes on to define the following sheaf: $\mathcal{F} \otimes k(y)$ where $k(y)$ is the constant sheaf on the closed subset $\overline{ \{ y \} }$ of $Y$.

First of all, is $k(y)$ with this definition a sheaf on $Y$ via extension by zero?

Also, I thought to take the tensor product of two sheaves they must both be modules over the same structure sheaf. However when Hartshorne writes $\mathcal{F} \otimes k(y)$ he never designates what sheaf we are tensoring over.

Answer 1

The notation $\mathcal F_y$ stands for the pullback $i^\ast \mathcal F$ under the immersion $i:X_y\to X$. But there is no reason why it should be locally free. By $k(y)$ one usually means the skyscraper sheaf of a point $y\in Y$.

As for $\mathcal F\otimes k(y)$, the tensor product is over $\mathcal O_Y=A$. Here $k(y)$ is an $A$-module via $A\to A/\mathfrak m_y=k(y)$ and $\mathcal F$ is viewed as an $A$-module via the map $f$.

To better see this, assume $X=\textrm{Spec }B$ is affine and $\mathcal F=M$ is a $B$-module. Let $\phi:A\to B$ be the ring map corresponding to $f$. You can view $M$ as an $A$-module via $a\cdot m=\phi(a)\cdot m$. If you want, you can denote this $A$-linear structure by $\phi_\ast M$. Then $\mathcal F\otimes k(y)$ means $\phi_\ast M\otimes_AA/\mathfrak m_y$.