What is the restriction of a function $f$ to a closed subscheme?

Question

I am confused by the following definition (Definition 7.7.3 in the book “Algebraic Geometry II” by Mumford and Oda (in page 258 in its draft)):

Definition. If $X$ is an irreducible reduced scheme, $Y\subset X$ an irreducible reduced subscheme and $D$ is a Cartier divisor on $X$, then if $Y\not\subseteq\operatorname{Supp} D$, define $\operatorname{Tr}_Y D$ to be the Cartier divisor on $Y$ whose local equations at $y\in Y$ are just the restrictions to $Y$ of its local equations at $y\in X$.

My questions are:

1. What word does Tr come from?
2. What are the local equations of $D$ at $x\in X$?
3. (Main question) What are the restrictions to $Y$ of the local equations of $D$ at $y\in Y$?

For the definition of Cartier divisors in the book, see section 3.6 (page 109 in the draft).

The following are my thought:

1. Trace?
2. I think that a local equation of $D$ at $x\in X$ is an $f$ such that $(U,f)$ represents $D$ and such that $x\in U$. In this case (i.e., $X$ integral) $f$ is an element of the function field $\mathbf R(X)$.
3. Then the restriction to $Y$ of a local equation $f$ at $y$ must be an element of $\mathbf R(Y)$. I think this has something to do with the surjection $\mathcal O_X \to\mathcal O_Y$, but does it induce a map $\mathbf R(X)\to\mathbf R(Y)$ between function fields?

Thanks in advance.

Answer 1

1. Probably trace, but this is the first time I've seen this construction called that.

2. Yes, that's correct.

3. No, there's no $k$-map $k(X)\to k(Y)$ - any map of fields must be an injection, but the target has smaller transcendence degree than the source (assuming $Y$ is a strict subvariety) which is impossible. Instead, locally on $U\subset X$ write $f=g/h$ where $g,h\in\mathcal{O}_X(U)$. Writing $\overline{g}$ and $\overline{h}$ for the images of $g$ and $h$ under the natural map $\mathcal{O}_X(U)\to \mathcal{O}_Y(Y\cap U)$, the restriction of $f$ to $Y$ is $\overline{g}/\overline{h}$.