The affine part of the Picard scheme
THE AFFINE PART OF THE PICARD SCHEME (CORRECTED).
THOMAS GEISSER
Abstract. We describe the maximal torus and maximal unipotent subgroup of the
Picard variety of a proper scheme over a perfect field. (This is a corrected and
improved version of the article originally published in Comp. Math. 145 (2009)).
1. Introduction
For a proper scheme p : X → k over a perfect field, the Picard scheme PicX representing the
functor T 7→ H0(Tet, R
1p∗Gm) exists, and its connected component Pic
X is separated and of
finite type [Mu64, II 15]. By Chevalley’s structure theorem [Chev60], the reduced connected
component Pic
0,red
X is an extension of an abelian variety AX by a linear algebraic group LX :
(1) 0 → LX → Pic
0,red
X → AX → 0.
The commutative, smooth affine group scheme LX is the direct product of a torus TX and a
unipotent group UX . The following theorem completely characterizes TX :
Theorem 1. If X is proper over a perfect field, then the cocharactermodule Homk̄(Gm, TX) of
the maximal torus of PicX is isomorphic to H
et(X̄,Z) as a Galois-module.
To analyze the unipotent part, we let Pic(X[t])[1] be the typical part, i.e. the subgroup of elements
x of Pic(X[t]) such that the map X[t] → X[t], t 7→ nt sends x to nx.
Theorem 2. Let X be proper over a perfect field. Then Pic(X[t])[1] is isomorphic to the group
of morphisms of schemes f : Ga → UX satisfying f(nx) = nf(x) for every n ∈ Z. In particular,
Homk(Ga, UX) ⊆ Pic(X[t])[1], and this is an equality in characteristic 0.
To get another description of UX , we assume that X is reduced (the map on the Picard scheme
induced by the map Xred → X is well understood by the work of Oort [Oort62]). The semi-
normalization X+ → X is the largest scheme between X and its normalization which is strongly
universally homeomorphic to X in the sense that the map X+ → X induces an isomorphism on
all residue fields. A Theorem of Traverso [Tra70] implies that Pic(X[t])[1], hence UX , vanishes if
X is reduced and seminormal. We use this to show
Theorem 3. Let X be reduced and proper over a perfect field.
a) We have a short exact sequence
(2) 0 → KX → Pic
0,red
X → Pic
0,red
and inclusions of unipotent group schemes
UX ⊆ KX ⊆ p∗(Gm,X+/Gm,X)
with quotients finite p-primary group schemes.
2010 Mathematics Subject Classification. 14K30.
Key words and phrases. Picard scheme, torus, unipotent subgroup, semi-normalization, etale cohomology.
Supported in part by NSF grant No.0556263.
http://arxiv.org/abs/0704.0479v4
2 THOMAS GEISSER
b) The group scheme p∗(Gm,X+/Gm,X) represents the functor
T 7→ {OX×T -line bundles L ⊆ OX+×T which are invertible in OX+×T }.
Notation: For a field k, we denote by k̄ its algebraic closure, and for a scheme X over k we let
X̄ = X ×k k̄. Unless specified otherwise, all extension and homomorphism groups are considered
on the fpqc site. 1
Acknowledgements:
This (original) paper was written while the author was visiting T. Saito at the University of Tokyo,
whom we thank for his hospitality. We are indebted to G. Faltings for pointing out a mistake
in a previous version, and the referee, whose comments helped to improve the exposition and to
give more concise proofs. O. Gabber pointed out mistakes in the original version and suggested
improvements.
2. The torus
Proposition 4. If p : X → k is reduced, geometrically connected, and proper over a perfect field,
then Gm,k → p∗Gm,X is an isomorphism. Moreover, if f : X
′ → X is a universal homeomorphism
and X ′ is reduced as well, then f induces an isomorphism p∗Gm,X ∼= p∗Gm,X′ .
Proof. Since any scheme T over k is flat, we have by flat base change Rjq∗OXT = H
j(X,OX )⊗kOT ,
where q : XT → T is the projection. In particular,
p∗Gm,X(T ) := Γ(X × T,OX×T )
× = (Γ(X,OX )⊗ Γ(T,OT ))
and it suffices to show that Γ(X,OX) ∼= Γ(X
′,OX′) ∼= k. Since Γ(X̄,OX̄)
Gal(k̄/k) = Γ(X,OX ), we
can assume that k is algebraically closed and that X is connected, in which case the statement
follows because X and X ′ are reduced, proper, connected, and have a k-rational point. ✷
Lemma 5. For any scheme X we have isomorphisms
H1et(X,Z)
fl(X,Z)
∼= Ext
X(Gm,X ,Gm,X).
Proof. The first isomorphism is [Mi80, III Rem. 3.11(b)]. To prove the second isomorphism, we
note that HomX(Gm,X ,Gm,X) ∼= ZX by [SGA3, VIII Cor. 1.5], and that Ext
X(Gm,X ,Gm,X) is
isomorphic to the group of extensions of group schemes [Oort66, Cor. 17.5], which vanishes by
[SGA7, VIII Prop. 3.3.1].3 Hence we obtain the isomorphism from the spectral sequence [Mi80,
III Thm.1.22]
2 = H
fl(X, Ext
X(Gm,X ,GmX)) ⇒ Ext
X (Gm,X ,Gm,X).
Proof. (Theorem 1) Since the maps defined below are natural, we can assume that k is alge-
braically closed and X is connected. We can also assume that X is reduced, because H1et(X,Z)
H1et(X
red,Z), and the map PicX → PicXred has unipotent kernel and cokernel [Oort62, Cor. page
9]. It suffices to calculate Homk(Gm,k,PicX), because there are no homomorphisms from Gm to
commutative group schemes other than tori [Oort66, p. 81]. By Yoneda’s Lemma, the latter group
1In [Gei09] we used the étale topology
2This replaces [Gei09, Prop. 9 a)] which is incorrect as stated because the induction step in the proof does not
preserve the hypothesis on reducedness.
3This was claimed without proof in [Gei09].
THE AFFINE PART OF THE PICARD SCHEME (CORRECTED). 3
is isomorphic to the group of homomorphisms of sheaves on the fpqc site Homk(Gm,k, R
1p∗Gm,X).
The Leray spectral sequence
(3) E
2 = Ext
k(Gm,k, R
tp∗Gm,X) ⇒ Ext
X (Gm,X ,Gm,X).
gives an exact sequence
0 → Ext1k(Gm,k, p∗GmX) → Ext
X(Gm,X ,Gm,X) → Homk(Gm,k, R
1p∗Gm,X)
−→ Ext2k(Gm,k, p∗Gm,X).
By Proposition 4 the left term agrees with Ext1k(Gm,k,Gm,k), and this vanishes by [Oort66, Cor.
17.5]. Thus it suffices to show that δX is the zero map
4. Choose a closed point of X and let i :
Z → X be the corresponding closed subscheme. Since p∗ ◦ i∗ = id we have R
sp∗i∗ = R
s(p◦ i)∗ = 0
for s > 0. Hence we obtain a diagram
Homk(Gm,k, R
1p∗Gm,X) −−−−→ Homk(Gm,k, R
1p∗i∗Gm,Z) = 0
Ext2k(Gm,k, p∗Gm,X)
−−−−→ Ext2k(Gm,k,Gm).
By Proposition 4, the lower horizontal map is an isomorphism. ✷
Remark. The example in [Gei06, Prop. 8.2] shows that the map H iet(X̄,Z) → Ext
(Gm,Gm) is
not an isomorphism for i ≥ 2. One can ask if it is an isomorphism if one replaces H iet(X̄,Z) by
the eh-cohomology group H ieh(X̄,Z) of [Gei06].
Example. If X is the node over an algebraically closed field, then H1et(X,Z)
∼= Z, and TX ∼= Gm.
Let X be a node with non-rational tangent slopes at the singular point. Base changing to the
algebraic closure, one sees that H1et(X̄,Z)
∼= Z, with Galois group acting as multiplication by −1,
hence TX is an anisotropic torus.
Using the theorem, we are able to recover the torsion of TX , AX and the diagonalizable part of
NSX := PicX /Pic
0,red
X in terms of etale cohomology:
Corollary 6. Let X be proper over a perfect field k. Then we have canonical isomorphisms
H1et(X̄,Z)⊗Q/Z
∼= colimHomk̄(µm, TX);
Div(torH
et(X̄,Z))
∼= colimHomk̄(µm, AX);
et(X̄,Z)/Div
∼= colimHomk̄(µm, NSX).
Proof. Taking the colimit of the isomorphismH1et(X̄,Z/m)
∼= Homk̄(µm,PicX) of [Mi80, Prop.4.16]
or [Ray70, §6.2], we obtain H1et(X̄,Q/Z)
∼= colimHomk̄(µm,PicX). Since Ext
(Gm, TX) = 0, The-
orem 1 implies that Homk̄(µm, TX)
∼= Homk̄(Gm, TX)/m
∼= H1et(X̄,Z)/m. Consider the commu-
tative diagram:
colimHomk̄(µm, TX) H
et(X̄,Z)⊗Q/Z
colimHomk̄(µm,Pic
0,red
X ) −−−−→ colimHomk̄(µm,PicX) −−−−→ colimHomk̄(µm, NSX)
colimHomk̄(µm, AX)
−−−−→ torH
et(X̄,Z) −−−−→ coker f.
The middle column is the short exact coefficient sequence. The left column and middle row are
short exact because Ext1
(µm, TX) = Ext
(µm,Pic
0,red
X ) = 0 by [Oort66, Cor. 17.5, II 14.2]. A dia-
gram chase shows that f is injective, and the right vertical map is an isomorphism. The Corollary
4The remainder of the proof is a simplification suggested by O. Gabber.
4 THOMAS GEISSER
follows because colimHomk̄(µm, AX) is divisible and colimHomk̄(µm, NSX) is finite. ✷
The above result should be compared to [Gei10, Prop.6.2], where we show that, for every proper
scheme over an algebraically closed field, the higher Chow group of zero-cycles CH0(X, 1,Z/m)
is the Pontrjagin dual of H1et(X,Z/m). This implies a short exact sequence
0 → torA
X(k) → CH0(X, 1,Q/Z) → χ(TX)⊗Q/Z → 0,
for AtX the dual abelian variety of AX , and χ(TX) the character module of TX . However, in this
case the contribution from the torus and from the abelian variety are not compatible with the
coefficient sequence
0 → CH0(X, 1) ⊗Q/Z → CH0(X, 1,Q/Z) → torCH0(X) → 0
as in Corollary 6.
Looking at tangent spaces, the previous Corollary gives a dimension formula:
Corollary 7. Let l be a prime different from char k. Then
dimk H
1(X,OX ) = dimUX + dimk Lie(NS
X) + rankH
et(X,Z) +
corankl H
et(X̄,Ql/Zl).
3. The unipotent part
Let N Pic(X) := ker
Pic(X[t])
−→ Pic(X)
. Since t 7→ 0t induces x 7→ 0x on the typical part,
Pic(X[t])[1] is a subgroup of N Pic(X). In [Wei91], Weibel shows that for every scheme there is a
direct sum decomposition
Pic(X[t, t−1]) ∼= Pic(X) ⊕N Pic(X) ⊕N Pic(X)⊕H
et(X,Z).
Proof. (Theorem 2). We show first that N Pic(X) = ker
1) → UX(k)
. Since there are
no non-trivial morphisms of schemes from A1k to an abelian variety, a torus, an infinitesimal
group, or a discrete group, we see that the kernel of UX(A
k) → UX(k) agrees with the kernel of
PicX(A
k) → PicX(k). Let p : X → k and p
′ : X × A1k → A
k be the structure morphisms. Then
the Leray spectral sequence gives a commutative diagram
0 −−−−→ H1et(A
Gm) −−−−→ Pic(X × A
k) −−−−→ PicX(A
k) −−−−→ H
0 −−−−→ H1et(k, p∗Gm) −−−−→ Pic(X) −−−−→ PicX(k) −−−−→ H
et(k, p∗Gm),
and it suffices to show that the outer vertical maps are isomorphisms. Let X
−→ L → k be the
Stein factorization of p, such that OL ∼= g∗OX and L is the spectrum of an Artinian k-algebra.
Since A1k → k is flat, p
OX×A1
= OA1
⊗kp∗OX , and X×A
−→ A1L −→ A
k is the Stein factorization
of p′. We obtain
H iet(A
Gm) ∼= H
Gm) ∼= H
L,Gm),
and H iet(k, p∗Gm)
∼= H iet(L,Gm). Hence the terms on the left vanish because Pic(L) = Pic(A
0. To show that H2et(A
L,Gm) → H
et(L,Gm) is an isomorphism, we can assume that L is a local
Artinian k-algebra with (perfect) residue field k′. By [Mi80, III Rem.3.11] we are reduced to
showing that H2et(A
k′ ,Gm) → H
′,Gm) is an isomorphism, and this can be found in [Mi80, IV
Ex.2.20].
Given an element x of N Pic(X), the condition x ∈ Pic(X[t])[1] implies that the corresponding f ∈
HomSch(A
1, UX) satisfies f(nx) = nf(x) for all n. If k has characteristic 0, then UX ∼= G
a for some
THE AFFINE PART OF THE PICARD SCHEME (CORRECTED). 5
r, and the map f : Ga → UX corresponds to a morphism of Hopf algebras f
∗ : k[x1, · · · , xr] → k[t].
If f∗(xi) =
j ajt
j, then
aj(nt)
j = nf∗(xi) = f
∗(nxi) = n
only if nj = n for all n, hence j = 1. ✷
Example. If k has characteristic p, then t 7→ t2p−1 induces a map Ga → Ga which is compatible
with multiplication by n, but not a homomorphism of group schemes.
Corollary 8. We have UX = 0 if and only if N Pic(X) = 0.
Proof. This follows from N Pic(X) = ker
1) → UX(k)
, because any unipotent, connected,
smooth affine group is an affine space as a scheme, hence admits a non-trivial morphism from A1
which sends 0 to 0 if it is non-trivial. ✷
The kernel and cokernel of PicX → PicXred has been described in [Oort62], hence we will from
now assume that X is reduced. If X+ is the semi-normalization of X, then the map OX → OX+
is an injection of sheaves on the same topological space. For X+ reduced and semi-normal,
N Pic(X+) = 0 by Traverso’s theorem [Tra70] together with [Wei91, Thm. 4.7]. Hence the
Corollary implies that UX+ = 0, and that
UX = ker(Pic
0,red
X → Pic
0,red
(For curves, this recovers [BLR90, Prop.9.2/10].) Indeed, by Corollary 6, the map Pic
0,red
0,red
induces an isomorphism on the torus and abelian variety part, because it induces an
isomorphism on etale cohomology.
Proof. (Theorem 3) a) We have isomorphims H iet(X,Z)
∼= H iet(X
+,Z), which combined with
Corollary 6 shows that the canonical map Pic
0,red
X → Pic
0,red
induces an isomorphism on the torus
components, and is an isogeny with kernel a unipotent group scheme PX on the abelian variety
parts. 5 Hence the map is surjective and the kernel KX is an extension of PX by UX . Applying
the Proposition to the exact sequence of etale sheaves
0 → p∗Gm,X → p∗Gm,X+ → p∗(Gm,X+/Gm,X) → PicX → PicX+
on Spec k, we obtain the diagram with exact columns
0 −−−−→ KX −−−−→ Pic
0,red
X −−−−→ Pic
0,red
−−−−→ 0
0 −−−−→ p∗(Gm,X+/Gm,X) −−−−→ PicX −−−−→ PicX+
0 −−−−→ coker u −−−−→ NSX −−−−→ NSX+ .
Since v is injective, so is u. The Neron-Severi group schemes are extensions of finitely gener-
ated étale group schemes by a finite connected group scheme. The isomorphism H2et(X,µm)
H2et(X
′, µm) implies that PicX(k̄)/m → PicX+(k̄)/m is injective for any m prime to p, and since
0,red
X (k̄) and Pic
0,red
(k̄) are m-divisible, the same holds for NSX(k̄)/m → NSX+(k̄)/m, and
consequently for NSX [
] → NSX+ [
]. Thus coker u is contained in the extension of the p-primary
torsion subgroup NSX{p} by the finite connected group scheme NS
X . Finally, the isomorphism
5O.Gabber [Gab20] showed that, conversely, any finite unipotent commutative group scheme can appear as PX .
6 THOMAS GEISSER
Homk̄(µm,PicX)
∼= H1et(X̄,Z/m) from [Mi80, III Prop. 4.16] together with the isomorphism
H iet(X,Z)
∼= H iet(X
+,Z) and the result on KX shows that the three right maps in the diagram
induce isomorphisms on Homk̄(µm,−) for all m, hence the three groups on the left are unipotent.
b) Recall that q : XT → T , and consider the diagram
0 −−−−→ H1et(T, q∗Gm,X×T ) −−−−→ Pic(X × T ) −−−−→ PicX/k(T ) −−−−→ H
et(T, q∗Gm,X×T )
0 −−−−→ H1et(T, q∗Gm,X+×T ) −−−−→ Pic(X
+ × T ) −−−−→ PicX+/k(T ) −−−−→ H
et(T, q∗Gm,X+×T ).
Since q∗Ga,X×T = H
0(X,OX )⊗OT = H
0(X+,OX+)⊗OT = q∗Ga,X+×T is an isomorphism as in
Proposition 4a), the outer maps are isomorphisms, and it suffices to calculate ker r. Let Y = X×T
and Y ′ = X+ × T , and consider the tautological map
f : {OY -line bundles L ⊆ OY ′ which are invertible in OY ′} → Pic(Y ).
It suffices to show the following statements:
a) The image of f is contained in ker
Pic(Y ) → Pic(Y ′)
b) f surjects onto ker
Pic(Y ) → Pic(Y ′)
c) f is injective.
a) We claim that the map L ⊗OY OY ′ → OY ′ ⊗OY OY ′
−→ OY ′ is an isomorphism. We can check
this on an affine covering, and in this case it is proved in [RS93, Lemma 2.2(4)].
b) Let L ∈ Pic(Y ) with L⊗OY OY ′
∼= OY ′ . Since L is flat, we get an injection L = L⊗OY OY →
L⊗OY OY ′
∼= OY ′ . We claim that the inverse of L in OY ′ is the sheaf associated to the presheaf
U 7→ {x ∈ OY ′(U)|xL(U) ⊆ OY (U)} ⊆ OY ′(U). This can be checked on an affine covering, and
then it is [RS93, Lemma 2.2(2)].
c) Let L and L′ be subsheaves of OY ′ which are invertible in OY ′ and isomorphic as abstract
invertible sheaves. Multiplying with the inverse of L′ inside OY ′ , it suffices to show that if L is
a subsheaf of OY ′ , and f : OY → L an isomorphism, then L = OY ⊆ OY ′ . But f(1) is a global
unit of OY ′(Y ), and by Proposition 4a), OY (Y )
× = OY ′(Y )
×. Hence L = f(1)−1L = OY . ✷
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Dep. of Math., Rikkyo University, Japan
	1. Introduction
	2. The torus
	3. The unipotent part
	References
