Alexandre Grothendieck's EGA V

112 Pages·2004·0.65 MB·English

Alexandre Grothendieck’s EGA V
Translation and Editing of his ‘prenotes’
by
Piotr Blass and Joseph Blass
1
Summary
This formulation gives p^ele m^ele a detailed summary of the set of results that should
appear in a (cid:12)nal formulation. To arrive at the latter we need to reorganize thoroughly the
present stagezero. The (cid:12)rst step should probably be tomake a new plan (in which without
a doubt the present sections 11, 12, 14, 15 will come much earlier). I have not even written
section 16 which should neither in principle cause any di(cid:14)culty nor does it in(cid:13)uence in
any way the previous Nos. since what is involved is a simple matter of translation.
You will notice the presence of a proposition 10.3 which should appear in a previous
paragraph.
I would like to tell you in this connection that I have several other results quite diverse
but all dealing with birational mappings that I would love to include somewhere.
Itseemstomethatthereisnotenoughtomakeaparagraph. Doyouhaveasuggestion
where to place them?
I plan to send them to you soon as well as section 16 of the present notes.*
In addition, the present paragraph 20 will probably blow up into two paragraphs, one
consisting of results of the type \elementary geometry" on grassmanians.
If need be, could one include there also (lacking a better place) the supplements that
I told you about dealing with birational transformations?
*Ask AG if No. 16 has ever been written. [Tr]
2
Hyperplane Sections and Conic Projections
1) Notations.
2) Generic hyperplane section, local properties.
3) Generic hyperplane section geometric irreducibility and connectedness.
4) Variable hyperplane section: \su(cid:14)ciently general" sections.
5) Theorems of Seidenberg type.
6) Connectednes sof any hyperplane section.
7) Application to the constructions of special hyperplane sections and multisections.
8) Dimension of the set of exceptional hyperplanes.
9) Change of projective immersion.
10) Pencils of hyperplane sections and (cid:12)brations of blown up varieties.
11 ) Grassmanians.
12 ) Generalization of the previously mentioned results to linear sections.
13) Elementary morphisms and a thoerem of M. Artin.
14) Conic projections.
15) Axiomatization of some of the previous results.
16) Translation into the language of linear systems.
3
Hyperplane Sections and Conic Projections
1 Preliminaries and Notation
x
Let S be a prescheme, let E be a locally free module of (cid:12)nite type over S, let Ev be
its dual. We denote by P = P(E) the projective (cid:12)bration de(cid:12)ned by E and by Pv the
projective (cid:12)bration de(cid:12)ned by Ev. We shall call Pv the scheme of hyperplanes of p. This
terminology can be justi(cid:12)ed as follows. Let (cid:24) be a section of Pv over S which is threfore
determined by an invertiblel quotient module L of Ev. From it we obtain an invertible
quotient module L of (Ev) = (E )v, on the other hand, we have the invertible quotient
P P P
module Op(1) of Ep. Passing to duals we may take LP(cid:0)1 resp. OP( 1) to be invertile
(cid:0)
submodules (locallydirect factors)of E (resp. of (E )v)and the pairing E E O
P P P Pv P
(cid:10) !
deifnes therefore a natural pairing
( ) OP( 1) LP(cid:0)1 OP
(cid:3) (cid:0) (cid:10) (cid:0)!
or also the transpose homomorphism
( ) O O (1) L = L (1)
P P P P
(cid:3)(cid:3) (cid:0)! (cid:10)
i.e. a section of L (1) canonically de(cid:12)ned by (cid:24). The \divisor" of that section, i.e. the
P
closed subscheme H of P de(cid:12)ned by the image ideal of ( ), is called the hyperplane in
(cid:24)
(cid:3)
P de(cid:12)ned by the element (cid:24) PV(S). We could describe it by noting that locally over S,
2
(cid:24) is given by a section (cid:30) of E such that (cid:30)(s) = 0 for all s ((cid:30) is determined by (cid:24) up to
6
multiplication by an invertible section of O ); since E = p (O (1)), (p:p S being the
S (cid:3) p
!
projection), (cid:30) can be considered as a section of O (1), the divisor of which is nothing else
p
but H .
(cid:24)
Of course, if we consider L 1 as an invertible submodule of E locally a direct factor
(cid:0)
in E then the correspondence between (cid:24) (i.e. L or L 1 E) and (cid:30) is obtained by taking
(cid:0)
(cid:26)
for (cid:30) a section of L 1 which does not vanish at any point, i.e. by a trivialization of L 1
(cid:0) (cid:0)
(which exists in every case locally). Let us note that H is nothing else by P(E=L 1)
(cid:24) (cid:0)
(canonical isomorphism) that is a third way of describing H (N.B. P(E=L 1) is indeed
(cid:24) (cid:0)
canonically embedded in P = P(E) which has the advantage of proving in addition that
H is a projective (cid:12)bration over S and is a fortiori smooth over S. (Again it would be
(cid:24)
necessary to say in par. 17 of EGA IV that a projective (cid:12)bration is smooth.). Without a
doubt it would be better to begin with this one.
Remarks. The formation of H is compatible with base change, in other words one
(cid:24)
(cid:12)nds a homomorphism of functors (Sch=S0) (Ens), Pv Div(P=S) where the sec-
! !
ond term denotes the functor of \relative divisors" of P=S whose values at S (an arbi-
0
4
trary S prescheme) is the set of closed subschemes of PS0 which are complete intersections
transversal and of codimension 1 relative to S (compare Par. 19) [of EGA IV Tr.].1
0
It is easy to show that this homomorphism of functors is a monomorphism, in other
words that (cid:24) is determined of H . (This last fact justi(cid:12)es the terminology \scheme of
(cid:24)
hyperplanes" used above.) We shall see that the functor Div(P=S) is representable by
the prescheme (direct) sum of P(Symmk(Ev)) so that Pv can be identi(cid:12)ed to an open
and closed subscheme of Div(P=S):::2 (N.B. to tell the truth, the determination of the
relative divisors of P=S could be done with the means available right now, using results
on Ch. II and could be added as an example to Par. 19 of EGA IV [Tr.].)
Let us now make the base change S = Pv S and let us consider the diagonal
0
!
section (or \generic section") of Pv = P(Ev ) over S : we (cid:12)nd a closed subscheme H of
S0 S0 0 S
PS0 = P SPv which is called sometimes the incidence scheme between P and Pv de(cid:12)ned
(cid:2)
by the image ideal of the canonical homomorphism
O ( 1) O ( 1) O
P (cid:0) (cid:10)S Pv (cid:0) (cid:0)! PxSPv
from which one sees that it is a projective (cid:12)bration over Pv, and by symmetry it is also
a projective (cid:12)bration over P. Let us note that one recovers the \special" hyperplanes H
(cid:24)
(for (cid:24) a section of Pv over S) by starting out from the \universal hyperplane" H and by
taking its inverse image for the base change S (cid:24) Pv.
(cid:0)!
The same remark holds for every point of Pv with values in an arbitrary S-prescheme
S0 which (considered as a section of PS0 over S0) allows us to de(cid:12)ne an H(cid:24) PS0; the
(cid:26)
latter is nothing else but the inverse image of H by the base change S (cid:24) Pv.
0
(cid:0)!
In what follows we assume a prescheme X of (cid:12)nite type over P [Tr]3 and an S mor-
phism f:X P. One of the main objectives of this paragraph is to study for every
!
hyperplane H of P its inverse image Y = f 1(H ) = XX H and especially to relate
(cid:24) (cid:24) (cid:0) (cid:24) P (cid:24)
the properties of X and Y . As usual one also has to consider the P(S ), S an arbitrary
(cid:24) 0 0
S scheme, in this case H(cid:24) is a hyperplane in PS0 and we put again
Y(cid:24) = fS(cid:0)01(H(cid:24)) = XS0 (cid:2)pS0 H(cid:24) = XXPH(cid:24)
where the subscript S denotes as usual the e(cid:11)ect of the base change S S and where
0 0
!
in the last expression we consider H as a P scheme via the combined morphism H
(cid:24) (cid:24)
!
PS0 P. It is therefore again convenient to consider the case where (cid:24) is \universal" i.e.
!
1
Uses notation of new edition of EGA I [Tr.]
2
Compare with Mumford’s: ‘Lectures on curves on an algebraic surface.’ [Tr]
3or over S, I am not sure [Tr]
5
where S = Pv and (cid:24) is the diagonal section so that H = H, in this case one observes (up
0 (cid:24)
to better notations to be suggested by Dieudonn(cid:19)e) that Y = Y . In the general case of a
(cid:24)
(cid:24):S Pv, one has therefore also Y = Yx S . Finally if F is a sheaf of modules4 over
0 ! (cid:24) vP 0
X we denote by G its inverse image over Y by G its inverse image over H so that one
(cid:24) (cid:24)
also has G(cid:24) = G Ov OS0.
(cid:10) P
Let us summarize in a small diagram the essentials of the constructions and notations
considered.
F G G
(cid:17)
X Xx Pv Y Y
S (cid:17)
(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)
? ? ? ?
? ? ? ?
Py PxySPv Hy Hy(cid:17)
(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)
? ? . ?
? ? ?
yS Pyv Sy0
(cid:0)(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)(cid:0)
(The squares and diamonds appearing in this diagram are Cartesian). In the next section
we will study systematically the following case: S is the spectrum of a (cid:12)eld K and its
0
image in Pv is generic in the corresponding (cid:12)ber Pv. After making the base change
s
Speck(s) S we are reduced to the case where S is the spectrum of a (cid:12)eld k, which is
!
what as assume in the next section. Also the majority of properties studied for X and Y
(cid:24)
are of \geometric nature" and therefore invariant under base change, which allows us also
(without loss of generality) to limit ourselves to the case where K is algebraically closed
or to the case where K = k((cid:17)), (cid:17) being the generic point of Pv and :(cid:24):Spec(K) Pv is
!
of course the canonical morphism. We also note that for geometric questions concerning
X, Y we can (after making a base change) restrict ourselves to the case of k algebraically
(cid:24)
closed.
A terminological reminder. If f is an immersion we usually call Y a hyperplane
(cid:24)
section of X (relative to the projective immersion f and the hyperplane H [Tr.]). There
(cid:24)
is no reason not to extend this terminology to the case of an arbitrary f.
2 Study of a generic hyperplane section: local properties
x
Let us recall that now S = Spec(k), k is a (cid:12)eld. If (cid:17) is a point of Pv and if
(cid:24):Speck((cid:17)) Pv is the canonical morphism we also write H , Y , G in place of H ,
(cid:17) (cid:17) (cid:17) (cid:24)
!
Y , G .
(cid:24) (cid:24)
In this section (numero) (cid:17) denotes always the generic point of Pv.
4
Ask A.G. If module always means coherent or quasi-coherent sheaf of modules.
6
Proposition 2.1. Let us assume that X is irreducible. Then Y is irreducible or empty
(cid:17)
and in the (cid:12)rst case it dominates X [illegible, ask AG]5 Y [Tr.] is irreducible.
Indeed, since H P is a projective (cid:12)bration that is also true for Y X which
! !
implies that Y is irreducible if X is irreducible. So the generic (cid:12)ber Y [Tr.] of Y over Pv
(cid:17)
is irreducible or empty in the (cid:12)rst case its generic point is the generic point of Y which
therefore lies over the generic point of X. q.e.d.
Proposition 2.2. Let Z be a subset of P. Then its inverse image Z in H is empty if
(cid:17) (cid:17)
and only if every point of Z is closed. In particular if Z is constructible then Z = (cid:30) if
(cid:17)
and only if Z is (cid:12)nite.
We may suppose that Z is reduced to a single point z and we only have to prove
that the image of H in P consists exactly of the non-closed points of P. Denoting by
(cid:17)
X the closure of z and using 2.1 we only have to prove that Z = (cid:30) if and only if X is
(cid:17)
(cid:12)nite (X being a closed subscheme of P). Replacing X by X , P the ‘only if’
k((cid:17)) k((cid:17))
!
[French ‘il faut’] part results from the following fact for which we have to have a reference
and which fact deserves to be restated here as a lemma: if Y is any hyperplane section of
X and if Y = (cid:30) then X is (cid:12)nite (indeed X P H is a(cid:14)ne and projective:::). The
(cid:17)
(cid:26) (cid:0)
‘is su(cid:14)cient’ part one needs is immediate, for example, by noting that Y is a projective
(cid:12)bration of relative dimension (n 1) over X (n being the relative dimension of P and Pv
(cid:0)
over S), thus X being (cid:12)nite over k, Y is of absolute dimension n 1, n = dim P thus
(cid:0) h i
the morphism Y Pv cannot be dominant thus its generic (cid:12)ber Y is empty.
(cid:17)
!
Corollary 2.3. Let f:X P be a morphism of (cid:12)nite type and let Z be a constructible
!
subset of X. In order for its inverse image in Y to e empty it is necessary and su(cid:14)cient
(cid:17)
tha the image f(Z)should be (cid:12)nite. In particular, in order for Y to e empty it is necessary
(cid:17)
and su(cid:14)cient that f(X) should be (cid:12)nite.
Corollary 2.4. Let Z;Z be two closed subsets of X with Z irreducible, and let Z and
0 (cid:17)
Z be their inverse image in Y . In order to have Z Z it is necessary and su(cid:14)cient
(cid:17)0 (cid:17) (cid:17) (cid:26) (cid:17)0
that f(Z) should be (cid:12)nite or that we have Z Z .
0
(cid:26)
In order that Z = Z it is necessary and su(cid:14)cient that f(Z) and f(Z ) should be
(cid:17) (cid:17)0 0
(cid:12)nite or that Z = Z .
0
This is an immediate consequence of 2.3 because we see that f(Z Z Z ) can only
0
(cid:0) \
be (cid:12)nite if Z Z or if f(Z) is (cid:12)nite (because if we do not have z Z then z Z Z is
0 0 0
(cid:26) (cid:26) (cid:0) \
dense in Z thus f(Z Z Z ) is dense in f(Z), and if the former is (cid:12)nite and thus closed,
0
(cid:0) \
being constructible, so is also the latter.
5
Ask Grothendieck: What is the meaning and role of underlined capital letters, in Section One for
example
7
Corollary 2.5. To every irreducible component X of X such that dim f(X ) > 0 we
i i
assignitsinverseimageY inY . ThenY isanirreduciblecomponentofY andweobtain
i(cid:17) (cid:17) i(cid:17) (cid:17)
in this manner a one-to-one correspondence between the set of irreducible components X
i
of X such that dim f(X ) > 0 and the set of irreducible components of Y .
i (cid:17)
Indeed, it follows from 2.3 that Y is union of Y de(cid:12)ned above and that the latter
(cid:17) i(cid:17)
are closed and non-empty subsets of Y; they are also irreducible because of 2.1. Finally,
they are mutually without an inclusion relation because of 2.4, hence the conclusion.
Let us notice that if dim X = d we have dim Y = d 1. More generally:
i i i i
(cid:0)
Proposition 2.6. Let us suppose that for every irreducible component X of X we have
i
dim f(X ) > 0, i.e. Y = or that6 f isan immersionand dim f(X) > 0[slightlyillegible,
i i(cid:17)
6 ;
(ask AG)]. Then we have dim Y = dim X 1.
(cid:17)
(cid:0)
We are reduced tothe case where X is irreducible due to2.5. Bythe veryconstruction
Y is de(cid:12)ned starting from X as the divisor of a section of an invertible module over
(cid:17) k((cid:17))
X (being the inverse image of O (1)). On the other hand X is irreducible (because
k((cid:17)) P k((cid:17))
X is such and k((cid:17)) is a pure transcendental extension of k which one should have indicated
at the beginning of the N0:::) and Y = X since the image of Y in X (contrary to
(cid:17) k((cid:17)) (cid:17)
6
that of X , which is faithfully (cid:13)at over X) is not equal to X, indeed it does not contain
k((cid:17))
the closed points of X because of 2.3. It follows that dim Y = dim X 1 = dim X 1
(cid:17) k((cid:17))
(cid:0) (cid:0)
(reference needed for the last equality.) q.e.d.
Proposition 2.7. Let F be a quasi coherent module over X, hence G over Y . Let
(cid:17) (cid:17)
Z be the associated prime cycles of F such that dim f(Z ) > 0. Let Z be the inverse
i i i(cid:17)
image of Z in Y then the Z are exactly all the prime cycles associated to G . Also,
i (cid:17) i(cid:17) (cid:17)
their relations of inclusion are the same as among the Z .
i
The last assertion is contained in 2.4. On the other hand, since Y X is a projective
!
(cid:12)bration, thus (cid:13)at with (cid:12)bers (S ) and irreducible, it follows from par. 3 of EGA IV that
1
the associated prime cycles to the inverse image G of F over Y are the inverse images of
the associated prime cycles of F. Hence they are induced on the generic (cid:12)ber Y of Y over
(cid:17)
Pv, the fact that the associated prime cycles to G are the non-empty inverse images of
(cid:17)
the Z which proves 2.7 by means of 2.3.
i
To tell the truth, the passage through Y is unnecessary (not used) (useless), and we
can use directly the fact that Y X is (cid:13)at with (cid:12)bers (S ) (and also geometrically
(cid:17) 1
!
regular, i.e. the morphism is regular) and with irreducible (cid:12)bers (and even geometrically
irreducible: they are localizations of projective schemes) remark for the proof of 2.1.
6
in French, ou que. I think the proper translation is, where [Tr.].
8
Proposition 2.8. Let F be coherent over X and let y Y , x is its image in X.
(cid:17)
2
Let P(M) be one of the following properties for a module of (cid:12)nite type N over a local
noetherian ring A:
(i) coprof M n (ref)
(cid:20)
(ii) M satis(cid:12)es (S ) (ref)
k
(iii) M is Cohen-Macaulay
(iv) M is reduced (ref)
(v) M is integral (ref
Then for G to satisfy the property P it is necessary and su(cid:14)cient that F should
(cid:17);y x
satisfy it.
This follows immediately from results of paragraph 67 taking into account that Y
(cid:17)
!
Y is a regular morphism so that O O should be regular. Taking into account
X;x Y(cid:17);y
!
2.3, we obtain thus:
Corollary 2.9. With the notations for 2.8, let Z be the set of x X such that P(F )
x
2
is not satis(cid:12)ed. Then in order for G to satisfy the condition P at all the points it is
(cid:17)
necessary and su(cid:14)cient that f(Z) should be a (cid:12)nite subset of P, or that dim f(Z) = 0.
Indeed, 2.8 tells us that h 1(Z) is a P-singular subset of G and it is empty if and
(cid:0) (cid:17)
only if f(Z) is (cid:12)nite by 2.3 (N.B. h denotes the morphism Y X; I have just realized
(cid:17)
!
that the letter P in 2.8 has been used double).
Corollary 2.10. Condition for Y to be reduced respectively locally integral.
(cid:17)
Corollary 2.11. Let y Y , in order that Y should be regular, respectively should
(cid:17) (cid:17)
2
satisfy the property R (reference) at y (respectively should be normal at y) it is necessary
k
and su(cid:14)cient that X should satisfy the same property at x. Let Z be the set of those
points of X where X is not regular, resp. E (resp. normal); for Y to be regular resp. R
k (cid:17) k
(resp. normal) it is necessary and su(cid:14)cient that f(Z) should be (cid:12)nite, i.e. dim f(Z) = 0.
The same proof as 2.8 and 2.9. One must give the di(cid:11)erent references assuring that
Z is closed (because we must know that it is constructible to apply 2.3).
Let us note that in 2.10 and 2.11 we do not talk at all about the corresponding
geometric properties; the results described are of ‘absolute’ nature. We now examine the
properties of geometric nature. (One could, if one wanted to, take the opportunity to
change the n0.)
Geometric Properties
[Tr.]
7
Tr: clear up this reference. Is it EGAIV ?
9
Theorem 2.12. Suppose that f:X P is unrami(cid:12)ed. Let y Y , let x be its image in
(cid:17)
! 2
X. In order for X to be smooth over k at x it is necessary and su(cid:14)cient that Y should
(cid:17)
be smooth over k((cid:17)) at y.
We may assume that k is algebraically closed. If Y is smooth over k((cid:17)) at y it is
(cid:17)
regular, thus since Y is (cid:13)at over X, X is regular at x (reference), therefore it is smooth
(cid:17)
over k at x since k is algebraically closed and thus perfect (reference).
For the converse we can (after replacing X by an open neighborhood of x) assume that
X is smooth, and (due to the jacobian criterion of smoothness) to be de(cid:12)ned in an open
subset U of P by p equations as X = V(f ;::: ;f ), where the di(cid:11)erentials df ;::: ;df are
1 p 1 p
everywhere linearly independent. By introducing the a(cid:14)ne coordinatesS ;::: ;S in Pv
1 n
and the a(cid:14)ne coordinates T ;T ;::: ;T in a neighborhood of x (by choosing a hyperplane
1 2 n
H [at in(cid:12)nity] not containing x) Y [Tr.] , U is then equal to V(f ;::: ;f ; S T
1 (cid:17) k((cid:17)) 1 p i i
! (cid:0)
1)anditsu(cid:14)ces toverifythatthedi(cid:11)erentials(relativetok((cid:17)))off ;::: ;f ; ( PS T ) 1
1 p i i
(cid:0)
are linearly independent. However, these di(cid:11)erentials are nothing else but thePsections of
((cid:10)1 ) k((cid:17)) [Tr.] [illegible] as follows: df ;::: ;df ; S dT . Since the df are linearly
U=k (cid:10)k 1 p i i i
independent at every point of U and since the dT formPa basis of (cid:10)1 at every point of
i U=S
U and a fortiori a system of generators, we conclude immediately the linear independence
of the written down quantities at every point of U at least when p n, i.e. if
k((cid:17))
(cid:20)
E = (cid:10)1 O df = 0
U=k(cid:30) U i 6
X
1 i n
(cid:20) (cid:20)
this is a small lemma about the family of generators a , 1 i n of a non-zero locally fre
i
(cid:20) (cid:20)
module E, thus S a considered as a section of E k((cid:17)) does not vanish at any point.
i i k
(cid:2)
On the other hanPd, the case p = n is trivial because then Y = .
(cid:17)
;
Corollary 2.13. Let Z be the set of points of X where X is not smooth over k. In order
that Y should be smooth over k((cid:17)) it is necessary and su(cid:14)cient that Z should be (cid:12)nite.
(cid:17)
In particular, if X is smooth, the same is true about Y .
(cid:17)
Follows from 2.12 and 2.3. More generally we obtain:
Theorem 2.14. Let y be a point of Y , x its image in X. Let P(A;K) be one of the
(cid:17)
following properties for an algebra A local and noetherian over a (cid:12)eld K:
(i) A is geometrically regular over K.
(ii) A is geometrically (R ) over K.
k
(iii) A is separable over K.
(iv) A is geometrically normal over K.
10