The Stacks project

Example 88.3.13. Let $\Lambda = k[[x]]$ be the power series ring in $1$ variable over $k$. Set $A = k$ and $B = \Lambda /(x^2)$. Then $B \to A$ is an essential surjection by Lemma 88.3.12 because it is a small extension and the map $B \to A$ does not have a right inverse (in the category $\mathcal{C}_\Lambda $). But the map

\[ k \cong \mathfrak m_ B/\mathfrak m_ B^2 \longrightarrow \mathfrak m_ A/\mathfrak m_ A^2 = 0 \]

is not an isomorphism. Thus in Lemma 88.3.12 (3) it is necessary to consider the map of relative cotangent spaces $\mathfrak m_ B/(\mathfrak m_\Lambda B + \mathfrak m_ B^2) \to \mathfrak m_ A/(\mathfrak m_\Lambda A + \mathfrak m_ A^2)$.


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