The Stacks project

Exercise 111.50.3. Suppose that $R$ is a ring and

\[ A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n). \]

Note that we are assuming that $A$ is presented by the same number of equations as variables. Thus the matrix of partial derivatives

\[ ( \partial f_ i / \partial x_ j ) \]

is $n \times n$, i.e., a square matrix. Assume that its determinant is invertible as an element in $A$. Note that this is exactly the condition that says that $\Omega _{A/R} = (0)$ in this case of $n$-generators and $n$ relations. Let $\pi : B' \to B$ be a surjection of $R$-algebras whose kernel $J$ has square zero (as an ideal in $B'$). Let $\varphi : A \to B$ be a homomorphism of $R$-algebras. Show there exists a unique homomorphism of $R$-algebras $\varphi ' : A \to B'$ such that $\varphi = \pi \circ \varphi '$.

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View Exercise 111.50.3 as pdf