Lemma 49.11.1. Let $f : Y \to X$ be a morphism of schemes. The following are equivalent

$f$ is finite and syntomic,

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

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

$f$ is finite and for every $x \in X$ there is an affine open $x \in U = \mathop{\mathrm{Spec}}(A) \subset X$ an integer $n$ and $f_1, \ldots , f_ n \in A[x_1, \ldots , x_ n]$ such that $f^{-1}(U)$ is isomorphic to the spectrum of $A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$,

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

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

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