Exposé XIII. Problems and conjectures

1. Relations between global and local results. Affine problems related to duality

It is well known that many statements concerning a projective scheme $X$ can be formulated in terms of statements concerning a certain graded ring, or better a complete local ring, namely the homogeneous coordinate ring of $X$ (i.e. the affine ring of the projecting cone $X$ of $X$), or its completion (i.e. the completion of the local ring of the vertex of $X$). The interest of this reformulation is that it often allows one, starting from known global results, to conjecture, and even to prove, analogous results for complete noetherian local rings more general than those which really appear in the global statement, for instance for local rings that are not necessarily of equal characteristic. Thus, Serre's duality theorem for projective space (XII 1.1) suggested the useful local duality theorem (V 2.1). Serre's fundamental theorem on the cohomology of coherent Modules on projective space (finiteness, asymptotic behavior for large $n$ of $H^i(X,F(n))$, cf. EGA III 2.2.1) generalizes to a structure theorem for the local invariants $H_{\mathfrak{m}}^i(M)$, see V 3. Likewise, the Lefschetz theorems for the fundamental group, and for the Picard group ("equivalence criteria"), well familiar in the classical case and subsequently extended to an arbitrary base field, suggested the "local" Lefschetz theorems of Exposés X and XI. Of course, the local theorems are in turn precious tools for obtaining global statements. For example, local duality permits one to formulate a global asymptotic property (XII 1.3 (i)) by the vanishing of certain local invariants $H^i(F_x)$. More substantially, the local Lefschetz theorems, implying for instance the "purity" or the parafactoriality of certain local rings that are complete intersections (X 3.4 and XI 3.13), allow one, in the global Lefschetz theorems, to dispose of certain non-singularity hypotheses, as in X 3.5, 3.6, 3.7.

Another useful generalization of the theorems concerning projective schemes over a field $k$ consists in replacing $k$ by a general base scheme. Thus, the sequel of EGA III will give¹ a generalization in this direction of Serre duality²; the theorems on finiteness and asymptotic behavior of the $H^i(X,F(n))$ were stated in EGA III 2.2.1 over a general base scheme, and finally the Lefschetz theorems can equally be developed for a projective morphism, as we saw in XII 4.9, thanks to the local theorem XII 4.7. Of course, working over a general base scheme also leads to essentially new statements, such as the "comparison theorem" EGA III 4.15 and the existence theorem for sheaves EGA III 5.1.4 (which, as we saw moreover in IX, derive from the same key cohomological theorems as the Lefschetz theorems for ${\pi }_1$ and Pic).

It then becomes necessary to extract theorems that simultaneously encompass the two generalizations we have just indicated of statements concerning projective schemes over a field. The natural objects for such a common generalization are noetherian rings that are separated and complete for an $I$-adic topology. Their study, from this point of view, has not yet been seriously addressed, and seems to me at the present time the most interesting subject in the local theory of coherent sheaves. Here is a typical problem in this direction:

Conjecture 1.1 ("Second affine finiteness theorem"[^XIII-1-2]³).

Let $M$ be a finitely generated module over a noetherian ring $A$ (which one will, if necessary, assume to be a quotient of a regular ring), and let $J$ be an ideal of $A$. Prove that the modules $H_J^i(M)$ are

"$J$-cofinite", i.e. that the modules

$${\mathrm{Hom}}_A(A/J,H_J^i(M))$$

are finitely generated.

Recall that $H_J^i(M)$ denotes the module $H_Y^i(X,M)$ (where $X=\mathrm{Spec}(A)$, $Y=V(J)$) of Exposé I, interpreted in II in terms of a direct limit of cohomologies of Koszul complexes, or again for $i⩾2$ the module $H^{i-1}(X-Y,M)$. Actually, 1.1 should be a consequence of a more precise statement, implying that the $H_J^i(M)$ lie in a suitable abelian subcategory ${\mathcal{D}}_J$ of the category ${\mathcal{C}}_J$ of $A$-modules of support $\subset Y=V(J)$, such that $H\in Ob{\mathcal{D}}_J$ implies that $H$ is $J$-cofinite. (N.B. The category of modules $H$ of support contained in $V(J)$ that are $J$-cofinite is unfortunately not stable under passage to a quotient!). The essential problem would then consist in defining ${\mathcal{D}}_J$. More precisely, the solution of problem 1.1 should follow (at least if $A$ is a quotient of a regular ring) from a duality theory, generalizing both local duality and the duality theory of projective morphisms to which we alluded above, and which would be of the following kind:

Conjecture 1.2 ("Affine duality"⁴).

Suppose $A$ is regular, separated and complete for the $J$-adic topology. Let $C\bullet (A)$ be an injective resolution of $A$.

(i) Prove that the functor

D_J : L• ↦ Hom_J(L•, C•(A))

from the category of complexes of $A$-modules that are free of finite type in each dimension and bounded above in degree (where morphisms are homomorphisms of complexes up to homotopy) into the category of complexes of $A$-modules $K\bullet$ that are injective in each dimension and bounded above in degree (where the morphisms are defined similarly) is fully faithful.

(ii) Prove that for every $K\bullet$ of the form $D_J(L\bullet )$, the $H^i(K\bullet )(=Ex{t}_Y^i(X;L\bullet ,{\mathcal{O}}_X))$ are $J$-cofinite.

(iii) More precisely, prove that the $K\bullet$ that are homotopic to a complex of the form $D_J(L\bullet )$ can be characterized by finiteness properties of the $H^i(K\bullet )$, stronger than the one envisaged in (ii), for example by the property $H^i(K\bullet )\in Ob{\mathcal{D}}_J$,

where ${\mathcal{D}}_J$ is a suitable abelian category, as envisaged above.

Note that the problem is resolved in the affirmative when $A$ is local and $J$ is an ideal of definition of it (cf. Exp. IV), and also when $J$ is the zero ideal. In these two cases, exceptionally, one can confine oneself to taking for ${\mathcal{D}}_J$ the category of Modules with support $V(J)$ that are $J$-cofinite, (which in the second case signifies simply that one takes the category of finitely generated Modules over $A$). An affirmative solution of

conjecture 1.2 in general would give one for 1.1, by taking for $L\bullet$ the dual of a free finitely generated resolution of $M$. On the other hand, an affirmative solution of 1.1 would give an affirmative answer to the first part of the following conjecture, which we formulate in "global" form:

Conjecture 1.3.

Let $X\subset {\mathbf{P}}_k^r$ be a closed subscheme of standard projective space that is locally a complete intersection and every irreducible component of which is of codimension $⩾s$. Let $U={\mathbf{P}}_k^r-X$.

(i) Prove that for every coherent Module $F$ on $U$, one has

dim Hⁱ(U, F) < +∞   for i ⩾ s.

⁵

(ii) Give an example, with $X$ connected and regular, where one has

$$H^s(U,F)\ne 0.$$

To see that (i) is a particular case of 1.1, one considers

Hⁱ(U, F(·)) = ⊕_n Hⁱ(U, F(n)) = Hⁱ(𝐄^{r+1} − X̃, F̃)

as a module over the affine ring $k[t_0,\cdots ,t_r]$ of the projecting cone ${\mathbf{E}}^{r+1}$ of ${\mathbf{P}}^r$. This module is none other than $H_J^{i+1}(M)$, where $J$ is the ideal of the projecting cone $\tilde{X}$ of $X$ in ${\mathbf{E}}^{r+1}$.

On the other hand, from the hypothesis made on $X$, which implies that $\tilde{X}$ is also a complete intersection of codimension $⩾s$ at every point of ${\mathbf{E}}^{r+1}$ distinct from the origin, it follows that $H_J^{i+1}(M)$ is zero outside the origin for $i⩾s$. If it is therefore $J$-cofinite as 1.1 demands, it is a fortiori $\mathfrak{m}$-cofinite, which easily implies that it is finite-dimensional in each degree⁶. Note moreover that conjecture 1.3 is already posed for a non-singular irreducible curve $X$ in ${\mathbf{P}}^3$; one does not know in this case whether the $H^2({\mathbf{P}}^3-X,{\mathcal{O}}_X(n))$ are finite-dimensional, or whether they are necessarily zero⁷. One does not even know whether there exists an irreducible curve in ${\mathbf{P}}^3$ that is not set-theoretically the intersection of two hypersurfaces⁸.

Problem 1.4.

Give an affine variant of the "comparison theorem" EGA III 4.1.5 as a theorem of commutation of the functors $H_J^i$ with certain inverse limits.

Finally, in the present order of ideas, I had posed the following problem: let $A$ be a complete regular noetherian local ring, $K$ its fraction field; prove that $Ex{t}_A^i(K,A)=0$ for every $i$. An affirmative answer was given on the spot by M. Auslander: the regularity hypothesis can be replaced by the assumption that $A$ is integral, and in fact it is true that $Ex{t}_A^i(K,M)=0$ for every $i$, as soon as $M$ is finitely generated over $A$. This follows immediately from the following statement, due to Auslander: if $A$ is a complete noetherian local ring, then for every finitely generated module $M$ over $A$, the functors $Ex{t}_A^i(\cdot ,M)$ transform direct limits into inverse limits⁹.

2. Problems related to ${\pi }_0$: local Bertini theorems

Let $A$ be a complete noetherian local ring, $f$ an element of its maximal ideal, $X=\mathrm{Spec}(A)$, $Y=\mathrm{Spec}(A/fA)$. The use of the local "Lefschetz" technique allows one to give criteria for $Y^{\prime }=X^{\prime }\cap Y$ (where $X^{\prime }=X-\mathfrak{m}$) to be connected, in terms of hypotheses on $X^{\prime }$. Thus, it suffices that one have: a) $X^{\prime }$ connected, b) $prof{\mathcal{O}}_{X^{\prime },x}⩾2$

for every closed point $x$ of $X^{\prime }$, c) $f$ is $A$-regular. One notes however that hypotheses b) and c) are not of purely topological nature; for instance, they are not invariant under replacing $X$ by $X_{red}$. In the analogous situation for a projective scheme $X^{\prime }$ over a field and a hyperplane section $Y^{\prime }$ of $X^{\prime }$, the use of "Bertini's theorem" and Zariski's "connection theorem" allows one in fact to obtain results of distinctly more satisfactory appearance, which had led me in the oral seminar to state a conjecture, which I have since resolved in the affirmative. Let us therefore state here:

Theorem 2.1.

Let $A$ be a complete noetherian local ring, $X$ its spectrum, $a$ the closed point of $X$, $X^{\prime }=X-a$. Suppose that $X$ satisfies the conditions (where $k$ denotes an integer $⩾1$):

$a_k$) The irreducible components of $X^{\prime }$ are of dimension $⩾k+1$.

$b_k$) $X^{\prime }$ is connected in dimension $⩾k$, i.e. one cannot disconnect $X^{\prime }$ by a closed part of dimension $<k$ (cf. III 3.8).

Let $m$ be an integer, $0⩽m⩽k$, and let $f_1,\cdots ,f_m\in \mathfrak{r}(A)$; set $B=A/{\sum }_i{f}_{i}A$, $Y=\mathrm{Spec}(B)=V(f_1)\cap \cdots \cap V(f_m)$, $Y^{\prime }=X^{\prime }\cap Y=Y-a$. Then $Y$ satisfies the conditions $a_{k-m}$), $b_{k-m}$). In particular, for every sequence of $m⩽k$ elements $f_1,\cdots ,f_m$ of $\mathfrak{r}(A)$, $Y^{\prime }=X^{\prime }\cap V(f_1)\cap \cdots \cap V(f_m)$ is connected.

It is moreover easy to see that if the last conclusion holds (it evidently suffices to take $m=k$ there), and excluding the case where $X$ would be irreducible of dimension 0 or 1, it follows that the irreducible components of $X^{\prime }$ are of dimension $⩾k+1$, and $X^{\prime }$ is connected in dimension $⩾k$, so that in a sense, 2.1 is a "best possible" result.

Let us give the principle of the proof of 2.1. Only condition $b_{k-m}$) for $Y$ poses a problem. One reduces easily, for given $k$, to the case where $X$ is integral, and even (by passing to the normalization, which is finite over $X$)

to the case where $X$ is normal. If $k=1$, hence $\dim X^{\prime }⩾2$, then $X^{\prime }$ is of depth $⩾2$ at its closed points, and one can apply the result recalled at the beginning of the section, which shows that $Y^{\prime }=X^{\prime }\cap V(f)$ is connected. In the case $k⩾1$, one supposes the theorem proved for $k^{\prime }<k$. By induction on $m$, one is reduced to the case where $m=1$, i.e. to verifying that for $f_1\in \mathfrak{r}(A)$, $X^{\prime }\cap V(f_1)$ is connected in dimension $⩾k-1$. If it were not, i.e. if it were disconnected by a $Z^{\prime }$ of dimension $<k-1$, there would exist a sequence $f_2,\cdots ,f_k$ such that $X^{\prime }\cap V(f_1)\cap \cdots \cap V(f_k)$ is disconnected, and in this sequence one can choose $f_2\in \mathfrak{r}(A)$ arbitrarily, subject to the sole condition of not vanishing at any point of a certain finite part $F$ of $X^{\prime }$ (namely the set of maximal points of $Z^{\prime }$). Moreover, one verifies easily, using the fact that $X^{\prime }$ is normal, hence satisfies Serre's condition ($S_2$)¹⁰, that there exists a finite part $F^{\prime }$ of $X^{\prime }$ such that $f\in \mathfrak{r}(A)$, $V(f)\cap F^{\prime }=\varnothing$ implies that $V(f)\cap X^{\prime }$ also satisfies condition ($S_2$). One can then choose $f_2$ in such a way that $f_2$ vanishes neither on $F$ nor on $F^{\prime }$, hence such that $X^{\prime }\cap V(f_2)$ satisfies ($S_2$). But then, by virtue of Hartshorne's theorem III 3.6, $X^{\prime }\cap V(f_2)$ is connected in codimension 1, hence (since every component of $X^{\prime }\cap V(f_2)$ is of dimension $⩾k$) it is connected in dimension $⩾k-1$. Applying the induction hypothesis to $V(f_2)=\mathrm{Spec}(A/f_{2}A)$, it follows that $X^{\prime }\cap V(f_2)\cap V(f_1)\cap V(f_3)\cap \cdots \cap V(f_k)$ is connected, whereas it had been constructed disconnected — absurd.

Let us point out some interesting corollaries:

Corollary 2.2.

Let $f:X\to Y$ be a proper morphism of locally noetherian preschemes, with $Y$ integral, $y_0\in Y$, $y_1$ the generic point of $Y$. Suppose

a) $Y$ is unibranch at $y_0$, and every irreducible component of $X$ dominates $Y$.

b) The irreducible components of $X_{y_1}$ are of dimension $⩾k+1$, and $X_{y_1}$ is connected in dimension $⩾k$.

Then the irreducible components of $X_{y_0}$ are of dimension $⩾k+1$, and $X_{y_0}$ is connected in dimension $⩾k$.

Indeed, Zariski's connection theorem (cf. EGA III 4.3.1) implies that $X_{y_0}$ is connected; to show that it is not disconnected by a closed part of dimension $<k$, one is reduced to showing that the local rings at points $x\in X_{y_0}$ such that $\dim x<k$ have a spectrum not disconnected by $x$. Now this is true without assuming either $f$ proper, or $Y$ unibranch at $y_0$. One reduces, to see this, to the case where $X$ is integral dominating $Y$, and if one wishes $Y$ affine of finite type over $\mathbb{Z}$, so that one is under the conditions of the dimension formula for ${\mathcal{O}}_{X,x}$ over ${\mathcal{O}}_{Y,y_0}$. Using in this case the finiteness of the normal closure, one can even suppose $X$ normal, hence by virtue of a theorem of Nagata¹¹, the completion of a local ring ${\mathcal{O}}_{X,x}$ of $X^{\prime }$ is again normal; hence (if ${\mathcal{O}}_{X,x}$ is of dimension $N$) $\mathrm{Spec}({\hat{\mathcal{O}}}_{X,x})$ is connected in dimension $⩾N-1$. Let $n=\dim {\mathcal{O}}_{Y,y_0}$; then $degtrk(x)/k(y)<k$ implies $\dim {\mathcal{O}}_{X,x}>n+(k+1)-k=n+1$, taking into account $\dim X_{y_1}⩾k+1$, and taking a system $f_1,\cdots ,f_n$ of parameters of ${\mathcal{O}}_{Y,y_0}$ which one lifts to elements of ${\mathcal{O}}_{X,x}$, one sees by 2.1 that $\mathrm{Spec}({\hat{\mathcal{O}}}_{X,x}/\sum f_i{\hat{\mathcal{O}}}_{X,x})$ is connected in dimension $⩾1$, i.e. is not disconnected by its closed point, or equivalently, $\mathrm{Spec}({\hat{\mathcal{O}}}_{X_{y_0},x})$ is not disconnected by its closed point; a fortiori the same holds for $\mathrm{Spec}({\mathcal{O}}_{X_{y_0},x})$.

As in the case of the ordinary connection theorem, one can vary 2.2 by taking geometric fibers (over the algebraic closures of the residue fields), provided one supposes $Y$ geometrically unibranch at $y_0$, or (without other hypothesis than $Y$ noetherian) that $f$ is universally open. Applying this to the case where $Y$ is the dual scheme of a projective scheme ${\mathbf{P}}_k^r$ over a field, one recovers a strengthened form of the global result that had inspired 2.1, namely:

Corollary 2.3¹².

Let $X$ be a closed subscheme of ${\mathbf{P}}_k^r$ ($k$ a field); suppose the irreducible components of $X$ are of dimension $⩾l+1$, and $X$ geometrically connected in dimension $⩾l$.

Then for every sequence $H_1,\cdots ,H_m$ of $m$ hyperplanes of ${\mathbf{P}}_k^r$ ($0⩽m⩽l-1$), $X\cap H_1\cap \cdots \cap H_m$ satisfies the same condition with $l-m$, in particular is geometrically connected in dimension $⩾l-1$.

One can moreover modify this statement in an obvious way for the case where one is given a proper morphism $X\to {\mathbf{P}}_k^r$, which is not necessarily an immersion; an analogous extension is possible for 2.1 (by considering a proper scheme over $X^{\prime }$).

These statements are moreover formally deduced from the statements given here, taking into account the ordinary connection theorem which reduces us to the case of a finite morphism.

Corollary 2.4.

Let $A$ be a complete noetherian normal local ring of dimension $⩾k+2$. Let $X=\mathrm{Spec}(A)$, $X^{\prime }=X-a$, and $f_1,\cdots ,f_k$ elements of $\mathfrak{r}(A)$; then $Y^{\prime }=X^{\prime }\cap V(f_1)\cap \cdots \cap V(f_k)$ is connected, and ${\pi }_1(Y^{\prime })\to {\pi }_1(X^{\prime })$ is surjective.

One proceeds as in SGA 1 X 2.11.

In all this, only questions of connectedness were at issue. Now in the global case, well-known theorems assert that for an irreducible projective variety $X\subset {\mathbf{P}}_k^r$, $k$ algebraically closed, its intersection with a sufficiently "general" hyperplane $H$ is irreducible (and not merely connected): this is Bertini's theorem, proved by Zariski, which in turn implies, by Zariski's connection theorem, that for every $H$, $H\cap X$ is connected (although not necessarily irreducible). One can moreover proceed in the reverse direction, proving this latter result by a Lefschetz-type technique, and deducing Bertini's theorem, reducing to the case where $X$ is normal, and using the following result: for $H$ "sufficiently general", $X\cap H$ is also normal. This suggests:

Conjecture 2.5¹³.

Let $A$ be a complete noetherian normal local ring. Show that there exists a nonzero $f\in \mathfrak{r}(A)$ such that $Y^{\prime }=X^{\prime }\cap V(f)=Y-a$ (where $Y=\mathrm{Spec}(A/fA)$) is normal (hence irreducible by 2.1 if $\dim A⩾3$).

To do things properly, one would have to show that, in a suitable sense, there exist even "many" elements $f$ having the property in question, for example that one can choose $f$ in an arbitrary power of the maximal ideal. Using Serre's normality criterion and the remark made above for Serre's property ($S_2$), one sees that one would have an affirmative answer to 2.5 if one had one to:

Conjecture 2.6¹⁴.

Let $A$ be a complete noetherian local ring, $U$ an open part of its spectrum $X$, $F$ a finite part of $X^{\prime }=X-a$. Suppose $U$ is regular. Prove that there exists $f\in \mathfrak{r}(A)$ such that $V(f)\cap U$ is regular, and $V(f)\cap F=\varnothing$.

For a "local Bertini"-type result, see Chow [2].

3. Problems related to ${\pi }_1$

Here again, one has numerous questions, suggested by the global results or by the transcendental results.

Conjecture 3.1¹⁵.

Let $A$ be a complete noetherian local ring with algebraically closed residue field, $X=\mathrm{Spec}(A)$, $X^{\prime }=X-a$, $a$ the closed point. Suppose the irreducible components of $X$ are of dimension $⩾2$, and $X^{\prime }$ connected.

(i) Prove that ${\pi }_1(X^{\prime })$ is topologically finitely generated.

(ii) If $p$ is the characteristic exponent of the residue field $k$ of $A$, prove that the largest topological quotient group of ${\pi }_1(X^{\prime })$ that is "of order prime to $p$" is finitely presented.

For part (i), using the theory of descent SGA 1 IX 5.2 and theorem 2.4, one is reduced to the case where $A$ is normal of dimension 2. In this case, a systematic method for studying the fundamental group of $X^{\prime }$, inaugurated by Mumford [5] in the transcendental setting, consists in desingularizing $X$, i.e. in considering a projective birational morphism $Z\to X$, with $Z$ integral regular, inducing an isomorphism $Z^{\prime }=Z{|}_{X^{\prime }}\to X^{\prime }$; it is plausible that such a $Z$ always exists, this is in any case what Abhyankar's method [1] demonstrates in the case of "equal characteristics"¹⁶. Let $C$ be the fiber of the closed point of $X$ by $Z\to X$; it is an algebraic curve over the residue field $k$, connected by virtue of the connection theorem. The solution of 3.1 then seems linked to:

Problem 3.2.

With the preceding notation, put ${\pi }_1(X^{\prime })$ in relation with the topological invariants of $C$, in particular its fundamental group, (in order to bring out the topological finite generation of ${\pi }_1(X^{\prime })$, by using for instance SGA 1, theorem X 2.6).

Another method would be to consider $A$ as a finite algebra over a complete regular local ring $B$ of dimension 2, ramified along a curve $C$ contained in $\mathrm{Spec}(B)=Y$. One is thus led to:

Problem 3.3.

Let $A$ be a complete regular local ring of dimension 2, with algebraically closed residue field $k$, $X$ its spectrum, $C$ a closed part of $X$ of dimension 1. Define local invariants of the embedded curve $C$, having a sense independent of the residual characteristic, and the knowledge of which permits one to calculate the

fundamental group of $X-C$ by generators and relations when $k$ is of characteristic zero. Prove that when $k$ is of characteristic $p>0$, the "tame" fundamental group of $X-C$ is a quotient of the preceding one, and that the two fundamental groups (in characteristic 0, and in characteristic $p>0$) have the same maximal quotient of order prime to $p$.

Of course, 3.3 shows us that in 3.1, there is also occasion to replace $X^{\prime }$ by a scheme of the form $X-Y$, where $Y$ is a closed part of $X$ that is of codimension $⩾2$ in every component of $X$ containing it.

When one abandons this restriction on $Y$, there must still exist an analogous finiteness result, on condition of imposing "tame"-type restrictions on the ramification at the maximal points of the irreducible components of $Y$ that are of codimension 1.

Problem 3.4.

Let $A$ be a complete noetherian local ring of dimension 2, with algebraically closed residue field. Let again $X=\mathrm{Spec}(A)$, $X^{\prime }=X-a$. Find particular structural properties of ${\pi }_1(X^{\prime })$ in the case where $A$ is a complete intersection.

A satisfactory solution of this problem would perhaps permit one to resolve the following old problem:

Conjecture 3.5¹⁷.

Find an irreducible curve in ${\mathbf{P}}_k^3$ ($k$ algebraically closed field), preferably non-singular, that is not set-theoretically the intersection of two hypersurfaces.

(Kneser [4] shows that one can always obtain it as the intersection of three hypersurfaces).

4. Problems related to higher ${\pi }_i$: local and global Lefschetz theorems for complex analytic spaces18

Let $X$ be a scheme locally of finite type over the field of complex numbers $\mathbb{C}$; one can associate to it an analytic space $X_h$ over $\mathbb{C}$, whence homotopy and homology invariants ${\pi }_i(X_h)$, $H_i(X_h)$, $H^i(X_h)$ etc. One knows moreover that $X$ is connected if and only if $X_h$ is, hence one has a bijection

$${\pi }_0(X_h)\to {\pi }_0(X).$$

Likewise, since every étale covering $X^{\prime }$ of $X$ defines an étale covering $X_h^{\prime }$

of $X_h$, one has a canonical homomorphism

$${\pi }_1(X_h)\to {\pi }_1(X),$$

which one knows, using a theorem of Grauert-Remmert, identifies the second group with the completion of the first for the topology of subgroups of finite index (which simply expresses the fact that $X^{\prime }\mapsto X_h^{\prime }$ is an equivalence of the category of étale coverings of $X$ with the category of finite étale coverings of $X_h$). It follows that the results of this seminar (by purely algebraic means) on ${\pi }_0(X)$ and ${\pi }_1(X)$ imply results for ${\pi }_0(X_h)$ and ${\pi }_1(X_h)$ (which are of transcendental nature). Moreover, if $X$ is proper, the well-known exact sequence $0\to \mathbb{Z}\to \mathbb{C}\to \mathbb{C}\ast \to 0$ allows one to show that the Néron-Severi group of $X$ (the quotient of its Picard group by the connected component of the identity) is isomorphic to a subgroup of $H^2(X_h,\mathbb{Z})$; in the non-singular Kähler case, it is the subgroup denoted $H^{(1,1)}(X_h,\mathbb{Z})$ (classes of type (1, 1)):

Pic(X) / Pic⁰(X) ⊂ H²(X, ℤ).

Consequently, the information we have obtained on Picard groups implies information, very partial it is true, about the groups $H^2(X_h,\mathbb{Z})$. It is tempting to complete all these fragmentary results by conjectures.

Very precise indications, going in the same direction as those just mentioned, are furnished by a classical theorem of Lefschetz [7]. It asserts that if $X$ is a non-singular irreducible projective analytic space of dimension $n$, and if $Y$ is a non-singular hyperplane section, then the injection

$$Y_{n-1}\to X_n$$

induces a homomorphism

$${\pi }_i(Y_{n-1})\to {\pi }_i(X_n)$$

which is an isomorphism for $i⩽n-2$, an epimorphism for $i=n-1$. The analogous statement follows for the homomorphisms

$$H_i(Y_{n-1})\to H_i(X_n)$$

on homology (integral, to fix ideas), while in cohomology,

$$H^i(X_n)\to H^i(Y_{n-1})$$

is an isomorphism in dimension $i⩽n-2$, a monomorphism in dimension $i=n-1$. We have obtained variants of these results in the framework of schemes, for ${\pi }_0$, ${\pi }_1$, Pic, valid moreover without non-singularity hypotheses to a large extent, cf. Exp. XII. Moreover, in the elimination of non-singularity hypotheses, we have used in an essential way "local" variants of these global Lefschetz theorems. All this suggests the following problems, which doubtless will have to be attacked simultaneously¹⁹.

Problem 4.1.

Let $X$ be an analytic space, $Y$ a closed analytic part of $X$ (or simply a closed part?²⁰) such that for every $x\in Y$, the local ring ${\mathcal{O}}_{X,x}$ is a complete intersection. Let $n$ be the complex codimension of $Y$ in $X$. Is the canonical homomorphism

$${\pi }_i(X-Y)\to {\pi }_i(X)$$

an isomorphism for $i⩽n-2$, and an epimorphism for $i=n-1$?

In this problem and the following ones, one supposes evidently implicitly a base-point chosen to define the homotopy groups. To state the next problem, one must define, for an analytic space $X$ (more generally, for a locally path-connected space) and $x\in X$, local invariants ${\Pi }_i^x(X)$²¹.

To do this, one chooses a non-constant map $f$ from the interval [0, 1] into $X$, such that $f(0)=x$ and $f(t)\ne x$ for $t\ne 0$ (such maps exist if $x$ is not an isolated point). Then for every neighborhood $U$ of $x$, there exists an $ϵ>0$ such that $0<t<ϵ$ implies $f(t)\in U$, and the homotopy groups ${\pi }_i(U-x,f(t))$ are essentially independent of $t$ (they are, for varying $t$, related by a transitive system of isomorphisms); one can denote them ${\pi }_i(U-x,f)$. One then sets

Π^x_i(X) = lim_{← U} πᵢ₋₁(U − x, f),

the inverse limit being taken over the system of open neighborhoods $U$ of $x$. Strictly speaking, this limit depends on $f$, and should be denoted ${\Pi }_i^x(X,f)$, but one verifies that for varying $f$, these groups are isomorphic to each other²²; more precisely, they form a local system on the space of paths of the type envisaged issuing from $x$. These invariants are the homotopical version of the local cohomology invariants $H_i^x(F)$ for a sheaf $F$ on $X$, introduced in I, and should play the role of relative local homotopy groups of $X$ modulo $X-x$. Their vanishing for $i⩽n$ and for every $x\in Y$, where $Y$ is a closed part of $X$ of topological dimension $⩽d$, should entail that the homomorphisms

$${\pi }_i(X-Y)\to {\pi }_i(X)$$

are bijective for $i<n-d$, and surjective for $i=n-d$²³. From this point of view, 4.1 would imply (for $Y$ reduced to a point) a conjecture of purely local nature, expressing itself by

Π^x_i(X) = 0   for i ⩽ n − 1

when $X$ is a complete intersection of dimension $n$ at $x$.

As an example of local invariants ${\Pi }_i^x(X)$, note that if $x$ is a non-singular point of $X$ of complex dimension $n$, then

$${\Pi }_i^x(X)={\pi }_{i-1}(S^{2n-1}),$$

where $S^{2n-1}$ denotes the sphere of dimension $2n-1$. In particular in this case ${\Pi }_i^x(X)=0$ for $i⩽2n-1$, which corresponds to the fact that if from a topological manifold $X$ one removes a closed part $Y$ of codimension $⩾m$, then ${\pi }_i(X-Y)\to {\pi }_i(X)$ is an isomorphism for $i⩽m-2$ and an epimorphism for $i=m-1$.

This being said:

Problem 4.2.

Let $X$ be an analytic space, $x\in X$, $t$ a section of ${\mathcal{O}}_X$ vanishing at $x$, $Y$ the set of zeros of $t$. Suppose the following conditions satisfied:

a) $t$ is regular at $x$ (i.e. not a zero-divisor at $x$, a hypothesis perhaps superfluous, moreover).

b) At the points $x^{\prime }$ of $X-Y$ near $x$, ${\mathcal{O}}_{X,x^{\prime }}$ is a complete intersection (a hypothesis which should be replaceable by the following more general one if 4.1 is true: for $x^{\prime }$ as above, ${\Pi }_i^{x^{\prime }}(X)=0$ for $i⩽n-1$).

c) At the points $y$ of $Y-x$ near $x$, one has

$$prof{\mathcal{O}}_{X,y}⩾n$$

(it suffices for example that one have $prof{\mathcal{O}}_{X,x}⩾n$).

Under these conditions, is the canonical homomorphism

$${\Pi }_i^x(Y)\to {\Pi }_i^x(X)$$

an isomorphism for $i⩽n-2$, an epimorphism for $i=n-1$?

Here finally is a global variant of 4.2, which should be deduced from it by consideration of the projecting cone at its origin, and which would generalize the classical Lefschetz theorems:

Problem 4.3.

Let $X$ be a projective analytic space, equipped with an ample invertible Module $L$, $t$ a section of $L$, $Y$ the set of zeros of $t$. Suppose:

a) $t$ is a regular section (hypothesis perhaps superfluous).

b) For every $x\in X-Y$, ${\mathcal{O}}_{X,x}$ is a complete intersection (should be replaceable by ${\Pi }_i^x(X)=0$ for $i⩽n-1$).

c) For every $x\in Y$, $prof{\mathcal{O}}_{X,x}⩾n$.

Under these conditions, is the homomorphism

$${\pi }_i(Y)\to {\pi }_i(X)$$

an isomorphism for $i⩽n-2$, an epimorphism for $i=n-1$?

We shall leave it to the reader to state analogous conjectures of cohomological nature²⁴, the hypotheses and conclusions then bearing on local cohomological invariants (with coefficients in a given group). In any case, the key result seems bound to be 4.2, when hypothesis b) there is taken in the form just discussed, — whether one places oneself from the point of view of homology, or of homotopy.

We have stated these conjectures in the transcendental setting, in the hope of interesting topologists in them and convincing them that "Lefschetz"-type questions are far from being closed. Of course, now that we are about to dispose of a good theory of cohomology of schemes (with finite coefficients), thanks to the recent work of M. Artin, the same questions arise in the framework of schemes, and it is difficult to doubt that they will not receive a positive answer, in the near future²⁵.

5. Problems related to local Picard groups

A first fundamental problem, signaled for the first time by Mumford [5] in a particular case, is the following. Let $A$ be a complete local ring with residue field $k$, $X=\mathrm{Spec}(A)$, $U=\mathrm{Spec}(A)-a$, where $a$ is the maximal ideal of $A$, i.e. the closed point of $\mathrm{Spec}(A)$. One proposes to construct a strict projective system $G$ of locally algebraic groups Gᵢ over $k$, and a natural isomorphism

$$\mathrm{Pic}(U)\simeq G(k)$$

where one of course sets G(k) = lim_{← } Gᵢ(k). Heuristically, one proposes to "put an algebraic group structure" (or, at least, pro-algebraic, in a suitable sense) on the group $\mathrm{Pic}(U)$.

It is evident that as it stands, the problem is not precise enough, for the datum of an isomorphism (+) is far from characterizing the pro-object $G$. If $A$ contains a subfield, still denoted $k$, that is a field of representatives, one can make the problem precise by requiring that for a variable extension $k^{\prime }$ of $k$, one have an isomorphism, functorial in $k^{\prime }$:

$$\mathrm{Pic}(U^{\prime })\simeq G(k^{\prime })$$

where $U^{\prime }$ is the open part analogous to $U$ in $\mathrm{Spec}(A^{\prime })$, $A^{\prime }=A{\hat{\otimes }}_k{k}^{\prime }$. One can proceed in an analogous way even if $A$ has no field of representatives, provided that $k$ is perfect, which then permits one to construct functorially an $A^{\prime }$ "by residual extension $k^{\prime }/k$". Moreover, when $A$ admits a field of representatives, the algebraic structure that one will find on $\mathrm{Pic}(U)$ will depend essentially on the choice of this field of representatives (as one sees already on the projecting cone of an elliptic curve); it seems therefore that one must start from a "pro-algebraic ring over $k$" in the sense of Greenberg [3], in order to arrive at

defining the pro-object $G$. It is moreover conceivable that in the case where there is no given field of representatives, one finds only a projective system of quasi-algebraic groups in the sense of Serre, or rather quasi-locally-algebraic groups

(the groups Gᵢ obtained will not in general be of finite type over $k$, but only locally of finite type over $k$). It is even possible that one will find in general only a still weaker structure on $\mathrm{Pic}(U)$, of the kind encountered by Néron [6] in his theory of degeneration of abelian varieties defined over local fields.

A method for attacking the problem, also introduced by Mumford, consists in desingularizing $X$, i.e. in considering a projective birational morphism $Y\to X$ with $Y$ regular. When $U$ is regular (i.e. $a$ is an isolated singular point), one can often find $Y$ in such a way that $Y{|}_U=V\to U$ is an isomorphism. In this case, one will therefore have

Pic(U) ≃ Pic(V) ≃ Pic(Y) / Im ℤ^I,

where $I$ is the set of irreducible components of the fiber $Y_a$ (each of these defining an element of $\mathrm{Pic}(Y)$, being a locally principal divisor, thanks to $Y$ regular). On the other hand, using the technique of formal geometry EGA III 4 and 5, notably the existence theorem, one finds

Pic(Y) ≃ lim_{← } Pic(Y_n),

where $Y_n=Y{\otimes }_A{A}_n$, $A_n=A/{\mathfrak{m}}^{n+1}$. When $A$ admits a field of representatives $k$, one has at one's disposal the theory of Picard schemes of the projective schemes $Y_n$ over $k$, hence one has

$$\mathrm{Pic}(Y_n)\simeq {\mathrm{Pic}}_{Y_n/k}(k).$$

This therefore furnishes a construction of a projective system of locally algebraic groups ${\mathrm{Pic}}_{Y_n/k}/Im{\mathbb{Z}}^I$, which is the desired system²⁶. In the case envisaged here, one can moreover see (using that $a$ is an isolated singular point) that the connected components of the universal-image subgroups in this projective system form an essentially constant projective system, so that in this case one finds a locally algebraic group $G$ as solution of the problem. If one supposes furthermore $A$ normal of dimension 2, then a remark of Mumford (stating that the intersection matrix of the components of $Y_a$ in $X$ is negative definite²⁷) implies that $G$ is even

an algebraic group, i.e. of finite type over $k$ (the number of its connected components being moreover equal to the determinant of the intersection matrix envisaged a moment ago).

If on the contrary $a$ is not an isolated singularity, one convinces oneself by examples (with $A$ of dimension 2) that one finds a projective system of algebraic groups, not reducing to a single algebraic group.

Once one had at one's disposal a good notion of "local Picard scheme", there would be occasion to strengthen the notion of parafactoriality, by saying that $A$ is "geometrically parafactorial" when not only $A$ and even Â are parafactorial, but the local Picard scheme $G(\hat{A})$ is the trivial group (which is stronger, when the residue field is not algebraically closed, than saying that $G$ has no other rational point over $k$ than the unit). One realizes the necessity of a strengthened notion of parafactoriality by recalling that there exist complete normal local rings of dimension 2 that are factorial, but that admit finite étale algebras that are not²⁸. A "geometrically factorial" local ring would then be a normal ring $A$ such that all the localizations of dimension $⩾2$ are geometrically parafactorial, or better, such that the localizations of Â are parafactorial²⁹. Of course, it would be interesting to find a "good" definition of these notions, independent of the theory, still to be done, of local Picard schemes³⁰.

It is in any case plausible that one will need these notions if one wishes to obtain statements of the following type: Let $A$ be a "good ring" (for example, an algebra of finite type over $\mathbb{Z}$, or over a complete local ring, for example over a field). Let $U$ be the set of $x\in X=\mathrm{Spec}(A)$ such that ${\mathcal{O}}_{X,x}$ is "geometrically factorial"; is $U$ open? Or again: Let $f:X\to Y$ be a flat morphism of finite type with $Y$ locally noetherian, let $U$ be the set of $x\in X$ such that ${\mathcal{O}}_{X_{f(x)},x}$ is "geometrically factorial"; is $U$ open, at least under sympathetic supplementary conditions on $f$? I doubt that with the usual notion of factorial ring, there exist true statements of this type.

We have raised here, in a particular case, the question of the study of geometric properties of "variable" local rings, for example the ${\mathcal{O}}_{X,x}$ as $x$ ranges over a prescheme $X$. When $X$ is a scheme of finite type over a field, for example, one knows³¹ that there exists on $X$ a projective system of finite algebras $P_n^{X/k}$ (obtained by completing $X{\times }_{k}X$ along the diagonal), whose fiber at every point $x\in X$ rational over $k$ is isomorphic to the projective system of ${\mathcal{O}}_{X,x}/{\mathfrak{m}}_X^{n+1}$. It is then natural to relate the study of the completions of the local rings ${\mathcal{O}}_{X,x}$, for varying $x$, to that of the "algebraic family of complete local rings" given by the $P_n$, by noting that for every $x\in X$ (rational over $k$ or not), one obtains a complete local ring

P_∞(x) = lim_{← } P_n(x)

(where $P_n(x)$ = reduced fiber $P_n{\otimes }_{{\mathcal{O}}_{X,x}}k(x)$). A particular interest will attach for example to the complete ring thus associated to the generic point, and one will expect that its algebraic-geometric properties (expressing themselves for instance by its local Picard groups, or homotopy groups, or homology groups), will be essentially those of the completions ${\hat{\mathcal{O}}}_{X,x}$ for $x$ in a suitable dense open $U$.

One can, in general, propose to make the simultaneous study of the complete local rings obtained in this way from an adic projective system $(P_n)$ of finite algebras over a given scheme $X$. It is plausible that one will find, subject to certain regularity conditions (such as the flatness of the $P_n$), that the local homotopy groups arise from a projective system of finite group schemes over $X$,

and that one will have analogous results for the local Picard groups. As regards the latter, a first interesting case that deserves to be investigated is that where one starts from an algebraic surface $X$ having singular curves, and one proposes to study the local Picard schemes at variable points on them, in terms of a suitable pro-group scheme defined on the singular locus.

6. Comments32

The point of view of "étale cohomology" of schemes and recent progress in this theory lead us to make precise and at the same time to broaden certain of the problems posed. For the notion of "topology" and of "étale topology of a scheme", I refer to M. Artin, Grothendieck Topologies, Harvard University 1962 (mimeographed notes)³³.

This theory, by a finer notion of localization than that furnished by the traditional "Zariski topology", leads one to attach a particular interest to strictly local rings, i.e. henselian local rings with separably closed residue field. For every local ring $A$ with residue field $k$, and every separable closure $k^{\prime }$ of $k$, one can find a local homomorphism of $A$ into a strictly local ring $A^{\prime }$, the strictly local closure of $A$, with residue field $k^{\prime }$, having an obvious universal property. $A^{\prime }$ is henselian, flat over $A$, and $A^{\prime }{\otimes }_{A}k\simeq k^{\prime }$; it is noetherian if and only if $A$ is. (Cf. loc. cit. Chap. III, section 4)³⁴.

If $X$ is a prescheme, and $x$ a point of $X$, $x^{\prime }$ a point above $x$, the spectrum of a separable closure $k^{\prime }$ of $k=k(x)$, one is led to define the strictly local ring of $X$ at $x^{\prime }$, ${\mathcal{O}}_{X,x^{\prime }}^{\prime }$, as the strictly local closure of the usual local ring ${\mathcal{O}}_{X,x}$, relatively to the residual extension $k^{\prime }/k$. It is the strictly local rings at the "geometric" points of $X$ that, from the point of view of the étale topology, are supposed to reflect the local properties of the prescheme $X$. They also play, in many respects, the role that one used to assign to the completions of the local rings of $X$ (say, at the points with algebraically closed residue field), while remaining "closer" to $X$ and permitting an easier passage to "neighboring points".

It is then in order to take up again a good number of questions, that one generally poses for complete local rings (eventually restricted to having an algebraically closed residue field), for noetherian henselian local rings (resp.

noetherian strictly local rings). Thus the topological problems raised in nos 2 and 3 are posed more generally for strictly local rings. One can moreover state conjecturally, for "good" strictly local rings, certain properties of simple connectedness and acyclicity for the geometric fibers of the canonical morphism $\mathrm{Spec}(\hat{A})\to \mathrm{Spec}(A)$, which would show that for many "topological"-nature properties, it amounts to the same to prove them for the ring $A$, or for its completion Â. Certain results already obtained in this direction³⁵ allow one to hope that one will soon have at one's disposal complete results in this direction.

The notion of étale localization furnishes a definition that seems reasonable of the notion of "geometrically parafactorial" or "geometrically factorial" local ring (the need for which was indicated in no 5, p. 150): one will call thus a local ring whose strictly local closure is parafactorial, resp. factorial. Hypotheses of this nature introduce themselves effectively in a natural way in the study of the étale cohomology of preschemes³⁶. Thus, if $X$ is a locally noetherian prescheme whose strictly local rings are factorial (i.e. whose ordinary local rings are "geometrically factorial"), one shows that the $H^i(X_{\acute{e}}t,{\mathbf{G}}_m)$

are torsion groups for $i⩾2$ (which allows one sometimes to express these groups in terms of cohomology groups with coefficients in the groups ${\mu }_n$ of $n$-th roots of unity), and if $X$ is integral with fraction field $K$, the natural homomorphism $H^2(X_{\acute{e}}t,{\mathbf{G}}_m)\to H^2(K,{\mathbf{G}}_m)=Br(K)$ is injective³⁷; examples show that these conclusions can fail, even for $X$ local, if one supposes only $X$ factorial instead of geometrically factorial³⁸.

Concerning the problems of local and global Lefschetz type raised in 3.4[^TRANSLATOR-NOTE-XIII-1], and their analogues in scheme theory, the homological version of these questions has been considerably clarified, all resulting formally from three general theorems: one concerning the cohomological dimension of certain affine schemes (resp. of Stein spaces), such as affine schemes $X$ of finite type over an algebraically closed field: their cohomological dimension is $⩽\dim X$ ("affine Lefschetz theorem")³⁹;

the other being a duality theorem for the cohomology (with discrete coefficients) of a projective morphism⁴⁰, and finally the last a local duality theorem of analogous nature⁴¹. In Algebraic Geometry, only this last is not proved at the time of writing these lines (it is however proved in characteristic 0, using Hironaka's resolution of singularities). Moreover, in the transcendental setting, one disposes from now on of global and local duality, recently demonstrated by Verdier⁴². Let us limit ourselves to indicating that in the statement of the homological versions of problems 4.2 and 4.3 (which from now on deserve the name of conjectures), the conditions a) and c) "at infinity" are certainly superfluous; only the local cohomological structure of $X-Y$ is important, which one will suppose for example locally a complete intersection of dimension $⩾n$. Moreover, in 4.3 say, the fact that $Y$ is a hyperplane section should not play a role, and should be replaceable by the sole hypothesis that $X$ is compact and $X-Y$ is Stein (i.e. in the case of Algebraic Geometry, $X$ is proper over $k$ and $X-Y$ affine; as we said, the homological version of this conjecture is demonstrated for algebraic spaces over the field $\mathbb{C}$)⁴³.

In the definition (p. 146) of the ${\Pi }_i^x(X)$, one must suppose $i⩾2$. For $i=0,1$, there is no reasonable definition of the ${\Pi }_i^x(X)$; one should replace them by $H_0^x(X)$ and $H_1^x(X)$, defined respectively as the cokernel and the kernel in the natural homomorphism

lim_{← } H⁰(U − {x}, ℤ) → lim_{← } H⁰(U, ℤ).

At a pinch, and for convenience of formulation, one can set ${\Pi }_i^x(X)=H_i^x(X)$ for $i⩽1$; otherwise one must complete the subsequent assertions concerning the ${\Pi }_i^x$ by the corresponding assertions for $H_0^x$, $H_1^x$. If $x$ is an isolated point of $X$, it is appropriate to set ${\Pi }_i^x(X)=0$ for $i\ne 0$, ${\Pi }_0^x(X)=H_0^x(X)=\mathbb{Z}$.

The assertion that the ${\Pi }_i^x(U,f)$ be isomorphic to each other is true only when $X$ is not disconnected by $x$ in a neighborhood of $x$, i.e. if ${\Pi }_i^x(X)=0$ for $i=0,1$. In the general case ${\Pi }_i^x(X)$ can designate only a family of groups, not necessarily isomorphic to each other; however the expression ${\Pi }_i^x(X)=0$ retains an obvious sense.

Page 146, where I predict that the vanishing of the local homotopy invariants ${\Pi }_i^x(X)$ for $x\in Y$, $i⩽n$, should entail the bijectivity of ${\pi }_i(X-Y)\to {\pi }_i(X)$ for $i<n-d$, the surjectivity for $i=n-d$, it is appropriate to be cautious, failing to be able to dispose in the present context (as in Algebraic Geometry) of "general" points at which the local conditions will also have to apply. It will doubtless be necessary, for this reason, to call upon relative local homotopy invariants

Π^Y_i(X, f) = Π^Y_i(X, x) = lim_{← U} πᵢ₋₁(U − U ∩ Y, f(t))   for i ⩾ 2,

(and an ad hoc definition as above for $i=0,1$), where $Y$ is a closed part of $X$; or to make up for the absence of general points by expressing the hypotheses on $X$ in terms of properties of topological nature (for the étale topology) of the spectra of the local rings of $X$, which allows one to recover general points. The same reservation applies to the generalization of conjectures 4.2 and 4.3 to the case where $X-Y$ is not assumed locally a complete intersection, a generalization suggested in the statement of conditions b) of these conjectures.

To formulate the expurgated versions of conjectures 4.2 and 4.3 suggested by the results to which we alluded above, it is appropriate to pose:

Definition 1.

Let $X$ be a topological space, $Y$ a locally closed part of $X$, and $n$ an integer. One says that $X$ is of homotopical depth $⩾n$ along $Y$, and one writes $profht{p}_Y(X)⩾n$, if for every $x\in Y$, one has ${\Pi }_i^Y(X,x)=0$ for $i<n$.

It should be equivalent to say that for every open $X^{\prime }$ of $X$, and every $x\in X^{\prime }\cap U=U^{\prime }$ (where $U=X-Y$), the canonical homomorphism

πᵢ(U', x) → πᵢ(X', x)

is an isomorphism for $i<n-1$, a monomorphism for $i=n-1$⁴⁴.

Definition 2.

Let $X$ be a complex analytic space, $n$ an integer; one says that the rectified homotopical depth of $X$ is $n$⁴⁵, if for every locally closed analytic part $Y$ of $X$, one has

prof htp_Y(X) ⩾ n − dim Y

(where, of course, $\dim Y$ denotes the complex dimension of $Y$).

It should be equivalent to say that for every irreducible analytic part $Y$ locally closed in $X$, there exists a closed analytic part $Z$ of $Y$, of dimension $<\dim Y$, such that the relation (x) is valid for $Y-Z$ in place of $Y$. This would permit one for example in definition 2 to confine oneself to the case where $Y$ is non-singular⁴⁶.

The following conjecture, of purely topological nature, is in the nature of a "local Hurewicz theorem".

Conjecture A ("local Hurewicz theorem"⁴⁷).

Let $X$ be a topological space, $Y$ a locally closed part, subject if necessary to "smoothness"-type conditions such as local triangulability of the pair $(X,Y)$, $n$ an integer $⩾3$. For one to have $profht{p}_Y(X)⩾n$, it is necessary and sufficient that one have

H^Y_i(ℤ_X) = 0   for i < n

(one then says that $X$ is of cohomological depth $⩾n$ along $Y$), and that the local fundamental groups

Π^Y_2(X, x) = lim_{← U ∋ x} π₁(U − U ∩ Y)

be zero (one then says that $X$ is "pure along $Y$").

One notes that if $X$ is an analytic space, $Y$ an analytic subspace, and if $X$ is pure along $Y$, then for every $x\in Y$, the local ring ${\mathcal{O}}_{X,x}$, as well as its localizations with respect to prime ideals containing the ideal defining the germ $Y$ at $x$ (i.e. in the inverse image $Y_x$ of $Y$ by $\mathrm{Spec}({\mathcal{O}}_{X,x})=X_x\to X$), are pure in the sense of Exp. X; it seems plausible that the converse is also true. Analogous remarks hold for cohomological depth, it being understood that one works with the étale topology on the $\mathrm{Spec}({\mathcal{O}}_{X,x})$.

Conjecture 4.1 then generalizes to:

Conjecture B ("Purity"⁴⁸).

Let $E$ be an analytic space, $X$ an analytic part of $E$. Suppose that $E$ is non-singular of dimension $N$ at $x\in X$, and that $X$ can be described by $p$ analytic equations in a neighborhood of every point. Then the rectified homotopical depth of $X$ is $⩾N-p$.

In particular, a local complete intersection of dimension $n$ at every point would be of rectified homotopical depth $⩾n$, which is none other than conjecture 4.1.

Conjectures 4.2 and 4.3 generalize respectively to:

Conjecture C ("Local Lefschetz"⁴⁹).

Let $X$ be an analytic space, $Y$ a closed analytic part, $x$ a point of $Y$; suppose that $X-Y$ is Stein in a neighborhood of $x$ (for example $Y$ defined by an equation at $x$), and that $X-Y$ is of rectified homotopical depth $⩾n$ in a neighborhood of $x$ (for example, is at every point of $X-Y$ near $x$ a complete intersection of dimension $⩾n$, cf. conjecture B). Then the canonical homomorphism

$${\Pi }_i^x(Y)\to {\Pi }_i^x(X)$$

is an isomorphism for $i<n-1$, an epimorphism for $i=n-1$.

Conjecture D ("Global Lefschetz"⁵⁰).

Let $X$ be a compact analytic space, $Y$ an analytic subspace of $X$ such that $U=X-Y$ is Stein, and is of

rectified homotopical depth $⩾n$ (for example a complete intersection of dimension $⩾n$ at every point). Then the canonical homomorphism

$${\pi }_i(Y)\to {\pi }_i(X)$$

is an isomorphism for $i<n-1$, an epimorphism for $i=n-1$.

Remark.

When, in statements C and D, one replaces the hypothesis that $X-Y$ is Stein by the hypothesis that $X-Y$ is the union of $c+1$ Stein opens (which will play the role of a hypothesis of topological "concavity"), the conclusions must be modified simply by replacing $n$ there by $n-c$⁵¹.

Let us make explicit, finally, in the "global case" D, the conjecture concerning the fundamental group (obtained by taking $n=3$):

Conjecture D' (Global Lefschetz for the fundamental group⁵²).

Let $X$ be a compact analytic space over the field of complex numbers, $Y$ a closed analytic part such that $U=X-Y$ is Stein. Suppose moreover the following conditions satisfied:

(i) For every $x\in U$, the local fundamental group ${\Pi }_2^x(X,x)$ is zero (i.e. $X$ is "pure at $x$"), or only the local ring ${\mathcal{O}}_{X,x}$ is pure.

(ii) The local rings of points of $U$ are "connected in dimension $⩾2$".

(iii) The local rings of points of $U$ are of dimension $⩾3$.

Under these conditions, for every $x\in Y$, the homomorphism

π₁(Y, x) → π₁(X, x)

is an isomorphism (and ${\pi }_2(Y,x)\to {\pi }_2(X,x)$ an epimorphism).

One notes that the local conditions (i) (ii) (iii) on $U$ are satisfied if $U$ is locally a complete intersection of dimension $⩾3$. From the point of view of Algebraic Geometry, (when $U$ comes from a scheme, still denoted $U$), the conditions (i) to (iii) correspond to hypotheses on the local invariants ${\Pi }_i^x(U)$, namely ${\Pi }_i^x(U)=0$ for $i<3-degtrk(x)/k$, for points $x$ such that one has respectively $degtrk(x)/k=0,1,2$. The global condition on $U$ ($U$ Stein) will be satisfied if $X$ is projective and $Y$ a hyperplane section.

Bibliography

1. S. Abhyankar — "Local uniformisation on algebraic surfaces over ground fields of characteristics $p\ne 0$", Annals of Math. 63 (1956), p. 491–526.
2. W.L. Chow — "On the theorem of Bertini for local domains", Proc. Nat. Acad. Sci. U.S.A. 44 (1958), p. 580–584.
3. M. Greenberg — "Schemata over local rings", Annals of Math. 73 (1961), p. 624–648.
4. M. Kneser — "Über die Darstellung algebraischer Raumkurven als Durchschnitte von Flächen", Archiv der Math. XI (1960), p. 157–158.
5. D. Mumford — "The topology of normal singularities of an algebraic surface, and a criterion for simplicity", Publ. Math. Inst. Hautes Études Sci. 9 (1961), p. 5–22.
6. A. Néron — "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux", Publ. Math. Inst. Hautes Études Sci. 21 (1964), p. 5–128.
7. A.H. Wallace — Homology Theory of algebraic Varieties, Pergamon Press, 1958.
8. S. Abhyankar — "Resolution of singularities of arithmetical surfaces", in Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, p. 111–152.
9. [HL] H.A. Hamm & Lê Dũng Tráng — "Rectified homotopical depth and Grothendieck conjectures", in The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, 1990, p. 311–351.

Footnotes

Translation ledger delta

πᵢ(X, X ∩ H) → πᵢ(𝐏^n_ℂ, H)