The Stacks project

Lemma 47.23.3. Properties (A), (B), (C), (D), and (E) of More on Algebra, Section 15.50 hold for $P(k \to R) =$“$R$ is a local complete intersection”. See Divided Power Algebra, Definition 23.8.5.

Proof. Part (A). Let $k \subset K$ be a finitely generated field extension. Let $R$ be a $k$-algebra which is a local complete intersection. We can find a global complete intersection $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ over $k$ such that $K$ is isomorphic to the fraction field of $A$, see Algebra, Lemma 10.156.11. Then $R \to R \otimes _ k A$ is a relative global complete intersection. It follows that $R \otimes _ k A$ is a local complete intersection by Divided Power Algebra, Lemma 23.8.9.

Proof of (B). This is clear because a ring is a local complete intersection if and only if all of its local rings are complete intersections.

Part (C). Let $A \to B \to C$ be flat maps of Noetherian rings. Assume the fibres of $A \to B$ are local complete intersections and $B \to C$ is regular. We have to show the fibres of $A \to C$ are local complete intersections. Clearly, we may assume $A = k$ is a field. Then we may assume that $B \to C$ is a regular local homomorphism of Noetherian local rings. Then $B$ is a complete intersection and $C/\mathfrak m_ B C$ is regular, in particular a complete intersection (by definition). Then $C$ is a complete intersection by Divided Power Algebra, Lemma 23.8.9.

Part (D). This follows by the same arguments as in (C) from the other implication in Divided Power Algebra, Lemma 23.8.9. Part (E) is immediate as the condition does not refer to the ground field. $\square$


Comments (2)

Comment #4673 by Bogdan on

A typo. The sentence ''We have to show the fibres of A→C are Gorenstein'' should read ''We have to show the fibres of A→C are local complete intersections''.


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