The Stacks project

37.56 Local complete intersection morphisms

In Divisors, Section 31.21 we have defined 4 different types of regular immersions: regular, Koszul-regular, $H_1$-regular, and quasi-regular. In this section we consider morphisms $f : X \to S$ which locally on $X$ factor as

\[ \xymatrix{ X \ar[rr]_ i \ar[rd] & & \mathbf{A}^ n_ S \ar[ld] \\ & S } \]

where $i$ is a $*$-regular immersion for $* \in \{ \emptyset , Koszul, H_1, quasi\} $. However, we don't know how to prove that this condition is independent of the factorization if $* = \emptyset $, i.e., when we require $i$ to be a regular immersion. On the other hand, we want a local complete intersection morphism to be perfect, which is only going to be true if $* = Koszul$ or $* = \emptyset $. Hence we will define a local complete intersection morphism or Koszul morphism to be a morphism of schemes $f : X \to S$ that locally on $X$ has a factorization as above with $i$ a Koszul-regular immersion. To see that this works we first prove this is independent of the chosen factorizations.

Lemma 37.56.1. Let $S$ be a scheme. Let $U$, $P$, $P'$ be schemes over $S$. Let $u \in U$. Let $i : U \to P$, $i' : U \to P'$ be immersions over $S$. Assume $P$ and $P'$ smooth over $S$. Then the following are equivalent

  1. $i$ is a Koszul-regular immersion in a neighbourhood of $x$, and

  2. $i'$ is a Koszul-regular immersion in a neighbourhood of $x$.

Proof. Assume $i$ is a Koszul-regular immersion in a neighbourhood of $x$. Consider the morphism $j = (i, i') : U \to P \times _ S P' = P''$. Since $P'' = P \times _ S P' \to P$ is smooth, it follows from Divisors, Lemma 31.22.9 that $j$ is a Koszul-regular immersion, whereupon it follows from Divisors, Lemma 31.22.12 that $i'$ is a Koszul-regular immersion. $\square$

Before we state the definition, let us make the following simple remark. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $x \in X$. Then there exist an open neighbourhood $U \subset X$ and a factorization of $f|_ U$ as the composition of an immersion $i : U \to \mathbf{A}^ n_ S$ followed by the projection $\mathbf{A}^ n_ S \to S$ which is smooth. Picture

\[ \xymatrix{ X \ar[rd] & U \ar[l] \ar[d] \ar[r]_-i & \mathbf{A}^ n_ S = P \ar[ld]^\pi \\ & S } \]

In fact you can do this with any affine open neighbourhood $U$ of $x$ in $X$, see Morphisms, Lemma 29.39.2.

Definition 37.56.2. Let $f : X \to S$ be a morphism of schemes.

  1. Let $x \in X$. We say that $f$ is Koszul at $x$ if $f$ is of finite type at $x$ and there exists an open neighbourhood and a factorization of $f|_ U$ as $\pi \circ i$ where $i : U \to P$ is a Koszul-regular immersion and $\pi : P \to S$ is smooth.

  2. We say $f$ is a Koszul morphism, or that $f$ is a local complete intersection morphism if $f$ is Koszul at every point.

We have seen above that the choice of the factorization $f|_ U = \pi \circ i$ is irrelevant, i.e., given a factorization of $f|_ U$ as an immersion $i$ followed by a smooth morphism $\pi $, whether or not $i$ is Koszul regular in a neighbourhood of $x$ is an intrinsic property of $f$ at $x$. Let us record this here explicitly as a lemma so that we can refer to it

Lemma 37.56.3. Let $f : X \to S$ be a local complete intersection morphism. Let $P$ be a scheme smooth over $S$. Let $U \subset X$ be an open subscheme and $i : U \to P$ an immersion of schemes over $S$. Then $i$ is a Koszul-regular immersion.

Proof. This is the defining property of a local complete intersection morphism. See discussion above. $\square$

It seems like a good idea to collect here some properties in common with all Koszul morphisms.

Lemma 37.56.4. Let $f : X \to S$ be a local complete intersection morphism. Then

  1. $f$ is locally of finite presentation,

  2. $f$ is pseudo-coherent, and

  3. $f$ is perfect.

Proof. Since a perfect morphism is pseudo-coherent (because a perfect ring map is pseudo-coherent) and a pseudo-coherent morphism is locally of finite presentation (because a pseudo-coherent ring map is of finite presentation) it suffices to prove the last statement. Being perfect is a local property, hence we may assume that $f$ factors as $\pi \circ i$ where $\pi $ is smooth and $i$ is a Koszul-regular immersion. A Koszul-regular immersion is perfect, see Lemma 37.55.7. A smooth morphism is perfect as it is flat and locally of finite presentation, see Lemma 37.55.5. Finally a composition of perfect morphisms is perfect, see Lemma 37.55.4. $\square$

Lemma 37.56.5. Let $f : X = \mathop{\mathrm{Spec}}(B) \to S = \mathop{\mathrm{Spec}}(A)$ be a morphism of affine schemes. Then $f$ is a local complete intersection morphism if and only if $A \to B$ is a local complete intersection homomorphism, see More on Algebra, Definition 15.33.2.

Proof. Follows immediately from the definitions. $\square$

Beware that a base change of a Koszul morphism is not Koszul in general.

Lemma 37.56.6. A flat base change of a local complete intersection morphism is a local complete intersection morphism.

Proof. Omitted. Hint: This is true because a base change of a smooth morphism is smooth and a flat base change of a Koszul-regular immersion is a Koszul-regular immersion, see Divisors, Lemma 31.21.3. $\square$

Lemma 37.56.7. A composition of local complete intersection morphisms is a local complete intersection morphism.

Proof. Let $g : Y \to S$ and $f : X \to Y$ be local complete intersection morphisms. Let $x \in X$ and set $y = f(x)$. Choose an open neighbourhood $V \subset Y$ of $y$ and a factorization $g|_ V = \pi \circ i$ for some Koszul-regular immersion $i : V \to P$ and smooth morphism $\pi : P \to S$. Next choose an open neighbourhood $U$ of $x \in X$ and a factorization $f|_ U = \pi ' \circ i'$ for some Koszul-regular immersion $i' : U \to P'$ and smooth morphism $\pi ' : P' \to Y$. In fact, we may assume that $P' = \mathbf{A}^ n_ V$, see discussion preceding and following Definition 37.56.2. Picture:

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[r]_-{i'} & P' = \mathbf{A}^ n_ V \ar[d] \\ Y \ar[d] & & V \ar[ll] \ar[r]_ i & P \ar[d] \\ S & & & S \ar[lll] } \]

Set $P'' = \mathbf{A}^ n_ P$. Then $U \to P' \to P''$ is a Koszul-regular immersion as a composition of Koszul-regular immersions, namely $i'$ and the flat base change of $i$ via $P'' \to P$, see Divisors, Lemma 31.21.3 and Divisors, Lemma 31.21.7. Also $P'' \to P \to S$ is smooth as a composition of smooth morphisms, see Morphisms, Lemma 29.34.4. Hence we conclude that $X \to S$ is Koszul at $x$ as desired. $\square$

slogan

Lemma 37.56.8. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

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

  2. $f$ 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 37.56.4). 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. $\square$

Lemma 37.56.9. A regular immersion of schemes is a local complete intersection morphism. A Koszul-regular immersion of schemes is a local complete intersection morphism.

Proof. Since a regular immersion is a Koszul-regular immersion, see Divisors, Lemma 31.21.2, it suffices to prove the second statement. The second statement follows immediately from the definition. $\square$

Lemma 37.56.10. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume $Y \to S$ smooth and $X \to S$ is a local complete intersection morphism. Then $f : X \to Y$ is a local complete intersection morphism.

Proof. Immediate from the definitions. $\square$

Lemma 37.56.11. Let $f : X \to Y$ be a morphism of schemes. If $f$ is locally of finite type and $X$ and $Y$ are regular, then $f$ is a local complete intersection morphism.

Proof. We may assume there is a factorization $X \to \mathbf{A}^ n_ Y \to Y$ where the first arrow is an immersion. As $Y$ is regular also $\mathbf{A}^ n_ Y$ is regular by Algebra, Lemma 10.163.10. Hence $X \to \mathbf{A}^ n_ Y$ is a regular immersion by Divisors, Lemma 31.21.12. $\square$

The following lemma is of a different nature.

Lemma 37.56.12. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume

  1. $S$ is locally Noetherian,

  2. $Y \to S$ is locally of finite type,

  3. $f : X \to Y$ is perfect,

  4. $X \to S$ is a local complete intersection morphism.

Then $X \to Y$ is a local complete intersection morphism and $Y \to S$ is Koszul at $f(x)$ for all $x \in X$.

Proof. In the course of this proof all schemes will be locally Noetherian and all rings will be Noetherian. We will use without further mention that regular sequences and Koszul regular sequences agree in this setting, see More on Algebra, Lemma 15.30.7. Moreover, whether an ideal (resp. ideal sheaf) is regular may be checked on local rings (resp. stalks), see Algebra, Lemma 10.68.6 (resp. Divisors, Lemma 31.20.8)

The question is local. Hence we may assume $S$, $X$, $Y$ are affine. In this situation we may choose a commutative diagram

\[ \xymatrix{ \mathbf{A}^{n + m}_ S \ar[d] & X \ar[l] \ar[d] \\ \mathbf{A}^ n_ S \ar[d] & Y \ar[l] \ar[ld] \\ S } \]

whose horizontal arrows are closed immersions. Let $x \in X$ be a point and consider the corresponding commutative diagram of local rings

\[ \xymatrix{ J \ar[r] & \mathcal{O}_{\mathbf{A}^{n + m}_ S, x} \ar[r] & \mathcal{O}_{X, x} \\ I \ar[r] \ar[u] & \mathcal{O}_{\mathbf{A}^ n_ S, f(x)} \ar[r] \ar[u] & \mathcal{O}_{Y, f(x)} \ar[u] } \]

where $J$ and $I$ are the kernels of the horizontal arrows. Since $X \to S$ is a local complete intersection morphism, the ideal $J$ is generated by a regular sequence. Since $X \to Y$ is perfect the ring $\mathcal{O}_{X, x}$ has finite tor dimension over $\mathcal{O}_{Y, f(x)}$. Hence we may apply Divided Power Algebra, Lemma 23.7.6 to conclude that $I$ and $J/I$ are generated by regular sequences. By our initial remarks, this finishes the proof. $\square$

Lemma 37.56.13. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume $S$ is locally Noetherian, $Y \to S$ is locally of finite type, $Y$ is regular, and $X \to S$ is a local complete intersection morphism. Then $f : X \to Y$ is a local complete intersection morphism and $Y \to S$ is Koszul at $f(x)$ for all $x \in X$.

Proof. This is a special case of Lemma 37.56.12 in view of Lemma 37.55.6 (and Morphisms, Lemma 29.15.8). $\square$

Lemma 37.56.14. Let $i : X \to Y$ be an immersion. If

  1. $i$ is perfect,

  2. $Y$ is locally Noetherian, and

  3. the conormal sheaf $\mathcal{C}_{X/Y}$ is finite locally free,

then $i$ is a regular immersion.

Proof. Translated into algebra, this is Divided Power Algebra, Proposition 23.11.3. $\square$

Lemma 37.56.15. Let $f : X \to Y$ be a local complete intersection homomorphism. Then the naive cotangent complex $\mathop{N\! L}\nolimits _{X/Y}$ is a perfect object of $D(\mathcal{O}_ X)$ of tor-amplitude in $[-1, 0]$.

Proof. Translated into algebra this is More on Algebra, Lemma 15.84.4. To do the translation use Lemmas 37.56.5 and 37.13.2 as well as Derived Categories of Schemes, Lemmas 36.3.5, 36.10.4 and 36.10.7. $\square$

Lemma 37.56.16. Let $f : X \to Y$ be a perfect morphism of locally Noetherian schemes. The following are equivalent

  1. $f$ is a local complete intersection morphism,

  2. $\mathop{N\! L}\nolimits _{X/Y}$ has tor-amplitude in $[-1, 0]$, and

  3. $\mathop{N\! L}\nolimits _{X/Y}$ is perfect with tor-amplitude in $[-1, 0]$.

Proof. Translated into algebra this is Divided Power Algebra, Lemma 23.11.4. To do the translation use Lemmas 37.56.5 and 37.13.2 as well as Derived Categories of Schemes, Lemmas 36.3.5, 36.10.4 and 36.10.7. $\square$

Lemma 37.56.17. Let $f : X \to Y$ be a flat morphism of finite presentation. The following are equivalent

  1. $f$ is a local complete intersection morphism,

  2. $f$ is syntomic,

  3. $\mathop{N\! L}\nolimits _{X/Y}$ has tor-amplitude in $[-1, 0]$, and

  4. $\mathop{N\! L}\nolimits _{X/Y}$ 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 37.56.5 and 37.13.2 as well as Derived Categories of Schemes, Lemmas 36.3.5, 36.10.4 and 36.10.7. $\square$

The following lemma gives a characterization of smooth morphisms as flat morphisms whose diagonal is perfect.

Lemma 37.56.18. Let $f : X \to Y$ be a finite type morphism of locally Noetherian schemes. Denote $\Delta : X \to X \times _ Y X$ the diagonal morphism. The following are equivalent

  1. $f$ is smooth,

  2. $f$ is flat and $\Delta : X \to X \times _ Y X$ is a regular immersion,

  3. $f$ is flat and $\Delta : X \to X \times _ Y X$ is a local complete intersection morphism,

  4. $f$ is flat and $\Delta : X \to X \times _ Y X$ is perfect.

Proof. Assume (1). Then $f$ is flat by Morphisms, Lemma 29.34.9. The projections $X \times _ Y X \to X$ are smooth by Morphisms, Lemma 29.34.5. Hence the diagonal is a section to a smooth morphism and hence a regular immersion, see Divisors, Lemma 31.22.8. Hence (1) $\Rightarrow $ (2). The implication (2) $\Rightarrow $ (3) is Lemma 37.56.9. The implication (3) $\Rightarrow $ (4) is Lemma 37.56.4. The interesting implication (4) $\Rightarrow $ (1) follows immediately from Divided Power Algebra, Lemma 23.10.2. $\square$

Lemma 37.56.19. The property $\mathcal{P}(f) =$“$f$ is a local complete intersection morphism” is fpqc local on the base.

Proof. Let $f : X \to S$ be a morphism of schemes. Let $\{ S_ i \to S\} $ be an fpqc covering of $S$. Assume that each base change $f_ i : X_ i \to S_ i$ of $f$ is a local complete intersection morphism. Note that this implies in particular that $f$ is locally of finite type, see Lemma 37.56.4 and Descent, Lemma 35.20.10. Let $x \in X$. Choose an open neighbourhood $U$ of $x$ and an immersion $j : U \to \mathbf{A}^ n_ S$ over $S$ (see discussion preceding Definition 37.56.2). We have to show that $j$ is a Koszul-regular immersion. Since $f_ i$ is a local complete intersection morphism, we see that the base change $j_ i : U \times _ S S_ i \to \mathbf{A}^ n_{S_ i}$ is a Koszul-regular immersion, see Lemma 37.56.3. Because $\{ \mathbf{A}^ n_{S_ i} \to \mathbf{A}^ n_ S\} $ is a fpqc covering we see from Descent, Lemma 35.20.32 that $j$ is a Koszul-regular immersion as desired. $\square$

Lemma 37.56.20. The property $\mathcal{P}(f) =$“$f$ is a local complete intersection morphism” is syntomic local on the source.

Proof. We will use the criterion of Descent, Lemma 35.23.4 to prove this. It follows from Lemmas 37.56.8 and 37.56.7 that being a local complete intersection morphism is preserved under precomposing with syntomic morphisms. It is clear from Definition 37.56.2 that being a local complete intersection morphism is Zariski local on the source and target. Hence, according to the aforementioned Descent, Lemma 35.23.4 it suffices to prove the following: Suppose $X' \to X \to Y$ are morphisms of affine schemes with $X' \to X$ syntomic and $X' \to Y$ a local complete intersection morphism. Then $X \to Y$ is a local complete intersection morphism. To see this, note that in any case $X \to Y$ is of finite presentation by Descent, Lemma 35.11.1. Choose a closed immersion $X \to \mathbf{A}^ n_ Y$. By Algebra, Lemma 10.136.18 we can find an affine open covering $X' = \bigcup _{i = 1, \ldots , n} X'_ i$ and syntomic morphisms $W_ i \to \mathbf{A}^ n_ Y$ lifting the morphisms $X'_ i \to X$, i.e., such that there are fibre product diagrams

\[ \xymatrix{ X'_ i \ar[d] \ar[r] & W_ i \ar[d] \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

After replacing $X'$ by $\coprod X'_ i$ and setting $W = \coprod W_ i$ we obtain a fibre product diagram of affine schemes

\[ \xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^ h \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

with $h : W \to \mathbf{A}^ n_ Y$ syntomic and $X' \to Y$ still a local complete intersection morphism. Since $W \to \mathbf{A}^ n_ Y$ is open (see Morphisms, Lemma 29.25.10) and $X' \to X$ is surjective we see that $X$ is contained in the image of $W \to \mathbf{A}^ n_ Y$. Choose a closed immersion $W \to \mathbf{A}^{n + m}_ Y$ over $\mathbf{A}^ n_ Y$. Now the diagram looks like

\[ \xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^ h \ar[r] & \mathbf{A}^{n + m}_ Y \ar[ld] \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

Because $h$ is syntomic and hence a local complete intersection morphism (see above) the morphism $W \to \mathbf{A}^{n + m}_ Y$ is a Koszul-regular immersion. Because $X' \to Y$ is a local complete intersection morphism the morphism $X' \to \mathbf{A}^{n + m}_ Y$ is a Koszul-regular immersion. We conclude from Divisors, Lemma 31.21.8 that $X' \to W$ is a Koszul-regular immersion. Hence, since being a Koszul-regular immersion is fpqc local on the target (see Descent, Lemma 35.20.32) we conclude that $X \to \mathbf{A}^ n_ Y$ is a Koszul-regular immersion which is what we had to show. $\square$

Lemma 37.56.21. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume both $X$ and $Y$ are flat and locally of finite presentation over $S$. Then the set

\[ \{ x \in X \mid f\text{ Koszul at }x\} . \]

is open in $X$ and its formation commutes with arbitrary base change $S' \to S$.

Proof. The set is open by definition (see Definition 37.56.2). Let $S' \to S$ be a morphism of schemes. Set $X' = S' \times _ S X$, $Y' = S' \times _ S Y$, and denote $f' : X' \to Y'$ the base change of $f$. Let $x' \in X'$ be a point such that $f'$ is Koszul at $x'$. Denote $s' \in S'$, $x \in X$, $y' \in Y'$ , $y \in Y$, $s \in S$ the image of $x'$. Note that $f$ is locally of finite presentation, see Morphisms, Lemma 29.21.11. Hence we may choose an affine neighbourhood $U \subset X$ of $x$ and an immersion $i : U \to \mathbf{A}^ n_ Y$. Denote $U' = S' \times _ S U$ and $i' : U' \to \mathbf{A}^ n_{Y'}$ the base change of $i$. The assumption that $f'$ is Koszul at $x'$ implies that $i'$ is a Koszul-regular immersion in a neighbourhood of $x'$, see Lemma 37.56.3. The scheme $X'$ is flat and locally of finite presentation over $S'$ as a base change of $X$ (see Morphisms, Lemmas 29.25.8 and 29.21.4). Hence $i'$ is a relative $H_1$-regular immersion over $S'$ in a neighbourhood of $x'$ (see Divisors, Definition 31.22.2). Thus the base change $i'_{s'} : U'_{s'} \to \mathbf{A}^ n_{Y'_{s'}}$ is a $H_1$-regular immersion in an open neighbourhood of $x'$, see Divisors, Lemma 31.22.1 and the discussion following Divisors, Definition 31.22.2. Since $s' = \mathop{\mathrm{Spec}}(\kappa (s')) \to \mathop{\mathrm{Spec}}(\kappa (s)) = s$ is a surjective flat universally open morphism (see Morphisms, Lemma 29.23.4) we conclude that the base change $i_ s : U_ s \to \mathbf{A}^ n_{Y_ s}$ is an $H_1$-regular immersion in a neighbourhood of $x$, see Descent, Lemma 35.20.32. Finally, note that $\mathbf{A}^ n_ Y$ is flat and locally of finite presentation over $S$, hence Divisors, Lemma 31.22.7 implies that $i$ is a (Koszul-)regular immersion in a neighbourhood of $x$ as desired. $\square$

Lemma 37.56.22. Let $f : X \to Y$ be a local complete intersection morphism of schemes. Then $f$ is unramified if and only if $f$ is formally unramified and in this case the conormal sheaf $\mathcal{C}_{X/Y}$ is finite locally free on $X$.

Proof. The first assertion follows immediately from Lemma 37.6.8 and the fact that a local complete intersection morphism is locally of finite type. To compute the conormal sheaf of $f$ we choose, locally on $X$, a factorization of $f$ as $f = p \circ i$ where $i : X \to V$ is a Koszul-regular immersion and $V \to Y$ is smooth. By Lemma 37.11.13 we see that $\mathcal{C}_{X/Y}$ is a locally direct summand of $\mathcal{C}_{X/V}$ which is finite locally free as $i$ is a Koszul-regular (hence quasi-regular) immersion, see Divisors, Lemma 31.21.5. $\square$

Lemma 37.56.23. Let $Z \to Y \to X$ be formally unramified morphisms of schemes. Assume that $Z \to Y$ is a local complete intersection morphism. The exact sequence

\[ 0 \to i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0 \]

of Lemma 37.7.12 is short exact.

Proof. The question is local on $Z$ hence we may assume there exists a factorization $Z \to \mathbf{A}^ n_ Y \to Y$ of the morphism $Z \to Y$. Then we get a commutative diagram

\[ \xymatrix{ Z \ar[r]_{i'} \ar@{=}[d] & \mathbf{A}^ n_ Y \ar[r] \ar[d] & \mathbf{A}^ n_ X \ar[d] \\ Z \ar[r]^ i & Y \ar[r] & X } \]

As $Z \to Y$ is a local complete intersection morphism, we see that $Z \to \mathbf{A}^ n_ Y$ is a Koszul-regular immersion. Hence by Divisors, Lemma 31.21.6 the sequence

\[ 0 \to (i')^*\mathcal{C}_{\mathbf{A}^ n_ Y/\mathbf{A}^ n_ X} \to \mathcal{C}_{Z/\mathbf{A}^ n_ X} \to \mathcal{C}_{Z/\mathbf{A}^ n_ Y} \to 0 \]

is exact and locally split. Note that $i^*\mathcal{C}_{Y/X} = (i')^*\mathcal{C}_{\mathbf{A}^ n_ Y/\mathbf{A}^ n_ X}$ by Lemma 37.7.7 and note that the diagram

\[ \xymatrix{ (i')^*\mathcal{C}_{\mathbf{A}^ n_ Y/\mathbf{A}^ n_ X} \ar[r] & \mathcal{C}_{Z/\mathbf{A}^ n_ X} \\ i^*\mathcal{C}_{Y/X} \ar[u]^{\cong } \ar[r] & \mathcal{C}_{Z/X} \ar[u] } \]

is commutative. Hence the lower horizontal arrow is a locally split injection. This proves the lemma. $\square$


Comments (4)

Comment #4860 by Matthieu Romagny on

Section sentence of introduction : "factors" --> "factor" (In this section we consider morphisms f:X→S which locally on X factors as...)

Comment #6239 by Fred Diamond on

X/Y in Lemma 0FK1 (3)


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