The Stacks project

Lemma 37.59.12. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume

  1. $S$ is locally Noetherian,

  2. $Y \to S$ is locally of finite type,

  3. $f : X \to Y$ is perfect,

  4. $X \to S$ is a local complete intersection morphism.

Then $X \to Y$ is a local complete intersection morphism and $Y \to S$ is Koszul at $f(x)$ for all $x \in X$.

Proof. In the course of this proof all schemes will be locally Noetherian and all rings will be Noetherian. We will use without further mention that regular sequences and Koszul regular sequences agree in this setting, see More on Algebra, Lemma 15.30.7. Moreover, whether an ideal (resp. ideal sheaf) is regular may be checked on local rings (resp. stalks), see Algebra, Lemma 10.68.6 (resp. Divisors, Lemma 31.20.8)

The question is local. Hence we may assume $S$, $X$, $Y$ are affine. In this situation we may choose a commutative diagram

\[ \xymatrix{ \mathbf{A}^{n + m}_ S \ar[d] & X \ar[l] \ar[d] \\ \mathbf{A}^ n_ S \ar[d] & Y \ar[l] \ar[ld] \\ S } \]

whose horizontal arrows are closed immersions. Let $x \in X$ be a point and consider the corresponding commutative diagram of local rings

\[ \xymatrix{ J \ar[r] & \mathcal{O}_{\mathbf{A}^{n + m}_ S, x} \ar[r] & \mathcal{O}_{X, x} \\ I \ar[r] \ar[u] & \mathcal{O}_{\mathbf{A}^ n_ S, f(x)} \ar[r] \ar[u] & \mathcal{O}_{Y, f(x)} \ar[u] } \]

where $J$ and $I$ are the kernels of the horizontal arrows. Since $X \to S$ is a local complete intersection morphism, the ideal $J$ is generated by a regular sequence. Since $X \to Y$ is perfect the ring $\mathcal{O}_{X, x}$ has finite tor dimension over $\mathcal{O}_{Y, f(x)}$. Hence we may apply Divided Power Algebra, Lemma 23.7.6 to conclude that $I$ and $J/I$ are generated by regular sequences. By our initial remarks, this finishes the proof. $\square$


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