Lemma 37.62.17 (0FK3)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.62: Local complete intersection morphisms
Lemma 37.62.17 (cite)

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Lemma 37.62.17. Let $f : X \to Y$ be a flat morphism of finite presentation. The following are equivalent

$f$ is a local complete intersection morphism,
$f$ is syntomic,
$\mathop{N\! L}\nolimits _{X/Y}$ has tor-amplitude in $[-1, 0]$, and
$\mathop{N\! L}\nolimits _{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.

Proof. Translated into algebra this is Divided Power Algebra, Lemma 23.11.5. To do the translation use Lemmas 37.62.5 and 37.13.2 as well as Derived Categories of Schemes, Lemmas 36.3.5, 36.10.4 and 36.10.7. $\square$

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