PROCEEDINGS OF THE
AMERICANMATHEMATICAL SOCIETY
Volume 53, Number 1, November 1975
THE PICARD SEQUENCE OF A FIBRATION
ANDY R. MAGID1
ABSTRACT.
A fibration,
leads
to an exact
Under
suitable
sequence
in the Zariski
sequence
hypotheses,
including
the
topology,
of cohomology
the exact
Picard
sequence
groups
of algebraic
of the relative
can
of the base
varieties
units
functor.
be interpreted
variety,
the
as a
total
space,
and the fibre.
Let
X —> Y be a fibration
algebraically
closed
Y. If V and
F are smooth
sition
field)
5, p. 8] constructed
Pic(F).
The purpose
of the cohomology,
sides
explaining
isolate
otherwise
an exact
J" a sheaf
call
on V, then
then
that
trivial,
(A)v
the higher
and,
constant.
{f)\U
denotes
hence
the
of ((k ),, —> G
V
Lemma
this
sheaf
to
Be-
helps
which
leads
to
topology,
Unless
and all sheaf
subset
of V and
to U. If A is an abelian
V associated
of constant
on
an algebraically
k and irreducible.
U an open
on
all locally
sheaf
and
in terms
sheaf.
interpretation
k denotes
J" restricted
groups
that
Pic(X),
sequence
units
on
[2, Propo-
result.
are over
constant
the units
,,). We refer
2. Let
of the relative
are for the Zariski
V be a variety.
772, V
Pic(Y),
conventions:
denotes
cohomology
1. Let
relating
that
an
topology
and Iverson
If V is a variety,
by descent,
G m, ,,V denotes
Definition
in the Zariski
of the sequence,
considered
all sheaves
(over
Fossum
sequence,
of the original
is in that topology.
varieties
is to interpret
topology,
the following
specified,
trivial
sequence
note
for the existence
and all varieties
cohomology
group
F is rational,
in the Zariski
We fix throughout
field
is locally
and
the Fossum-Iverson
the hypotheses
F of algebraic
which
of the present
some mild generalizations
closed
with fibre
to
sheaves
constant
on
A. We re-
V are
sheaves
on
V are
V.
U, v denotes
U, ,, as the sheaf
fZ ,V
'
the presheaf
of relative
J
units
cokernel
on
V.
V be a variety.
(i) U, v is a sheaf on V.
(ii)tf'(V, UkiV) = H'(V, GmV) for all i > 1.
(iii) Pic(V)=V(V, Uky). '
Received
by the editors
October
25, 1974.
AMS(MOS)subject classifications (1970). Primary 14D99, 14F10, 14F20.
Key words and phrases.
Sheaf cohomology,
group, smooth variety,
rational
variety.
iPartially
supported
Leray
spectral
sequence,
Picard
by NSF Grant GP-37051.
Copyright © 1975. American Mathematical
37
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Society
38
A. R. MAGID
Proof.
an open
hence
Let D denote
subset
there
the sheaf cokernel
of V. Then
is an exact
(+)
((k*)y)\W
and
W be
(G^ y)\W = Gm w, and
sequence
l-+(k\-*GmW
of sheaves
of (k ),,V —*Gm, ...
Let
V
= (k*)w
on W. The long
exact
-^(D\W) ^1
cohomology
sequence
begins
0 — (k*)w — G ,.(W) -> D(W)— f/Hw, (A*)™)= 0,
and hence
sequence
H1
D(W) = (/,
■/(M/). This
proves
part (i).
For part (ii),
(+) for W =-.V and D = U, y and the fact that
(V, (k )y) - 0 for all
i > 1. Part (iii) is immediate
we use
the
H'(V, (k*)y) =
from (ii) and the fact
that Pic(V) = HHV, Gm , ,.).
V
Definition
with fibre
isomorphic
3. A morphism
F if there
with
E —> B of varieties
is an open
F x W. (over
cover
is a Zariski
\W.\ of B such that
W.) for each
fibration
E xR W. is
i.
Lemma 4. Let f: E —. B be a Zariski fibration with fibre F. Then
there
is an exact
of sheaves
Proof.
sequence
on B.
The first
for an element
a, send a to class
Let
in Definition
3, then
in some
U, B(W)
as follows:
and for an element
for an open set
b of G
W of B,
g(W)
representing
of b ° f in f*Uk B(W) = Uk E(f~ (W)). This is trivially
an injection.
tained
map is defined
a of
C denote
f^U,
its
cokernel.
If \W .\ is an open cover
E(W) = U, £(F x W) for every
W.. By Rosenlicht's
lemma
[2, Lemma
open set
3, p.7],
map U, p(F) x U, „(W) —> U, E(F x W) is an isomorphism.
of B as
W con-
the natural
It follows
that
y*Uh,B>\Wi = ^k.B^i
X (Ufe,F(F))W; and' henCe' that C\Wi = (C7fe,F(F)VThus C is locally constant, and it follows that C = (U. p(F))B.
Theorem 5. Let f: E —>B be a Zariski
that,
for all sufficiently
Pic(W)
—> Pic(F
small
open
sets
x W) is an isomorphism.
fibration
with fibre F. Suppose
W of B the natural map
Then
there
is an exact
Pic(F)
sequence
0 -♦ U,k,Ba(B) — U,fe,cP(E) -* U,fe,rAF) -. Pic(B) -> Pic(F)
-Pic(F)
Proof.
H"(E,
Consider
U, p).
the Leray
The exact
0 - HHB, UUk<E) -
sequence
-
spectral
H2(B, GmB) sequence
of low degree
H2(E, GnE).
HP(B,
terms
Rqf^.Uk E)=>
is
Hl(E, UhfB) -. H°(B, R7*tft>B)
-H2iB,UUhtB)-+H2{E,
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UkE).
x
PICARD SEQUENCE OF A FIBRATION
We consider
the sheaf
2(i+),
the terms
associated
this
is the presheaf
—> /*Pic.
Let
/*Pic
associated
isomorphism
beginning
f^Pic.
= Pic
x (Pic(F))u/
constant
R f%U, E is
U, U, E).
By Lemma
On sufficiently
Pic
open
sets
on
W. The
sheaf
—>C —>0 then yields
to C, since
and locally
on B,
small
as presheaves
Pic —>/^Pic
E and the sheaf associated
to Pic is 0. But C is locally
the third.
a map of presheaves
cokernel.
to the exact sequence
of R f^k
with
U —> H (f~
We have
C be the presheaf
W of B we have
sequence
separately,
to the presheaf
39
isomorphic
an
the sheaf associated
to (Pic(F)).,,
hence
Rlf*Vk E = (Pic(F))fi, and H°(B, Rlf^k E) = Pic(F). By Lemma 2(ii) and
(iii) we have Hl(E, Uk E) = Pic(F) and H2'(E, Ifk E) = H2(E, Gm R). The long
exact
sequence
of cohomology
arising
from the sheaf
sequence
of Lemma
4
shows that H2iB, f^UktE) = H2iB, Uk^B) since
H^B, (Ukp(F))B) =
= 0,' and hence, by Lemma 2(ii), H2(B,'f^U fe>E)=
H2(B, (UkF(F))B)
H (B, G
„).
The first
0 - UkJB)
By Lemma
-
2(iii),
that
of that long exact
Uk,E{E) -
Uk,F(p)-*Hl{B>
H (B, U, „) = Pic(B),
to the sequence
Remark.
part
obtained
Let
—> Pic(F
Then
the map
Proof.
zero.
from the Leray
small
open
—> Pic(F)
Uk,B) - "I(B> f*,Jk,E] - 0-
spectral
sets
this
sequence
fibration
and that
of Theorem
F and
sequence
yields
with fibre
W of B the natural
x W) is an isomorphism,
Pic(F)
is
and now splicing
/: E —• B be a Zariski
for all sufficiently
Pic(W)
sequence
F.
map
the result.
Suppose
Pic(F)
x
B are smooth.
5 is onto.
We need to show that the kernel of H2(B, ' G772) — H2(E, ' G772) is
Let
W be an open
set
of B with
/"
(W) = F x W. We have
a commuta-
tive diagram
H2(B,G
f/2(W'
and the bottom
maps
The exact
sequence
X W' Gm,FxW^
injective.
and the result
sequence,
p)
G772,^->//2(F
map is clearly
are injective
Iverson
_)-,H2(E,G
of Theorem
and the above
By [l,
1.8, p. 104], the vertical
follows.
5 is the generalization
remark
explains
of the Fossum-
why their
sequence
stops
W are smooth
varieties,
and
with Pic(F).
Fossum
and Iverson
show
that if F and
one is rational,
then Pic(F x W) = Pic(F)
This
some
then
gives
(We remark
quence
then
obtains
rational
that
argument
and
this
conditions
product
given
here,
the Fossum-Iverson
x Pic(W) [2, Corollary
for the hypotheses
formula
does
not follow
but is a hypothesis
sequence
of Theorem
under
from the spectral
for it.)
For example,
the assumptions
F smooth.
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6, p. 11].
5 to obtain.
that
se-
one
B is
40
A. R. MAGID
REFERENCES
la
M. Artin,
Grothendieck
Univ., Cambridge,
2. R. Fossum
Aarhus
Universitet
Mass.,
topologies,
Mathematics
Department
Lecture
Notes,Harvard
1962.
and B. Iverson,
On Picard
groups of algebraic
fibre
Matematisk
Institut
Preprint
Series,
no. 19, Aarhus,
spaces,
Denmark,
1972/73.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF OKLAHOMA, NORMAN, OKLAHOMA73069
Current
address:
Department
of Mathematics,
University
Illinois 61801
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of Illinois,
Urbana,