Cote n° 57, pages 1–20

Cote n° 57 · batch 1 · pages 1–20 · Transcription · Picard : tapuscrit annoté (s.d.), lettres (s.d., 1962).
Datation de l’inventaire : 1962-[vers 1968]
Édition de démonstration

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Local Picard schemes

2tapuscrit en anglais, annoté à l'encre bleue de sa main ; les numéros d'équation (1) à (3) sont ajoutés à la main

Let $A$ be a complete local ring, $S = \mathrm{Spec}(A)$, $U = S - \{s\}$ the complement of the closed point of $S$. We wish to interprete the Picard group $\mathrm{Pic}(U)$ as the group $P(k)$ of $k$-valued points of a group-scheme $P$ defined over the residue field $k$ of $A$. Assume for simplicity that $U$ is regular, and that there exists a resolution of singularities for $S$ not changing $U$, i.e. a proper morphism \[f \colon X \longrightarrow S\] inducing an isomorphism $f^{-1}(U) \simeq U$, such that $X$ be regular and equal to the closure of $f^{-1}(U)$ (this open set will be identified with $U$). Let $X_0$ be the special fiber of $X$, so that \[U = X - X_0 .\] Because $X$ is regular, we have an exact sequence \[\begin{equation*} \tag{1} D \longrightarrow \mathrm{Pic}(X) \longrightarrow \mathrm{Pic}(U) \longrightarrow 0 , \end{equation*}\] where $D$ is the group of divisors on $X$ concentrated on $X_0$ (which is a free group of finite type $\underline{Z}^I$, $I$ being the set of irreducible components of $X_0$). On the other hand, by the existence theorem for coherent sheaves, it is known that \[\begin{equation*} \tag{2} \mathrm{Pic}(X) \simeq \varprojlim_n \mathrm{Pic}(X_n) , \end{equation*}\] where $X_n = X \times_S S_n$, $S_n = \mathrm{Spec}(A/\underline{m}^{n+1})$. Assume we can find group schemes $P_n$ over $k$ and isomorphisms \[\begin{equation*} \tag{3} \mathrm{Pic}(X_n) \xrightarrow{\ \sim\ } P_n(k) , \end{equation*}\] and transition morphisms $P_m \to P_n$ ($m \geq n$) compatible with the transition morphisms for the $\mathrm{Pic}(X_n)$. Denoting by $P$ the pro-group $(P_n)$, or if one prefers, in case the transition morphisms $P_m \to P_n$ are affine, the inverse limit (EGA IV 8) of the $P_n$, we get a pro-group scheme resp. a group-scheme $P$, and an isomorphism

3\[\begin{equation*} \tag{4} \mathrm{Pic}(X) \xrightarrow{\ \sim\ } P(k) . \end{equation*}\] The map $D \to \mathrm{Pic}(X) \simeq P(k)$ défines a morphism of (pro) groups \[\begin{equation*} \tag{5} D_k \longrightarrow P , \end{equation*}\] where $D_k$ is the constant group-scheme with value $D$. Assuming that the cokernel of (5) exists in any reasonable sense a a pro-group scheme or a group-scheme, let $Q$ be this kernel.sic : « kernel » pour le conoyau qui vient d'être nommé We get, in virtue of (1), an injective homomorphism \[\begin{equation*} \tag{6} \mathrm{Pic}(U) \hookrightarrow Q(k) , \end{equation*}\] the obstruction for an element of $Q(k)$ to be in the image being in $H^1(k, D'_k)$, where $D'_k$ is the image of $D_k$ in $P$ (it is again a constant group, with value a quotient $D'$ of $D$). If ⌜If $k$ is sep. closed, or if⌝ $D'$ is torsion-free, for example is $D'$ is equal to $D$ i.e. (5) is injective, then this obstruction vanishes, and Q is a good candidate (6) is an isomorphism. Thus $Q$ looks a good candidate for a ``local Picard scheme''.

To construct the $P_n$'s, assume first that $A$ is a $k$-algebra, i.e. that a lifting of the ⌜the⌝ residue field $k$ exists, and has been choosen. Then the natural candidate for $P_n$ (the only one I could think of, indeed!) would be \[\begin{equation*} \tag{7} P_n = \underline{\mathrm{Pic}}_{X_n/k} . \end{equation*}\] We know from Murre that the right had side is indeed representable by a scheme locally of finite type over $k$. There is however the usual trouble that we have indeed a canonical injective morphism \[\begin{equation*} \tag{8} \mathrm{Pic}(X_n) \hookrightarrow P_n(k) , \end{equation*}\] but in general we are not sureit is bijective. However it is if $k$ is separably closed; more specifically, the obstruction for an element of $P_n(k)$ to be in the image of $\mathrm{Pic}(X_n)$ lies in the Brauer group $\mathrm{Br}(k)$ of […]

4of the Artin ring $B_n = H^0(X_n, \mathcal{O}_{X_n})$. Assuming $U$ hence $X$ hence $X_n$ to be connected, this Artin ring is local, and it's Brauer group coincides with the Brauer group of its residue field $k_n$. Of course, for an element of $P(k) = \lim P_n(k)$, these various obstructions match together and come from an element of the Brauer group of the residue field $k' = \bigcap_n k_n$ of the ring $H^0(X, \mathcal{O}_X) = \varprojlim_n H^0(X_n, \mathcal{O}_{X_n})$, of which is just the normalisation of $A$. If for instance $A$ is normal, ⌜then⌝ $k' = k$ and the obstruction lies in $\mathrm{Br}(k)$.

Whatever this be, defining $Q$ as before, we still do get an injective homomorphism (6), but even if (5) is injective, we cannot be sure that (6) is bijective. We are however if $k$ is separable closed. Therefore, we still can more or less describe $Q$ (or at least $Q(k)$) in termes of local Picard groups, in terms of ⌜Galois⌝ descent from the completion $A'$ of the strict henselization of $A$: if $U' = S' - \{s'\}$, where $S' = \mathrm{Spec}(A')$, and if \[\begin{equation*} \tag{9} \pi = \mathrm{Gal}(\bar{k}/k) , \end{equation*}\] we will have an isomorphism \[\begin{equation*} \tag{10} Q(k) \simeq \mathrm{Pic}(U')^{\pi} , \end{equation*}\] This is perfectly true provided $D'$ is torsion free, for instance (5) injective.ligne tapée biffée d'un trait This is perfectly natural and in accordance with the familiar phenomena when dealing with global Picard schemes. Therefore $Q$ looks like the natural object to be called a local Picard scheme.

To make sure it makes a sense, one will have however to analyze somewhat the inverse system $(P_n)$, and the morphism (5). Moreover, to feel really secure, one should check that up to canonical isomorphism, the group scheme $Q$ together with (10) does not depend on the choice of

5the resolution; this should not be hard whenever we have existence of resolutions in the strong sense of Hironaka⌜, so that two resolutions can be dominated by a third one⌝, for instance in char. 0 of if $\dim A = 2$. I should point out that, in any case, the algebraic structure thus defined on $\mathrm{Pic}(U)$ depends in an essential way on the choice of the lifting of the residue field; an instructive example on which to study the dependance of this structure on the lifting is the one where $A$ is the completion of the local ring at the origin of the projecting cone of an elliptic curve (here $Q$ is isomorphic to the elliptic curve in question, for the given lifting of $k$ …).

I have to correct an inaccuracy that has slipped in, when defining $Q$ in the case when $k$ is not separably closed, so as to get (10). One will then have to replace the constant group scheme $D_k$ by the twisted constant group scheme $\underline{D}$, deduced by descent from the constant group-scheme occuring after passage from $k$ to the separable closure. In terms of the components $X_0^i$ of $X_0$, this can be defined as the direct image, under the morphism $\mathrm{Spec}(B_{0\,\mathrm{sep}}) \to \mathrm{Spec}(k)$, of the constant sheaf $\underline{Z}$ on the spectrum of the largest separable subalgebra $B_{0\,\mathrm{sep}}$ of $B_0$.passage encadré et barré à la main ; « the components $X_0^i$ of $X_0$ » est tapé au-dessus d'une ligne biffée à la machine, et la flèche de $\mathrm{Spec}(B_{0\,\mathrm{sep}}) \to \mathrm{Spec}(k)$ manque au tapuscrit This can be made explicit, in terms of the separable algebraic closures $k_i$ of $k$ in the function fields of the irreducible components $X_0^i$ of $X_0$, as the product of the schemes direct image of the constant sheaf $\underline{Z}$ under the projection $\mathrm{Spec}(\prod k_i) \to \mathrm{Spec}(k)$.

To study the inverse system $(P_n)$, use the fact (SGA 6 XII) that the transition morphisms are affine, which gives a sense to $P = \varprojlim P_n$ as an actual group-scheme. It's Lie algebra is the inverse limit of the Lie algebras of the $P_n$, which are $H^1(X_n, \mathcal{O}_{X_n})$, therefore we get

6\[\begin{equation*} \tag{11} \underline{\mathrm{Lie}}(P) \simeq H^1(X, \mathcal{O}_X) . \end{equation*}\] As $R^1 f_*(\mathcal{O}_X)$ is coherent and concentrated at the origin, it follows that its group of sections $H^1(X, \mathcal{O}_X)$ is of finite dimension. In fact, it seems, with a little care, that replacing if necessary $X$ by a suitable model dominating it, we may assume that there exists an ⌜invertible⌝ ideal $J$ on $X$, defining a subscheme $Z$ of $X$ having same support as $X_0$, and such that ⌜$\underline{L} =$⌝ $J/J^2$ be an ample sheaf on $Z$. This will imply that, replacing the system $(X_n)_{n}$ by the system $(Z_n = V(J^{n+1}))_{n}$, which defines an isomorphic pro-group ⌜object⌝ of Picard-schemes, we obtain an inverse system of group-schemes $P'_n = \underline{\mathrm{Pic}}_{Z_n/k}$ where for large $n$ the transition morphisms $P'_{n+1} \to P'_n$ are isomorphisms (because $H^1(Z_0, \underline{L}^{\otimes n}) = H^2(Z_0, \underline{L}^{\otimes n}) = 0$).la flèche de $P'_{n+1} \to P'_n$ manque au tapuscrit Thus it seems that, under the assumptions made ⌜from the start⌝, one can prove that the pro-group $(P_n)$ is essentially constant, hence its inverse limit $P$ is in fact a group scheme locally of finite type over $k$. This is certainly so, at least, if $\dim A = 2$, $A$ normal, as follows from the negative definiteness of the intersection matrix of the components of $X_0$.

This $P$ seems to be definable in quite a reasonable way, and one still has to investigate the morphism (5) (or more accurately, the morphisme $\underline{D} \to P$ defined by descent from the case where $k$ is separably closed; but then we may as well assume $k$ separably closed from the start, ⌜which we will do⌝). We would be particularly happy if this were a closed immersion, or what amounts to the same, if the morphism \[\begin{equation*} \tag{12} D \longrightarrow \mathrm{NS}(X_0) \end{equation*}\] is injective, were $\mathrm{NS}(X_0) = \underline{\mathrm{NS}}_{X_0/k}(k)$, $\underline{\mathrm{NS}} = \underline{\mathrm{Pic}} / \underline{\mathrm{Pic}}^0$. (The equivalence comes from the fact that $P = \underline{\mathrm{Pic}}_{Z_n/k}$ for some large $n$, and that Pic

7as recalled above, the morphism $\underline{\mathrm{Pic}}_{Z_n/k} \to \underline{\mathrm{Pic}}_{X_0/k}$ is affine, and hence $\mathrm{NS}(Z_n) \to \mathrm{NS}(X_0)$ has finite kernel.) I expect that this is always the case, […] In case $A$ is normal of dimension 2, this follows from Du-Val – Mumford's negative definiteness. It would he nice to have a proof in general (perhaps through a negative definiteness of an intersection matric $X^i . C^j$, where for every irreducible component $X^i$ of $X_0$, $C^i$ denotes a ⌜suitable⌝ curve on $X^i$, for instance defined in terms of a suitable relatively ample sheaf on $X$ by taking intersections of corresponding hypersurface sections on the $X^i$'s ?). In case (12) is indeed injective, we have no trouble defining \[\begin{equation*} \tag{13} Q = P/\underline{D} \end{equation*}\] as a group-scheme locally of finite type over $k$. We then have \[\begin{equation*} \tag{14} P^0 \xrightarrow{\ \sim\ } Q^0 , \end{equation*}\] an isomorphism for the connected components, and \[\begin{equation*} \tag{15} Q/Q^0 = (P/P^0)/\underline{D} \simeq \underline{\mathrm{NS}}_{Z_n/k} / \underline{D} , \end{equation*}\] where $n$ is chosen large enough. Of course (14) and (11) yield an isomorphism \[\begin{equation*} \tag{16} \underline{\mathrm{Lie}}(Q) \simeq H^1(X, \mathcal{O}_X) . \end{equation*}\] Let us remark by the way that we have an injective restriction morphism \[\begin{equation*} \tag{17} H^1(X, \mathcal{O}_X) \hookrightarrow H^1(U, \mathcal{O}_U) = H^2_s(S) \end{equation*}\] as one sees by remarking that, with the above notations for $J$, $\underline{L}$, ⌜the sheaf⌝ $i_*(\mathcal{O}_U)/\mathcal{O}_X$ ($i \colon U \to X$ the inclusion) has a composition series whose quotients are the $\underline{L}^{-n}$, $n \geq 1$, whose $H^0$ is zero, and the map (17) is just the canonical map $H^1(X, \mathcal{O}_X) \hookrightarrow H^1(X, i_*(\mathcal{O}_U))$.l'astérisque de $i_*$ et la crosse de $\hookrightarrow$ sont ajoutés à la main Thus, the Lie algebra of the local Picard scheme is identified with a subspace of $H^1(U, \mathcal{O}_U)$. (This suggests that if $A$ is Cohen-Macaulay of dim. $\geq 3$, $Q$ is discrete, hence $\mathrm{Pic}(U)$ countable etc.)

8The previous constructions rely heavily on the existence and choice of a field of representatives. Even this being granted, the question of caracterising $Q$ as an actual scheme ⌜over $k$⌝, i.e. as a functor on the category of $k$-algebras, has not been touched. In fact, the description I can give of this functor is ⌜(tautological, and is)⌝ in terms of a particular choice of a resolution $X$, and it really gives rather a description of $P$ rather than $Q$. One then still has to define $Q$ as the cokernel of the silly morphism $\underline{D} \to P$. As a matter of fact, even if we restrict to arguments $k'$ which are, say, finite extensions of $k$ (but not necessarily separable ones), I cannot describe $Q(k')$ in terms of local Picard groups of local rings as $A' = A \hat{\otimes}_k k'$. The trouble, of course, is that $X \otimes_k k'$ will in general no longer be a resolution of $S'$, i.e. it will no longer be regular. However, such geometric descriptions will be possible provided we take as arguments algebras which are smooth of formally smooth ⌜geometrically regular⌝ over $k$, so that tensoring with them will not destroy the regularity of $X$. (Thus, the case when $k'$ is a power series ring, for instance the completion of the local ring of $Q$ at the origin, $Q$ being smooth, is of interest …) Let's now try to be specific. By defintion ⌜more or less,⌝ for any ⌜affine⌝ scheme $T$ over $k$, we have \[\begin{equation*} \tag{18} P(T) = \varprojlim P_n(T) \hookleftarrow \varprojlim_n \mathrm{Pic}(X_{n_T}) \text{\add{$/\mathrm{Pic}(T)$}} \qquad (X_{n\,T} = X_n \times_k T) . \end{equation*}\] affineness of $T$ used to have $H^1(T, \mathcal{O}_T) = H^2(T, \mathcal{O}_T) = 0$ In fact, it is not hard to check that the left hand side functor is the étale sheaf (on $(\mathrm{Sch})_{/k}$) associated to the right-hand side functor. This comes from the more precise statement that the obstruction, for an element of $P(T)$, to come from an element of $\varprojlim_n \mathrm{Pic}(X_{nT})$, lies in $H^2(T_{\text{ét}}, \underline{G}_m)$ (at least for $T$ affine, which is used to insure that it's $H^1$ and $H^2$ with values in $\mathcal{O}_T$ vanish. For simplicity, I have assumed here that $A$ is normal). As we have seendans (18), « (18) », la flèche $\hookleftarrow$, le $n$ sous la seconde limite et « $/\mathrm{Pic}(T)$ » sont de sa main, ce dernier remplaçant une fin de formule biffée ; dans la parenthèse finale, les exposants 2 et 3 tapés sont corrigés à la main en 1 et 2

9on the other hand that $P$ is locally of finite presentation over $k$, it commutes (as a functor) to filtering direct limits of rings, and therefore it is known when we know it's restriction to arguments which are of finite type over $k$, and a fortiori noetherian. For such ⌜(i.e. noetherian)⌝ arguments, using the existence theorem for coherent sheaves, we can give a more geometric expression for the right hand side of (18), namely Assume for simplicity $T = \mathrm{Spec}(B)$ affine, and let \[C = A \hat{\otimes}_k B = \lim A_n \otimes_k B ,\] which is an adic noetherian ring augmented to $B$ (the augmentation ideal being an ideal of definition). Consider […] ⌜$X \hat{\otimes}_k B$⌝ $= X \otimes_A C$, which is a proper scheme over $C$. We then have \[\begin{equation*} \tag{19} \mathrm{Pic}(X \hat{\otimes}_k B) \simeq \varprojlim_n \mathrm{Pic}(X_{nB}) , \end{equation*}\] […] ⌜$\mathrm{Pic}(X \hat{\otimes}_k B)/\mathrm{Pic}(B) \hookrightarrow P(B)$⌝ and thus on noetherian arguments $B$, $P$ appears as the étale sheaf associated to the functor $B \mapsto \mathrm{Pic}(X \hat{\otimes}_k B)$⌜$/\mathrm{Pic}(B)$⌝, ⌜which is a separated presheaf,⌝ this statement being complemented by the description given above of the group for obstructions for an element of $P(B)$ to belong to $\mathrm{Pic}(X \hat{\otimes}_k B)$.

As for $B \mapsto Q(B)$, this appears as the étale sheaf associated with the functor \[\begin{equation*} \tag{20} B \longmapsto \mathrm{Pic}(X \hat{\otimes}_k B)/\bigl(\text{\add{$\mathrm{Pic}(B) +$}}\, \underline{D}(B)\bigr) \quad (\hookrightarrow Q(B)) \end{equation*}\] where $\underline{D}$ is the twisted constant group over $k$ defined above. For an element of $Q(B)$ to belong to the first hand side there are two successive obstructions, the first is in $H^1(\mathrm{Spec}(B), \underline{D})$, which vanishes ⌜for instance⌝ if $\underline{D}$ is constant (for instance $k$ separably closed) and $B$ normal, the second is in the Brauer group of $B$, and vanishes ⌜for instance⌝ if $X_0$ has a zero-cycle of degree 1.

Assume for simplicity that the irreducible components of $X_0$ are

10geometrically irreducible, i.e. $\underline{D}$ is constant, and assume $B$ geometrically regular over $k$. This second condition implies that $X \hat{\otimes}_k B$ is regular, and denoting by $U \hat{\otimes}_k B$ the inverse image of $U$ in this scheme, the first condition this together with the first condition implies that the left hand side of (20) ⌜$\mathrm{Pic}(X \hat{\otimes}_k B)/D(B)$⌝ is isomorphic to $\mathrm{Pic}(U \hat{\otimes}_k B)$. This shows that on the sub-category of arguments $B$ which are noetherian and geometrically regular, $Q$ is the étale sheaf associated to the presheaf \[\begin{equation*} \tag{21} B \longmapsto \mathrm{Pic}(U \hat{\otimes}_k B)\text{\add{$/\mathrm{Pic}(B)$}} \quad (\hookrightarrow Q(B)) . \end{equation*}\] The obstruction for an element of $Q(B)$ to belong to $\mathrm{Pic}(U \hat{\otimes}_k B)$⌜$/\mathrm{Pic}(B)$⌝ is in the Brauer group $\mathrm{Br}(B)$. Thus, under very stringent conditions on $B$ at least (being geometrically regular) we get a description of $Q(B)$ in terms of actual local Picard groups. I should have stated that $U \hat{\otimes}_k B$ can also be interpreted as the inverse image of $U$ in the scheme $\mathrm{Spec}(A \hat{\otimes}_k B)$, and thus makes a sense independently of the choice of a particular resolution.

One may wish to construct a local Picard scheme without also in case $A$ is not equi-caracteristic, which implies that $\mathrm{car}.\,k > 0$. It seems likely that this can be done if $k$ is perfect. The key-point here is the construction of schemes $P_n$, which leads us to the following

Problem. Let $A$ be a local Artin ring with perfect residue field $k$ of car. $p > 0$, $X$ a proper scheme over $A$. Give a ``natural'' construction of a group scheme locally of finite type $P$ over $k$, together with an imbedding \[\begin{equation*} \tag{22} \mathrm{Pic}(X) \hookrightarrow P(k) , \end{equation*}\] the obstruction for an element of $P(k)$ to beling to $\mathrm{Pic}(X)$ ⌜(⌝being in $\mathrm{Br}\, H^0(X, \mathcal{O}_X)$, and hence⌜)⌝ vanishing if $k$ is algebraically closed.l'énoncé du problème est marqué d'un trait vertical dans la marge de gauche

11Here is a candidate for a functor, which may turn out to be representable, in which case this would be $P$. Let $\underline{A}$ be the ring-scheme over $k$ defined by $A$ (see Greenberg's papers, or Serre's in Bulletin Soc. Math.), so that \[\begin{equation*} \tag{23} A \simeq \underline{A}(k) . \end{equation*}\] For every algebra $B$ over $k$, consider the ring $\underline{A}(B)$, which is an algebra over $\underline{A}(k) = A$. We may thus consider \[X_B \overset{\mathrm{dfn}}{=} X \otimes_A \underline{A}(B) ,\] and consider the functor \[\begin{equation*} \tag{24} B \longmapsto \mathrm{Pic}(X_B) . \end{equation*}\] le numéro tapé « (34) » est corrigé à la main en « (24) » This of course may not be representable, even if $A = k$. However, it may turn out that the fppf sheaf associated to this functor is representable, and this then would give the looked for candidate for $P$. In any case, Artin's work gives us a very handy set of necessary and sufficient conditions for a group functor to be representable by a scheme locally of finite type, and in principle it should be possible to decide wether or not the functor sheaf just considered satisfies to these conditions or not. If so, one may expect that the developments on local Picard schemes given for the case when there is a field of representatives will carry over to the case when the residue field is perfect with car. $> 0$.

12In the construction of a local Picard scheme $Q$, we have seemed to use in a technically essential way the fact ⌜assumption⌝ that $U$ is regular (plus resolution). I would conjecture that a reasonable local Picard scheme $Q$, ⌜locally of finite type over $k$,⌝ can be constructed also without this assumption (provided that a field of representatives is given, or the residue field is perfect of char. $p > 0$, of course). Instead of a resolution of singularities, one may then wish to find a proper surjective morphism \[f \colon X \longrightarrow S\] inducing an isomorphism $f^{-1}(U) \xrightarrow{\ \sim\ } U$, and such that the corresponding map \[\mathrm{Pic}(X) \longrightarrow \mathrm{Pic}(U)\] is surjective. We then could repeat the construction of $P$, as $\lim \underline{\mathrm{Pic}}_{X_n/k}$, as above, and of $Q$ as $\mathrm{Coker}(\underline{D} \to P)$, where $\underline{D}$ is a group scheme corresponding to divisors concentrated on $X_0$ (possibly no longer discrete ?). Another way of approach would be to take a resolution of singularities $f \colon X \to S$, but where now we cannot assume any longer that $f^{-1}(U) \to U$ is an isomorphism, and to describe a scheme $P$ in terms of invertible sheaves on $X$, together with descent data of $\underline{L}|f^{-1}(U)$ to $U$. A nice test for the yoga of local Picard schemes without regularity assumptions on $U$ would be to see if it is true in general that for $A$ Cohen-Macaulay of $\dim \geq 3$ (more generally, for $H^2_{\underline{m}}(A) = 0$), the group $\mathrm{Pic}(U)$ is indeed countable, and for separably closed residue field […] ⌜is invariant under⌝ separable field extension $k \to k'$, $A$ being replaced by $A \hat{\otimes}_k k'$ (which would express the expected discrete structure of the local Picard scheme).

Carré, cube, rigidité…

titre de sa main, en haut à droite d'une feuille de garde par ailleurs blanche (p. 13), suivi de deux croix

14feuillet de calculs, encres noire et bleue \[\boxed{\, n^*(L) \simeq L^{\otimes n}\, (\delta L)^{\frac{n(n-1)}{2}} \,} \qquad\qquad n + 2\,\frac{n(n-1)}{2} = n^2\] \[\delta L = L^{\otimes 2}\, \delta'(L) \qquad \delta' L_x = L_{2x}\, L_x^{-4}\] \[\boxed{\, n^*(L) \simeq L^{n^2}\, (\delta' L)^{\frac{n(n-1)}{2}} \,}\] l'exposant de $L$ dans la seconde formule encadrée est noyé sous une surcharge ; on y lit $n^2$ sous réserve \[A \longrightarrow A \times A \qquad A \longrightarrow \hat{A}\] \[\boxed{\, \underline{\underline{\mathrm{Pic}}}^{\tau}_{(\prod X_i)/S} \simeq \prod_i \underline{\underline{\mathrm{Pic}}}^{\tau}_{X_i/S} \,}\] ⌜sans torsion⌝ajouté en travers, souligné deux fois, à côté de l'encadré \[0 \to \prod_i \underline{\underline{\mathrm{Pic}}}_{X_i/S} \longrightarrow \underline{\underline{\mathrm{Pic}}}_{(\prod X_i)/S} \longrightarrow \prod_{\{i,j\}} \mathcal{C}_S(X_i, X_j) \longrightarrow 0\] $\{i,j\}$ parcourant les parties à deux éléments de l'ensemble d'indices $I$la note est écrite sous le produit, dans une bulle qui l'entoure

[Scindée si les $X_i/S$ ont tous une section]

Suite exacte de faisceaux [pour la […] essent. fid. plate qu.-compacte]

Les $X_i$ sont propres ⌜et plats⌝, $f_*(\mathcal{O}_{X_i}) \simeq \mathcal{O}_S$ universellement.

explications […]

15feuillet surchargé : à gauche, un schéma vertical $X \to Y$ et une lettre $G$ noircie ; sur la droite, des traits de crayon en diagonale et un encadrement ondulé traversent le texte

$\mathrm{Pic}(X/Y)$

$\mathrm{Pic}_{\mathrm{sp}}(X/Y)$ : classes à isom. près de faisceaux inversibles $\underline{L}$, sections marquées, telles que […] soit (i) l'on ait $H^i_f(\underline{L}) = 0$ pour $i > 0$, […]

donc $\underline{H}^0_f(\underline{L})$ est loc. libre de rang $\chi_{X/Y}(\underline{L})$. (ii) $\chi_{X/Y}(\underline{L})$ […] $\mapsto 1$, i.e. l'[…] […] équivaut à 2 sections marquées […] $H^0_f(\underline{L}) \to$ […]

$\underline{H}^0_f(\mathcal{O}_X) = \mathcal{O}_Y$ « universellement »

[…] sections marquées […]

un diagramme à demi biffé, en partie écrit tête-bêche : des produits $P^1 \times P$, $P^0 \times_P P \rightrightarrows P_1 \to P$, des flèches verticales et une flèche courbe ; les termes ne se lisent pas assez pour être composés

Une telle donnée équivaut à […] : la donnée […]

16\[\begin{array}{ccc} & X_i \times X_j & \longrightarrow\ \prod X_i \\ \nearrow & & \nearrow \\ X_{[ijkl]} \longrightarrow & X_k \times X_l & \end{array}\] diagramme recomposé : $X_{[ijkl]}$ (sur un $X$ surchargé) s'envoie dans $X_i \times X_j$ et $X_k \times X_l$, qui s'envoient dans $\prod X_i$ \[X_{ijkl} = \begin{cases} X_{\emptyset} & \text{si } \{i,j\} \cap \{k,l\} = \emptyset \\ X_{\alpha} & \text{si } \{i,j\} \cap \{k,l\} = \{\alpha\} \end{cases}\] \[L_{m_1 x_1 + \cdots + m_n x_n}\] \[L_{x,y} \simeq L_{xe}\, L_{ye}\, \mathcal{M}_{xy} \qquad \mathcal{M}_{xy}\, \mathcal{M}_{yx}\] au-dessus de $L_{x,y}$, $L_{xe}$, $L_{ye}$, trois petits exposants surchargés, illisibles

$L_{xy}\, L_{yz}$ $L_{x+y+z}$ \[L_{x_1 + \cdots + x_n} = \Bigl(\prod_{i<j} L_{x_i + x_j}\Bigr) \Bigl(\prod L_{x_i}^{-n+2}\Bigr) L_{\emptyset}^{\alpha_n}\] \[L_{x+y} = L_x\, L_y\, (\delta L)_{x,y}\] symétriqueécrit près de $(\delta L)_{x,y}$, entouré \[\prod_{i \neq j} L_{x_i + x_j} = \Bigl(\prod L_{x_i}^{n-1}\Bigr) \prod_{i \neq j} (\delta L)_{x_i, x_j}\] \[\boxed{\, L_{x_1 + \cdots + x_n} = \prod_{i<j} (\delta L)_{x_i, x_j} \prod_i L_{x_i}\, L_{\emptyset}^{\alpha_n} \,}\] dans l'encadré, un facteur surchargé avant le premier produit et un autre, biffé, avant $\prod_i L_{x_i}$ ; sous l'encadré, une accolade et l'annotation « si $x_i = 0$ pour […] » mènent au calcul suivant \[L_{\emptyset}^{-\frac{n(n-1)}{2}}\, L_{\emptyset}^{\frac{(n-1)(n-2)}{2}} = L_{\emptyset}^{-n+1} \qquad (\delta L)_{0,0} = L_{\emptyset}^{-1}\] \[1 = -\frac{n(n-1)}{2} + n + \frac{(n-1)(n-2)}{2}\] en marge droite, deux autres essais de cette identité, biffés

17\[1 + 1 - 1 = 1 \qquad -(n-2) - (n-2) + (n-3) = -(n-1)\] \[\alpha_n + \alpha_n - \alpha_{n-1} + 1 = \text{\struck{\ill{}}}\quad 2\alpha_n - \alpha_{n-1} + 1 = \alpha_{n+1}\] \[\alpha_3 = 1 \quad \alpha_4 = 3 \quad \alpha_5 = 6 \quad \alpha_6 = 10 \quad \alpha_7 = 15\] $\alpha_n = an + b$, $2\alpha_n - \alpha_{n-1} + 1$ $= 2an + 2b - an + a - b + 1$ $= an + (\dots)$, $\alpha_{n+1} = an + a + b$essai linéaire barré de deux grandes croix ; la parenthèse du terme constant est surchargée \[\begin{align*} \alpha_n &= an^2 + bn + c \\ 2\alpha_n - \alpha_{n-1} + 1 &= 2an^2 + 2bn + 2c - a(n^2 - 2n + 1) - b(n-1) - c + 1 \\ &= an^2 + (b + 2a)n + (c - a + b + 1) \\ \alpha_{n+1} &= a(n^2 + 2n + 1) + b(n+1) + c \\ &= an^2 + (2a + b)n + (a + b + c) \end{align*}\] \[a + b + c = -a + b + c + 1 \qquad a = -a + 1 \qquad a = \tfrac12\] \[\alpha_n = \tfrac12 n^2 + bn + c \qquad \alpha_2 = 0,\ \alpha_3 = 1\] \[\tfrac12 \cdot 4 + 2b + c = 0 \qquad \tfrac12 \cdot 9 + 3b + c = 1\] \[2b + c = -2 \qquad 6b + 2c = -7 \qquad 2b = -3 \qquad c = +1\] plusieurs chiffres de ce calcul sont surchargés ($\alpha_3$ sur $\alpha_2$, $-7$, $-3$, $+1$, $\pm 1$ dans $c - a + b + 1$) ; on donne la dernière lecture \[\tfrac12 n^2 - \tfrac32 n + 1 = \tfrac12 [n^2 - 3n + 2] \qquad \boxed{\, \alpha_n = \frac{(n-2)(n-1)}{2} \,}\] $L_{x_1 \dots x_n}$ $\simeq \bigl(\prod_{i \neq j} L_{x_i x_j}\bigr) \bigl(\prod_i L_{x_i}^{-1}\bigr) L_{\emptyset}^{\frac{(n-1)(n-2)}{2}}$ \[\boxed{\, L_{x_1 \dots x_n} \simeq \Bigl(\prod_{i \neq j} L_{x_i x_j}\Bigr) \Bigl(\prod_i L_{x_i}\Bigr)^{-n+2} L_{\emptyset}^{\frac{(n-1)(n-2)}{2}} \,}\]

18\[\begin{matrix} x & y & z \\ a & b & c \end{matrix} \qquad \boxed{\, L_{x,y,z} \simeq L_{xy}\, L_{y,z}\, L_{x,z}\, L_x^{-1} L_y^{-1} L_z^{-1} L_{\emptyset} \,}\] « Th. du cube » — \[\begin{align*} L_{xyzt} &= L_{(x,y)z}\, L_{z,t}\, L_{(x,y),t}\, L_{(x,y)}^{-1} L_z^{-1} L_t^{-1} L_{\emptyset} \\ &= L_{xy} L_{y,z} L_{xz} L_x^{-1} L_y^{-1} L_z^{-1} L_{\emptyset}\, L_{zt} \\ &\qquad L_{xy} L_{y,t} L_{x,t} L_x^{-1} L_y^{-1} L_t^{-1} L_{\emptyset}\, L_{xy}^{-1} L_z^{-1} L_t^{-1} L_{\emptyset} \end{align*}\] les facteurs sont cochés un à un et ceux qui se compensent biffés ; on n'a pas reproduit ces marques \[\boxed{\, L_{xyzt} = L_{xy} L_{xz} L_{xt} L_{yz} L_{yt} L_{zt}\, L_x^{-2} L_y^{-2} L_z^{-2} L_t^{-2}\, L_{\emptyset}^{3} \,}\] \[L_{xyztu} = L_{xyzt}\, L_{tu}\, L_{xyzu}\, L_{xyz}^{-1} L_t^{-1} L_u^{-1} L_{\emptyset}\] suivent trois lignes de développement, cochées et biffées de même \[\boxed{\, \begin{array}{l} L_{xyztu} = L_{xy} L_{xz} L_{xt} L_{xu} L_{yz} L_{yt} L_{yu} L_{zt} L_{zu} L_{tu} \\ \qquad L_x^{-3} L_y^{-3} L_z^{-3} L_t^{-3} L_u^{-3}\, L_{\emptyset}^{6} \end{array}\,}\] \[L_{x_1 \dots x_n} = \prod_{i \neq j} L_{x_i x_j} \prod L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n}\] \[\begin{align*} L_{x_1 \dots x_{n+1}} &= L_{(x_1 \dots x_{n-1}) x_n x_{n+1}} \\ &= L_{x_1 \dots x_n}\, L_{x_1 \dots x_{n-1} x_{n+1}}\, L_{x_n x_{n+1}}\, L_{x_1 \dots x_{n-1}}^{-1} L_{x_n}^{-1} L_{x_{n+1}}^{-1} L_{\emptyset} \\ &= \prod_{1 \leq i < j \leq n} L_{x_i x_j} \prod_{1 \leq i \leq n} L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n} \prod_{\substack{1 \leq i < j \leq n+1 \\ i, j \neq n}} L_{x_i x_j} \prod_{\substack{1 \leq i \leq n+1 \\ i \neq n}} L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n} \\ &\qquad L_{x_n x_{n+1}} \prod_{1 \leq i < j \leq n-1} L_{x_i x_j}^{-1} \prod_{1 \leq i \leq n-1} L_{x_i}^{+(n-3)} L_{\emptyset}^{-\alpha_{n-1}}\, L_{x_n}^{-1} L_{x_{n+1}}^{-1} L_{\emptyset} \end{align*}\] les bornes des produits sont lues sous réserve ; un « $1 \leq i, j \leq n-1$, $i \neq j$ » est écrit à part, à gauche \[\Bigl(\prod_{i \neq j} L_{x_i x_j}\Bigr) \Bigl(\prod L_{x_i}^{-(n-1)}\Bigr) L_{\emptyset}\] la dernière ligne s'arrête là ; sous $\prod$, « $i \neq j$ » surcharge un autre indice

Lettre à Murre

19lettre tapée, non datée sur ces deux pages ; elle se poursuit au-delà de la p. 20, hors de ce lot

Dear Murre,

I am glad to hear that you are still willing to give the talk on unramified functors. Here what I can say to your questions.

1. The theorem about passage to quotient I alluded to is the following:

Theorem. Let $f \colon X \to Y$ be a morphisme of $S$-preschemes, assume either $X$ and $Y$ of ⌜locally⌝ finite presentation over $Y$ $S$lecture du $S$ surchargé sous réserve, or $Y$ loc noeth and $X$ ⌜locally⌝ of finite type over $Y$. Assume that the equivalence relation ⌜$R =$⌝ $X \times_Y X$ defined by $f$ is flat over $X$ i.e. $\mathrm{pr}_1 \colon X \times_Y X \to X$ is flat. Then the quotient $X/R$ exists in the strongest reasonable sense, i.e. one can factor $f$ into a compositum $X \to Z \to Y$, with $X \to Z$ faithfully flat locally of finite presentation, $Z$ locally of finite type ⌜pres.⌝ over $S$ $Y$lecture sous réserve de la lettre surchargée (in fact of finite presentation over $S$ if $X$ is so) and $Z \to Y$ a monomorphism.les flèches de $X \to Z \to Y$ et de $X \to Z$ manquent au tapuscrit

Of course the factorization is unique, andthe theorem can be expressed by saying that the quotient sheaf (for the fpqc topology) $X/R$ is representable. That is in fact how the theorem is proved.

Raynaud has recently made a very nice (and non trivial) application of this theorem, by proving the following: if $S$ is the spectrum of a discrete valuation ring, $G$ a group prescheme of finite type over $S$, $H$ a closed ⌜and flat⌝ sub-group scheme, such that $G_t/H_t$ is affine quasi-affine (where $t$ is the generic point of $S$) then $G/H$ is representable as a quasi-affine and flat $S$-scheme, which is even affine if $H$ is invariant (i.e. if $G$ is a flat group scheme of finite type with affine generic fibre, than $G$ is affine). This extends immediately to a base which is regular of dim one. Raynaud is now trying to extend his construction to the case when he drops the quasi-affinness assumption, namely to construct still $G/H$

20as a quasi-projective scheme over $S$.

2. Theorem of the cube. I believe we discussed about it time ago, but maybe the proof I told you was valid only if one assumes the Pic functor of one of the […] ⌜factors⌝ involved representable. To prove unramifiedness of the functor $\underline{\mathrm{Corr}}$ however you need only a week infinitesimal form of the theorem of the square, for which you will find a proof in the manuscript notes I am joining on correspondance classes, containing also the proofs of the statements you were recalling in your question 4. I hope you will be able to read them, I agree the handwriting is wretched and the notes moreover very sketchy. – On the other hand, I recall you that the theorem of the cube follows rather formally once one knows separatedness of $\underline{\mathrm{Corr}}_S(X,Y)$ for two of the three factors involved, and using the usual formal properties of the Picard functor (among which commutation with inverse limits of Artin rings is the less trivial).

3. As for the separatedness of $\underline{\mathrm{Corr}}_S(X,Y)$, this is about trivial whenever the Pic functor of one of the factors $X,Y$ is separated? Now this is certainly the case for $X$ if its geometric fibers are integral, (a fortiori if $X$ is an abelian scheme over $S$ !). To show this, one may assume $S$ the spectrum of a ⌜discrete⌝ valuation ring, and one is reduced to show that if $\underline{L}$ is an invertible sheaf on $X$ whose restriction to the general fiber $X_t$ is trivial, then $\underline{L}$ is trivial. Now $X_1$ is an open subset of $X$, and the assumption on $\underline{L}$ can be expressed by saying that $\underline{L}$ is defined by a Cartier divisor whose support is contained in the special fiber $X_0$. Now $X_0$ itself is already a Cartier divisor (defined by a global equation $t = 0$) and moreover is an integral subscheme of $X$, from this follows that the divisor $D$ is a multiple of $X_0$ (assume for la phrase se poursuit sur la page suivante, hors de ce lot