The Stacks project

Lemma 15.105.4. Let $L/K$ be an extension of fields. Let $A$ be a $K$-algebra. Let $B \subset A$ be the maximal weakly étale $K$-subalgebra of $A$ as in Lemma 15.105.2. Then $B \otimes _ K L$ is the maximal weakly étale $L$-subalgebra of $A \otimes _ K L$.

Proof. For an algebra $A$ over $K$ we write $B_{max}(A/K)$ for the maximal weakly étale $K$-subalgebra of $A$. Similarly we write $B_{max}(A'/L)$ for the maximal weakly étale $L$-subalgebra of $A'$ if $A'$ is an $L$-algebra. Since $B_{max}(A/K) \otimes _ K L$ is weakly étale over $L$ (Lemma 15.104.7) and since $B_{max}(A/K) \otimes _ K L \subset A \otimes _ K L$ we obtain a canonical injective map

\[ B_{max}(A/K) \otimes _ K L \to B_{max}((A \otimes _ K L)/L) \]

The lemma states that this map is an isomorphism.

To prove the lemma for $L$ and our $K$-algebra $A$, it suffices to prove the lemma for any field extension $L'$ of $L$. Namely, we have the factorization

\[ B_{max}(A/K) \otimes _ K L' \to B_{max}((A \otimes _ K L)/L) \otimes _ L L' \to B_{max}((A \otimes _ K L')/L') \]

hence the composition cannot be surjective without $B_{max}(A/K) \otimes _ K L \to B_{max}((A \otimes _ K L)/L)$ being surjective. Thus we may assume $L$ is algebraically closed.

Reduction to finite type $K$-algebra. We may write $A$ is the filtered colimit of its finite type $K$-subalgebras. Using Lemma 15.105.3 we see that it suffices to prove the lemma for finite type $K$-algebras.

Assume $A$ is a finite type $K$-algebra. Since the kernel of $A \to A_{red}$ is nilpotent, the same is true for $A \otimes _ K L \to A_{red} \otimes _ K L$. Then

\[ B_{max}((A \otimes _ K L)/L) \to B_{max}((A_{red} \otimes _ K L)/L) \]

is injective because the kernel is nilpotent and the weakly étale $L$-algebra $B_{max}((A \otimes _ K L)/L)$ is reduced (Lemma 15.105.1). Since $B_{max}(A/K) = B_{max}(A_{red}/K)$ by Lemma 15.105.3 we conclude that it suffices to prove the lemma for $A_{red}$.

Assume $A$ is a reduced finite type $K$-algebra. Let $Q = Q(A)$ be the total quotient ring of $A$. Then $A \subset Q$ and $A \otimes _ K L \subset Q \otimes _ A L$ and hence

\[ B_{max}(A/K) = A \cap B_{max}(Q/K) \]

and

\[ B_{max}((A \otimes _ K L)/L) = (A \otimes _ K L) \cap B_{max}((Q \otimes _ K L)/L) \]

by Lemma 15.105.3. Since $-\otimes _ K L$ is an exact functor, it follows that if we prove the result for $Q$, then the result follows for $A$. Since $Q$ is a finite product of fields (Algebra, Lemmas 10.25.4, 10.25.1, 10.31.6, and 10.31.1) and since $B_{max}$ commutes with products (Lemma 15.105.3) it suffices to prove the lemma when $A$ is a field.

Assume $A$ is a field. We reduce to $A$ being finitely generated over $K$ by the argument in the third paragraph of the proof. (In fact the way we reduced to the case of a field produces a finitely generated field extension of $K$.)

Assume $A$ is a finitely generated field extension of $K$. Then $K' = B_{max}(A/K)$ is a field separable algebraic over $K$ by Lemma 15.105.3 part (6). Hence $K'$ is a finite separable field extension of $K$ and $A$ is geometrically irreducible over $K'$ by Algebra, Lemma 10.47.13. Since $L$ is algebraically closed and $K'/K$ finite separable we see that

\[ K' \otimes _ K L \to \prod \nolimits _{\sigma \in \mathop{\mathrm{Hom}}\nolimits _ K(K', L)} L,\quad \alpha \otimes \beta \mapsto (\sigma (\alpha )\beta )_\sigma \]

is an isomorphism (Fields, Lemma 9.13.4). We conclude

\[ A \otimes _ K L = A \otimes _{K'} (K' \otimes _ K L) = \prod \nolimits _{\sigma \in \mathop{\mathrm{Hom}}\nolimits _ K(K', L)} A \otimes _{K', \sigma } L \]

Since $A$ is geometrically irreducible over $K'$ we see that $A \otimes _{K', \sigma } L$ has a unique minimal prime. Since $L$ is algebraically closed it follows that $B_{max}((A \otimes _{K', \sigma } L)/L) = L$ because this $L$-algebra is a field algebraic over $L$ by Lemma 15.105.3 part (6). It follows that the maximal weakly étale $K' \otimes _ K L$-subalgebra of $A \otimes _ K L$ is $K' \otimes _ K L$ because we can decompose these subalgebras into products as above. Hence the inclusion $K' \otimes _ K L \subset B_{max}((A \otimes _ K L)/L)$ is an equality: the ring map $K' \otimes _ K L \to B_{max}((A \otimes _ K L)/L)$ is weakly étale by Lemma 15.104.11. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CKU. Beware of the difference between the letter 'O' and the digit '0'.