Dimension of fiber product of subvarieties. Let $X,Y \subset \mathbb{A}^n_k$ be be subvarieties of pure dimension r,s respectively, K a field. How could I show that $X \times_k Y$ is of pure dimension $r+s$? I am self-studying so anything is welcome. I have tried Noether normalization but I couldn't make it work.
 A: I agree with the OP that there is a bit of an issue here: if you use the argument with Noether normalization, then you have to show there aren't irreducible components of $X \times Y$ of lower dimension.
If $k$ is algebraically closed you can first prove the product is irreducible, see Lemma Tag 05P3). In the case that $k$ is not algebraically closed, you can work around the issue, by extending $k$ to the algebraic closure and working with one irreducible component at a time (for each $X$ and $Y$). This will work and it is the right thing to do for most students of this material.
However, if you want, you can try to prove something about minimal primes of $A \otimes_k B$ for domains $A$ and $B$ over $k$. For example, since $k \to B$ is flat, we see that $A \to A \otimes_k B$ is flat. Hence, by going down for flat ring maps (Lemma Tag 00HS), any minimal prime of $A \otimes_k B$ lies over $(0) \subset A$. Similarly, for $B \to A \otimes_k B$. Thus now we reduce to thinking about minimal primes of $K \otimes_k L$ where $k \subset K$ and $k \subset L$ are field extensions (they are the fraction fields of $A$ and $B$). In this case, if $K' \subset K$ and $L' \subset L$ are subextensions, then $K' \otimes_k L' \to K \otimes_k L$ is flat and we conclude that minimal primes of $K \otimes_k L$ lie over minimal primes of $K' \otimes_k L'$.
Finally, it is easy to see that $k(x_1, \ldots, x_n) \otimes_k k(y_1, \ldots, y_m)$ has a unique minimal prime with residue field $k(x_1, \ldots, x_n, y_1, \ldots, y_m)$.
