Étale morphism induces isomorphism of tangent spaces

by Feryll   Last Updated October 10, 2019 03:20 AM - source

Let $(A, m) \to (B, n)$ be a flat map of local Noetherian rings with $mB = n$, $B$ of finite type over $A$, and $k(B) = B / n$ a finite separable field extension of $k(A) = A / m$. Then, I want to show that the map $m / m^2 \to n / n^2$ induces an isomorphism of (base-changed) tangent spaces: $$\text{Hom}_{k(B)}(n/n^2, k(B)) \cong \text{Hom}_{k(A)}(m/m^2, k(B))$$ But I'm running into some problems trying to manipulate the objects in question... So, I of course have a short exact sequence $0 \to m^2 \to m \to m / m^2 \to 0$ of $A$-modules, to which I apply the exact functor $- \otimes_A B$ to obtain $n / n^2 \cong (m / m^2) \otimes_A B \cong (m / m^2) \otimes_{k(A)} k(B)$. But now in applying the tensor-hom adjunction, I get $$\text{Hom}_{k(B)}(n/n^2, k(B)) \cong \text{Hom}_{k(A)}(m/m^2, \text{Hom}_{k(A)}(k(B),k(B)))$$ which is seemingly a bigger module than I want. What's going wrong?



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