Exposé XI. Application to the Picard group

This Exposé is modeled on the preceding one, but this time the result of no. 1 is weaker.

Throughout this Exposé, $X$ will denote a locally noetherian prescheme, $I$ a quasi-coherent ideal of O_X (so that $Y=V(I)$ is a closed part of $X$), $U$ a variable open neighborhood of $Y$ in $X$, and $\hat{X}$ the formal completion of $X$ along $Y$. For every ringed space $(Z,O_Z)$, we denote by $P(Z)$ the category of invertible O_Z-Modules — in other words, locally free of rank 1 — and by $\mathrm{Pic}(Z)$ the group of isomorphism classes of invertible Modules on $Z$.

1. Comparison of Pic(X̂) and Pic(Y)

For every $n\in \mathbb{N}$, set $X_n=(Y,O_X/I^{n+1})$ and $P_n=I^{n+1}/I^{n+2}$. The sequence of sheaves of abelian groups on $Y$

0 ⟶ P_n ──u──→ O*_{X_{n+1}} ──v──→ O*_{X_n} ⟶ 1

is exact. Let us be precise: the group structure on $P_n$ is the additive structure, $u(x)=1+x$ for every $x\in P_n$, and $v$ is the homomorphism deduced from the injection $I^{n+2}\to I^{n+1}$. We see that $v$ is surjective by remarking that, for every $y\in Y$, $O_{X_n,y}$ is a local ring, the quotient of $O_{X_{n+1},y}$ by a nilpotent ideal; the rest is equally trivial. From (1.1) we deduce an exact cohomology sequence:

(∗)   H¹(Y, P_n) ──u¹──→ H¹(Y, O*_{X_{n+1}}) ──v¹──→ H¹(Y, O*_{X_n}) ──d──→ H²(Y, P_n).

On the other hand, for every $n\in \mathbb{N}$, one knows how to identify $\mathrm{Pic}(X_n)$ with $H^1(Y,O{\ast }_{X_n})$; moreover, if $E$ is an invertible $O_{X_{n+1}}$-Module corresponding to a cohomology class $c(E)$, the cohomology class corresponding to the inverse image of $E$ on $X_n$ is equal to $v^1(c(E))$.

Whence the following proposition:

Proposition.

Retain the notations introduced above. Let $p\in \mathbb{N}$. The map $\mathrm{Pic}(\hat{X})\to \mathrm{Pic}(Y_n)$:

1. is injective for $n⩾p$, if $H^1(Y,P_n)=0$ for $n⩾p$;
2. is an isomorphism for $n⩾p$, if $H^i(Y,P_n)=0$ for $n⩾p$ and $i=1,2$.

Of course, the exact sequence (∗) contains more information than the proposition above. The reader will have noticed that we have said nothing about the functor $P(\hat{X})\to P(Y)$. Given two invertible $O_{\hat{X}}$-Modules $E$, $F$, the sheaf $H=\mathrm{Hom}(E,F)$ is also invertible. If we indicate reduction modulo $I^{n+1}$ by a subscript $n$, we find an exact sequence:

0 ⟶ H_0 ⊗ P_n ⟶ Hom(E_{n+1}, F_{n+1}) ⟶ Hom(E_n, F_n) ⟶ 0.

Whence an exact cohomology sequence that we shall not write down and whose interpretation is evident; one may use this remark to study the functor $P$.

2. Comparison of Pic(X) and Pic(X̂)

The reader will find in Exposé X, no. 2, the proof of what follows:

Proposition.

Suppose that $Lef(X,Y)$ holds; then for every open neighborhood $U$ of $Y$ in $X$, the functor

$$P(U)⟶P(\hat{X})$$

is fully faithful, so that the map

$$\mathrm{Pic}(U)⟶\mathrm{Pic}(\hat{X})$$

is injective. If $Leff(X,Y)$ holds, then the map (2.3) is an isomorphism:

lim→_U Pic(U) ⟶ Pic(X̂).

Corollary.

Suppose that $Lef(X,Y)$ holds and that $H^1(Y,P_n)=0$ for every integer $n⩾p$; then for every open $U\supset Y$, the maps

Pic(X) ⟶ Pic(U) ⟶ Pic(Y_n)

are injective for $n⩾p$. If $Leff(X,Y)$ holds and if, moreover, $H^i(Y,P_n)=0$ for every integer $n⩾p$ and $i=1,2$, then the map

lim→_U Pic(U) ⟶ Pic(Y_n)

is an isomorphism for $n⩾p$.

3. Comparison of P(X) and P(U)

A definition:

Definition. ¹

Let $X$ be a prescheme and let $Z$ be a closed part of $X$. Set $U=X-Z$. We say that $X$ is parafactorial at the points of $Z$ if, for every open set $V$ of $X$, the functor $P(V)\to P(V\cap U)$ is an equivalence of categories. We also say that the pair $(X,Z)$ is parafactorial.

Recall that $P(Z)$ denotes the category of Modules locally free of rank 1 on $Z$.

Definition.

A noetherian local ring is said to be parafactorial if the pair $(\mathrm{Spec}(A),r(A))$ is parafactorial.

One proves the following proposition, which shows that the notion is "pointwise":

Proposition.

Suppose $X$ is locally noetherian. In order that the pair $(X,Z)$ be parafactorial, it is necessary and sufficient that, for every $z\in Z$, the local ring $O_{X,z}$ be so.

Note that in "parafactorial" there is "fully faithful". One proves, as in Lemma 3.5 of Exposé X, the:

Lemma.

If $X$ is a locally noetherian prescheme and if $Z=X-U$ is a closed part of $X$, the following conditions are equivalent:

1. for every open set $V$ of $X$, the functor $P(V)\to P(V\cap U)$ is fully faithful;
2. the homomorphism $O_X\to i_{\ast }(O_U)$ is an isomorphism;
3. for every $z\in Z$, one has $prof(O_{X,z})⩾2$.

Thus "parafactorial" means that the conditions of 3.4 are satisfied and that, for every open set $V$ of $X$, the homomorphism $\mathrm{Pic}(V)\to \mathrm{Pic}(V\cap U)$ is surjective. In particular, if $X$ is the spectrum of a noetherian local ring, we find:

Proposition.

Let $A$ be a noetherian local ring; in order that it be parafactorial, it is necessary and sufficient that $profA⩾2$ and $\mathrm{Pic}(X^{\prime }-x)=0$, where we have set $X^{\prime }=\mathrm{Spec}(A)$ and $x$ is the unique closed point of $X^{\prime }$.

Note that a local ring of dimension $⩽1$ is never parafactorial, since its depth is $⩽1$. Hence "factorial" does not imply "parafactorial"; however, the converse holds for noetherian local rings of dimension $⩾2$, as we shall see below.

Lemma.

Let $X$ be a locally noetherian prescheme and let $Z$ be a closed part of $X$. Let $f:X_1\to X$ be a faithfully flat and quasi-compact morphism. Set $Z_1=f^{-1}(Z)$. If $(X_1,Z_1)$ is parafactorial, then so is $(X,Z)$.

We first remark that, if $i:(X-Z)\to X$ denotes the canonical immersion of $U=X-Z$ into $X$, the formation of the direct image by $i$ of a quasi-coherent O_U-Module commutes with the base change $f$, since the latter is flat. It is therefore equivalent to assume the equivalent conditions of Lemma 3.5 for $(X,Z)$ or for $(X_1,Z_1)$, since $f$ is a morphism of descent for the category of quasi-coherent sheaves. It remains to prove that, for every open set $V$ of $X$, $\mathrm{Pic}(V)\to \mathrm{Pic}(V\cap U)$ is surjective. We make the base change $V\to X$, which changes nothing (sic), and we are reduced to the case $V=X$. We then remark that, if $L$ is an invertible O_U-Module and if $L$ admits a locally free prolongation, this prolongation is isomorphic to $i_{\ast }(L)$, because of what has just been seen. It remains to prove that $i_{\ast }(L)$ is invertible. Using once more the fact that the direct image by $i$ commutes with flat base change, and that "locally free of rank 1" is a property that descends by faithfully flat and quasi-compact morphism, we are done.

Corollary.

Let $A$ be a noetherian local ring; if Â is parafactorial, so is $A$.

Do not believe that, if $A$ is parafactorial, so is Â.²

Before stating the principal theorem of this section, let us make the connection with the theory of divisors and the notion of factorial ring.³

Let $X$ be a noetherian and normal prescheme. Let $Z^1(X)$ be the free abelian group generated by the $x\in X$ such that $\dim O_{X,x}=1$. The local ring of such a point is a discrete valuation ring. We shall write $v_x$ for the corresponding normalized valuation. Let $K(X)$ be the ring of rational functions on $X$ and let

$$p:K(X)\ast ⟶Z^1(X)$$

be the map that to every $f\in K(X)\ast$ associates the codimension-one cycle:

(f) = Σ_{x ∈ X, dim O_{X,x} = 1} v_x(f) · x.

The image of $p$ is denoted $P(X)$, and its elements are called principal divisors.⁴ We set

$$Cl(X)=Z^1(X)/P(X).$$

Let $Z^{\prime 1}(X)$ be the subgroup of $Z^1(X)$ whose elements are the locally principal divisors. One knows that

$$\mathrm{Pic}(X)\simeq Z^{\prime 1}(X)/P(X),$$

and consequently $\mathrm{Pic}(X)$ is identified with a subgroup of $Cl(X)$.

Note that if $U$ is a dense open of $X$, then $K(X)\to K(U)$ is an isomorphism, and that if $codim(X-U,X)⩾2$, i.e. if every $x\in X$ such that $\dim O_{X,x}⩽1$ belongs to $U$, the homomorphism $Z^1(X)\to Z^1(U)$, and consequently $Cl(X)\to Cl(U)$, is also an isomorphism. Finally, if every $x\in U$ is factorial — i.e. $O_{X,x}$ is so — then $Z^1(U)=Z^{\prime 1}(U)$, and so $\mathrm{Pic}(U)\simeq Cl(U)$.

Proposition.

Let $X$ be a noetherian and normal prescheme. Let $(U_i{)}_{i\in I}$ be a family of open sets of $X$ such that:

1. the $U_i$ form a filter base;⁵
2. if one sets $Y_i=X-U_i$, then $codim(Y_i,X)⩾2$ for every $i$;
3. if $x\in U_i$ for every $i\in I$, then $O_{X,x}$ is factorial.

Then one has an isomorphism:

lim→_{i ∈ I} Pic(U_i) ──≅──→ Cl(X).

Note that b) implies that every $x\in X$ such that $\dim O_{X,x}⩽1$ belongs to $U_i$ for every $i$. Hence the $U_i$ are dense, and moreover the homomorphism $Z^1(U_i)\to Z^1(X)$ is an isomorphism, as is $K(U_i)\to K(X)$. So $\mathrm{Pic}(U)\subset Cl(U_i)\simeq Cl(X)$. To prove what is desired, it therefore suffices to show that every $D\in Z^1(X)$ belongs to $Z^{\prime 1}(U_i)$ for a suitable $i$. It suffices to do this for irreducible positive "divisors". Let then $x\in X$ be such that $\dim O_{X,x}=1$. It suffices to prove that there exists $i\in I$ such that $x$ is locally principal at the points of $U_i$. Let $I$ be the largest ideal of definition of the closed set $x$. The set of points in whose neighborhood $I$ is free is an open set $U$. Now $U\supset {\bigcap }_{i\in I}{U}_i$ by c). If we set $Y=X-U$, then $Y\subset {\bigcup }_{i\in I}{Y}_i$ with $Y_i=X-U_i$; now $Y$ is closed, so admits a finite number of generic points, so is contained in the union of finitely many $Y_i$, hence in some $Y_j$ for a $j\in I$, because the $U_i$ form a filter base. Thus $U\supset U_j$. QED.

Corollary.

Let $X$ be a noetherian and normal prescheme and let $Y$ be a closed part of codimension $⩾2$. Suppose that, for every $p\in X-Y$, $O_{X,p}$ is factorial; then

Pic(X − Y) ⟶ Cl(X − Y) ⟶ Cl(X)

are isomorphisms.

Corollary.

Let $A$ be a noetherian, normal local ring. Set $X^{\prime }=\mathrm{Spec}(A)$ and $x=r(A)$. In order that $A$ be factorial, it is necessary and sufficient that $\mathrm{Pic}(X^{\prime }-x)=0$ and that $p\in X^{\prime }-x$ implies $A_p$ factorial.

Indeed, in order that $A$ be factorial, it is necessary and sufficient that $Cl(X^{\prime })=0$.⁶

Corollary.

Let $A$ be a noetherian local ring of dimension $⩾2$. Set $X^{\prime }=\mathrm{Spec}(A)$ and let $x=r(A)$. Set $X=X^{\prime }-x$. The following conditions are equivalent:

1. $A$ is factorial;
2. a) for every $y\in X$, $O_{X,y}$ is factorial, and b) $A$ is parafactorial, i.e. $profA⩾2$ and $\mathrm{Pic}(X)=0$.

Before proving this corollary, let us state the:

Serre's criterion of normality. ⁷

Let $A$ be a noetherian local ring. In order that $A$ be normal, it is necessary and sufficient that

1. for every prime ideal $p$ of $A$ such that $\dim A_p⩽1$, $A_p$ be normal;
2. for every prime ideal $p$ of $A$ such that $\dim A_p⩾2$, one have $prof{A}_p⩾2$.

Let us prove 3.10.

(i) ⇒ (ii). Knowing that a localization of a factorial ring is factorial, we have (ii) a). Moreover $A$ is normal, so $profA⩾2$, since $\dim A⩾2$ (3.11 (ii)). Finally $A$ is parafactorial; indeed $\mathrm{Pic}(X)\simeq Cl(X^{\prime })=0$ (cf. 3.9).

(ii) ⇒ (i). We first prove that $A$ is normal by applying Serre's criterion. Since $\dim A⩾2$, condition (i) of the criterion is among the hypotheses. Moreover, for every $p\in X$, $A_p$ is factorial, hence normal, hence of depth $⩾2$, at least if $\dim A_p⩾2$. Finally $profA⩾2$ by (ii) b). It remains to apply 3.9.

Let us summarize the preceding:

Proposition.

Let $X$ be a locally noetherian prescheme and let $I$ be a quasi-coherent ideal of $X$. Set $Y=V(I)$. Let $p\in \mathbb{N}$. Suppose that:

1. $Leff(X,Y)$ holds (Exposé X);
2. $H^i(X,I^{n+1}/I^{n+2})=0$ if $i=1$ or 2 and if $n⩾p$;
3. for every open neighborhood $U$ of $Y$ in $X$ and every $x\in X-U$, the ring $O_{X,x}$ is parafactorial.

Then, for every $n⩾p$ and every open neighborhood $U$ of $Y$, the homomorphisms

Pic(X) ⟶ Pic(U) ⟶ Pic(X_n)

(with the canonical commutative triangle) are isomorphisms.

One knows some parafactorial rings:

Theorem.

1. (Auslander–Buchsbaum)⁸ A regular noetherian local ring is factorial (hence parafactorial if its dimension is $⩾2$).
2. A noetherian local ring of dimension $⩾4$ that is a complete intersection is parafactorial.

Corollary (Samuel conjecture)⁹.

A noetherian local ring $A$ that is a complete intersection and that is factorial in codimension $⩾3$ (i.e. $\dim A_p⩽3$ implies that $A_p$ is factorial) is factorial.

Proof of the corollary. We argue by induction on the dimension of $A$. If $\dim A⩽3$, then $A$ is factorial by hypothesis. If $\dim A>3$, by the induction hypothesis, and remarking that a localization of a complete intersection is also a complete intersection, all localizations of $A$ other than $A$ itself are factorial. By Theorem 3.13 (ii), $A$ is parafactorial, hence factorial by 3.10.

Proof of 3.13 (i) (following Kaplansky).¹⁰

Let $A$ be a regular noetherian local ring; set $\dim A=n$. If $n=0$ or 1, the result is known. Suppose $n⩾2$, and argue by induction on $n$: suppose $n⩾2$ and the theorem proved for rings of dimension $<n$. Set $X^{\prime }=\mathrm{Spec}(A)$ and $X=X^{\prime }-x$, where $x=r(A)$. The localizations of $A$ other than $A$ are regular and of dimension $<n$, hence factorial. Moreover $profA=\dim A⩾2$. It therefore suffices to prove that $\mathrm{Pic}(X)=0$ (Cor. 3.10). Let then $L$ be an invertible O_X-Module; one knows that one can prolong it to a coherent $O_{X^{\prime }}$-Module $L^{\prime }$. There exists a resolution of $L^{\prime }$ by free $O_{X^{\prime }}$-Modules:

$$0⟵L^{\prime }⟵L_1^{\prime }⟵\cdots ⟵L_n^{\prime }⟵0,$$

since the cohomological dimension of $A$ is finite. By restriction to $X$ one obtains a finite free resolution. It therefore suffices to prove the following lemma:

Lemma.

Let $(X,O_X)$ be a ringed space and let $L$ be a locally free O_X-Module that admits a finite resolution by free modules of finite type. Then $det(L)\simeq O_X$.

Recall that one defines $det(L)$ as the maximal exterior power of $L$. In the case envisaged, $det(L)\simeq L$ since $L$ is invertible, so the lemma allows us to conclude. Let us prove this lemma. Let

0 ⟵ L_0 ⟵ L_1 ⟵ L_2 ⟵ ⋯ ⟵ L_n ⟵ 0

be the announced exact sequence, where $L_0=L$. Since everything is locally free, one has:

⨂_{0 ⩽ i ⩽ n} (det(L_i))^{(−1)^i} ≃ O_X;

now all the $L_i$ for $i>0$ are free, so their determinants are free as well, hence so is the determinant of $L_0=L$. QED.

It remains to prove (ii) of the theorem. Beforehand, let us prove a lemma that will permit us to proceed by induction:

Lemma.

Let $A$ be a noetherian local ring that is a quotient of a regular ring. Let $t\in r(A)$ be an $A$-regular element. Suppose that $A$ is complete for the $t$-adic topology. Set $X^{\prime }=\mathrm{Spec}(A)$, $Y^{\prime }=V(t)\simeq \mathrm{Spec}(B)$, $B=A/tA$, $X=X^{\prime }-x$, $Y=Y^{\prime }-x$, $x=r(A)$. Suppose that:

1. for every $y\in X$ closed in $X$, one has $prof{O}_{X,y}⩾2$,
2. $profA/tA⩾3$,

then the map $\mathrm{Pic}(X)\to \mathrm{Pic}(Y)$ is injective. In particular, if $B$ is parafactorial, then so is $A$.

One knows that a) implies $Lef(X,Y)$ thanks to X 2.1. If we prove that $H^1(Y,P_n)=0$ for every $n⩾0$, we shall know thanks to (2.2) that $\mathrm{Pic}(X)\to \mathrm{Pic}(Y)$ is injective. If, moreover, $B$ is parafactorial, we shall know that $\mathrm{Pic}(Y)=0$ (3.5), hence $\mathrm{Pic}(X)=0$; now $prof(A)⩾3+1⩾2$ since $t$ is $A$-regular, so $A$ will be parafactorial by 3.5.¹¹

Let $I=(tA{)}^{\sim }$ be the $O_{X^{\prime }}$-Module associated with the ideal tA. In no. 1, we set $P_n=(I^{n+1}/I^{n+2})|Y$ for every $n⩾0$. Now $t$ is $A$-regular, so $P_n\simeq O_Y$. It therefore remains to prove that $H^1(Y,O_Y)=0$. Now $Y=Y^{\prime }-x$ is an open subset of $Y^{\prime }$, so we have an exact sequence (I (27)):

H¹(Y′, O_{Y′}) ⟶ H¹(Y, O_Y) ⟶ H²_x(Y′, O_{Y′}),

whose right-hand term is zero by virtue of hypothesis b), and whose left-hand term is zero because $Y^{\prime }$ is affine. QED.

Lemma.

Retaining the hypotheses of 3.16, suppose moreover that:

1. (c) for every $y$ closed in $Y$, $prof{O}_{X,y}⩾3$,
2. (d) $profA/tA⩾4$ (stronger than b),
3. (e) for every $y$ closed in $X$ with $y\in Y$, the ring $O_{X,y}$ is parafactorial.

Then the map $\mathrm{Pic}(X)\to \mathrm{Pic}(Y)$ is an isomorphism; in particular, in order that $A$ be parafactorial, it is necessary and sufficient that $B$ be so.

One knows (X 2.1) that a) and c) imply $Leff(X,Y)$. Moreover, by the reasoning just made, d) implies that $H^i(Y,P_n)=0$ for every $n⩾0$ and $i=1$ or $i=2$. Furthermore, for every open neighborhood $U$ of $Y$ in $X$, the complement of $U$ in $X$ consists of a finite number of closed points. Thanks to e) and Theorem 3.12, we deduce that $\mathrm{Pic}(X)\to \mathrm{Pic}(Y)$ is an isomorphism. On the other hand, $profA⩾profB⩾2$; by criterion 3.5, we deduce that $A$ is parafactorial if and only if $B$ is so.

Let us now prove 3.13 (ii). Let $R$ be a regular noetherian local ring. Let $(t_1,\cdots ,t_k)$ be an $R$-sequence. Set $B=R/(t_1,\cdots ,t_k)$ and suppose $\dim B⩾4$. We must prove that $B$ is parafactorial. We argue by induction on $k$. If $k=0$, then $B$ is regular, hence factorial by 3.13 (i), hence parafactorial by 3.10. Suppose $k⩾1$ and the theorem proved for $k^{\prime }<k$. Set $A=R/(t_1,\cdots ,t_{k-1})$, so $B=A/t_{k}A$. We may suppose $B$ complete by 3.7. By the induction hypothesis, $A$ is parafactorial. Let us prove that we may apply Lemma 3.17. We have supposed $B$ complete, hence so is $A$, and therefore $A$ is complete for the $t_k$-adic topology. If $x\in X$, and if $x$ is closed in $X$, then $A_x$ is a complete intersection of dimension $⩾4$, with $k^{\prime }<k$. By the induction hypothesis, $A_x$ is parafactorial, and moreover of depth $⩾4$. This gives a), c), and e). Moreover $\dim A⩾5$, whence d). QED.

Theorem.

Let $X$ be a locally noetherian prescheme and let $I$ be a coherent sheaf of ideals on $X$. Set $Y=V(I)$. Let $n$ be an integer. Suppose that:

1. $Leff(X,Y)$ holds (cf. examples X 2.1 and X 2.2);
2. for every $p⩾n$, one has $H^i(Y,I^{p+1}/I^{p+2})=0$ for $i=1$ and $i=2$;
3. for every open $U\supset Y$ and every $x\in X-U$, the ring $O_{X,x}$ is regular of dimension $⩾2$ or a complete intersection of dimension $⩾4$.

Then, for every open $U\supset Y$ and every integer $p⩾n$, the homomorphisms

Pic(X) ⟶ Pic(U) ⟶ Pic(Y_p)

are isomorphisms, where $Y_p$ denotes the prescheme $(Y,O_X/I^{p+1})$.

It suffices to combine 3.12 and 3.13.