Comptes Rendus Mathématique

Ananyo Dan

Agustín Romano-Velázquez

Comptes Rendus
Mathématique
Ananyo Dan and Agustín Romano-Velázquez
Examples of non-flat bundles of rank one
Volume 361 (2023), p. 965-968
Published online: 7 September 2023
https://doi.org/10.5802/crmath.459
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 965-968
https://doi.org/10.5802/crmath.459
Complex analysis and geometry / Analyse et géométrie complexes
Examples of non-flat bundles of rank one
Exemples de fibrés en droites qui ne sont pas plats
Ananyo Dan∗, a and Agustín Romano-Velázquezb, c
a School of Mathematics and Statistics, University of Sheffield, Hicks building,
Hounsfield Road, S3 7RH, UK
b Alfréd Rényi Institute Of Mathematics, Hungarian Academy Of Sciences, Reáltanoda
Utca 13-15, H-1053, Budapest, Hungary
c Universidad Nacional Autónoma de México Avenida Universidad s/n, Colonia
Lomas de Chamilpa CP 62210, Cuernavaca, Morelos Mexico
E-mails: a.dan@sheffield.ac.uk, agustin@renyi.hu, agustin.romano@im.unam.mx
Abstract. It is expected that there exist line bundles on a quasi-affine non-singular surface which do not admit
a flat connection. However, to the best of our knowledge there is no known example of such a line bundle.
In this article we give several explicit examples of line bundles on certain non-singular, quasi-affine surfaces
that cannot be equipped with a flat connection.
Résumé. On s’attend à ce qu’il existe des fibrés en droites sur une surface non-singulière quasi-affine qui
n’admettent pas de connexion plate mais, à notre connaissance, aucun exemple d’un tel fibré n’est connu.
Dans cet article, nous en donnons plusieurs exemples explicites.
2020 Mathematics Subject Classification. 14B05, 13C14.
Funding. The first author is funded by EPSRC grant number EP/T019379/1. The second author is funded by
OTKA 126683 and Lendület 30001.
Manuscript received 12 November 2022, accepted 21 December 2022.
1. Introduction
Existence of connections on modules defined over isolated surface singularities have been exten-
sively studied (see [1, 3–6]). However, in these literatures the authors stress that they cannot pro-
duce a single example of a maximal Cohen–Macaulay (MCM) module over a surface singularity
that does not admit a flat connection (see [5, p. 106], [4, p. 1562], [1, p. 903]), even with the help
of a computer (see [3, p. 322]). By flat connection, we mean a connection with zero curvature. In
this article we produce numerous examples of maximal Cohen–Macaulay modules over certain
isolated surface singularities that cannot be equipped with a flat connection. In particular, we
prove:
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
966 Ananyo Dan and Agustín Romano-Velázquez
Theorem 1. Let (X , x) be the germ of a normal surface singularity such that the fundamental
group of the link is perfect. Suppose that (X , x) contains a smooth curve passing through x. Then,
there exists a line bundle on X \ {x} that cannot be equipped with a flat connection. Moreover, one
can associate to any such smooth curve (i.e., passing through x), an unique (up to isomorphism)
line bundle on X \ {x} that cannot be equipped with a flat connection.
Note that, given a line bundle L on X \ {x}, the pushforward i∗L is a reflexive sheaf on
X , where i is the inclusion of X \ {x} into X (see [7, Proposition 1.6]). Furthermore, if X is a
normal, integral surface, then reflexive sheaves on it are maximal Cohen–Macaulay. Therefore,
as a corollary we produce explicit examples of maximal Cohen–Macaulay modules that cannot
be equipped with a flat connection.
Corollary 2. Let (p, q,r ) be a triple of positive integers that are pairwise coprime and r > pq.
Denote by G(p, q,r ) the surface in C3 defined by the polynomial X p +Y q + Z r and by U (p, q,r )
the regular locus of G(p, q,r ). Then, there exists line bundles on U (p, q,r ) that cannot be equipped
with a flat connections.
The study of the obstruction to the existence of flat connection on MCM modules has appli-
cations in Lie–Rinehart cohomology (see [6]) and Chern–Simmons theory (see [1]).
Acknowledgement
We thank Prof. Javier F. de Bobadilla and Prof. Duco van Straten for their interest in this problem
and helpful comments. The second author thanks CIRM, Luminy, for its hospitality and for
providing a perfect work environment. He also thanks Prof. Javier F. de Bobadilla, the 2021
semester 2 Jean-Morlet Chair, for the invitation.
2. Example of non-flat invertible sheaves
We give examples of rank 1 invertible sheaves which cannot be equipped with a flat connection.
We will assume familiarity with reflexive sheaves. See [7] for basic properties of reflexive sheaves.
2.1. Brieskorn–Pham surfaces
Given positive integers (p, q,r ), denote by G(p, q,r ) ⊂ C3 the zero locus of the polynomial
X p +Y q + Z r . The resulting surface is the Brieskorn–Pham type surface. The origin 0 is the only
singularity of the surface. Denote by
S := {C ⊂G(p, q,r ) |C is a non-singular curve passing through the origin 0}.
The following theorem proves the existence of smooth curves through the singularity of G(p, q,r ).
Theorem 3. Assume (p, q,r ) = 1 and p ≤ q ≤ r . Then, S ̸= ; if and only if at least one of the
following conditions hold:
(1) two of the three integers (p, q,r ) are coprime and the other one is divisible by at least one
of the two coprime numbers,
(2) the inequality r > pq/gcd(p, q) holds.
Proof. See [8, Theorem 3]. □
Ananyo Dan and Agustín Romano-Velázquez 967
2.2. Proof of Theorem 1
Denote by U the regular locus of (X , x), i.e. U = X \ {x}. Note that the fundamental group π1(U )
is the same as the fundamental group of the link L of (X , x). By hypothesis, the fundamental
group of L is perfect. This means that the abelianization π1(U )ab of the fundamental group
π1(U ) is trivial. Since GL1(C) = C∗ is abelian, any 1-dimensional group representation of π1(U )
factors through π1(U )ab, which is trivial. Therefore, every 1-dimensional representation of π1(U )
is trivial. By the Riemann–Hilbert correspondence, this implies that there does not exist any non-
trivial line bundle on U that can be equipped with a flat connection. Therefore, it suffices to show
the existence of non-trivial line bundles on U .
By hypothesis, there exists C ⊂ X a smooth curve contained in the surface X passing through
the singular point. Denote by I C the ideal sheaf of C . We now show that the restriction MU
of M := H omX (I C ,O X ) to U is a non-trivial line bundle. Indeed, MU is trivial if and only if
i∗(MU ) ∼= M is trivial (M is reflexive, and reflexive sheaves are uniquely determined by their
restriction to the open subset U ), where i : U → X is the natural inclusion. Furthermore, since
M is a reflexive module, M is trivial if and only if M∨ is trivial (double dual of a reflexive module
is isomorphic to itself). Consider the short exact sequence:
0 −→I C −→O X −→OC −→ 0 (1)
By the depth comparison in exact sequences (see [2, Proposition 1.2.9]), we have that I C is a
reflexive O X -module. This implies that
M∨ ∼=I∨∨C ∼=I C .
This implies, MU is trivial if and only if I C is trivial. By (1), I C is trivial if and only if C is a Cartier
divisor. So, it suffices to show that C is a Weil divisor which is not Cartier.
We prove this by contradiction. In particular, suppose that C is a Cartier divisor. We are going
to give a contradiction to the non-smoothness of X . Let f ∈ O X be a function defining C (the
existence of f is guaranteed as C is Cartier) and m (resp. m′) the maximal ideal of O X (resp.
O X /( f )). We then have the following short exact sequence of O X -modules:
0 −→ ( f ) −→m−→m′ −→ 0. (2)
Since C is a smooth curve, the dimension as a complex vector space of m′/(m′)2 is one. Let c ′ ∈m′
be a generator of m′/(m′)2. By the surjectivity of (2), there exists an element c ∈m that is mapped
onto c ′. We will now show that m/(m)2 is generated as a C-vector space by f and c. Indeed, let g
be any element in m. If g maps to 0 in m′ then by the exact sequence (2), g =β f for some β ∈O X .
If g maps to a non-zero element in m′, then there exists α ∈ O X such that g −αc is mapped to
zero in m′. Hence by the exactness of (2), there exists β ∈ O X such that β f = g −αc. Therefore, g
is a O X -linear combination of f and c in both cases. Since g was arbitrary, this implies that the
dimension as a complex vector space of m/(m)2 is two (generated by f and c). But this will mean
X is smooth at the origin, which is a contradiction. Hence, C cannot by a Cartier divisor. This
proves the theorem. □
Proof of Corollary 2. Let (p, q,r ) be a triple of positive integers that are pairwise coprime. By [9,
p. 7], the link of (G(p, q,r ),0) is an integer homology sphere. Hence, the fundamental group of the
link is perfect. By Theorem 3 there exists a smooth curve in G(p, q,r ) passing through the origin.
The corollary then follows from Theorem 1. □
Example 4. Let f = x2+y3+z7. By Corollary 2, the hypersurface in C3 defined by f admits MCM
modules of rank one that cannot be equipped with a flat connection.
968 Ananyo Dan and Agustín Romano-Velázquez
References
[1] S. Bloch, H. Esnault, “Algebraic Chern–Simons theory”, Am. J. Math. 119 (1997), no. 4, p. 903-952.
[2] W. Bruns, J. Herzog, Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge
University Press, 1998.
[3] E. Eriksen, T. S. Gustavsen, “Computing obstructions for existence of connections on modules”, J. Symb. Comput. 42
(2007), no. 3, p. 313-323.
[4] ——— , “Connections on modules over singularities of finite CM representation type”, J. Pure Appl. Algebra 212 (2008),
no. 7, p. 1561-1574.
[5] ——— , “Connections on modules over singularities of finite and tame CM representation type”, in Generalized Lie
Theory in Mathematics, Physics and Beyond, Springer, 2009, p. 99-108.
[6] ——— , “Lie–Rinehart cohomology and integrable connections on modules of rank one”, J. Algebra 322 (2009), no. 12,
p. 4283-4294.
[7] R. Hartshorne, “Stable reflexive sheaves”, Math. Ann. (1980), p. 121-176.
[8] G. Jiang, M. Oka, D. T. Pho, D. Siersma, “Lines on Brieskorn–Pham surfaces”, Kodai Math. J. 23 (2000), no. 2, p. 214-223.
[9] A. Némethi, “Five lectures on normal surface singularities”, in Low dimensional topology. Proceedings of the summer
school, Budapest, Hungary, August 3–14, 1998, Bolyai Society Mathematical Studies, vol. 8, János Bolyai Mathematical
Society, 1998, p. 269-351.


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Comptes Rendus
Mathématique
Zev Rosengarten
Picard Groups of Algebraic Groups and an Affineness Criterion
Volume 361 (2023), p. 559-564
Published online: 1 February 2023
https://doi.org/10.5802/crmath.419
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 559-564
https://doi.org/10.5802/crmath.419
Algebraic geometry / Géométrie algébrique
Picard Groups of Algebraic Groups and an
Affineness Criterion
Zev Rosengarten∗, a
a Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J.
Safra Campus, 91904, Jerusalem, Israel
E-mail: zev.rosengarten@mail.huji.ac.il
Abstract. We prove that an algebraic group over a field k is affine precisely when its Picard group is torsion,
and show that in this case the Picard group is finite when k is perfect, and the product of a finite group of
order prime to p and a p-primary group of finite exponent when k is imperfect of characteristic p.
Résumé. Nous prouvons qu’un groupe algébrique sur un corps k est affine si et seulement si son groupe de
Picard est de torsion, et que dans ce cas, le groupe de Picard est fini si k est parfait, et le produit d’un groupe
fini d’ordre premier à p par un p-groupe d’exposant fini lorsque k est imparfait de caractéristique p.
Mathematical subject classification (2010). 14L10, 14L15, 14L17, 14L40, 20G15.
Manuscript received 10 May 2022, revised 6 September 2022, accepted 7 September 2022.
The main purpose of this short paper is to prove the following two results on algebraic groups
over arbitrary fields.
Theorem 1. Let G be a finite type group scheme over a field k. Then G is affine if and only if Pic(G)
is torsion.
Theorem 2. Let G be an affine group scheme of finite type over a field k.
(i) If k is perfect, then Pic(G) is finite.
(ii) If char(k) = p > 0, then Pic(G) is the product of a finite group and a p-primary group of
finite exponent.
In general, affine algebraic groups over imperfect fields need not have finite Picard groups.
Indeed, [7, Prop. 5.9] shows that, when k is either separably closed (and imperfect) or a local
function field, then this fails for almost every nontrivial k-form of Ga. (For examples of nontrivial
forms of Ga over arbitrary imperfect fields, see, for example, [7, Ex. 5.8].)
We now turn to the proofs of the theorems above.
Lemma 3. Let X be a scheme over a field k, and let k ′/k be a field extension such that (k ′)p
n ⊂ k.
Then for all i , the kernel of the pullback map
Hiét(X ,Gm) → Hiét (Xk ′ ,Gm)
is pn-torsion.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
560 Zev Rosengarten
Proof. Let f : Spec(k ′) → Spec(k) denote the obvious morphism. The composition
(Gm)k → f∗ ((Gm)k ′ ) [
pn]−−−→ (Gm)k (1)
is the pn-th power map, hence induces multiplication by pn on Hi (X ,Gm). But pushforward
through a radicial map is an exact functor on the category of étale sheaves, hence we have a
canonical identification
Hi
(
X , f∗(Gm)
)= Hi (Xk ′ ,Gm)
which identifies the map on cohomology induced by the first map in (1) with the pullback map
of the Lemma 3. It follows that anything killed by pullback is killed by pn . □
Proof of Theorem 2. We are free to pass to a finite field extension. Indeed, it suffices to show that
we may pass to a finite separable extension or (when k is imperfect) a finite purely inseparable
extension. The separable case follows from [5, Th. 1.13], while the purely inseparable case, which
is only relevant when k is imperfect, follows from Lemma 3. We may in particular assume that
Gred ⊂ G is a smooth k-subgroup scheme. By [8, Lem. 2.2.9], the map Pic(G) → Pic(Gred) is an
isomorphism, so we may assume that G is smooth. Choose a finite extension k ′/k such that
Gk ′ =
∐
Gi , where the Gi are the connected components of Gk ′ , and such that Gi (k
′) ̸= ; for
each i . Renaming k ′ as k, we have that Gi ≃G0 as k-schemes, hence Pic(G) ≃∏i Pic(G0). We may
therefore replace G by G0, so we may assume that G is smooth and connected. Finally, passing to
a finite purely inseparable extension (in the imperfect case) and applying [1, Ch. V, Th. 18.2(ii)],
we may assume that G is unirational, so Pic(G) is finite by [7, Th. 1.3]. □
Let G be a k-group scheme, and let m,πi : G ×G → G (i = 1,2) denote the multiplication
and projection maps, respectively. We say that an element L ∈ Pic(G) is primitive if m∗(L ) =
π∗1 (L )+π∗2 (L ).
Lemma 4. Given an exact sequence
1 −→G ′ j−→G −→G ′′ −→ 1 (2)
of smooth connected group schemes over a field k, and a k-character χ : G ′ → Gm, consider the
pushout E of the sequence along χ, which is a Gm-torsor over G ′′. Then E is a primitive torsor.
Proof. We first claim that the conjugation action of G on Ĝ ′(k) := Homk−gps(G ′,Gm) is trivial.
Indeed, DG ′ ⊂ G ′ is a universally characteristic k-subgroup through which the character group
factors, so in order to prove the claim we may replace the sequence (2) by the sequence obtained
by forming the quotient by this subgroup, and we may assume that G ′ is commutative. Let T ⊂G ′
be the maximal k-torus. Then T is a characteristic k-subgroup of G ′, hence normal in G . Further,
G ′/T has no nontrivial characters, hence has no nontrivial k-characters, so Ĝ ′(k) ⊂ T̂ (k), and it
suffices to check that the G-conjugation action on T̂ (k) is trivial. But in fact we claim that T is
central in G , which is sufficient. Indeed, the automorphism functor AutT /k of T is represented by
an étale k-scheme. (This is well-known, but is easily seen by using Galois descent to pass to the
case when T ≃ Gnm, in which case the automorphism functor is GLn(Z).) Conjugation induces a
k-homomorphism G → AutT /k from a connected to an étale k-group scheme, which is therefore
constant. This proves that the G-conjugation action on Ĝ ′(k) is trivial.
Now suppose we are in the situation of the lemma. We must show that there is an isomorphism
m∗(E) ≃ π∗1 (E) +π∗2 (E) of Gm-torsors over G ′′ ×G ′′. This is equivalent to constructing a map
f : E ×E → E such that f (t1 · e1, t2 · e2) = (t1t2) · f (e1,e2), and such that the following diagram
commutes:
E ×E E
G ′′×G ′′ G ′′.
f
m
Zev Rosengarten 561
The torsor E may be described explicitly as E = (Gm × G)/i (G ′), where i is the map g ′ 7→
(χ(g ′), j (g ′)−1) with the obvious Gm-action and map to G ′′. Then one may define a map g : (Gm ×
G)× (Gm ×G) → Gm ×G by the formula(
t1, g1
)× (t2, g2) 7→ (t1t2, g1g2) .
We claim that this descends to a map f : E ×E → E . Once this is verified, the map clearly satisfies
the required conditions above. A straightforward check reveals that what one needs to verify
is that χ(g−1g ′g ) = χ(g ′) for all g ∈ G and g ′ ∈ G ′, and this is exactly the triviality of the G-
conjugation action on Ĝ ′(k) proved above. □
Lemma 5. Given an exact sequence
1 −→G ′ −→G q−→G ′′ −→ 1 (3)
of smooth connected group schemes over a field k, then every element of
ker
(
q∗ : Pic(G ′′) → Pic(G))
is primitive.
Proof. By descent theory, ker(q∗) may be identified with the Čech cohomology group Ȟ1(q,Gm)
associated to the fppf cover q . This group is the middle cohomology of the sequence
Γ(G ,Gm)
p∗1 −p∗2−−−−−→ Γ(G ×G ′′ G ,Gm)
p∗23−p∗13+p∗12−−−−−−−−−→ Γ(G ×G ′′ G ×G ′′ G ,Gm), (4)
where the pi and pi j denote the obvious projections. In particular, we obtain a surjection from
the kernel of the second map above to ker(q∗). We have isomorphisms
G ×k G ′ ∼−→G ×G ′′ G , (g , g ′) 7→ (g , g g ′)
G ×k G ′×k G ′ ∼−→G ×G ′′ G ×G ′′ G , (g , g ′1, g ′2) 7→ (g , g g ′1, g g ′1g ′2).
Under these isomorphisms, the second map in the sequence (4) is transported to the map
Γ
(
G ×k G ′,Gm
) f ∗23− f ∗13+ f ∗12−−−−−−−−→ Γ(G ×k G ′×k G ′,Gm) , (5)
where the fi j are defined by the following formulas:
(g , g ′1, g
′
2) 7→

(g g ′1, g
′
2), f23
(g , g ′1g
′
2), f13
(g , g ′1), f12.
Now suppose given h ∈ Γ(G×k G ′,Gm) which lies in the kernel of the map (5). Then restricting the
equality (h ◦ f23)(h ◦ f13)−1(h ◦ f12) = 1 to the point (1,1,1) of G ×k G ′×k G ′ shows that h(1,1) = 1.
It follows from the Rosenlicht Unit Theorem (more precisely, a corollary of that theorem [2,
Cor. 1.2]) that h is a character of G×k G ′: say h =χχ′, where χ is a character of G and χ′ a character
of G ′. A simple calculation shows that the fact that h lies in the kernel of the map (5) is equivalent
to the identity χ(g g ′) = 1 for all g ∈G , g ′ ∈G ′. Thus χ= 1 – that is, h = χ′ ∈ Ĝ ′(k) is a character of
G ′. We thus obtain a surjection
Ĝ ′(k)↠ ker(q∗).
This surjection is functorial in the exact sequence, hence, via the identification of ker(q∗) with
Ȟ1(q,Gm), it is none other than the pushout map which sends a character χ : G ′ → Gm to the Gm-
torsor over G ′′ obtained by pushing out the sequence (3) along χ. (Actually, it could also be the
inverse of this map, since the identification of ker(q∗) with Ȟ1(q,Gm) is only well-defined up to
a universal choice of sign.) It follows that any element of ker(q∗) is obtained by such a pushout,
and Lemma 4 implies that such a pushout is primitive. □
562 Zev Rosengarten
Lemma 6. Suppose given an exact sequence
1 −→ H −→G q−→ A −→ 1
of smooth connected group schemes over a field k, with A a nonzero abelian variety. Then for any
ample L ∈ Pic(A), the pullback q∗(L ) ∈ Pic(G) is not torsion.
Proof. Replacing L with a power of itself, it suffices to show that q∗(L ) ̸= 0. But if it were 0,
then Lemma 5 would imply that L is primitive. In particular, the homomorphism φL : A → A
defined by the formula a 7→ t∗a (L )⊗L −1 – where ta denotes translation by a – is the 0 map. But
by [6, Ch. II, §6, p. 60, Application I], the ampleness of L is equivalent to the map φL being an
isogeny, and this yields a contradiction because A is nonzero. □
Lemma 7. Let k be a field, and suppose given an isogenyφ : G1 →G2 of finite type k-group schemes
(that is, a faithfully flat k-homomorphism with finite kernel). If, in the diagram below, the rows are
exact sequences of finite type k-group schemes with Li affine and Ai abelian varieties (i = 1,2), and
L1 smooth and connected, then the diagram of solid arrows extends uniquely to a commutative
diagram with dotted arrows, and the map f is an isogeny.
1 L1 G1 A1 1
1 L2 G2 A2 1.
φ
π
f
Proof. That the diagram extends uniquely, if it extends at all, is clear. That it extends is equivalent
to the claim that the composition L1 ,→G1 φ−→G2 → A2 is trivial, and this follows from the fact that
any k-homomorphism from a smooth connected affine k-group to an abelian variety is trivial.
Finally, it remains to check that f is an isogeny. It is surjective becauseφ is, so if it is not an isogeny
then its kernel contains a nonzero abelian variety B ⊂ ker( f ). Then π−1(B) → L2 has finite kernel
F , hence letting G ′ := π−1(B)/F and L′ the image of π−1(B) in L2, we obtain a k-isomorphism
ψ : G ′ ∼−→ L′ with the former an extension of a nonzero abelian variety C by a finite type affine
k-group M , and the latter an affine k-group of finite type. Then G ′/M ≃ L′/ψ(M), with the former
a nonzero abelian variety and the latter affine. This is a contradiction, so f is an isogeny. □
Proof of Theorem 1. Theorem 2 implies that affine algebraic groups have torsion Picard groups,
so it only remains to show that finite type k-groups which are not affine have non-torsion
elements in their Picard groups. Because G is affine if and only if G0 is, we may assume that G
is connected.
First we treat the case in which k is perfect. Then Gred ⊂G is a smooth connected k-subgroup
scheme, and G is affine precisely when Gred is, so we may assume that G is smooth and connected.
Chevalley’s Theorem (which again uses the perfection of k) then furnishes an exact sequence
1 −→ L −→G −→ A −→ 1
with L a smooth connected affine k-group scheme, and A an abelian variety. Because G is not
affine by assumption, A is nonzero. Lemma 6 implies that any ample line bundle on A pulls back
to a non-torsion element of Pic(G).
Next we treat the case in which char(k) = p > 0. By [3, VIIA, Prop. 8.3], there is a normal
infinitesimal k-subgroup scheme I ⊴G such that H :=G/I is smooth (and necessarily connected,
because G is). We claim that H is not affine. Indeed, π : G ↠ H has affine kernel I . The I -torsor
over H defined by π becomes trivial under pullback along the faithfully flat quasi-compact map
π : G → H . Therefore, after a faithfully flat quasi-compact base change, π becomes an affine
morphism. It follows that π is itself affine [4, § 2.7.1(xiii)]. Therefore, if H were affine, then G
would also be, in violation of our assumption.
Zev Rosengarten 563
By Chevalley’s Theorem, Hkperf can be written as an extension of an abelian variety by a smooth
connected affine kperf-group scheme. It follows that there is such an extension over k
1/pn for
some integer n ≥ 0. Extending scalars via the pnth-power isomorphism k1/pn ∼−→ k, we obtain an
exact sequence
1 −→ L −→ H(pn) −→ A −→ 1
with L a smooth connected affine k-group and A/k a nonzero abelian variety (because H , hence
also H (p
n ), is not affine). Denote by q the composite map
G → H F
(pn )
H/k−−−−→ H(pn) → A,
where the map in middle is the n-fold relative Frobenius map. Let L be an ample line bundle on
A. We claim that the pullback q∗(L ) ∈ Pic(G) is not torsion.
It suffices to check this after extending scalars to k, hence we are now free to assume that k = k.
Then Gred ⊂ G is a smooth connected k-subgroup scheme, hence by Chevalley there is an exact
sequence
1 −→ M −→Gred g−→ B −→ 1
with M a smooth connected affine k-group and B/k an abelian variety. The composition
Gred ⊂G → H
F (
pn )
H/k−−−−→ H(pn)
is an isogeny: each map in this sequence is set-theoretically surjective, and Gred and H
(pn ) are
smooth, so the map is fppf, and its kernel is finite because that is true of each map in the above
composition. Lemma 7 therefore implies that we have a unique commutative diagram with exact
rows
1 M Gred B 1
1 L H(p
n) A 1,
g
f
and that f is an isogeny. In particular, f ∗(L ) is still ample, hence by Lemma 6, g∗ f ∗(L ) ∈
Pic(Gred) is not torsion. The commutative diagram
Gred G
B A
g q
f
then shows that q∗(L ) is not torsion either. □
Acknowledgements
It is a pleasure to thank Nguyễn Quốc Th´̆ang – for pointing me to [5], and in particular Theo-
rem 1.13 of that paper – and Christophe Cornut, whose question inspired this paper.
References
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564 Zev Rosengarten
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Picard Groups of Algebraic Groups and an Affineness Criterion

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