For a finite etale map $X\rightarrow T$ of degree $d$, and a $U$-point $t\in T(U)$, are there at most $d$ points in $X(U)$ lying over $t$?

Question

If $f : X\rightarrow T$ is a finite etale morphism of connected schemes, and $U$ is another connected scheme and we're given a map $t : U\rightarrow T$, then must it be true that there are at most $d$ morphisms $x : U\rightarrow X$ such that $f\circ x = t$?

Answer 1

Yes. (And you don't need $X$ or $T$ to be connected, just $U$.)

To see this, pull-back $X$ over $U$, and so reduce to the case $U = T$.

So the question becomes: if $T$ is connected and $X \to T$ is finite etale of degree $d$, are there at most $d$ sections $T \to X.$

Suppose that $f$ and $g$ are two such sections: then these induce a morphism $f\times g: T \to X\times_{T} X.$
Now note that since $X$ is finite etale over $T$, the diagonal section $\Delta:X \to X\times_T X$ is both an open (this follows from etaleness, or more generally from unramifiedness) and a closed (this follows from finiteness, which implies separatedness) immersion. Thus, since $T$ is connected, we see that the image of $f\times g$ either lies in the image of $\Delta$, or is disjoint from it.

Now if $t\in T$ is a point for which $f(t)$ and $g(t)$ coincide, then $f\times g$ maps $t$ into the image of $\Delta$, and so the image of $f\times g$ lies in the image of $\Delta$, implying that $f = g$.

Thus the number of possible choices of $f$ is bounded above by the number of possible choices of $f(t)$, which is $d$.

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These kinds of arguments are developed carefully in SGA I (and probably in EGA IV as well, and likely Milne's book and other texts on the etale fundamental group and etale cohomology).