Annales de l’institut Fourier (AIF)

Tange, Rudolf

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L’INSTITUT FOURIER
Université Grenoble Alpes
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Rudolf Tange
Corrigendum: On the rst restricted cohomology of a
reductive Lie algebra and its Borel subalgebras
Tome 70, no 1 (2020), p. 169-170.
<http://aif.centre-mersenne.org/item/AIF_2020__70_1_169_0>
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Ann. Inst. Fourier, Grenoble
70, 1 (2020) 169-170
CORRIGENDUM: ON THE FIRST RESTRICTED
COHOMOLOGY OF A REDUCTIVE LIE ALGEBRA
AND ITS BOREL SUBALGEBRAS
Annales de l’institut Fourier, vol. 69 (2019), n°3, 1295–1308
by Rudolf TANGE
Abstract. — We correct a mistake in the proof of the results for the cohomol-
ogy of the coordinate rings of the groups G and B.
Résumé. — Nous corrigeons une erreur dans la preuve des résultats pour la
cohomologie des anneaux de coordonnées des groupes G et B.
All references below to results where the source isn’t mentioned are to [2].
(1). — The second reference to Lemma 1.2(2) in the proof of Lem-
ma 1.3(1) should be to Lemma 1.2(1).
(2). — The “local version of Lemma 1.3” mentioned after the proof
of Lemma 1.3 and in the proof of Theorem 3.1 would need the ordinary
Nakayama Lemma which only holds for finitely generated modules, so we
cannot use this local version in the proof of Theorem 3.1. Instead we can
use the following version of Lemma 1.3.
Lemma. — Let S be a Noetherian commutative ring, let R be a Noe-
therian subring of S which is a domain such that R ∩ n ∈ Maxspec(R) for
all n ∈ Maxspec(S), and let N be a finitely generated S-module which is
flat over R.
(i) Let M be an S-submodule of N with (mN) ∩ M ⊆ mM for all
m ∈ Maxspec(R). Then N/M is a torsion-free R-module.
(ii) Let M be an S-module, let ϕ : M → N be an S-linear map. Assume
that for all m ∈ Maxspec(R) the canonical map M → M/mM
maps Ker(ϕ) onto the kernel of the induced R/m-linear map ϕ :
M/mM → N/mN . Then N/ Im(ϕ) is a torsion-free R-module.
Keywords: Cohomology, Frobenius kernel, reductive group.
2020 Mathematics Subject Classification: 20G05, 20G10.
170 Rudolf TANGE
Proof.
(i). — It is enough to show that (N/M)n = Nn/Mn is Rm-torsion-free,
where m = n ∩ R, for all n ∈ Maxspec(S). Now Nn is flat over R by the
associativity of the tensor product, and since it is an Rm-module, it is also
flat over Rm. Furthermore, (mmNn) ∩Mn ⊆ mmMn. So we may assume
that (R,m) is a local Noetherian domain, and (S, n) a local Noetherian
R-algebra with mS ⊆ n.
The assumption on M means that the R/m-module map M/mM →
N/mN is injective. Since TorR1 (R/m, N) = 0 we get that TorR1 (R/m,
N/M) = 0. By the local criterion for flatness (see e.g. [1, Thm. 6.8]) this
means that N/M is a flat R-module. So it is torsion-free by [1, Cor. 6.3].
(ii). — Follows from (i) as in the proof of Lemma 1.3. 
Now we can correct the proof of Theorem 3.1 as follows.
Let R = k[G]G be as in that proof. To show that H1(G1, k[G]) is R-
torsion-free it suffices to show that the cokernel of ϕ = f 7→ (x 7→ x · f) :
k[G] → Homk(g, k[G]) is R-torsion-free. This follows by applying the sec-
ond assertion of the above lemma with S = k[G]g. The point is that
k[G]g contains (R and) the p-th powers, so Homk(g, k[G]) is a finitely
generated k[G]g-module. The proofs of Theorem 3.2 and 4.2(2) are cor-
rected similarly. We take (R, S) =
(
k[B]B , k[B]b
)
for Theorem 3.2 and
(R, S) =
(
k[G]G, k[G]Gr
)
and (R, S) =
(
k[B]B , k[B]Br
)
for the two state-
ments in Theorem 4.2(2).
BIBLIOGRAPHY
[1] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad-
uate Texts in Mathematics, vol. 150, Springer, 1995, xvi+785 pages.
[2] R. Tange, “On the first restricted cohomology of a reductive Lie algebra and its
Borel subalgebras”, Ann. Inst. Fourier 69 (2019), no. 3, p. 1295-1308.
Manuscrit reçu le 4 octobre 2019,
révisé le 20 novembre 2019,
accepté le 20 décembre 2019.
Rudolf TANGE
University of Leeds
School of Mathematics
LS2 9JT, Leeds (UK)
R.H.Tange@leeds.ac.uk
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