On the injectivity and non-injectivity of the l-adic cycle class maps
Bruno Kahn
Conférence inaugurale du Laboratoire de Mathématiques franco-japonais
KomabaApril 5, 2024

1. C HOW GROUPS
X algebraic variety over a field kn ≥ 0: Zn(X) free abelian group on closed integral subvarieties of X
Too coarse, e.g. for intersection theory
Solution: impose rational equivalence (≡ homotopy)
Get Chow group CHn(X).
X non singular (= smooth): can number by codimension 7→ CH n(X).
e.g. CH 1(X) = Pic(X).

2. C YCLE CLASS MAPSTo understand algebraic cycles, use cycle class maps
Most classical: k = C, singular cohomology:CH n(X) → H 2n(X(C), Z)
Here: use étale cohomology
SGA 4 1/2: l prime number invertible in k, ν ≥ 12n(X, µ⊗n)clnν : CH n(X)/lν → Hétlν

Surjectivity? ↔ Hodge conjecture (over C), Tate conjecture (over finitelygenerated k).

To express the Tate conjecture, pass from k to k̄ (algebraic closure) and2n(X, µ⊗n) byreplace Hétlν
2n(X̄, µ⊗n)
HH 2n(X̄, Zl (n) := ←limlν
− étν

finitely generated Zl -module.
Over k, can do the same, but get an awkward theory: refined by U. Jannsen
2n (X, Z (n)): Milnor exact sequences
to continuous étale cohomology Hcontl
1 H 2n−1(X̄, µ⊗n) → H 2n (X, Z (n)) → lim H 2n(X̄, µ⊗n) → 00 → lim
lcontlνlν
←
− ét
←
− ét

and Jannsen’s l-adic cycle class map
2n (X, Z (n)).
clnX : CH n(X) ⊗ Zl → Hcontl3. T HE IMAGE OF clnX
Theorem 1 (S. Saito, essentially). X smooth over k finitely generated. For
any n ≥ 0, the image of clnX is a finitely generated Zl -module.

4. T HE KERNEL OF clnX
Question 1. When is clnX injective?
Completely false in general: e.g. k = k̄, n = 1, X smooth projective curve:
0 → Pic0(X) → CH 1(X) → NS(X) → 0 but Pic0(X) = J(X)(k) is
divisible hence killed by cl1X .
Theorem 2 (Jannsen). If k is finitely generated, cl1X is injective.
(N.B. Pic(X) f.g. abelian group in this case.)
Important conjecture on Chow groups: Bloch-Beilinson–Murre (BBM): forX smooth projective, ∃ filtration on CH n(X) ⊗ Q with very good properties.
Theorem 3 (Jannsen). If clnX ⊗ Q is injective for X smooth projective overfinitely generated k, the BBM conjecture is true.

People have studied the injectivity of clnX restricted to CH n(X)tors: S. Saito(positive cases), Scavia-Suzuki, Alexandrou-Schreieder, Colliot-ThélèneScavia (negative cases). But what about ⊗Q?
Question 2. Is the converse to Jannsen’s theorem 3 true?
I don’t know!
Question 3. Are there known conjectures which would imply the injectivityof clnX ⊗ Q for X smooth projective over f.g. k’s?
Theorem 4. Yes in positive characteristic.
(I don’t know in characteristic 0.)

5. W ITH Q COEFFICIENTS
k finite field, X smooth projective: Tate-Beilinson conjecture
Conjecture 1. clnX ⊗ Q is bijective for all n ≥ 0.
Theorem 5. Conjecture 1 implies that clnX ⊗ Q is injective for any smoothprojective X over any k finitely generated over Fp (p ̸= l).
Sketch. Extend Conjecture 1 to a statement on general smooth Fq -varieties;
if k = Fq (U ) with U smooth small enough, spread X to smooth projective morphism X → U , apply Deligne’s degeneration criterion for Leray
spectral sequence and pass to the limit.
□In characteristic 0 I don’t know any similar conjecture implying the injectivity of cl∗X ⊗ Q.

6.1.

6. R ESTRICTION TO TORSION
Positive results: decomposition of the diagonal.

Theorem 6. k finitely generated of char. 0, X smooth projective. Assumedeg ⊗Qthat CH0(Xk(X)) ⊗ Q −−−−−→ Q is an isomorphism. Let δ = | Coker deg |.
Then δ Ker cl2X = 0, and Ker cl2X is finite if k is of Kronecker dimension ≤ 2.
If char k = p > 0, this is true up to p-primary groups of finite exponent.
(Kronecker dimension: in characteristic p, transcendence degree over Fp;
in characteristic 0, transcendence degree over Q +1.)
Sketch. All is due to S. Saito (in char. 0), except the case of Kronecker dimension 2. In this case, use finite generation of CH0 of arithmetic surfaces(Bloch, Kato-Saito). (Tricky!)□6.2. Counterexamples. Of two types: 1) for CH 2; for CH n, n > 2.
To understand these examples, refine cycle class map using étale Chowgroups:nαXnβXn (X) ⊗ Z −−→ H 2n (X, Z (n))
CH n(X) ⊗ Zl −−→ CHétll
contn (X) = H 2n(X, Z(n)) is étale motivic cohomology (cln =where CHét
XétnnβX ◦ αX ).

For n = 2, short exact sequence
2αX

2 (X) → H 3 (X, Q/Z(2)) → 00 → CH 2(X) −−→ CHétnrn not injective for n > 2 in general. In fact:but αX

Theorem 7. In all examples of Scavia-Suzuki and Alexandrou-Schreiedern.
with n > 2, the cycle in Ker clnX is already killed by αX
Trivialises their proofs, e.g. no need of refined Bloch maps in AlexandrouSchreieder.

For n = 2, two examples: Scavia-Suzuki and Colliot-Thélène-Scavia. First
one uses a norm variety (used in proof of Bloch-Kato conjecture) and corresponding Rost motive R, with CH 2(R) = Z/l. One way to understand itisProposition 1. a) The canonical map
2 (k) ⊗ Z → CH 2 (R)CHét(l)étis an isomorphism.
2 = 0.
b) βRa) is basically because R becomes trivial over k̄. In view of a), the reason
4 (k, Z (2)) = 0 because k is chosen with cohomologicalfor b) is that Hcont
ldimension 3!
Second counterexample much more delicate to “explain” in this way...

T he End