Lemma 37.59.8. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

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

$f$ is syntomic.

** A morphism is flat and lci if and only if it is syntomic. **

Lemma 37.59.8. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

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

$f$ is syntomic.

**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 10.136.13 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 37.59.4). 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. $\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #825 by Johan Commelin on

There are also: