I am reading the Gortz's Algebraic Geometry, Proposition 5.49 and stuck at some point.
First, I propose a question.
Q. Let $Y = \operatorname{Spec}B$ is affine reduced $k$-scheme ( $k$ is a field ). Let $\xi$ be a maximal point ; i.e., the corresponding prime ideal $\mathfrak{p}_{\xi}$ is minimal prime ideal of $B$. Then the residue field $k(\xi) = B_{\mathfrak{p}_{\xi}}/ \mathfrak{p_{\xi}}B_{\mathfrak{p}_{\xi}} = \operatorname{Frac}(B/\mathfrak{p}_{\xi})$ is finite purely inseparable extension of $k$?
This question originates from following proof of theorem 5.49 in the Gortz's book.
Proposition 5.49. Let $X$ be a $k$-scheme. Then the following assertions are equivalent.
(i) $X$ is geometrically reduced. (ii) For every reduced $k$-scheme $Y$ the product $X \times_k Y$ is reduced. (iii) $X$ is reduced and for every maximal point $\eta$ of $X$ the residue field $\kappa(\eta)$ is a separable extension of $k$. (iv) There exists a perfect extension $\Omega$ of $k$ such that $X_{\Omega}$ is redueced.
(v) For every finite purely inseparable extension $K$ of $k$, $X_K$ is reduced.
Proof. We may assume that $X=\operatorname{Spec}A$ is affine, where $A$ is a $k$-algebra. By Corolary 5.45 (4) each assertion implies that $X$ is reduced. Thus we may assume that $A$ is reduced. Let $(\eta_i) _{i\in I}$ be the family of maximal points of $X$. As $A$ is reduced, the canonical homomorphism $$ A \to \prod_{i \in I} \kappa(\eta_i) \tag{5.13.1}$$
is injective ( since a ring is reduced iff the only element contained in all primes is zero )and $\kappa(\eta_i)$ is a localization of $A$ for all $i\in I$ (C.f. https://stacks.math.columbia.edu/tag/00EU). Let $L$ be a field extension of $k$. If $A \otimes_k L$ is reduced, then its localization $\kappa(\eta_i) \otimes_k L$ is redueced ( C.f. Tensor product and localization ). Conversely, we have an injective homomorphism
$$ A \otimes_k L \hookrightarrow ( \prod_{i\in I} \kappa(\eta_i) ) \otimes _k L \hookrightarrow \prod_{i\in I}(\kappa(\eta_i) \otimes_k L). \tag{5.13.2}$$
Therefore $A\otimes_k L$is reduced if and only if $\kappa(\eta_i) \otimes_k L $ is reduced for all $i\in I$. Thus by Proposition B.93 assertion (iii) is equivalent to (i), to (iv), and to (v). Here, the Proposition B.93 is,
Q. First question : Why (iii) is equivalent to (i) ? To show $X_L$ is reduced for 'any' field extension $L/k$, it suffices to show that $A\otimes_k L $ is reduced, which is implied (as proof) from the reducibility of $\kappa(\eta_i) \otimes_k L$ for each $i\in I$. However, there are difficulties in applying the Proposition B.93 directly, since there is possibility that the $L$ is not finite purely inseparable extension of $k$. Perhaps next more strong statement also holds?
$k \to K$ is separable $\Leftrightarrow$ For every field extension $L/k$, $K \otimes_k L$ is reduced ?
I found associated data from wiki-pedia : https://en.wikipedia.org/wiki/Separable_extension
In item "Separability of transcendental extensions" in the link, the Wikipedia states that
Let $E \supseteq F$ be a field extension of characteristic exponent $p$ (that is $p=1$ in characteristic zero, otherwise $p$ is the characteristic) The following properites are equivalent.
- E is separable extension of $F$,
- $E^p$ and $F$ are linearly disjoint over $F^p$,
- $F^{1/p}\otimes_F E$ is reduced,
- $L\otimes_F E$ is reduced for every field extension $L$ of $E$.
The theorem is really true? How can we prove $1) \Leftrightarrow 4) $ ?
(Continuing proof) The implication "(ii) $\Rightarrow$ (i)" is trivial. Thus is suffices to show that (iii) implies (ii). For this we amy assume that $Y=\operatorname{Spec}B$ is affine. Let $(L_j)_{j\in J}$ be the family of residue fields of maximal points of $Y$. We obtain an injective homomorphism
$$ A \otimes_k ( \prod_{j\in J} L_j) \hookrightarrow ( \prod_{i\in I} \kappa(\eta_i) ) \otimes_k ( \prod_{j \in J} L_j) \hookrightarrow \prod_{i,j} (\kappa(\eta_i) \otimes_k L_j ) \tag{5.13.3}$$
Thus first using (5.13.1) for $B$ and then (5.13.3) shows that $A\otimes_k B$ is a $k$-subalgebra of a product of rings which are reduced ( again by Proposition B.93 ) because all $\kappa(\eta_i)$ are separable over $k$.
Q. Second question : I'm now stuck at understanding the bold statement. I'm trying to use the Proposition B.93, (i) $\Rightarrow $ (ii). If we use this, then it will be good if each $L_j$ is finite purely inseparable extension of $k$, as I asked in the beginning. Is it true?
Note that if the conjecture I proposed in the first question is true, then we can also get the reducibility of $\kappa(\eta_i)\otimes_k L_j$.
Or is there any other route to show that each $\kappa(\eta_i) \otimes_k L_j$ is reduced? Or is there any other route to show that $A\otimes_k B$ is a $k$-subalgebra of a product of rings which are reduced so that $A\otimes_k B$ is reduced?
Can anyone helps?
Best Answer
O.K. The conjecture that I posed in the first question is true for general separable extension ; See. Eisenbud, Commutative Algebra, p.558, Theorem A1.3
Definition. $K$ is separably generated over $k$ if there exists a transcendence base $\{ x_\lambda\}_{\lambda \in \Lambda}$ for $K$ such that $K$ is a separable algebraic extension of $k( \{ x_{\lambda} \}_{\lambda \in \Lambda})$. $K$ is (possibly nonalgebraic) is separable over $k$ if every subfield of $K$ that is finitely generated over $k$ is separably generated over $k$.