The Stacks project

Proof. Working affine locally this is More on Algebra, Lemma 15.33.5. We also give a more geometric proof.

Assume (2). By Morphisms, Lemma 29.30.10 for every point $x$ of $X$ there exist affine open neighbourhoods $U$ of $x$ and $V$ of $f(x)$ such that $f|_ U : U \to V$ is standard syntomic. This means that $U = \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)) \to V = \mathop{\mathrm{Spec}}(R)$ where $R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a relative global complete intersection over $R$. By Algebra, Lemma 10.136.12 the sequence $f_1, \ldots , f_ c$ is a regular sequence in each local ring $R[x_1, \ldots , x_ n]_{\mathfrak q}$ for every prime $\mathfrak q \supset (f_1, \ldots , f_ c)$. Consider the Koszul complex $K_\bullet = K_\bullet (R[x_1, \ldots , x_ n], f_1, \ldots , f_ c)$ with homology groups $H_ i = H_ i(K_\bullet )$. By More on Algebra, Lemma 15.30.2 we see that $(H_ i)_{\mathfrak q} = 0$, $i > 0$ for every $\mathfrak q$ as above. On the other hand, by More on Algebra, Lemma 15.28.6 we see that $H_ i$ is annihilated by $(f_1, \ldots , f_ c)$. Hence we see that $H_ i = 0$, $i > 0$ and $f_1, \ldots , f_ c$ is a Koszul-regular sequence. This proves that $U \to V$ factors as a Koszul-regular immersion $U \to \mathbf{A}^ n_ V$ followed by a smooth morphism as desired.

Assume (1). Then $f$ is a flat and locally of finite presentation (Lemma 37.62.4). Hence, according to Morphisms, Lemma 29.30.10 it suffices to show that the local rings $\mathcal{O}_{X_ s, x}$ are local complete intersection rings. Choose, locally on $X$, a factorization $f = \pi \circ i$ for some Koszul-regular immersion $i : X \to P$ and smooth morphism $\pi : P \to S$. Note that $X \to P$ is a relative quasi-regular immersion over $S$, see Divisors, Definition 31.22.2. Hence according to Divisors, Lemma 31.22.4 we see that $X \to P$ is a regular immersion and the same remains true after any base change. Thus each fibre is a regular immersion, whence all the local rings of all the fibres of $X$ are local complete intersections. $\square$