Lemma 91.2.3. Let $A$ be a ring. Let $B$ be an $A$-algebra. Let $N$ be a $B$-module. The set of isomorphism classes of extensions of $A$-algebras

\[ 0 \to N \to B' \to B \to 0 \]

where $N$ is an ideal of square zero is canonically bijective to $\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N)$.

**Proof.**
To prove this we apply the previous results to the case where (91.2.0.1) is given by the diagram

\[ \xymatrix{ 0 \ar[r] & N \ar[r] & {?} \ar[r] & B \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & A \ar[u] \ar[r]^{\text{id}} & A \ar[u] \ar[r] & 0 } \]

Thus our lemma follows from Lemma 91.2.2 and the fact that there exists a solution, namely $N \oplus B$. (See remark below for a direct construction of the bijection.)
$\square$

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