Lemma 49.10.1. Let $f : Y \to X$ be a morphism of schemes. The following are equivalent

$f$ is locally quasi-finite and syntomic,

$f$ is locally quasi-finite, flat, and a local complete intersection morphism,

$f$ is locally quasi-finite, flat, locally of finite presentation, and the fibres of $f$ are local complete intersections,

$f$ is locally quasi-finite and for every $y \in Y$ there are affine opens $y \in V = \mathop{\mathrm{Spec}}(B) \subset Y$, $U = \mathop{\mathrm{Spec}}(A) \subset X$ with $f(V) \subset U$ an integer $n$ and $h, f_1, \ldots , f_ n \in A[x_1, \ldots , x_ n]$ such that $B = A[x_1, \ldots , x_ n, 1/h]/(f_1, \ldots , f_ n)$,

for every $y \in Y$ there are affine opens $y \in V = \mathop{\mathrm{Spec}}(B) \subset Y$, $U = \mathop{\mathrm{Spec}}(A) \subset X$ with $f(V) \subset U$ such that $A \to B$ is a relative global complete intersection of the form $B = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$,

$f$ is locally quasi-finite, flat, locally of finite presentation, and $\mathop{N\! L}\nolimits _{X/Y}$ has tor-amplitude in $[-1, 0]$, and

$f$ is flat, locally of finite presentation, $\mathop{N\! L}\nolimits _{X/Y}$ is perfect of rank $0$ with tor-amplitude in $[-1, 0]$,

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