# Steenrod operations and algebraic classes

Olivier Benoist  Département de mathématiques et applications, École normale
supérieure, CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France
<EMAIL_ADDRESS>
###### Abstract.

Based on a relative Wu theorem in étale cohomology, we study the compatibility
of Steenrod operations on Chow groups and on étale cohomology. Using the
resulting obstructions to algebraicity, we construct new examples of non-
algebraic cohomology classes over various fields ($\mathbb{C}$, $\mathbb{R}$,
$\overline{\mathbb{F}}_{p}$, $\mathbb{F}_{q}$).

We also use Steenrod operations to study the mod $2$ cohomology classes of a
compact $\mathcal{C}^{\infty}$ manifold $M$ that are algebraizable, _i.e._
algebraic on some real algebraic model of $M$. We give new examples of
algebraizable and non-algebraizable classes, answering questions of Benedetti,
Dedò and Kucharz.

##  1\. Introduction

The theme of this article is the interaction between Steenrod operations and
algebraic cohomology classes of smooth projective varieties.

###  1.1. Action of Steenrod operations on algebraic classes

Our starting point is a theorem of Kawai [Kaw77] according to which mod $\ell$
Steenrod operations preserve algebraic cohomology classes with
$\mathbb{Z}/\ell$ coefficients on smooth projective complex varieties (see
also [Ful98, Example 19.1.8]). This implies in particular that mod $\ell$
algebraic classes on smooth projective complex varieties are killed by odd
degree Steenrod operations, and hence provides topological obstructions to the
algebraicity of such classes. One recovers in this way the first
counterexamples to the integral Hodge conjecture, constructed by Atiyah and
Hirzebruch [AH62].

These arguments were adapted by Colliot-Thélène and Szamuely with help from
Totaro [CTS10, Theorem 2.1] to construct counterexamples to the integral Tate
conjecture over $\overline{\mathbb{F}}_{p}$ (whose statement is recalled in
§1.4). We refer to [Qui11, PY15, Kam15, Ant16] for further examples. In
particular, the vanishing of odd degree Steenrod operations on mod $\ell$
algebraic classes over algebraically closed fields of characteristic
$\neq\ell$, proven in a particular case in [CTS10, Theorem 2.1 (1)], follows
in general from Quick’s work [Qui11, Theorem 1.1]. Our first goal is to extend
this result to arbitrary, not necessarily algebraically closed, fields.

To do so, we compare Brosnan’s Steenrod operations on Chow groups ([Bro03],
see also §3.1) and the Steenrod operations in (twisted) étale cohomology (see
[GW19] and §§2.1–2.2) by means of the mod $\ell$ étale cycle class map
$\mathrm{cl}$.

######  Theorem 1.1 (Theorem 3.4).

Let $X$ be a smooth quasi-projective variety over a field $k$. Let $\ell$ be a
prime number invertible in $k$. For $x\in\mathrm{CH}^{c}(X)/\ell$, one has

$\displaystyle\mathrm{Sq}(\mathrm{cl}(x))$ $\displaystyle=$
$\displaystyle\sum_{i\geq
0}(1+\omega)^{c-i}\cdot\mathrm{cl}(\mathrm{Sq}^{2i}(x))$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell=2,$ $\displaystyle
P(\mathrm{cl}(x))$ $\displaystyle=$ $\displaystyle\mathrm{cl}(P(x))$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell\textrm{ is odd.}\ $

In this theorem, $\mathrm{Sq}$ is the total Steenrod square (when $\ell=2$)
and $P$ is the total Steenrod $\ell$-th power (when $\ell$ is odd). When
$\ell=2$, the cohomology class $\omega$ is the class of $-1$ in
$k^{*}/(k^{*})^{2}\simeq
H^{1}_{\mathrm{\acute{e}t}}(\mathrm{Spec}(k),\mathbb{Z}/2)$. See also Remark
2.1 (ii).

It follows from Theorem 1.1 that algebraic cohomology classes are preserved by
all Steenrod operations as soon as $k$ contains a primitive $\ell^{2}$-th root
of unity, and by Steenrod’s $\ell$-th powers when $\ell$ is odd and $k$ is
arbitrary (see Proposition 3.7). It is however not true that Steenrod squares
always preserve algebraic classes (_e.g._ if $\ell=2$ and $k=\mathbb{R}$, see
Remark 3.8), and one can view Theorem 1.1 as providing the correct substitute
for this incorrect statement.

Theorem 1.1 is proved by comparing two relative Wu theorems (describing the
behaviour of Steenrod operations under proper pushforwards): one for Chow
groups and another one in étale cohomology. The former is due to Brosnan (see
Proposition 3.1) and the latter, due to Scavia and Suzuki [SS22, Proposition
A.1] over algebraically closed fields when $\ell=2$, is new in general. The
characteristic classes $w^{\mathrm{\acute{e}t}}$ appearing in its statement
are defined in §2.4.

######  Theorem 1.2 (Theorem 2.5).

Let $f:Y\to X$ be a proper morphism of regular Noetherian
$\mathbb{Z}[1/\ell]$-schemes admitting ample invertible sheaves, with virtual
normal bundle $N_{f}$. For $x\in
H_{\mathrm{\acute{e}t}}^{q}(Y,\boldsymbol{\mu}_{\ell}^{\otimes r})$, one has

$\displaystyle\mathrm{Sq}(f_{*}x)$ $\displaystyle=$ $\displaystyle
f_{*}(\mathrm{Sq}(x)\cdot w^{\mathrm{\acute{e}t}}(N_{f}))$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell=2,$ $\displaystyle P(f_{*}x)$
$\displaystyle=$ $\displaystyle f_{*}(P(x)\cdot
w^{\mathrm{\acute{e}t}}(N_{f}))$ $\displaystyle\textrm{ \hskip 5.0ptif
}\ell\textrm{ is odd.}\ $

Section 2 is devoted to the proof of Theorem 1.2. We rely on or take
inspiration from works of Urabe [Ura96], Guillou and Weibel [GW19], Feng
[Fen20] and Scavia and Suzuki [SS22].

###  1.2. Examples of non-algebraic cohomology classes

Our second goal, following the above-mentioned counterexamples to the integral
Hodge and Tate conjectures, is to exploit the obstructions to algebraicity
stemming from Theorem 1.1 to give interesting new examples of non-algebraic
cohomology classes.

#### 1.2.1.

We first apply classical topological obstructions to algebraicity going back
to Atiyah and Hirzebruch [AH62] (the vanishing of odd degree Steenrod
operations) to a variety constructed in [BO21, §5.1]. This yields a
$5$-dimensional counterexample to the integral Tate conjecture over
$\overline{\mathbb{F}}_{p}$ (see Remark 4.3 for its Hodge counterpart).

######  Theorem 1.3 (Theorem 4.2).

Let $p$ be an odd prime number. There exists a smooth projective variety of
dimension $5$ over $\overline{\mathbb{F}}_{p}$ on which the $2$-adic integral
Tate conjecture for codimension $2$ cycles fails.

The counterexamples to the integral Tate conjecture over
$\overline{\mathbb{F}}_{p}$ that have appeared so far in the literature all
have dimension $\geq 7$. By [Tat68, (5.10)] and [Sch98, Theorem 0.5], it would
follow from the (rational) Tate conjecture over $\overline{\mathbb{F}}_{p}$
that no $3$-dimensional counterexample exists. We do not know any
$4$-dimensional counterexample. In fact, we do not know any counterexample to
the integral Tate conjecture for $2$-cycles over $\overline{\mathbb{F}}_{p}$.

#### 1.2.2.

We then notice that the stability of algebraic classes by Steenrod operations
(over, say, algebraically closed fields) yields obstructions to algebraicity
that go beyond the vanishing of odd degree Steenrod operations. For instance,
over the complex numbers, it implies that even degree Steenrod operations send
mod $\ell$ algebraic classes to reductions mod $\ell$ of integral Hodge
classes (and, over $\overline{\mathbb{F}}_{p}$, to reductions mod $\ell$ of
integral Tate classes). This really may obstruct the integral Hodge and Tate
conjectures, because there are no reasons why reductions mod $\ell$ of
integral Hodge classes (or of integral Tate classes) should be preserved by
even degree Steenrod operations. We illustrate these new Hodge-theoretic (or
Galois-theoretic) obstructions to the integral Hodge and Tate conjectures by
using them to prove the following theorem (see Remark 4.10 for its Hodge
counterpart).

######  Theorem 1.4 (Theorem 4.9).

Choose $\ell\in\\{2,3,5\\}$ and let $p$ be a prime number distinct from
$\ell$. There exists a smooth projective variety $X$ of dimension $2\ell+2$
over $\overline{\mathbb{F}}_{p}$ with
$H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ torsion-free on which the
$\ell$-adic integral Tate conjecture for codimension $2$ cycles fails.

Theorem 1.4 gives the first counterexample to the integral Tate conjecture
over $\overline{\mathbb{F}}_{p}$ with a torsion-free cohomology ring. Indeed,
the topological obstructions used in the counterexamples appearing in the
literature force their $\ell$-adic cohomology rings to contain non-zero
torsion classes (although the non-algebraic integral Tate classes of [PY15]
live in torsion-free $\ell$-adic cohomology groups). We do not know how to
prove an analogue of Theorem 1.4 for other values of $\ell$.

Our proof builds upon [PY15]. In this article, Pirutka and Yagita use
algebraic approximations of classifying spaces of exceptional reductive groups
to construct interesting $(2\ell+3)$-dimensional counterexamples to the
$\ell$-adic integral Tate conjecture over $\overline{\mathbb{F}}_{p}$ by the
Atiyah–Hirzebruch method, for $\ell\in\\{2,3,5\\}$. We notice that a well-
chosen hypersurface in (a blow-up of) their variety has a torsion-free
cohomology ring, but may still fail the integral Tate conjecture by the
obstruction described above.

In [Tot97], Totaro explained how to better understand and generalize the
Atiyah–Hirzebruch topological obstructions to the integral Hodge conjecture
using complex cobordism. In §3.4, we show that our Hodge-theoretic enhancement
of these obstructions also has such an interpretation, in terms of the Hodge
classes in complex cobordism introduced by Hopkins and Quick [HQ15].

#### 1.2.3.

Finally, we give examples of non-algebraic cohomology classes on varieties
defined over non-closed fields, that become algebraic (and even vanish) over
the algebraic closure of the base field. This is particularly interesting
because their non-algebraicity is then a purely arithmetic phenomenon.

Over finite fields, the first examples of such classes were constructed very
recently by Scavia and Suzuki [SS22, Theorem 1.3]. As they relied on
obstructions to algebraicity induced by non-vanishing odd degree Steenrod
operations, they had to restrict to finite fields containing a primitive
$\ell^{2}$-th root of unity. Based on the subtler obstructions to algebraicity
stemming from Theorem 1.1 (see Proposition 3.12), we are able to remove this
hypothesis.

######  Theorem 1.5 (Theorem 4.12).

Let $p\neq\ell$ be prime numbers, and let $\mathbb{F}$ be a finite subfield of
$\overline{\mathbb{F}}_{p}$. There exist a smooth projective geometrically
connected variety $X$ of dimension $2\ell+3$ over $\mathbb{F}$ and a non-
algebraic class

$x\in\mathrm{Ker}\big{(}H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(2))\to
H^{4}_{\mathrm{\acute{e}t}}(X_{\overline{\mathbb{F}}_{p}},\mathbb{Z}_{\ell}(2))\big{)}.$

Scavia and Suzuki’s examples are algebraic approximations of classifying
spaces of semisimple algebraic groups. To be able to encompass all finite
fields, we instead resort to algebraic approximations of classifying spaces of
(not necessarily constant) finite étale group schemes.

Over the field $\mathbb{R}$ of real numbers, examples of non-algebraic classes
that are geometrically algebraic are harder to construct on real varieties
with no real points (as there are then no obstructions to algebraicity induced
by the topology of the set of real points). The example of lowest dimension
appearing in the literature is the $7$-dimensional anisotropic quadric [BW20a,
Example 2.5]. Another application of Proposition 3.12 allows us to produce a
$4$-dimensional example.

######  Theorem 1.6 (Theorem 4.16).

There exists a smooth projective variety $X$ of dimension $4$ over
$\mathbb{R}$ such that $X(\mathbb{R})=\varnothing$, and a non-algebraic class

$x\in\mathrm{Ker}\big{(}H^{4}_{\mathrm{Gal}(\mathbb{C}/\mathbb{R})}(X(\mathbb{C}),\mathbb{Z}(2))\to
H^{4}(X(\mathbb{C}),\mathbb{Z}(2))\big{)}.$

Theorem 1.6 gives a new counterexample to the real integral Hodge conjecture
introduced in [BW20a, Definition 2.2], which is not explained by a failure of
the usual (complex) integral Hodge conjecture (see Corollary 4.17). We refer
to Remarks 4.18 for a comparison with previously known counterexamples to the
real integral Hodge conjecture.

###  1.3. Algebraizability of cohomology classes of $\mathcal{C}^{\infty}$
manifolds

In the last section of this article, we consider questions specific to real
algebraic geometry, that are thematically related but logically independent of
the above.

Let $X$ be a smooth variety over $\mathbb{R}$. Borel and Haefliger ([BH61],
see also [BW20a, §1.6.2]) have constructed a cycle class map

$\mathrm{cl}_{\mathbb{R}}:\mathrm{CH}^{c}(X)\to
H^{c}(X(\mathbb{R}),\mathbb{Z}/2),$

associating with a codimension $c$ integral subvariety $Z\subset X$ the
fundamental class of its real locus. Computing or controlling the image
$H^{*}_{\mathrm{alg}}(X(\mathbb{R}),\mathbb{Z}/2)$ of
$\mathrm{cl}_{\mathbb{R}}$ is a central topic in real algebraic geometry (see
_e.g._ [BCR98, §11.3] or [BK98]).

Let $M$ be a compact $\mathcal{C}^{\infty}$ manifold of dimension $d$. The
Nash–Tognoli theorem [Tog73, Corollario p. 176] asserts that there exists a
smooth projective variety $X$ of dimension $d$ over $\mathbb{R}$ and a
diffeomorphism $\chi:M\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}X(\mathbb{R})$.
We call such a pair $(X,\chi)$ an algebraic model of $M$. A cohomology class
$x\in H^{*}(M,\mathbb{Z}/2)$ is said to be algebraizable if it belongs to
$\chi^{*}H^{*}_{\mathrm{alg}}(X(\mathbb{R}),\mathbb{Z}/2)$ for some algebraic
model $(X,\chi)$ of $M$. The problem of determining which cohomology classes
are algebraizable, a cohomological strengthening of the Nash–Tognoli theorem,
is still largely open.

It is known since Benedetti and Dedò [BD84, Theorem 1] that there exist non-
algebraizable cohomology classes. Teichner [Tei95, Theorem 1] has found
examples in dimension $6$ (the smallest dimension possible), and Kucharz
[Kuc08, Theorem 1.13] has discovered examples of all even degrees $\geq 2$. It
remained an open problem, raised by Benedetti and Dedò [BD84, p. 150] and
Kucharz [Kuc05, p. 194], to determine the possible degrees of non-
algebraizable classes, and in particular to construct non-algebraizable
classes of odd degree. We solve this problem entirely.

######  Theorem 1.7 (Theorem 5.3).

For all $c\geq 2$, there exists a compact $\mathcal{C}^{\infty}$ manifold $M$
and a class $x\in H^{c}(M,\mathbb{Z}/2)$ that is not algebraizable.

Examples of algebraizable cohomology classes include Stiefel–Whitney classes
of topological real vector bundles on $M$ and Poincaré-duals of fundamental
classes of $\mathcal{C}^{\infty}$ submanifolds of $M$ (see [BT80, Theorem 4.2
and Corollary 4.5] or [AK81, Theorem 2.10]). Following Kucharz [Kuc05], let us
denote by $A(M)\subset H^{*}(M,\mathbb{Z}/2)$ the subring generated by these
classes. It was asked by Benedetti and Dedò [BD84, Conjecture 2 p. 150] and
again by Kucharz in [Kuc05, Conjecture A] whether all algebraizable classes
belong to $A(M)$. We give a negative answer to this conjecture.

######  Theorem 1.8 (Theorem 5.6).

There exists a compact $\mathcal{C}^{\infty}$ manifold $M$ and an
algebraizable class $x\in H^{5}(M,\mathbb{Z}/2)$ that does not belong to
$A(M)$.

The proofs of Theorems 1.7 and 1.8 rely on the stability of algebraizable
classes by mod $2$ Steenrod operations, a result due to Akbulut and King
([AK85, Theorem 6.6], see also Corollary 5.2). To prove Theorem 1.7, we
construct cohomology classes whose images by a Steenrod operation can be shown
to be non-algebraizable by the method of Kucharz [Kuc08], hence are themselves
not algebraizable. As for the proof of Theorem 1.8, it relies on the
observation that the subring $A(M)$ of $H^{*}(M,\mathbb{Z}/2)$ is in general
not stable under the action of Steenrod operations.

###  1.4. Conventions

A variety over a field $k$ is a separated $k$-scheme of finite type.

Let $X$ be a smooth projective variety over the algebraic closure $k$ of a
finitely generated field $k_{0}$. A class in
$H^{2c}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(c))$ is said to be a Tate
class if it belongs to the subgroup
$H^{2c}_{\mathrm{\acute{e}t}}(X_{1}\times_{k_{1}}k,\mathbb{Z}_{\ell}(c))^{G_{k_{1}}}$
of $H^{2c}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(c))$ associated with some
model $X_{1}$ of $X$ over a finite extension $k_{1}$ of $k_{0}$ with absolute
Galois group $G_{k_{1}}$. The classes that lie in the image of the cycle class
map $\mathrm{CH}^{c}(X)\to
H^{2c}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(c))$ are said to be
algebraic, and are automatically Tate classes. In this article, we call the
integral Tate conjecture over $k$ (for codimension $c$ cycles on $X$) the
converse statement that all Tate classes are algebraic (see [CTS20, §1] for a
discussion of variants of this statement).

###  1.5. Acknowledgements

I thank Jean-Louis Colliot-Thélène, Burt Totaro and Olivier Wittenberg for
helpful comments on a previous version of this article.

##  2\. A relative Wu theorem in étale cohomology

In topology, the behaviour of Steenrod operations with respect to proper
pushforwards is controlled by a Grothendieck–Riemann–Roch-type theorem: the
relative Wu theorem of Atiyah and Hirzebruch [AH61, Satz 3.2]. In this
section, we prove an analogue of this theorem in the étale cohomology of
schemes.

We first recall, in §2.1 and §2.2, a construction of Steenrod operations on
the (twisted) étale cohomology of schemes. The characteristic classes
appearing in the statement of the relative Wu theorem are defined and studied
in §2.4, and the relative Wu theorem is proved in §2.5.

In this whole section we fix a prime number $\ell$ and we set
$S_{\ell}:=\mathrm{Spec}(\mathbb{Z}[\frac{1}{\ell}])$.

###  2.1. Steenrod operations in étale cohomology

Let $\mathcal{A}_{\ell}$ be the mod $\ell$ Steenrod algebra [Ste62, I §3, VI
§2]. It is a graded $\mathbb{Z}/{\ell}$-algebra generated by the Steenrod
squares $(\mathrm{Sq}^{i})_{i\geq 1}$ which have degree $i$ (when $\ell=2$),
or by Steenrod’s $\ell$-th powers $(P^{i})_{i\geq 1}$ which have degree
$2i(\ell-1)$ and by the Bockstein $\beta$ which has degree $1$ (when $\ell$ is
odd), subject to the Adem relations. By convention, $\mathrm{Sq}^{0}=P^{0}=1$.
If $\ell=2$, we set $\beta:=\mathrm{Sq}^{1}$. We write
$\mathrm{Sq}:=\sum_{i\geq 0}\mathrm{Sq}^{i}$ and $P:=\sum_{i\geq 0}P^{i}$.

The algebra $\mathcal{A}_{\ell}$ acts functorially on the mod $\ell$
(relative) cohomology of (pairs of) topological spaces [Ste62, VII, VIII].
There are several ways to construct analogous Steenrod operations on the mod
$\ell$ étale cohomology of schemes [Eps66, May70, Jar89, Fen20]. We follow
Feng [Fen20, §3] and Scavia and Suzuki [SS22, §2] in exploiting étale homotopy
theory [Fri82]. This choice will allow us to use Scavia and Suzuki’s analysis
of the behaviour of Steenrod operations in Hochschild–Serre spectral sequences
in §4.4.

Friedlander associates with any Noetherian scheme $X$ a pro-simplicial set
denoted by $\mathrm{\acute{E}t}(X)$ in [Fen20]: its étale topological type
[Fri82, Definition 4.4]. Let $f:Y\hookrightarrow X$ be a closed immersion of
Noetherian schemes with complement open immersion $j:U\to X$. Let
$M(\mathrm{\acute{E}t}(j))$ be the mapping cylinder in the sense of [Fri82, p.
140] of the map
$\mathrm{\acute{E}t}(j):\mathrm{\acute{E}t}(U)\to\mathrm{\acute{E}t}(X)$
induced by $j$. It is a pro-simplicial set endowed with an inclusion
$\mathrm{\acute{E}t}(U)\hookrightarrow M(\mathrm{\acute{E}t}(j))$. For any
abelian group $A$, it follows from [Fri82, Corollary 14.5, Proposition 14.6]
that there are natural isomorphisms

(2.1) $H^{*}_{\mathrm{\acute{e}t},Y}(X,A)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{*}(M(\mathrm{\acute{E}t}(j)),\mathrm{\acute{E}t}(U),A).$

When $A=\mathbb{Z}/\ell$, one can use the geometric realization of simplicial
sets to let $\mathcal{A}_{\ell}$ act on the right-hand side of (2.1). We may
then let $\mathcal{A}_{\ell}$ act on
$H^{*}_{\mathrm{\acute{e}t},Y}(X,\mathbb{Z}/\ell)$ functorially in $(X,Y)$,
using the isomorphism (2.1). By construction, this action has all the
properties of the action of $\mathcal{A}_{\ell}$ on the mod $\ell$ cohomology
of topological spaces (see [Ste62, §I.1 and §VI.1]). We will use them freely
in the sequel.

If $X$ is a complex variety, it follows from [Fri82, Theorem 8.4] that Artin’s
comparison isomorphism
$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/\ell)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{q}(X(\mathbb{C}),\mathbb{Z}/\ell)$
is $\mathcal{A}_{\ell}$-equivariant. If $X$ is a variety over $\mathbb{R}$,
this statement has a $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$-equivariant
analogue: the comparison isomorphism
$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/\ell)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{q}_{\mathrm{Gal}(\mathbb{C}/\mathbb{R})}(X(\mathbb{C}),\mathbb{Z}/\ell)$
of [Sch94, Corollary 15.3.1] is $\mathcal{A}_{\ell}$-equivariant as a
consequence of [Cox79, Theorem 1.1].

###  2.2. Steenrod operations in twisted étale cohomology

Guillou and Weibel have explained in [GW19, §§1–3] how to extend the Steenrod
operations of §2.1 to étale cohomology with twisted coefficients. (They work
with the definitions of Epstein [Eps66] and May [May70]). We have found it
convenient to give a self-contained treatment adapted to our needs. We follow
their arguments closely.

Consider the morphism of schemes $\pi:S^{\prime}_{\ell}\to S_{\ell}$, where
$S^{\prime}_{\ell}:=\mathrm{Spec}(\mathbb{Z}[\frac{1}{\ell},t]/(\frac{t^{\ell}-1}{t-1}))$
and $S_{\ell}=\mathrm{Spec}(\mathbb{Z}[\frac{1}{\ell}])$. It is a Galois
finite étale cover with Galois group $\Gamma:=(\mathbb{Z}/\ell)^{*}$, where
$n\in\Gamma$ sends $t$ to $t^{n}$. Decomposing $\mathbb{Z}/\ell[\Gamma]$ as a
direct sum of one-dimensional $\Gamma$-representations yields an isomorphism

(2.2)
$\pi_{*}\mathbb{Z}/\ell=\bigoplus_{r\in\mathbb{Z}/(\ell-1)}\boldsymbol{\mu}_{\ell}^{\otimes
r}$

of sheaves of algebras on the big étale site of $S_{\ell}$, where
$\boldsymbol{\mu}_{\ell}^{\otimes r}$ is the factor of
$\pi_{*}\mathbb{Z}/\ell$ on which $n\in\Gamma$ acts by multiplication by
$n^{r}$, and where the algebra structure on the right comes from the canonical
isomorphism

(2.3) $\mathbb{Z}/\ell\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\boldsymbol{\mu}_{\ell}^{\otimes\ell-1}$

sending $1$ to $\zeta^{\otimes\ell-1}$ for any primitive $\ell$-th root of
unity $\zeta$.

Let $f:Y\hookrightarrow X$ be a closed immersion of Noetherian
$S_{\ell}$-schemes, and denote by $f^{\prime}:Y^{\prime}\hookrightarrow
X^{\prime}$ its base change by $\pi:S^{\prime}_{\ell}\to S_{\ell}$. As the
projection $\pi_{X}:X^{\prime}\to X$ is finite, one has
$\mathrm{R}^{k}\pi_{X,*}\mathbb{Z}/\ell=0$ for $k>0$. It then follows from
(2.2) that one has a $\Gamma$-equivariant decomposition

(2.4)
$H_{\mathrm{\acute{e}t},Y^{\prime}}^{*}(X^{\prime},\mathbb{Z}/\ell)=H^{*}_{\mathrm{\acute{e}t},Y}(X,\pi_{X,*}\mathbb{Z}/\ell)=\bigoplus_{r\in\mathbb{Z}/(\ell-1)}H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r}),$

where $n\in\Gamma$ acts by multiplication by $n^{r}$ on the $r$-th factor of
the right-hand side.

The action of $\mathcal{A}_{\ell}$ on the left side of (2.4) defined in §2.1
commutes with the $\Gamma$-action by functoriality, and hence preserves the
factors of the right side of (2.4). Let $\mathcal{A}_{\ell}$ act on the
$H^{*}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes r})$ in this
way. For all $\alpha\in\mathcal{A}_{\ell}$, $x\in
H^{q}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes r})$ and
$x^{\prime}\in
H^{q^{\prime}}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes{r^{\prime}}})$,
the standard properties of Steenrod operations imply that $\alpha(x)\in
H^{q+\deg(\alpha)}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})$, that the identities

(2.5) $\displaystyle\mathrm{Sq}^{i}(x)$ $\displaystyle=0$ for $\displaystyle
q<i$ and $\displaystyle\mathrm{Sq}^{i}(x)=x^{2}$ for $\displaystyle q=i$ if
$\displaystyle\ell=2,$ $\displaystyle P^{i}(x)$ $\displaystyle=0$ for
$\displaystyle q<2i$ and $\displaystyle P^{i}(x)=x^{\ell}$ for $\displaystyle
q=2i$ if $\displaystyle\ell\textrm{ is odd,}$

are verified, and that the Cartan formulas

(2.6) $\displaystyle\beta(x^{\prime}\cdot x)$
$\displaystyle=\beta(x^{\prime})\cdot x+(-1$
$\displaystyle)^{q^{\prime}}x^{\prime}\cdot\beta(x),$
$\displaystyle\mathrm{Sq}(x^{\prime}\cdot x)$
$\displaystyle=\mathrm{Sq}(x^{\prime})\cdot\mathrm{Sq}(x)$ if
$\displaystyle\ell=2,$ $\displaystyle P(x^{\prime}\cdot x)$
$\displaystyle=P(x^{\prime})\cdot P(x)$ if $\displaystyle\ell\textrm{ is
odd,}$

hold in $H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r+r^{\prime}})$.

######  Remarks 2.1.

(i) By construction, the action of $\mathcal{A}_{\ell}$ on
$H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes r})$ only
depends on $r\in\mathbb{Z}/(\ell-1)$, if one uses the canonical isomorphism
(2.3) to trivialize $\boldsymbol{\mu}_{\ell}^{\otimes\ell-1}$.

(ii) We will often tacitly identify Tate twists that are congruent modulo
$\ell-1$ in our statements, by means of (2.3). For instance, Theorem 1.1 for
$\ell$ odd equates $P^{i}(\mathrm{cl}(x))\in
H^{2i(\ell-1)+2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
c})$ and $\mathrm{cl}(P^{i}(x))\in
H^{2i(\ell-1)+2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
i(\ell-1)+c})$.

(iii) If an isomorphism $\xi:\mathbb{Z}/\ell\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\mu_{\ell}^{\otimes
r^{\prime}}$ over $X$ is fixed, then the induced isomorphisms
$H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r+r^{\prime}})$ are $\mathcal{A}_{\ell}$-equivariant by (2.6) applied with
$x^{\prime}$ chosen to be the class $\xi\in
H^{0}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes r^{\prime}})$
of the fixed isomorphism.

(iv) As a consequence of (iii), if $X$ is a variety over a field $k$ in which
a primitive $\ell$-th root of unity $\zeta$ has been fixed, one obtains
$\mathcal{A}_{\ell}$-equivariant isomorphisms
$H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
s})$ for all $r$ and $s$ (which depend on the choice of $\zeta$).

###  2.3. The Bockstein

We now proceed to the computation of the Bockstein, following [GW19,
Proposition 3.3]. Let
$\delta_{r}:H^{q}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})\to H^{q+1}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes r})$
be the boundary map of the short exact sequence

(2.7) $0\to\boldsymbol{\mu}_{\ell}^{\otimes
r}\to\boldsymbol{\mu}_{\ell^{2}}^{\otimes
r}\to\boldsymbol{\mu}_{\ell}^{\otimes r}\to 0.$

(In [GW19], such morphisms are called $\beta$, but we reserve this notation
for the Bockstein element of $\mathcal{A}_{\ell}$.) Use a primitive $\ell$-th
root of unity $\zeta\in\mathcal{O}(S_{\ell}^{\prime})$ to identify
$\boldsymbol{\mu}_{\ell}$ and $\mathbb{Z}/\ell$ on $S_{\ell}^{\prime}$. The
short exact sequence (2.7) for $r=1$ then reads

(2.8) $0\to\mathbb{Z}/\ell\to\boldsymbol{\mu}_{\ell^{2}}\to\mathbb{Z}/\ell\to
0,$

and we let $\omega\in
H^{1}_{\mathrm{\acute{e}t}}(S_{\ell}^{\prime},\mathbb{Z}/\ell)$ be the image
of $1\in H^{0}_{\mathrm{\acute{e}t}}(S_{\ell}^{\prime},\mathbb{Z}/\ell)$ by
its boundary map. Since the isomorphism class of the extension (2.8) is
independent of the choice of $\zeta$, the class $\omega$ is
$\Gamma$-invariant. As the degree of $\pi:S_{\ell}^{\prime}\to S_{\ell}$ is
prime to $\ell$, the Hochschild–Serre spectral sequence shows that $\omega$
comes by pull-back from a class that we still denote by $\omega\in
H^{1}_{\mathrm{\acute{e}t}}(S_{\ell},\mathbb{Z}/\ell)$.

We still call $\omega\in H^{1}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/\ell)$ the
pull-back of $\omega$ to any $S_{\ell}$-scheme $X$. When $\ell=2$, the class
$\omega$ is the image of $-1$ by the boundary map of the Kummer short exact
sequence. (The class $\omega$ is called $z$ in [GW19], and $\alpha$ in [Fen20]
when $\ell=2$.)

######  Lemma 2.2.

The Bockstein morphism
${\beta:H^{q}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})\to H^{q+1}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})}$ is equal to $x\mapsto\delta_{r}(x)-r\,\omega\cdot x$.

###### Proof.

As the degree of the finite étale cover $\pi_{X}:X^{\prime}\to X$ is prime to
$\ell$, the pull-back map
$\pi_{X}^{*}:H^{*}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})\to
H^{*}_{\mathrm{\acute{e}t},Y^{\prime}}(X^{\prime},\boldsymbol{\mu}_{\ell}^{\otimes
r})$ is injective. By functoriality, it thus suffices to prove the lemma on
$X^{\prime}$. Choose an isomorphism $\mathbb{Z}/\ell\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\boldsymbol{\mu}_{\ell}$
on $X^{\prime}$. The morphism $\delta_{r}$ is then the boundary map of
$0\to\mathbb{Z}/\ell\to\boldsymbol{\mu}_{\ell^{2}}^{\otimes
r}\to\mathbb{Z}/\ell\to 0$, while the morphism $\beta$ is the boundary map of
$0\to\mathbb{Z}/\ell\to\mathbb{Z}/\ell^{2}\to\mathbb{Z}/\ell\to 0$ (this is
one of the standard properties of Steenrod operations). That
$\delta_{r}(x)=\beta(x)+r\,\omega\cdot x$ now follows from the results of
[Gre65]. (The reference [Gre65] deals with the cohomology of topological
spaces. By means of the mapping cone construction, they are still valid for
the relative cohomology of pairs of topological spaces. One can then apply
them in our setting using (2.1).) ∎

###  2.4. Étale Stiefel–Whitney classes

If $f:Y\hookrightarrow X$ is a closed immersion of codimension $c$ of regular
$S_{\ell}$-schemes, the Gysin morphism
$\mathrm{Cl}_{f}:\mathbb{Z}/\ell\to\mathrm{R}f^{!}\boldsymbol{\mu}_{\ell}^{\otimes
c}[2c]$ defined in [Rio14, Définition 2.3.1] is an isomorphism by Gabber’s
purity theorem [Rio14, Théorème 3.1.1]. We may then consider the Thom
isomorphisms

(2.9) $\phi_{f}:H^{q}_{\mathrm{\acute{e}t}}(Y,\boldsymbol{\mu}_{\ell}^{\otimes
r})\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{q+2c}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r+c})$

induced by $\mathrm{Cl}_{f}$, and we let $s_{f}:=\phi_{f}(1)\in
H^{2c}_{\mathrm{\acute{e}t},Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes c})$ be the
Thom class.

When $X=E$ is a vector bundle on $Y$ with structural morphism $g:E\to Y$ and
$f:Y\hookrightarrow E$ is the zero section, one has

(2.10) $\phi_{f}(x)=g^{*}(x)\cdot s_{f}\textrm{\hskip 5.0pt for all }x\in
H^{q}_{\mathrm{\acute{e}t}}(Y,\boldsymbol{\mu}_{\ell}^{\otimes r}).$

In this setting, we define the étale Stiefel–Whitney classes
$w^{\mathrm{\acute{e}t}}_{j}(E)\in
H^{j}_{\mathrm{\acute{e}t}}(Y,\mathbb{Z}/\ell)$ of $E$ by the formulas

$\displaystyle w^{\mathrm{\acute{e}t}}(E)$ $\displaystyle:=$
$\displaystyle\phi_{f}^{-1}(\mathrm{Sq}(s_{f}))$ $\displaystyle\textrm{ \hskip
5.0ptif }\ell=2,$ $\displaystyle w^{\mathrm{\acute{e}t}}(E)$ $\displaystyle:=$
$\displaystyle\phi_{f}^{-1}(P(s_{f}))$ $\displaystyle\textrm{ \hskip 5.0ptif
}\ell\textrm{ is odd.}$

In view of (2.10), they are characterized by the identities

(2.11) $\mathrm{Sq}(s_{f})=g^{*}w^{\mathrm{\acute{e}t}}(E)\cdot s_{f}\textrm{
if }\ell=2\textrm{\hskip 5.0pt and \hskip
5.0pt}P(s_{f})=g^{*}w^{\mathrm{\acute{e}t}}(E)\cdot s_{f}\textrm{ if
}\ell\textrm{ is odd.}$

When $\ell=2$, this definition appears in [Ura96, p. 569] for varieties over
algebraically closed fields and in [Fen20, Definition 5.2] for varieties over
arbitrary fields. (These classes are denoted by $w_{j}(E)$ in [Ura96, Fen20],
but we want to distinguish them from those defined in [Bro03], see (3.1).)

The following lemma extends [Ura96, Lemma 2.6 (4)] and [Fen20, Lemma 5.7]. It
allows us to define
$w^{\mathrm{\acute{e}t}}(\kappa):=w^{\mathrm{\acute{e}t}}(E)\cdot
w^{\mathrm{\acute{e}t}}(F)^{-1}$ for any class $\kappa=[E]-[F]\in K_{0}(X)$.

######  Lemma 2.3.

If $Y$ is a regular $S_{\ell}$-scheme and if

(2.12) $0\to E_{1}\to E\to E_{2}\to 0$

is a short exact sequence of vector bundles on $Y$, one has

(2.13) $w^{\mathrm{\acute{e}t}}(E)=w^{\mathrm{\acute{e}t}}(E_{1})\cdot
w^{\mathrm{\acute{e}t}}(E_{2}).$

###### Proof.

Let $\sigma_{0},\sigma_{1}:Y\to\mathbb{A}^{1}_{Y}$ be the sections of the
projection $\mathbb{A}^{1}_{Y}\to Y$ with values $0$ and $1$. Arguing as in
[Ura96, Proof of Lemma 2.7], one constructs a short exact sequence of vector
bundles $0\to F_{1}\to F\to F_{2}\to 0$ on $\mathbb{A}^{1}_{Y}$ whose pull-
backs by $\sigma_{0}$ and $\sigma_{1}$ are respectively isomorphic to the
split exact sequence ${0\to E_{1}\to E_{1}\oplus E_{2}\to E_{2}\to 0}$ and to
(2.12). Since the two pull-back morphisms
$\sigma_{0}^{*},\sigma_{1}^{*}:H^{*}_{\mathrm{\acute{e}t}}(\mathbb{A}^{1}_{Y},\mathbb{Z}/\ell)\to
H^{*}_{\mathrm{\acute{e}t}}(Y,\mathbb{Z}/\ell)$ are isomorphisms by [SGA43,
XVI, Corollaire 2.2], one can assume, to prove (2.13), that the exact sequence
(2.12) is split.

In this case, let $f_{1}:E_{1}\to E$ and $f_{2}:E_{2}\to E$ be the inclusions
of the factors, let $g_{1}:E\to E_{1}$ and $g_{2}:E\to E_{2}$ be the
projections and let $h_{1}:Y\to E_{1}$ and ${h_{2}:Y\to E_{2}}$ be the zero
sections. It then follows from [Rio14, Remarque 2.3.6] and [Rio14, Proposition
2.3.2] that $s_{f}=s_{f_{1}}\cdot s_{f_{2}}=g_{1}^{*}s_{h_{1}}\cdot
g_{2}^{*}s_{h_{2}}$. The multiplicativity (2.13) of $w^{\mathrm{\acute{e}t}}$
now follows from the Cartan formulas (2.6). ∎

If $E$ is a vector bundle on an $S_{\ell}$-scheme $Y$, we let
$c^{\mathrm{\acute{e}t}}_{j}(E)\in
H^{2j}_{\mathrm{\acute{e}t}}(Y,\boldsymbol{\mu}_{\ell}^{\otimes j})$ be the
mod $\ell$ Chern classes of $E$ (see [Rio14, Théorème 1.3] for a construction
following [Gro58]). We also let $c^{\mathrm{\acute{e}t}}(E):=\sum_{j\geq
0}c^{\mathrm{\acute{e}t}}_{j}(E)$ be the total mod $\ell$ Chern class of $E$.

The following lemma was proven in [Fen20, Theorem 5.10] when $\ell=2$ and $Y$
is a variety over a finite field. Recall the definition of $\omega$ given in
§2.2.

######  Lemma 2.4.

Let $E$ be a vector bundle on a regular $S_{\ell}$-scheme $Y$. Write formally
$c^{\mathrm{\acute{e}t}}(E)=\prod_{i}(1+\lambda_{i})$, where the $\lambda_{i}$
are the Chern roots of $E$. Then

(2.14) $\displaystyle w^{\mathrm{\acute{e}t}}(E)$ $\displaystyle=$
$\displaystyle\prod_{i}(1+\omega+\lambda_{i})$ $\displaystyle\textrm{ \hskip
5.0ptif }\ell=2,$ (2.15) $\displaystyle w^{\mathrm{\acute{e}t}}(E)$
$\displaystyle=$ $\displaystyle\prod_{i}(1+\lambda_{i}^{\ell-1})$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell\textrm{ is odd.}$

###### Proof.

By Grothendieck’s splitting principle, we may assume that $E$ is a successive
extension of line bundles. By Lemma 2.3, we may further assume that $E$ is a
line bundle. In this case, we let $g:E\to Y$ be the structural morphism and
$f:Y\hookrightarrow E$ be the zero section. The image of the Thom class
$s_{f}$ by the morphism
$H^{2}_{\mathrm{\acute{e}t},Y}(E,\boldsymbol{\mu}_{\ell})\to
H^{2}_{\mathrm{\acute{e}t}}(E,\boldsymbol{\mu}_{\ell})$ forgetting the support
is equal to
$c^{\mathrm{\acute{e}t}}_{1}(\mathcal{O}_{E}(Y))=c^{\mathrm{\acute{e}t}}_{1}(g^{*}E)=g^{*}c^{\mathrm{\acute{e}t}}_{1}(E)$
by [Rio14, §2.1]. As a consequence,

(2.16) $s_{f}^{2}=g^{*}c^{\mathrm{\acute{e}t}}_{1}(E)\cdot s_{f}\textrm{
\hskip 5.0ptin
}H^{4}_{\mathrm{\acute{e}t},Y}(E,\boldsymbol{\mu}_{\ell}^{\otimes 2}).$

If $\ell$ is odd, then $P(s_{f})=s_{f}+s_{f}^{\ell}$ by (2.5), hence
$P(s_{f})=(1+g^{*}c^{\mathrm{\acute{e}t}}_{1}(E)^{\ell-1})\cdot s_{f}$ by
(2.16). Using (2.11), we get
$w^{\mathrm{\acute{e}t}}(E)=1+c^{\mathrm{\acute{e}t}}_{1}(E)^{\ell-1}$, and
(2.15) holds.

If $\ell=2$, one has
$\mathrm{Sq}(s_{f})=s_{f}+\mathrm{Sq}^{1}(s_{f})+s_{f}^{2}$ by (2.5). As
$s_{f}$ lifts to $H^{2}_{\mathrm{\acute{e}t},Y}(E,\boldsymbol{\mu}_{4})$ by
[Rio14, Définition 2.3.1], one has $\mathrm{Sq}^{1}(s_{f})=\omega\cdot s_{f}$
by Lemma 2.2 and (2.7). Applying moreover (2.16) yields
$\mathrm{Sq}(s_{f})=(1+\omega+g^{*}c^{\mathrm{\acute{e}t}}_{1}(E))\cdot
s_{f}$. We deduce from (2.11) that
$w^{\mathrm{\acute{e}t}}(E)=1+\omega+c^{\mathrm{\acute{e}t}}_{1}(E)$, and
hence that (2.14) holds. ∎

###  2.5. A relative Wu theorem

Let $f:Y\to X$ be a morphism of finite type of regular Noetherian
$S_{\ell}$-schemes admitting ample invertible sheaves, which has virtual
relative dimension $-c$ in the sense of [SGA6, VIII, Définition 1.9]. As $f$
may be written as the composition of a regular closed immersion $g:Y\to Z$ and
of a smooth morphism $h:Z\to X$ (one may take $Z=\mathbb{P}^{n}_{X}$ for $n\gg
0$), one can define its virtual normal bundle $N_{f}\in K_{0}(Y)$ to be
$N_{f}:=-(T_{f})^{\vee}$, where $T_{f}\in K_{0}(Y)$ is defined in [SGA6, VIII,
Corollaire 2.5]. The class $N_{f}$ has rank $c$. It is equal to
$[(I/I^{2})^{\vee}]$ if $f$ is a regular closed immersion defined by an ideal
$I\subset\mathcal{O}_{X}$ and to $-[T_{Y/X}]$ if $f$ is smooth.

One defines as in [Rio14, Définition 2.5.11] a morphism
$\mathrm{Cl}_{f}:\mathbb{Z}/\ell\to Rf^{!}\boldsymbol{\mu}_{\ell}^{\otimes
c}[2c]$ which coincides with the one considered in §2.4 when $f$ is a closed
immersion and which is induced by the trace map when $f$ is smooth [Rio14,
Proposition 2.5.13]. As noted in [Rio14, Remarque 2.5.14], if $f$ is moreover
proper, the morphism $\mathrm{Cl}_{f}$ induces pushforward (or Gysin)
morphisms
$f_{*}:H^{q}_{\mathrm{\acute{e}t}}(Y,\boldsymbol{\mu}_{\ell}^{\otimes r})\to
H^{q+2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes r+c})$.

The following theorem is the main result of this section. The case of Steenrod
squares for varieties over algebraically closed fields is due to Scavia and
Suzuki [SS22, Proposition A.1], and we generalize their arguments.

######  Theorem 2.5.

Let $f:Y\to X$ be a proper morphism of regular Noetherian $S_{\ell}$-schemes
admitting ample invertible sheaves. For $x\in
H_{\mathrm{\acute{e}t}}^{q}(Y,\boldsymbol{\mu}_{\ell}^{\otimes r})$, one has

(2.17) $\displaystyle\mathrm{Sq}(f_{*}x)$ $\displaystyle=$ $\displaystyle
f_{*}(\mathrm{Sq}(x)\cdot w^{\mathrm{\acute{e}t}}(N_{f}))$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell=2,$ $\displaystyle P(f_{*}x)$
$\displaystyle=$ $\displaystyle f_{*}(P(x)\cdot
w^{\mathrm{\acute{e}t}}(N_{f}))$ $\displaystyle\textrm{ \hskip 5.0ptif
}\ell\textrm{ is odd.}\ $

###### Proof.

Write $f$ as the composition of a closed immersion
$g:Y\hookrightarrow\mathbb{P}^{n}_{X}$ and of the natural projection
$h:\mathbb{P}^{n}_{X}\to X$. Using the covariant functoriality of Gysin
morphisms (see [Rio14, Théorème 2.5.12]), the multiplicativity of
$w^{\mathrm{\acute{e}t}}$ (see Lemma 2.3), the Cartan formulas (2.6), and the
additivity of $T_{f}$ [SGA6, VIII, Corollaire 2.7], we are reduced to proving
the theorem for $g$ and $h$ separately. We may thus assume that $f$ is either
a closed immersion or a projection $\mathbb{P}^{n}_{X}\to X$.

Assume first that $f$ is a closed immersion of codimension $c$. Let
${f^{\prime}:Y\to N_{Y/X}}$ and $g^{\prime}:N_{Y/X}\to Y$ be the zero section
and the structural morphism of its normal bundle. Let
$\sigma_{0},\sigma_{1}:X\to\mathbb{A}^{1}_{X}$ be the sections of the
projection $\mathbb{A}^{1}_{X}\to X$ with values $0$ and $1$. Let
$\mathfrak{X}:=\mathrm{Bl}_{\sigma_{0}(Y)}\mathbb{A}^{1}_{X}\setminus\mathrm{Bl}_{\sigma_{0}(Y)}\sigma_{0}(X)$
be the deformation of $f$ to its normal cone and
$h:\mathbb{A}^{1}_{Y}\to\mathfrak{X}$ be the strict transform of the natural
closed immersion $\mathbb{A}^{1}_{Y}\to\mathbb{A}^{1}_{X}$. One gets a
cartesian diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{\mathbb{A}^{1}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{N_{Y/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{X}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\hbox
to0.0pt{,\hss}}$

in which the left (resp. right) column is cut out in the middle column by the
equation $t=0$ (resp. $t=1$), where $t$ is the coordinate of the affine line.
The induced diagram

(2.18) $\textstyle{H^{*}(Y,\boldsymbol{\mu}_{\ell}^{\otimes
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\scriptstyle{\phi_{f^{\prime}}}$$\textstyle{H^{*}(\mathbb{A}^{1}_{Y},\boldsymbol{\mu}_{\ell}^{\otimes
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\hskip
7.00002pt\sim}$$\scriptstyle{\wr}$$\scriptstyle{\phi_{h}}$$\textstyle{H^{*}(Y,\boldsymbol{\mu}_{\ell}^{\otimes
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\scriptstyle{\phi_{f}}$$\textstyle{H^{*+2c}_{Y}(N_{Y/X},\boldsymbol{\mu}_{\ell}^{\otimes{r+c}})}$$\textstyle{H^{*+2c}_{\mathbb{A}^{1}_{Y}}(\mathfrak{X},\boldsymbol{\mu}_{\ell}^{\otimes{r+c}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{*+2c}_{Y}(X,\boldsymbol{\mu}_{\ell}^{\otimes{r+c}})\hbox
to0.0pt{,\hss}}$

whose vertical arrows are Thom isomorphisms (2.9) and whose upper horizontal
arrows are isomorphisms by [SGA43, XVI, Corollaire 2.2] is commutative by
[Rio14, Proposition 2.3.2]. When $\ell=2$, one has

$\mathrm{Sq}(\phi_{f^{\prime}}(x))=g^{\prime*}\mathrm{Sq}(x)\cdot\mathrm{Sq}(s_{f^{\prime}})=g^{\prime*}(\mathrm{Sq}(x)\cdot
w^{\mathrm{\acute{e}t}}(N_{f^{\prime}}))\cdot
s_{f^{\prime}}=\phi_{f^{\prime}}(\mathrm{Sq}(x)\cdot
w^{\mathrm{\acute{e}t}}(N_{f^{\prime}}))$

in $H^{*}_{Y}(N_{Y/X},\boldsymbol{\mu}_{2}^{\otimes r+c})$ by (2.6), (2.10)
and (2.11). In view of (2.18) and since $N_{f}$ and $N_{f^{\prime}}$ are both
restrictions of $N_{h}$, we deduce that
$\mathrm{Sq}(\phi_{f}(x))=\phi_{f}(\mathrm{Sq}(x)\cdot
w^{\mathrm{\acute{e}t}}(N_{f}))$ in $H^{*}_{Y}(X,\boldsymbol{\mu}_{2}^{\otimes
r+c})$. Forgetting the support yields (2.17). The argument when $\ell$ is odd
is identical, replacing $2$ by $\ell$ and $\mathrm{Sq}$ by $P$.

Assume now that $f:\mathbb{P}^{n}_{X}\to X$ is the projection. Suppose first
that $\ell=2$. Let
$\lambda:=c^{\mathrm{\acute{e}t}}_{1}(\mathcal{O}_{\mathbb{P}^{n}_{X}}(1))\in
H^{2}_{\mathrm{\acute{e}t}}(\mathbb{P}^{n}_{X},\boldsymbol{\mu}_{2})$. As
$\lambda$ lifts to
$H^{2}_{\mathrm{\acute{e}t}}(\mathbb{P}^{n}_{X},\boldsymbol{\mu}_{4})$ by
[Rio14, Définition 2.3.1], Lemma 2.2 and (2.7) imply that
$\mathrm{Sq}^{1}(\lambda)=\omega\cdot\lambda$. Taking (2.5) into account, we
get

(2.19) $\mathrm{Sq}(\lambda)=\lambda+\omega\cdot\lambda+\lambda^{2}.$

The Euler exact sequence
$0\to\mathcal{O}_{\mathbb{P}^{n}_{X}}\to\mathcal{O}_{\mathbb{P}^{n}_{X}}(1)^{\oplus
N+1}\to T_{\mathbb{P}^{n}_{X}/X}\to 0$ and Lemmas 2.3 and 2.4 imply that
$w^{\mathrm{\acute{e}t}}(T_{\mathbb{P}^{n}_{X}/X})=(1+\omega+\lambda)^{n+1}/(1+\omega)$
hence that

(2.20)
$w^{\mathrm{\acute{e}t}}(N_{f})=w^{\mathrm{\acute{e}t}}(-[T_{\mathbb{P}^{n}_{X}/X}])=(1+\omega)/(1+\omega+\lambda)^{n+1}.$

As
$H^{*}_{\mathrm{\acute{e}t}}(\mathbb{P}^{n}_{X},\boldsymbol{\mu}_{2}^{\otimes*})=H^{*}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes*})[\lambda]/\lambda^{n+1}$
by the projective bundle formula (see for instance [Rio14, §1]), it suffices
to prove the identity (2.17) for $x$ of the form $f^{*}y\cdot\lambda^{m}$,
where $y\in H_{\mathrm{\acute{e}t}}^{*}(X,\boldsymbol{\mu}_{2}^{\otimes*})$
and $m\in\\{0,\dots,n\\}$. By the projection formula, the left-hand side of
(2.17) is equal to $\mathrm{Sq}(y)$ if $m=n$ and vanishes otherwise. Let us
compute the right-hand side of (2.17). One has

$\mathrm{Sq}(x)=\mathrm{Sq}(f^{*}y\cdot\lambda^{m})=f^{*}\mathrm{Sq}(y)\cdot\mathrm{Sq}(\lambda)^{m}=f^{*}\mathrm{Sq}(y)\cdot\lambda^{m}(1+\omega+\lambda)^{m}$

by the Cartan formula (2.6) and (2.19). It follows from (2.20) that

(2.21) $\mathrm{Sq}(x)\cdot
w^{\mathrm{\acute{e}t}}(N_{f})=f^{*}\mathrm{Sq}(y)\cdot\lambda^{m}\eta(\eta+\lambda)^{-k-1}=f^{*}\mathrm{Sq}(y)\cdot\lambda^{m}\eta^{m-n}(1+\frac{\lambda}{\eta})^{-k-1},$

where $\eta:=1+\omega$ and $k:=n-m$. By Lemma 2.6 below applied with
$T=\frac{\lambda}{\eta}$, the coefficient of $\lambda^{n}$ in (2.21) is equal
to $f^{*}\mathrm{Sq}(y)$ if $m=n$ and to $0$ otherwise. It then follows from
the projection formula that the right-hand side of (2.17) is equal to
$\mathrm{Sq}(y)$ if $m=n$ and vanishes otherwise. This concludes the proof
when $\ell=2$.

The argument when $\ell$ is odd is similar, only easier because $\omega$ is
not involved. Equations (2.19) and (2.20) are replaced with the identities
$P(\lambda)=\lambda+\lambda^{\ell}$ and
$w^{\mathrm{\acute{e}t}}(N_{f})=1/(1+\lambda)^{n+1}$. Both the left and the
right side of (2.17) for $x=f^{*}y\cdot\lambda^{m}$ with $y\in
H_{\mathrm{\acute{e}t}}^{q-2m}(\mathbb{P}^{n}_{X},\boldsymbol{\mu}_{\ell}^{\otimes
r-m})$ are then computed to be equal to $P(y)$ if $m=n$ and to vanish
otherwise (using Lemma 2.6 applied with $T=\lambda$). ∎

######  Lemma 2.6.

The coefficient of $T^{k}$ in the power series
$F(T)=(1+T)^{-k-1}\in\mathbb{Z}[[T]]$ is odd if $k=0$ and even if $k\geq 1$.

###### Proof.

Since $F^{(k)}(T)=(-1)^{k}\prod_{i=k+1}^{2k}i\cdot(1+T)^{-2k-1}$, the
coefficient of $T^{k}$ in $F(T)$ is
$\frac{F^{(k)}(0)}{k!}=(-1)^{k}\binom{2k}{k}$. It is equal to $1$ if $k=0$ and
it is even if $k\geq 1$. ∎

######  Remark 2.7.

In topology, the Wu theorem describes the interaction of Steenrod operations
and Poincaré duality. The same holds for Theorem 2.5, where Poincaré duality
appears in the guise of pushforwards. Feng’s Wu theorem over finite fields
[Fen20, Theorem 6.5], however, describes the interaction of Steenrod
operations and a subtler duality that combines Poincaré duality and duality in
Galois cohomology of finite fields. It is therefore not a particular case of
Theorem 2.5.

##  3\. Steenrod operations and algebraic classes

We now apply the relative Wu theorem proved in Section 2 to study the
compatibility of Brosnan’s Steenrod operations on Chow groups (recalled in
§3.1) and those in étale cohomology (defined in §§2.1–2.2). Our main result in
this direction is Theorem 3.4. In §3.3, we explain how to deduce obstructions
to the algebraicity of cohomology classes. We clarify their relation with
complex cobordism in §3.4.

Throughout this section, we fix a field $k$ and a prime number $\ell$
invertible in $k$.

###  3.1. Steenrod operations on Chow groups

Let $X$ be a smooth variety over $k$. Brosnan has defined in [Bro03,
Definition 8.11 and §11] an action of $\mathcal{A}_{\ell}$ on
$\mathrm{CH}^{*}(X)/\ell$ such that all odd degree elements of
$\mathcal{A}_{\ell}$ act by zero, and such that for all
$\alpha\in\mathcal{A}_{\ell}$ of even degree and all
$x\in\mathrm{CH}^{c}(X)/\ell$, one has
$\alpha(x)\in\mathrm{CH}^{c+\deg(\alpha)/2}(X)/\ell$.

Let $E$ be a vector bundle on a smooth variety $X$ over $k$. Let
$c_{j}(E)\in\mathrm{CH}^{j}(X)/\ell$ be the mod $\ell$ Chern classes of $E$
defined as in [Gro58] (see also [Ful98, p. 325]), and let $c(E):=\sum_{j\geq
0}c_{j}(E)$. Write formally $c(E)=\prod_{i}(1+\lambda_{i})$, where the
$\lambda_{i}$ are the Chern roots of $E$. Brosnan introduces in [Bro03, p.
1891] the characteristic classes $w_{2j}(E)\in\mathrm{CH}^{j}(X)/\ell$ defined
by

(3.1) $w(E)=\sum_{j\geq 0}w_{2j}(E):=\prod_{i}(1+\lambda_{i}^{\ell-1}).$

This definition is extended to $K_{0}(X)$ by setting $w([E]-[F]):=w(E)\cdot
w(F)^{-1}$.

Brosnan uses these classes to describe the behaviour of the Steenrod
operations on Chow groups with respect to proper pushforwards.

######  Proposition 3.1.

Let $f:Y\to X$ be a proper morphism between smooth varieties over $k$ with
virtual normal bundle $N_{f}$. For $x\in\mathrm{CH}^{*}(X)/\ell$, one has

$\displaystyle\mathrm{Sq}(f_{*}x)$ $\displaystyle=$ $\displaystyle
f_{*}(\mathrm{Sq}(x)\cdot w(N_{f}))$ $\displaystyle\textrm{ \hskip 5.0ptif
}\ell=2,$ $\displaystyle P(f_{*}x)$ $\displaystyle=$ $\displaystyle
f_{*}(P(x)\cdot w(N_{f}))$ $\displaystyle\textrm{ \hskip 5.0ptif }\ell\textrm{
is odd.}\ $

###### Proof.

This follows from [Bro03, Propositions 9.4 (iii) and 10.3]. ∎

######  Remark 3.2.

An earlier construction of Steenrod operations on Chow groups was given by
Voevodsky in the context of motivic cohomology [Voe03]. We do not know a
published proof of the fact that his operations are compatible with Brosnan’s.

###  3.2. Comparing Steenrod operations

Combining Proposition 3.1 and Theorem 2.5, we may compare the Steenrod
operations on Chow groups and in étale cohomology. To do so, we let
$\mathrm{cl}:\mathrm{CH}^{*}(X)/\ell\to
H^{2*}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes*})$ denote the
mod $\ell$ étale cycle class map of a smooth variety $X$ over $k$.

######  Lemma 3.3.

Let $X$ be a smooth variety over $k$ and let $\kappa\in K_{0}(X)$ be a
$\mathrm{K}$-theory class of rank $c\in\mathbb{Z}$. Then

$\displaystyle w^{\mathrm{\acute{e}t}}(\kappa)$ $\displaystyle=$
$\displaystyle\sum_{j\geq 0}(1+\omega)^{c-j}\cdot\mathrm{cl}(w_{2j}(\kappa))$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell=2,$ $\displaystyle
w^{\mathrm{\acute{e}t}}(\kappa)$ $\displaystyle=$
$\displaystyle\mathrm{cl}(w(\kappa))$ $\displaystyle\textrm{ \hskip 5.0ptif
}\ell\textrm{ is odd.}\ $

###### Proof.

It follows from Lemma 2.3 and the definition (3.1) that both sides of these
equalities are multiplicative in the class $\kappa$. By the splitting
principle, we may thus assume that $\kappa$ is the class of a line bundle $L$.
It then follows from Lemma 2.4 and (3.1) that, if $\ell$ is odd, one has

$w^{\mathrm{\acute{e}t}}(\kappa)=1+c_{1}^{\mathrm{\acute{e}t}}(L)^{\ell-1}=\mathrm{cl}(1+c_{1}(L)^{\ell-1})=\mathrm{cl}(w(\kappa)),$

and that, if $\ell=2$, one has

∎
$w^{\mathrm{\acute{e}t}}(\kappa)=1+\omega+c_{1}^{\mathrm{\acute{e}t}}(L)=1+\omega+\mathrm{cl}(c_{1}(L))=(1+\omega)\cdot\mathrm{cl}(w_{0}(\kappa))+\mathrm{cl}(w_{2}(\kappa)).$

######  Theorem 3.4.

Let $X$ be a smooth quasi-projective variety over $k$. For $c\geq 0$ and
$x\in\mathrm{CH}^{c}(X)/\ell$, one has

$\displaystyle\mathrm{Sq}(\mathrm{cl}(x))$ $\displaystyle=$
$\displaystyle\sum_{i\geq
0}(1+\omega)^{c-i}\cdot\mathrm{cl}(\mathrm{Sq}^{2i}(x))$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell=2,$ $\displaystyle
P(\mathrm{cl}(x))$ $\displaystyle=$ $\displaystyle\mathrm{cl}(P(x))$
$\displaystyle\textrm{ \hskip 5.0ptif }\ell\textrm{ is odd.}\ $

###### Proof.

By invariance of étale cohomology by inseparable extensions of the base field
[SGA42, VIII, Corollaire 1.2], we may assume that $k$ is perfect (after
replacing it with its perfect closure). We may assume that $x$ is the class of
integral subvariety $g:Z\hookrightarrow X$. As $k$ is perfect, it follows from
Gabber’s improvement on de Jong’s alteration theorem [IT14, Theorem 2.1] that
there exists a smooth variety $Y$ over $k$ and a projective morphism $\nu:Y\to
Z$ that is generically finite of degree $d$ prime to $\ell$. Set
$f:=g\circ\nu$. Replacing $x$ by $d\;\\!x$, which is legitimate because
$d\in(\mathbb{Z}/\ell)^{*}$, we may assume that $x=f_{*}[Y]$. If $\ell$ is
odd, we then compute

$\displaystyle P(\mathrm{cl}(x))$
$\displaystyle=P(\mathrm{cl}(f_{*}[Y]))=P(f_{*}1)=f_{*}w^{\mathrm{\acute{e}t}}(N_{f})\hskip
10.00002pt\textrm{and}$ $\displaystyle\mathrm{cl}(P(x))$
$\displaystyle=\mathrm{cl}(P(f_{*}[Y]))=\mathrm{cl}(f_{*}(P([\widetilde{Y}])\cdot
w(N_{f})))=\mathrm{cl}(f_{*}w(N_{f}))=f_{*}\mathrm{cl}(w(N_{f})),$

where the third equality is Theorem 2.5 and the fifth is Proposition 3.1. If
$\ell=2$, the same arguments show that
$\mathrm{Sq}(\mathrm{cl}(x))=f_{*}w^{\mathrm{\acute{e}t}}(N_{f})$ and
$\mathrm{cl}(\mathrm{Sq}(x))=f_{*}\mathrm{cl}(w(N_{f}))$. In both cases, the
theorem now follows from Lemma 3.3. ∎

One can complement Theorem 3.4 with a formula describing the action of the
Bockstein on a cycle class.

######  Lemma 3.5.

For a smooth quasi-projective variety $X$ over $k$ and
$x\in\mathrm{CH}^{c}(X)/\ell$,

$\beta(\mathrm{cl}(x))=-c\,\omega\cdot\mathrm{cl}(x).$

###### Proof.

This follows from Lemma 2.2 because $\mathrm{cl}(x)$ lifts to
$H^{2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell^{2}}^{\otimes c})$. ∎

######  Remark 3.6.

We do not think that Theorem 3.4 was previously known, even in the framework
of Voevodsky’s motivic Steenrod operations alluded to in Remark 3.2. We note
for instance that the class $\omega$ does not appear in the statement of
[BJ15, Theorem 1.1 (iii)]. We also note that the claim made in [GW19, Remark
7.3.1] implies that the Steenrod operations in étale cohomology considered in
[GW19] preserve algebraic classes, which is not true in general (see Remark
3.8 below).

###  3.3. Obstructions to algebraicity

Let
$H^{2*}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes*})_{\mathrm{alg}}\subset
H^{2*}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes*})$ be the
image of the cycle class map. We obtain stability results of these algebraic
classes under Steenrod operations.

######  Proposition 3.7.

Let $X$ be a smooth quasi-projective variety over $k$.

  1. (i)

If $k$ contains a primitive $\ell^{2}$-th root of unity, then
$H^{2*}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes*})_{\mathrm{alg}}$
is $\mathcal{A}_{\ell}$-stable.

  2. (ii)

If $\ell$ is odd, then
$H^{2*}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes*})_{\mathrm{alg}}$
is stable by $P$.

###### Proof.

By construction, $\omega\in
H^{1}_{\mathrm{\acute{e}t}}(\mathrm{Spec}(k),\mathbb{Z}/\ell)$ vanishes if $k$
contains a primitive $\ell^{2}$-th root of unity. Theorem 3.4 and Lemma 3.5
thus imply the proposition. ∎

######  Remark 3.8.

When $\ell=2$, the hypothesis that $-1$ is a square in Proposition 3.7 (i)
cannot be removed, even if one restricts to even degree Steenrod squares. For
instance, if $k=\mathbb{R}$ and $X=\mathbb{P}^{2}_{\mathbb{R}}$, then

$H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/2)=H^{*}_{\mathrm{\acute{e}t}}(\mathrm{Spec}(\mathbb{R}),\mathbb{Z}/2)[\lambda]/\lambda^{3}=\mathbb{Z}/2[\omega,\lambda]/\lambda^{3},$

where $\lambda:=c^{\mathrm{\acute{e}t}}_{1}(\mathcal{O}_{X}(1))$. The class
$\lambda$ is algebraic, hence so is $\lambda^{2}$. Using (2.5) and (2.6) shows
that $\mathrm{Sq}^{2}(\lambda^{2})=\mathrm{Sq}^{1}(\lambda)^{2}$. Since
$\lambda$ lifts to $H^{2}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{4})$,
Lemma 2.2 implies that $\mathrm{Sq}^{1}(\lambda)=\omega\cdot\lambda$. It
follows that $\mathrm{Sq}^{2}(\lambda^{2})=\omega^{2}\cdot\lambda^{2}$ is not
algebraic. This shows that Steenrod operations do not preserve algebraic
classes in general.

In this example, the algebraic class $\lambda^{2}$ is not sent to an algebraic
class by $\mathrm{Sq}^{2}$, but it is sent to an algebraic class by
$\mathrm{Sq}^{2}+\omega^{2}$. This is a general phenomenon, as we show in
Proposition 3.12 (i) below.

Traditionally, instances of Proposition 3.7 have been used to prove that some
cohomology classes are not algebraic, by showing that they are not killed by
an odd degree Steenrod operations.

######  Corollary 3.9.

Let $X$ be a smooth quasi-projective variety over $k$. Assume that $k$
contains a primitive $\ell^{2}$-th root of unity. Then
$H^{2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
c})_{\mathrm{alg}}$ is killed by all odd degree Steenrod operations.

Even over algebraically closed fields, Proposition 3.7 yields obstructions to
the algebraicity of cohomology classes that go beyond the vanishing of odd
degree Steenrod operations. Over the complex numbers, that algebraic mod
$\ell$ cohomology classes are reductions mod $\ell$ of integral Hodge classes
implies the following.

######  Corollary 3.10.

Let $X$ be a smooth projective variety over $\mathbb{C}$. Fix
$\alpha\in\mathcal{A}_{\ell}$ and $x\in
H^{2c}(X(\mathbb{C}),\mathbb{Z}/\ell)_{\mathrm{alg}}$.

  1. (i)

If $\deg(\alpha)$ is odd, then $\alpha(x)=0$.

  2. (ii)

If $\deg(\alpha)$ is even, then $\alpha(x)$ is the reduction mod $\ell$ of an
integral Hodge class.

As algebraic classes are reductions mod $\ell$ of Tate classes, we obtain
similarly:

######  Corollary 3.11.

Assume that $k$ is the algebraic closure of a finitely generated field. Let
$X$ be a smooth projective variety over $k$. Fix $\alpha\in\mathcal{A}_{\ell}$
and $x\in H^{2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
c})_{\mathrm{alg}}$.

  1. (i)

If $\deg(\alpha)$ is odd, then $\alpha(x)=0$.

  2. (ii)

If $\deg(\alpha)$ is even, then $\alpha(x)$ is the reduction mod $\ell$ of an
integral Tate class.

Corollary 3.10 (i) and 3.11 (i) are the well-known topological obstructions to
the integral Hodge and Tate conjectures going back to Atiyah and Hirzebruch
[AH62]. Corollary 3.10 (ii) and 3.11 (ii) yield new Hodge-theoretic (or
Galois-theoretic) obstructions to them.

When $k$ does not contain a primitive $\ell^{2}$-th root of unity, one can
still deduce from Theorem 3.4 and Lemma 3.5 stability properties of algebraic
cohomology classes by Steenrod operations, and hence obstructions to the
algebraicity of cohomology classes that go beyond Proposition 3.7 and
Corollary 3.9. We give an example of such an obstruction, to be used in §4.4
and §4.5, in the next proposition.

######  Proposition 3.12.

Assume that $\ell=2$ and let $X$ be a smooth quasi-projective variety over
$k$. Then

  1. (i)

$\mathrm{Sq}^{2}+\frac{c(c-1)}{2}\omega^{2}$ sends
$H^{2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
c})_{\mathrm{alg}}$ to
$H^{2c+2}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
c+1})_{\mathrm{alg}}$;

  2. (ii)

$\mathrm{Sq}^{3}+(c+1)\,\omega\cdot\mathrm{Sq}^{2}+\frac{c(c-1)}{2}\,\omega^{3}$
kills $H^{2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
c})_{\mathrm{alg}}$.

###### Proof.

If $x\in\mathrm{CH}^{c}(X)/2$, it follows from Theorem 3.4 that

$\mathrm{cl}(\mathrm{Sq}^{2}(x))=\mathrm{Sq}^{2}(\mathrm{cl}(x))+\frac{c(c-1)}{2}\omega^{2}\cdot\mathrm{cl}(x).$

This proves (i). Applying Lemma 3.5, we deduce that
$H^{2c}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
c})_{\mathrm{alg}}$ is killed by
$(\mathrm{Sq}^{1}+(c+1)\,\omega)(\mathrm{Sq}^{2}+\frac{c(c-1)}{2}\omega^{2})$.
Developing and applying Lemma 3.5 again, we see that this operator equals
$\mathrm{Sq}^{3}+(c+1)\,\omega\cdot\mathrm{Sq}^{2}+\frac{c(c-1)}{2}\,\omega^{3}$,
proving (ii). ∎

###  3.4. Relation with complex cobordism

In [Tot97], Totaro reinterpreted the Atiyah–Hirzebruch topological
obstructions to the integral Hodge conjecture using complex cobordism. His
point of view is that algebraic cohomology classes on a smooth projective
complex variety $X$ always lie in the image of the natural map
${\mu:\mathrm{MU}^{*}(X)\to H^{*}(X(\mathbb{C}),\mathbb{Z})}$. Consequently,
an integral Hodge class not in the image of $\mu$ cannot be algebraic. This
fact also explains the topological obstructions to algebraicity given by
Corollary 3.10 (i) (see [BO21, Proposition 3.6]). We now explain how to
similarly understand the obstruction of Corollary 3.10 (ii) using complex
cobordism.

Let $\mathbf{MU}$ and $\mathbf{H}(\mathbb{Z})$ be spectra representing complex
cobordism and ordinary cohomology with integral coefficients. The morphism
$\mu:\mathrm{MU}^{*}(X)\to H^{*}(X(\mathbb{C}),\mathbb{Z})$ is induced by a
map of spectra $\mu:\mathbf{MU}\to\mathbf{H}(\mathbb{Z})$. Let
$\mathbf{H}(\pi_{*}(\mathbf{MU})_{\mathbb{C}})$ denote the Eilenberg–MacLane
spectrum of the graded ring
$\pi_{*}(\mathbf{MU})_{\mathbb{C}}:=\pi_{*}(\mathbf{MU})\otimes_{\mathbb{Z}}\mathbb{C}$.
In [HQ15, §5], Hopkins and Quick define a map
$\iota:\mathbf{MU}\to\mathbf{H}(\pi_{*}(\mathbf{MU})_{\mathbb{C}})$ inducing
the complexification
$\pi_{*}(\mathbf{MU})\to\pi_{*}(\mathbf{MU})_{\mathbb{C}}$ on homotopy groups.
If $X$ is a smooth projective complex variety of dimension $n$ and
$c\in\mathbb{Z}$, the map $\iota$ induces morphisms

(3.2)
$\iota_{*}:\mathrm{MU}^{2c}(X)\to\bigoplus_{e=0}^{n-c}H^{2c+2e}(X(\mathbb{C}),\pi_{2e}(\mathbf{MU})_{\mathbb{C}}).$

One recovers the morphism $\mu:\mathrm{MU}^{2c}(X)\to
H^{2c}(X(\mathbb{C}),\mathbb{C})$ by projecting (3.2) onto the $e=0$ summand.
In [HQ15, (17)], Hopkins and Quick define $\mathrm{Hdg}_{\mathrm{MU}}^{2c}(X)$
to be the set of $y\in\mathrm{MU}^{2c}(X)$ such that all the components of
$\iota_{*}y$ are Hodge. They prove that any algebraic class $x\in
H^{2c}(X(\mathbb{C}),\mathbb{Z})$ belongs to
$\mu(\mathrm{Hdg}_{\mathrm{MU}}^{2c}(X))$ (see [HQ15, Corollary 7.12]). We
claim that this obstruction to algebraicity combining Hodge theory and complex
cobordism is finer than Corollary 3.10.

######  Proposition 3.13.

Let $X$ be a smooth projective complex variety and let $\ell$ be a prime
number. Fix $\alpha\in\mathcal{A}_{\ell}$ and let $x$ be a class in the image
of the composition

$\mathrm{Hdg}^{2c}_{\mathrm{MU}}(X)\xrightarrow{\mu}H^{2c}(X(\mathbb{C}),\mathbb{Z})\xrightarrow{\rho}H^{2c}(X(\mathbb{C}),\mathbb{Z}/\ell)$

of $\mu$ and of the reduction modulo $\ell$ morphism $\rho$.

  1. (i)

If $\deg(\alpha)$ is odd, then $\alpha(x)=0$.

  2. (ii)

If $\deg(\alpha)$ is even, then $\alpha(x)$ is the reduction mod $\ell$ of an
integral Hodge class.

###### Proof.

Assertion (i) follows from [BO21, Proposition 3.6] (the argument given there
when $\ell=2$ works as well when $\ell$ is odd). Landweber [Lan67, Theorem
8.1] has shown that $\alpha$ lifts to a stable cohomological operation
$\tilde{\alpha}$ in complex cobordism. Letting $2d$ denote the degree of
$\alpha$, we get a diagram of spectra

(3.3)
$\textstyle{\mathbf{H}(\pi_{*}(\mathbf{MU})_{\mathbb{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}(\tilde{\alpha})_{\mathbb{C}}\hskip
7.00002pt}$$\textstyle{\Sigma^{2d}\mathbf{H}(\pi_{*}(\mathbf{MU})_{\mathbb{C}})}$$\textstyle{\mathbf{MU}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\alpha}\hskip
6.29996pt}$$\scriptstyle{\iota}$$\scriptstyle{\rho\circ\mu}$$\textstyle{\Sigma^{2d}\mathbf{MU}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\rho\circ\mu}$$\textstyle{\mathbf{H}(\mathbb{Z}/\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\hskip
7.00002pt}$$\textstyle{\Sigma^{2d}\mathbf{H}(\mathbb{Z}/\ell)}$

where $\mathbf{H}(\mathbb{Z}/\ell)$ is the Eilenberg–MacLane spectrum with
coefficients $\mathbb{Z}/\ell$. We claim that (3.3) is commutative up to
homotopy. The lower square commutes by our choice of $\tilde{\alpha}$. For the
upper square, we need to show that $\iota\circ\tilde{\alpha}$ and
$\pi_{*}(\tilde{\alpha})_{\mathbb{C}}\circ\iota$ classify the same element of
$H^{*}(\mathbf{MU},\pi_{*-2d}(\mathbf{MU})_{\mathbb{C}})=\mathrm{Hom}(H_{*}(\mathbf{MU},\mathbb{Z}),\pi_{*-2d}(\mathbf{MU})_{\mathbb{C}})$.
As the Hurewicz morphism $\pi_{*}(\mathbf{MU})\to
H_{*}(\mathbf{MU},\mathbb{Z})$ is injective by [Ada74, II, Corollary 8.11], it
suffices to verify that the maps $\iota\circ\tilde{\alpha}$ and
$\pi_{*}(\tilde{\alpha})_{\mathbb{C}}\circ\iota$ induce the same morphism
$\pi_{*}(\mathbf{MU})\to\pi_{*-2d}(\mathbf{MU})_{\mathbb{C}}$, which is
tautological.

Now let $y\in\mathrm{Hdg}^{2c}_{\mathrm{MU}}(X)$ be such that
$x=\rho(\mu(y))$. The upper square of (3.3) shows that
$\tilde{\alpha}(y)\in\mathrm{Hdg}^{2c+2d}_{\mathrm{MU}}(X)$. By the lower
square of (3.3), the class $\alpha(x)$ is the reduction mod $\ell$ of the
integral Hodge class $\mu(\tilde{\alpha}(y))$. ∎

It should be possible to carry out a similar analysis in the setting of the
integral Tate conjecture, by relying on Quick’s work [Qui07, Qui11]. We do not
do it here.

##  4\. Non-algebraic cohomology classes

In this section, we construct new examples of non-algebraic cohomology
classes, over algebraically closed fields (in §4.1 and §4.3), finite fields
(in §4.4) and the field $\mathbb{R}$ of real numbers (in §4.5). Applications
to unramified cohomology are made explicit in §4.6. In addition, we review in
§4.2 the construction of algebraic approximations of classifying spaces that
we use in §4.3 and §4.4.

We denote by $\rho$ all the reduction modulo $\ell$ morphisms: in Betti
cohomology, in equivariant Betti cohomology, or in $\ell$-adic étale
cohomology, where the prime number $\ell$ is always clear from the context
(and equal to $2$ in §4.1 and §4.5).

###  4.1. A fivefold failing the integral Tate conjecture over
$\overline{\mathbb{F}}_{p}$

######  Proposition 4.1.

There exists a smooth projective complex variety $Y$ such that:

  1. (i)

The variety $Y$ has dimension $5$.

  2. (ii)

There exists $y\in H^{4}(Y(\mathbb{C}),\mathbb{Z})[2]$ such that
$\mathrm{Sq}^{3}(\rho(y))\neq 0$.

  3. (iii)

The variety $Y$ is defined over $\overline{\mathbb{Q}}$, with good reduction
at all places not above $2$.

###### Proof.

Let $Z$ be the complex fourfold defined in [BO21, Proposition 5.3]. It carries
a class $\sigma\in H^{3}(Z(\mathbb{C}),\mathbb{Z})[2]$ such that
$\rho(\sigma)^{2}\neq 0$ in $H^{6}(Z(\mathbb{C}),\mathbb{Z}/2)$. Let $E$ be a
complex elliptic curve. Choose $\tau\in H^{1}(E(\mathbb{C}),\mathbb{Z})$ not
divisible by $2$. Set $Y:=Z\times E$ and $y:=p_{1}^{*}\sigma\cdot
p_{2}^{*}\tau$. Then Cartan’s formula and the vanishing of
$\mathrm{Sq}^{1}(\rho(\tau))$ imply

$\mathrm{Sq}^{3}(\rho(y))=p_{1}^{*}\mathrm{Sq}^{3}(\rho(\sigma))\cdot
p_{2}^{*}\rho(\tau)=p_{1}^{*}\rho(\sigma)^{2}\cdot p_{2}^{*}\rho(\tau)\neq 0.$

That $Y$ may be chosen to have good reduction at odd places was explained in
[SS22, §4.1]. To give more details, choose the elliptic curve $E$, as well as
all the elliptic curves appearing in the construction of $Z$ given in [BO21,
§5.1] to have complex multiplication. Such elliptic curves are defined over a
number field and have potentially good reduction everywhere. Combined with the
fact that degree $2$ finite étale covers of smooth projective varieties
specialize in characteristic not $2$ (see [SGA1, X, Théorème 3.8]), this shows
that the construction of $Z$, hence also of $Y$, works in mixed characteristic
away from characteristic $2$. The variety $Y$ is thus defined over
$\overline{\mathbb{Q}}$ with good reduction away from $2$. ∎

######  Theorem 4.2.

Let $p$ be an odd prime number. There exists a smooth projective variety $X$
of dimension $5$ over $\overline{\mathbb{F}}_{p}$ on which the $2$-adic
integral Tate conjecture for codimension $2$ cycles fails.

###### Proof.

Let $Y$ and $y$ be as in Proposition 4.1. Let $X$ denote a smooth projective
variety over $\overline{\mathbb{F}}_{p}$ obtained by reduction of $Y$ modulo
$p$. Artin’s comparison theorem [SGA43, XI, Théorème 4.4], the invariance of
étale cohomology under extensions of algebraically closed fields [SGA43, XVI,
Corollaire 1.6], and the smooth and proper base change theorems [SGA43, XVI,
Corollaire 2.2] yield a commutative diagram

(4.1)
$\textstyle{H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim\hskip
4.20004pt}$$\scriptstyle{\rho}$$\textstyle{H^{*}(Y(\mathbb{C}),\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim\hskip
4.20004pt}$$\textstyle{H^{*}(Y(\mathbb{C}),\mathbb{Z}/2)}$

whose horizontal arrows are isomorphisms, and whose bottom row is
$\mathcal{A}_{2}$-equivariant (by functoriality of Steenrod operations and
$\mathcal{A}_{2}$-equivariance of Artin’s comparison isomorphism, see §2.1).
The class $x\in H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{2}(2))[2]$ induced
by $y$ and a fixed isomorphism $\mathbb{Z}_{2}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\mathbb{Z}_{2}(2)$
over $\overline{\mathbb{F}}_{p}$ is Tate since it is torsion. As
$\mathrm{Sq}^{3}(\rho(x))\neq 0$ by Proposition 4.1 (ii) and (4.1), the class
$x$ is not algebraic by Corollary 3.11 (i). ∎

######  Remark 4.3.

The same argument shows that the complex variety $Y$ of Proposition 4.1 is a
$5$-dimensional counterexample to the integral Hodge conjecture: the class $y$
is Hodge because it is torsion, and it is not algebraic by Corollary 3.10 (i).

###  4.2. Algebraic approximations of classifying spaces

Proposition 4.4 below is essentially due to Ekedahl [Eke09, Theorem 1.3 and
§1.1]. Simplifications to the proof were made by Pirutka and Yagita [PY15, §4]
with further additions by Antieau [Ant16, §3]. We give a proof for the
convenience of the reader, and claim no originality.

Recall that one may let any Lie group $H$ act freely and properly on a
contractible space $EH$. The resulting quotient $BH:=EH/H$ is the classifying
space of $H$.

We also recall that a continuous map $f:Y\to X$ between topological spaces is
an $N$-equivalence if $f_{*}:\pi_{q}(Y,y)\to\pi_{q}(X,f(y))$ is an isomorphism
for $q<N$ and a surjection for $q=N$, for all $y\in Y$. By the Hurewicz
theorem [Hat02, Theorem 4.37] and the universal coefficient theorem, the
morphism $f^{*}:H^{q}(X,A)\to H^{q}(Y,A)$ is then an isomorphism for $q<N$ and
an injection for $q=N$, for all abelian groups $A$.

######  Proposition 4.4.

Let $H$ be a connected reductive complex Lie group. Let $p$ be a prime number,
and fix $N\geq 1$. Then there exists a smooth projective complex variety $Y$
of dimension $N$ such that:

  1. (i)

There exists an $N$-equivalence $Y(\mathbb{C})\to B(H\times\mathbb{C}^{*})$.

  2. (ii)

The variety $Y$ is defined over $\mathbb{Q}$ and has good reduction at $p$.

The following lemma is [Eke09, Lemma 1.1 (ii)-(iv)].

######  Lemma 4.5.

Let $V$ be a $d$-dimensional vector space over an algebraically closed field
$K$. For $n\geq d$, let $U_{n}\subset V^{\oplus n}$ be the open subset of
$n$-tuples that span $V$.

  1. (i)

The complement of $U_{n}$ in $V^{\oplus n}$ has codimension $\geq n+1-d$.

  2. (ii)

The algebraic group $\mathrm{GL}(V)$ acts freely on $U_{n}$.

  3. (iii)

The image of $U_{n}$ in $\mathbb{P}(V^{\oplus n})$ consists of
$\mathrm{SL}(V)$-stable points.

###### Proof of Proposition 4.4.

Let $\mathcal{H}$ be a reductive group scheme over $\mathbb{Z}_{(p)}$ (in the
sense of [SGA33, XIX, Définition 2.7]) with $\mathcal{H}(\mathbb{C})\simeq H$
(see [SGA33, XXV, Corollaire 1.3]). By [SGA31, VIB, Proposition 13.2], there
exists a closed embedding of group schemes
$\mathcal{H}\hookrightarrow\mathrm{GL}_{d-2,\mathbb{Z}_{(p)}}$ for some $d\geq
2$. Composing it with

$M\mapsto\bigg{(}\begin{smallmatrix}M&0&0\\\ 0&\det(M)^{-1}&0\\\
0&0&1\end{smallmatrix}\bigg{)}$

yields embeddings $\mathcal{H}\hookrightarrow\mathrm{SL}_{d,\mathbb{Z}_{(p)}}$
and
$\mathcal{H}\times\mathbf{G}_{m}\hookrightarrow\mathrm{GL}_{d,\mathbb{Z}_{(p)}}$
(where $\mathbf{G}_{m}$ is embedded by homotheties). Fix $n\gg 0$. Let
$\mathcal{H}\times\mathbf{G}_{m}$ act linearly on
$\mathbb{A}^{nd}_{\mathbb{Z}_{(p)}}=(\mathbb{A}^{d}_{\mathbb{Z}_{(p)}})^{n}$,
in a diagonal way using $n$ times the above representation. By Lemma 4.5,
there exists an open subset
$\mathcal{U}\subset\mathbb{A}^{nd}_{\mathbb{Z}_{(p)}}$ on which
$\mathcal{H}\times\mathbf{G}_{m}$ acts freely, whose image
$\mathcal{V}\subset\mathbb{P}^{nd-1}_{\mathbb{Z}_{(p)}}$ consists only of
$\mathcal{H}$-stable points, and such that the complement of $\mathcal{U}$ in
$\mathbb{A}^{nd}_{\mathbb{Z}_{(p)}}$ has codimension $\geq N+1$ in the fibers
of $\mathbb{A}^{nd}_{\mathbb{Z}_{(p)}}\to\mathrm{Spec}(\mathbb{Z}_{(p)})$ (as
$n\gg 0$).

Let $\mathcal{Z}:=\mathbb{P}^{nd-1}_{\mathbb{Z}_{(p)}}/\\!\\!/\mathcal{H}$ be
the GIT quotient (see [Ses77, Theorem 4]). The complement of the smooth open
subset $\mathcal{W}:=\mathcal{V}/\mathcal{H}\subset\mathcal{Z}$ has
codimension $\geq N+1$ in the fibers of
$\mathcal{Z}\to\mathrm{Spec}(\mathbb{Z}_{(p)})$. Choosing a projective
embedding of the projective $\mathbb{Z}_{(p)}$-scheme $\mathcal{Z}$, one can
use Poonen’s Bertini theorem [Poo04, Theorem 1.1] to find an $N$-dimensional
complete intersection $\mathcal{Y}_{\mathbb{F}_{p}}$ of
$\mathcal{Z}_{\mathbb{F}_{p}}$ which is smooth and included in
$\mathcal{W}_{\mathbb{F}_{p}}$. Lifting over $\mathbb{Z}_{(p)}$ the equations
of the projective hypersurfaces defining $\mathcal{Y}_{\mathbb{F}_{p}}$ in
$\mathcal{Z}_{\mathbb{F}_{p}}$ yields a complete intersection $\mathcal{Y}$ of
$\mathcal{Z}$ which is smooth of relative dimension $N$ over
$\mathbb{Z}_{(p)}$, and included in $\mathcal{W}$.

Define $Y:=\mathcal{Y}_{\mathbb{C}}$, $V:=\mathcal{V}_{\mathbb{C}}$ and
$U:=\mathcal{U}_{\mathbb{C}}$. The variety $Y$ satisfies (ii) by construction.
Applying Hamm’s Lefschetz theorem [Ham83, Theorem 2 and the remark below]
several times shows that the natural map $Y(\mathbb{C})\to
V(\mathbb{C})/H=U(\mathbb{C})/(H\times\mathbb{C}^{*})$ is an $N$-equivalence.
In the diagram of fibrations

$U(\mathbb{C})/(H\times\mathbb{C}^{*})\leftarrow(U(\mathbb{C})\times
E(H\times\mathbb{C}^{*})/(H\times\mathbb{C}^{*})\to B(H\times\mathbb{C}^{*}),$

the left-hand side arrow is a weak equivalence because its fibres are
isomorphic to $E(H\times\mathbb{C}^{*})$ hence contractible, and the right-
hand side arrow is an $N$-equivalence as its fibres are isomorphic to
$U(\mathbb{C})$ hence are $(N-1)$-connected because the complement of
$U(\mathbb{C})$ in $\mathbb{C}^{nd}$ has high codimension when $n\gg 0$.
Assertion (i) follows. ∎

When we apply Proposition 4.4 to the simple simply connected complex Lie group
$E_{8}$ in §4.3, we will need the following lemma.

######  Lemma 4.6.

Fix $\ell\in\\{2,3,5\\}$. Then the following assertions hold.

  1. (i)

There exists a $16$-equivalence $f:BE_{8}\to K(\mathbb{Z},4)$.

  2. (ii)

The group $H^{q}(BE_{8},\mathbb{Z})$ has no $\ell$-torsion for $q\leq
2\ell+2$.

  3. (iii)

There exists $y\in H^{4}(BE_{8},\mathbb{Z})$ such that the classes
$\mathrm{Sq}^{3}(\rho(y))$ (if $\ell=2$) and $\beta P^{1}(\rho(y))$ (if $\ell$
is odd) are non-zero.

###### Proof.

As $\pi_{1}(E_{8})=0$ and $E_{8}$ is simple, Bott [Bot56, Theorem A and top of
p. 253] has shown that $\pi_{2}(E_{8})=0$ and $\pi_{3}(E_{8})=\mathbb{Z}$.
Moreover, Bott and Samelson [BS58, Theorem V p. 995] have proven that
$\pi_{q}(E_{8})=0$ for $4\leq q\leq 14$. It follows that $\pi_{q}(BE_{8})=0$
for $1\leq q\leq 3$ and $5\leq q\leq 15$ and that
$\pi_{4}(BE_{8})=\mathbb{Z}$. This proves (i).

Assertion (ii) follows from (i) and from the computation of
$H_{*}(K(\mathbb{Z},4),\mathbb{Z})$ given in [Car55, §4]. We now turn to
(iii). In view of (i), it suffices to construct such a class $y$ in the
cohomology of $K(\mathbb{Z},4)$, and by the universal property of
$K(\mathbb{Z},4)$, it suffices to construct it in the cohomology of a space of
our choice. The space $K(\mathbb{Z}/\ell,3)$ works because the Steenrod
operations $\mathrm{Sq}^{3}\mathrm{Sq}^{1}$ (when $\ell=2$) and $\beta
P^{1}\beta$ (when $\ell$ is odd) are non-zero in general on degree $3$ mod
$\ell$ cohomology classes. ∎

In §4.4, we will use the following variant of Proposition 4.4.

######  Proposition 4.7.

Let $p\neq\ell$ be prime numbers. Let $\mathcal{H}$ be a finite étale group
scheme over $\mathbb{Z}_{(p)}$. Fix $N\geq 1$. There exists a smooth
projective scheme $\mathcal{Y}$ of relative dimension $N$ over
$\mathbb{Z}_{(p)}$ and a line bundle $\mathcal{L}$ over $\mathcal{Y}$ such
that:

  1. (i)

There exists an $N$-equivalence $\mathcal{Y}(\mathbb{C})\to
B(\mathcal{H}(\mathbb{C})\times\mathbb{C}^{*})$.

  2. (ii)

The topological complex line bundle
$\mathcal{L}(\mathbb{C})\to\mathcal{Y}(\mathbb{C})$ is classified by the map
$\mathcal{Y}(\mathbb{C})\to B\mathbb{C}^{*}$ appearing in (i).

  3. (iii)

If $N\geq 2$, there is a
$\mathrm{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$-equivariant map

(4.2)
$\mathrm{Hom}(\mathcal{H}(\overline{\mathbb{F}}_{p}),\mathbb{Z}/\ell)=H^{1}(\mathcal{H}(\overline{\mathbb{F}}_{p}),\mathbb{Z}/\ell)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{1}_{\mathrm{\acute{e}t}}(\mathcal{Y}_{\overline{\mathbb{F}}_{p}}\mathbb{Z}/\ell).$

###### Proof.

Construct $\mathcal{Y}$ exactly as in the proof of Proposition 4.4. (As the
reference [Ses77] assumes that $\mathcal{H}$ has connected geometric fibers,
one has to replace it with [Ray67, Théorème 1 iv)].) Assertion (i) is verified
there.

By construction, there exists an $\mathcal{H}$-torsor
$\widetilde{\mathcal{Y}}\to\mathcal{Y}$ whose total space is a relative
complete intersection in $\mathbb{P}^{m}_{\mathbb{Z}_{(p)}}$ for some $m$. The
tautological bundle $\mathcal{O}_{\widetilde{\mathcal{Y}}}(1)$ is naturally
$\mathcal{H}$-linearized, and hence descends to a line bundle $\mathcal{L}$ on
$\mathcal{Y}$. Inspecting the proof of Proposition 4.4 shows that (ii) holds.

As $\widetilde{\mathcal{Y}}_{\overline{\mathbb{F}}_{p}}$ is simply connected
by [SGA2, XII, Corollaire 3.5], the Hochschild–Serre spectral sequence for
$\widetilde{\mathcal{Y}}_{\overline{\mathbb{F}}_{p}}\to\mathcal{Y}_{\overline{\mathbb{F}}_{p}}$
yields the isomorphism (4.2). It is canonical, hence
$\mathrm{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$-equivariant. ∎

###  4.3. A torsion-free counterexample to the integral Tate conjecture over
$\overline{\mathbb{F}}_{p}$

The proof of Proposition 4.8 below is to be compared with [Sch98]. In this
article, Schoen overcomes the difficulty of constructing hyperplane sections
of smooth projective varieties over $\overline{\mathbb{F}}_{p}$ whose
vanishing cohomology contains interesting Tate classes. In contrast, the main
ingredient of the proof of Proposition 4.8 is the construction a hyperplane
section whose vanishing cohomology contains no non-zero Tate classes.

######  Proposition 4.8.

Choose $\ell\in\\{2,3,5\\}$ and let $p$ be a prime number distinct from
$\ell$. There exist a smooth projective variety $X$ of dimension $2\ell+2$
over $\overline{\mathbb{F}}_{p}$ and a class $x\in
H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(2))$ such that the following
assertions hold.

  1. (i)

The ring $H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ is torsion-free.

  2. (ii)

A multiple of the class $x$ is algebraic.

  3. (iii)

Define $\alpha:=\mathrm{Sq}^{2}$ if $\ell=2$ and $\alpha:=P^{1}$ if $\ell$ is
odd. Then the class $\alpha(\rho(x))$ is not the reduction modulo $\ell$ of an
integral Tate class.

###### Proof.

We split the proof in several steps.

######  Step 1.

An algebraic approximation of the classifying space of $E_{8}$.

Let $Y$ be a variety obtained by applying Proposition 4.4 with $H=E_{8}$ and
$N=2\ell+3$. Let $Z$ be a smooth projective variety over a finite field
$\mathbb{F}$ obtained by reducing $Y$ modulo $p$. After maybe enlarging
$\mathbb{F}$, we may assume that $Z(\mathbb{F})\neq\varnothing$. Define
$\nu:W\to Z$ to be the identity if $\ell\in\\{2,5\\}$ and the blow-up of an
$\mathbb{F}$-point of $Z$ if $\ell=3$. Set
$\overline{Z}:=Z_{\overline{\mathbb{F}}_{p}}$ and
$\overline{W}:=W_{\overline{\mathbb{F}}_{p}}$.

Proposition 4.4 (i), Artin’s comparison theorem [SGA43, XI, Théorème 4.4], the
invariance of étale cohomology under extensions of algebraically closed fields
[SGA43, XVI, Corollaire 1.6] and the smooth and proper base change theorems
[SGA43, XVI, Corollaire 2.2] yield a commutative diagram

(4.3)
$\textstyle{H^{*}(BE_{8},\mathbb{Z}_{\ell})[t]=H^{*}(B(E_{8}\times\mathbb{C}^{*}),\mathbb{Z}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{H^{*}_{\mathrm{\acute{e}t}}(\overline{Z},\mathbb{Z}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\nu^{*}}$$\textstyle{H^{*}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Z}_{\ell})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{H^{*}(BE_{8},\mathbb{Z}/\ell)[t]=H^{*}(B(E_{8}\times\mathbb{C}^{*}),\mathbb{Z}/\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{*}_{\mathrm{\acute{e}t}}(\overline{Z},\mathbb{Z}/\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{*}}$$\textstyle{H^{*}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Z}/\ell)}$

whose left horizontal arrows are isomorphisms in degree $\leq 2\ell+2$ and
injective in degree $2\ell+3$, whose right horizontal arrows are injective by
computation of the cohomology of a blow-up, whose bottom row is
$\mathcal{A}_{\ell}$-equivariant, and where $t\in
H^{2}(B\mathbb{C}^{*},\mathbb{Z})$ is a generator of
$H^{*}(B\mathbb{C}^{*},\mathbb{Z})=H^{*}(\mathbb{P}^{\infty}(\mathbb{C}),\mathbb{Z})=\mathbb{Z}[t]$.

Since
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{Z},\mathbb{Q}_{\ell})=\mathbb{Q}_{\ell}^{\oplus
s}$ with $s=2$ if $\ell=2$, $s=3$ if $\ell=3$ and $s=4$ if $\ell=5$, by (4.3)
and Lemma 4.6, we deduce from the computation of the cohomology of a blow-up
that $H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Q}_{\ell})$
always has even rank.

######  Step 2.

A Lefschetz pencil.

Let $\mathcal{L}$ be a very ample line bundle on $W$ and fix $m\geq 3$. After
maybe enlarging the finite field $\mathbb{F}$, we may choose a Lefschetz
pencil $f:\widetilde{W}\to\mathbb{P}^{1}_{\mathbb{F}}$ on $W$ associated with
$\mathcal{L}^{\otimes 2m}$ (see [SGA72, XVII, Théorème 2.5]). Let
$U\subset\mathbb{P}^{1}_{\mathbb{F}}$ be the smooth locus of the pencil, and
let $\overline{\eta}$ be a geometric generic point of
$\mathbb{P}^{1}_{\mathbb{F}}$.

By the hard Lefschetz theorem [Del80, Corollaire 4.3.9], one has a
decomposition

(4.4)
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\widetilde{W}_{\overline{\eta}},\mathbb{Q}_{\ell}(\ell+1))=H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Q}_{\ell}(\ell+1))\oplus
H^{2\ell+2}_{\mathrm{\acute{e}t}}(\widetilde{W}_{\overline{\eta}},\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$

which is orthogonal with respect to the cup-product pairing. Let $O$ be the
orthogonal group of the vanishing cohomology
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\widetilde{W}_{\overline{\eta}},\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$
endowed with its cup-product pairing, and let $O^{+}\subset O$ be the special
orthogonal subgroup. We view $O$ and $O^{+}$ as $\ell$-adic Lie groups. By
[Sch98, Proposition 1.1 (iii)], the image of the geometric monodromy
representation $\pi_{1}(U_{\overline{\mathbb{F}}_{p}},\overline{\eta})\to O$
is infinite (in _loc. cit._ , the proof is written for a Lefschetz pencil on a
threefold, but the argument works as well in our situation). It follows from
[Del80, Théorème 4.4.1], that this image is open for the $\ell$-adic topology.
So is, a fortiori, the image $I\subset O$ of the arithmetic monodromy
representation $\pi_{1}(U,\overline{\eta})\to O$. We deduce that $I$ contains
a non-empty open subset $\Omega$ of $O^{+}$.

######  Step 3.

Applying the Čebotarev density theorem.

By [Sch99, proof of Lemma 9.2.1 (i)], the dimension of
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\widetilde{W}_{\overline{\eta}},\mathbb{Q}_{\ell}(\ell+1))$
is even. As we had checked at the end of Step 1 that
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Q}_{\ell}(\ell+1))$
has even rank, it follows from (4.4) that the dimension of
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\widetilde{W}_{\overline{\eta}},\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$
is also even, say, equal to $2d$. Let $n_{0}\in\mathbb{N}$ be such that
$\varphi(n)>2d$ for all $n\geq n_{0}$, where $\varphi$ is Euler’s totient
function. As there exists an element of
$\mathrm{SO}_{2d}(\overline{\mathbb{Q}}_{\ell})\simeq\mathrm{SO}_{2d}(\mathbb{C})$
none of whose eigenvalues are roots of unity, a nonempty Zariski-open subset
of $\mathrm{SO}_{2d}(\overline{\mathbb{Q}}_{\ell})$ consists of matrices with
no roots of unity of order $<n_{0}$ as eigenvalues. It follows that there
exists a nonempty open subset $\Omega^{\prime}\subset\Omega$ for the
$\ell$-adic topology containing only matrices none of whose eigenvalues are
roots of unity of order $<n_{0}$.

Čebotarev’s theorem now shows the existence of a closed point ${u\in U}$ such
that the image in $O$ of the Frobenius $F_{u}$ at $u$ belongs to
$\Omega^{\prime}$. Denoting by
$X:=(\widetilde{W}_{u})_{\overline{\mathbb{F}}_{p}}$ the geometric fiber of
$f$ above $u$, this implies that $F_{u}$ acts on the vanishing cohomology
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$
with no roots of unity of order $<n_{0}$ as eigenvalues. By [Del74, Théorème
1.6], the characteristic polynomials of $F_{u}$ acting on
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Q}_{\ell}(\ell+1))$
and $H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_{\ell}(\ell+1))$ have
rational coefficients. By hard Lefschetz [Del80, Corollaire 4.3.9], so is the
characteristic polynomial of $F_{u}$ acting on
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$.
By our choice of $n_{0}$, this characteristic polynomial, which has degree
$2d$, cannot annihilate any root of unity of order $\geq n_{0}$. Consequently,
none of the eigenvalues of $F_{u}$ acting on
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$
are roots of unity. In other words,
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$
contains no non-zero Tate classes.

######  Step 4.

The cohomology ring $H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ is
torsion-free.

By the weak Lefschetz theorem, the restriction map
$H^{q}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Z}_{\ell})\to
H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ is an isomorphism for $q\leq
2\ell+1$ and is injective with torsion-free cokernel for $q=2\ell+2$ (see
[Del80, (4.1.6)]). In view of (4.3) and Lemma 4.6, this implies that
$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ is torsion-free for $q\leq
2\ell+2$. Let us show by descending induction on $q$ that
$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ is also torsion-free for
$q\geq 2\ell+3$. The assertion holds for $q\geq 4\ell+5$ by cohomological
dimension [SGA43, X, Corollaire 4.3]. Fix $2\ell+3\leq q\leq 4\ell+4$. Since
both $H^{q+1}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ and
$H^{4\ell+5-q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ are torsion-free,
the long exact sequences of cohomology associated with
$0\to\mathbb{Z}_{\ell}\xrightarrow{\ell^{m}}\mathbb{Z}_{\ell}\to\mathbb{Z}/\ell^{m}\to
0$ yield isomorphisms

$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})/\ell^{m}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/\ell^{m})\textrm{
\,and
}H^{4\ell+4-q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})/\ell^{m}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{4\ell+4-q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/\ell^{m})$

for all $m\in\mathbb{N}$. As
$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/\ell^{m})$ and
$H^{4\ell+4-q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/\ell^{m})$ have the same
cardinality by Poincaré duality (see [SGA43, XVIII, (3.2.6.2)]), and as the
cardinality of
$H^{4\ell+4-q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})/\ell^{m}$ is a
linear function of $\ell^{m}$ because
$H^{4\ell+4-q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ is torsion-free, we
deduce that the cardinality of
$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})/\ell^{m}$ is a linear
function of $\ell^{m}$. It follows that
$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ is torsion-free. We have
thus proven assertion (i).

######  Step 5.

The cohomology class $x$.

Let $y\in H^{4}(BE_{8},\mathbb{Z})$ be the class given by Lemma 4.6, let $z\in
H^{4}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Z}_{\ell})$ be the class it
induces via (4.3), and let $x\in
H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(2))$ be the class obtained
from $z|_{X}\in H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ by choosing
an isomorphism $\mathbb{Z}_{\ell}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\mathbb{Z}_{\ell}(2)$
over $\overline{\mathbb{F}}_{p}$.

By Edidin and Graham [EG97, Theorem 1(c)] (see also [Tot14, Theorem 2.14]),
the cycle class map
$\mathrm{CH}^{2}(B(E_{8}\times\mathbb{C}^{*}))\otimes_{\mathbb{Z}}\mathbb{Q}\to
H^{4}(B(E_{8}\times\mathbb{C}^{*}),\mathbb{Q})$ is surjective. Consequently, a
multiple of the class $y$, hence also of $x$, is algebraic, proving (ii).

Assume for contradiction that $\alpha(\rho(x))\in
H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
2})=H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes\ell+1})$
is the reduction modulo $\ell$ of an integral Tate class $w\in
H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(\ell+1))$. By the weak
Lefschetz theorem (see [Del80, (4.1.6)]), there is an exact sequence

(4.5) $0\to
H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Z}_{\ell}(\ell+1))\to
H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(\ell+1))\to K\to 0$

with $K$ torsion-free. Since
$K\otimes_{\mathbb{Z}_{\ell}}\mathbb{Q}_{\ell}=H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\mathbb{Q}_{\ell}(\ell+1))_{\mathrm{van}}$
contains no non-zero Tate class by Step 3, the image of $w$ in $K$ is torsion,
hence zero because $K$ is torsion-free. By (4.5), we deduce the existence of a
Tate class $v\in
H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\mathbb{Z}_{\ell}(\ell+1))$
with $\alpha(\rho(x))=\rho(v|_{X})$. As the restriction map
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\boldsymbol{\mu}_{\ell}^{\otimes\ell+1})\to
H^{2\ell+2}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes\ell+1})$
is injective, again by weak Lefschetz [Del80, (4.1.6)], one has
$\alpha(\rho(z))=\rho(v)$ in
$H^{2\ell+2}_{\mathrm{\acute{e}t}}(\overline{W},\boldsymbol{\mu}_{\ell}^{\otimes\ell+1})$.
Applying $\beta$ to this identity shows that $\beta\alpha(\rho(z))=0$, because
$\beta$ kills the reductions mod $\ell$ of integral classes. This is a
contradiction because $\beta\alpha(\rho(z))$ is non-zero by (4.3) and Lemma
4.6. We have finally proved (iii). ∎

######  Theorem 4.9.

Choose $\ell\in\\{2,3,5\\}$ and let $p$ be a prime number distinct from
$\ell$. There exists a smooth projective variety $X$ of dimension $2\ell+2$
over $\overline{\mathbb{F}}_{p}$ with
$H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ torsion-free on which the
$\ell$-adic integral Tate conjecture for codimension $2$ cycles fails.

###### Proof.

Let $X$ and $x$ be as in Proposition 4.8. The class $x$ is Tate because a
multiple of it is algebraic by Proposition 4.8 (ii). It cannot be algebraic
itself by Corollary 3.11 (ii) and Proposition 4.8 (iii). ∎

######  Remark 4.10.

Fix $\ell\in\\{2,3,5\\}$. The proofs of Proposition 4.8 and Theorem 4.9 may be
adapted to show that, over the complex numbers, a $(2\ell+2)$-dimensional
complete intersection of very general sufficiently ample hypersurfaces in a
smooth projective variety obtained by applying Proposition 4.4 to $H=E_{8}$
fails the integral Hodge conjecture for codimension $2$ cycles (but their
cohomology ring has no $\ell$-torsion, and in fact no torsion at all when
$\ell=2$).

The main modifications to implement are the following. First, one has to use
Corollary 3.10 (ii) instead of Corollary 3.11 (ii). More importantly, instead
of verifying that the vanishing cohomology contains no non-zero Tate classes
as in Steps 2 and 3 of the proof of Proposition 4.8, one needs to verify that
it contains no non-zero Hodge classes. This is a consequence of a
Noether–Lefschetz theorem (see [Voi02, Théorème 18.28]).

###  4.4. Geometrically trivial non-algebraic classes over all finite fields

In this paragraph, we work over a finite field $\mathbb{F}$ of characteristic
$p$, with algebraic closure $\overline{\mathbb{F}}_{p}$. If $X$ is a variety
over $\mathbb{F}$, we set $\overline{X}:=X_{\overline{\mathbb{F}}_{p}}$. Let
$G:=\mathrm{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F})$ be the absolute Galois
group of $\mathbb{F}$, and $F\in G$ be the geometric Frobenius (see [Del74,
(1.15.1)]). Recall that $G$ has cohomological dimension $1$ and that, if $M$
is a finite $G$-module, one has a natural isomorphism

(4.6) $M/(F-\mathrm{Id})\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{1}(G,M).$

######  Proposition 4.11.

Let $p\neq\ell$ be prime numbers, and let $\mathbb{F}$ be a finite field of
characteristic $p$. There exist a smooth projective geometrically connected
variety $X$ of dimension $2\ell+3$ over $\mathbb{F}$ and a class $z\in
H_{\mathrm{\acute{e}t}}^{2}(\overline{X},\boldsymbol{\mu}_{\ell}^{\otimes 2})$
such that the class $\mathrm{Sq}^{3}\mathrm{Sq}^{1}(z)$ (if $\ell=2$), or
$\beta P^{1}\beta(z)$ (if $\ell$ is odd), does not belong to the image of
$F-\mathrm{Id}$.

###### Proof.

Consider the étale abelian sheaf $(\boldsymbol{\mu}_{\ell})^{2}$ over
$\mathbb{Z}_{(p)}$ and view it as a finite étale group scheme $\mathcal{H}$
over $\mathbb{Z}_{(p)}$. Let $\mathcal{Y}$ be a smooth projective scheme over
$\mathbb{Z}_{(p)}$ obtained by applying Proposition 4.7 to $\mathcal{H}$, with
$N=2\ell+3$. Define $X:=\mathcal{Y}_{\mathbb{F}}$.

Proposition 4.7 (i), Artin’s comparison theorem [SGA43, XI, Théorème 4.4], the
invariance of étale cohomology under extensions of algebraically closed fields
[SGA43, XVI, Corollaire 1.6] and the smooth and proper base change theorems
[SGA43, XVI, Corollaire 2.2] yield a morphism

$\Phi:H^{*}(B((\mathbb{Z}/\ell)^{2}\times\mathbb{C}^{*}),\mathbb{Z}/\ell)\to
H^{*}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}/\ell)$

which is an isomorphism in degree $\leq 2\ell+2$, and which is
$\mathcal{A}_{2}$-equivariant. That $\Phi$ is an isomorphism in degree $0$
implies that $X$ is geometrically connected.

It is well-known that
$H^{*}(B\mathbb{C}^{*},\mathbb{Z}/\ell)=\mathbb{Z}/\ell[t]$ with $t\in
H^{2}(B\mathbb{C}^{*},\mathbb{Z}/\ell)$, that
$H^{*}(B\mathbb{Z}/2,\mathbb{Z}/2)=\mathbb{Z}/2[x]$ with $x\in
H^{1}(B\mathbb{Z}/2,\mathbb{Z}/2)$, and that, when $\ell$ is odd,
$H^{*}(B\mathbb{Z}/\ell,\mathbb{Z}/\ell)=\mathbb{Z}/\ell[x,y]/\langle
x^{2}\rangle$ with $x\in H^{1}(B\mathbb{Z}/\ell,\mathbb{Z}/\ell)$ and
$y=\beta(x)$. It then follows from the Künneth formula that

$H^{*}(B((\mathbb{Z}/2)^{2}\times\mathbb{C}^{*}),\mathbb{Z}/2)=\mathbb{Z}/2[x_{1},x_{2},t],$

and that, when $\ell$ is odd, one has

$H^{*}(B((\mathbb{Z}/\ell)^{2}\times\mathbb{C}^{*}),\mathbb{Z}/\ell)=\mathbb{Z}/\ell[x_{1},y_{1},x_{2},y_{2},t]/\langle
x_{1}^{2},x_{2}^{2}\rangle.$

We now compute Galois actions. Let $|\mathbb{F}|=p^{a}$ be the cardinality of
$\mathbb{F}$. By $G$-equivariance of (4.2) and our choice of $\mathcal{H}$,
one has $F(\Phi(x_{i}))=p^{a}\Phi(x_{i})$ in
$H^{1}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}/\ell)$ for
$i\in\\{1,2\\}$. When $\ell$ is odd, one also has
$F(\Phi(y_{i}))=p^{a}\Phi(y_{i})$ in
$H^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}/\ell)$ for
$i\in\\{1,2\\}$ by $G$-equivariance of the Bockstein. In addition, since
$\Phi(t)\in H^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}/\ell)$ is the
first Chern class of a line bundle which is defined over $\mathbb{F}$, by
Proposition 4.7 (ii), one has $F(\Phi(t))=p^{a}\Phi(t)$ in
$H^{2}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}/\ell)$.

Define $w:=\Phi(x_{1}x_{2})$. When $\ell=2$, one has

$\mathrm{Sq}^{3}\mathrm{Sq}^{1}(w)=\Phi(\mathrm{Sq}^{3}\mathrm{Sq}^{1}(x_{1}x_{2}))=\Phi(\mathrm{Sq}^{3}(x_{1}^{2}x_{2}+x_{1}x_{2}^{2}))=\Phi(x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4}).$

When $\ell$ is odd, one computes similarly that

$\displaystyle\beta P^{1}\beta(w)$ $\displaystyle=\Phi(\beta
P^{1}\beta(x_{1}x_{2}))=\Phi(\beta P^{1}(y_{1}x_{2}-x_{1}y_{2}))$
$\displaystyle=\Phi(\beta(y^{\ell}_{1}x_{2}-x_{1}y^{\ell}_{2}))=\Phi(y_{1}^{\ell}y_{2}-y_{1}y_{2}^{\ell}).$

Choose a class $\xi\in
H^{0}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell}^{\otimes 2})$
and define $z:=\xi\cdot w\in
H_{\mathrm{\acute{e}t}}^{2}(\overline{X},\boldsymbol{\mu}_{\ell}^{\otimes
2})$. The action of $F$ on the group
$H_{\mathrm{\acute{e}t}}^{2\ell+2}(\overline{X},\boldsymbol{\mu}_{\ell}^{\otimes
2})$ is diagonal in a basis formed by multiples by $\xi$ of appropriate
monomials in $\Phi(t)$, $\Phi(x_{1})$, $\Phi(x_{2})$ (and, when $\ell$ is odd,
$\Phi(y_{1})$ and $\Phi(y_{2})$). Paying attention to the Tate twist, we see
that the elements $\xi\cdot\Phi(x_{1}^{4}x_{2}^{2})$ (when $\ell=2$) and
$\xi\cdot\Phi(y^{\ell}_{1}y_{2})$ (when $\ell$ is odd) of this group are
$F$-stable. Taking Remark 2.1 (iv) into account, we deduce that
$\mathrm{Sq}^{3}\mathrm{Sq}^{1}(z)=\xi\cdot\Phi(x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4})$
(when $\ell=2$) and $\beta
P^{1}\beta(z)=\xi\cdot\Phi(y_{1}^{\ell}y_{2}-y_{1}y_{2}^{\ell})$ (when $\ell$
is odd), are not in the image of $F-\mathrm{Id}$. ∎

The following theorem was proven by Scavia and Suzuki [SS22, Theorem 1.3] when
$\mathbb{F}$ contains a primitive $\ell^{2}$-th root of unity.

######  Theorem 4.12.

Let $p\neq\ell$ be prime numbers, and let $\mathbb{F}$ be a finite field of
characteristic $p$. There exist a smooth projective geometrically connected
variety $X$ of dimension $2\ell+3$ over $\mathbb{F}$ and a non-algebraic class

$x\in\mathrm{Ker}\big{(}H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(2))\to
H^{4}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(2))\big{)}.$

###### Proof.

Let $X$ and $z$ be as in Proposition 4.11. The inverse limit, when $m$ varies,
of the short exact sequences

(4.7) $0\to
H^{1}(G,H^{q-1}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r}))\to H^{q}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r})\to
H^{q}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r})^{G}\to 0$

stemming from Hochschild–Serre spectral sequences yield short exact sequences

(4.8) $0\to
H^{1}(G,H^{q-1}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(r)))\to
H^{q}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(r))\to
H^{q}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(r))^{G}\to 0$

because the groups
$H^{q}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r}))$ are finite and the $G$-cohomology groups of a finite $G$-module are
finite.

Denote by
$\psi_{X}:H^{q-1}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r})\to H^{q}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell^{m}}^{\otimes r})$
the compositions

$H^{q-1}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r})\to
H^{q-1}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r})/(F-\mathrm{Id})\to
H^{1}(G,H^{q-1}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r}))\to H^{q}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell^{m}}^{\otimes
r})$

of the quotient map, of the isomorphism (4.6), and of the left arrow of (4.7).
Consider the element $y:=\psi_{X}(\beta(z))$ of
$H^{4}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes 2})$. As
$\beta$ is the Bockstein, $\beta(z)$ lifts compatibly to
$H^{3}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell^{m}}^{\otimes
2})$ for all $m$. Applying $\psi_{X}$ to these compatible lifts, we deduce
from (4.8) that $y$ lifts to a class $x\in
H^{1}(G,H^{3}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(2)))\subset
H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(2))$. Moreover, it follows
from (4.8) that
$x\in\mathrm{Ker}\big{(}H^{4}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell}(2))\to
H^{4}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}_{\ell}(2))\big{)}$.

It remains to prove that $x$ is not algebraic. We will prove the stronger
statement that $y$ is not algebraic. Recall from §2.2 the definition of the
finite étale cover $\pi:S^{\prime}_{\ell}\to S_{\ell}$ with Galois group
$\Gamma:=(\mathbb{Z}/\ell)^{*}$, set
$X^{\prime}:=X\times_{S_{\ell}}S^{\prime}_{\ell}$ and
$\overline{X}^{\prime}:=\overline{X}\times_{S_{\ell}}S^{\prime}_{\ell}$, and
define $\psi_{X^{\prime}}$ in the same way as $\psi_{X}$. For $\alpha=\beta
P^{1}$ (if $\ell$ is odd), or $\alpha\in\\{\mathrm{Sq}^{2},\mathrm{Sq}^{3}\\}$
(if $\ell=2$), the diagram

$\textstyle{H^{3}_{\mathrm{\acute{e}t}}(\overline{X}^{\prime},\mathbb{Z}/\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\hskip
10.50002pt}$$\scriptstyle{\psi_{X^{\prime}}}$$\textstyle{H^{3+\deg(\alpha)}_{\mathrm{\acute{e}t}}(\overline{X}^{\prime},\mathbb{Z}/\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{X^{\prime}}}$$\textstyle{H^{4}_{\mathrm{\acute{e}t}}(X^{\prime},\mathbb{Z}/\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\hskip
10.50002pt}$$\textstyle{H^{4+\deg(\alpha)}_{\mathrm{\acute{e}t}}(X^{\prime},\mathbb{Z}/\ell)}$

commutes by [SS22, Corollary 3.5]. Retaining the appropriate isotypic
components for the action of $\Gamma$, we deduce the commutativity of the
following diagram:

(4.9)
$\textstyle{H^{3}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell}^{\otimes
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\hskip
10.50002pt}$$\scriptstyle{\psi_{X}}$$\textstyle{H^{3+\deg(\alpha)}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{\ell}^{\otimes
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{X}}$$\textstyle{H^{4}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\hskip
10.50002pt}$$\textstyle{H^{4+\deg(\alpha)}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
r})\hbox to0.0pt{.\hss}}$

Assume first that $\ell$ is odd. As $\beta P^{1}\beta(z)$ does not belong to
the image of ${F-\mathrm{Id}}$ by Proposition 4.11, one has $\psi_{X}(\beta
P^{1}\beta(z))\neq 0$, hence $\beta P^{1}(y)\neq 0$ by the commutativity of
(4.9) for $\alpha=\beta P^{1}$. It follows from Proposition 3.7 (ii) and Lemma
3.5 that $\beta P^{1}$ kills
$H^{4}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{\ell}^{\otimes
2})_{\mathrm{alg}}$. Hence $y$ cannot be algebraic.

Assume now that $\ell=2$. As $\mathrm{Sq}^{3}\mathrm{Sq}^{1}(z)$ does not
belong to the image of ${F-\mathrm{Id}}$ by Proposition 4.11, one has
$\psi_{X}(\mathrm{Sq}^{3}\mathrm{Sq}^{1}(z))\neq 0$, hence
$\mathrm{Sq}^{3}(y)\neq 0$ by the commutativity of (4.9) for
$\alpha=\mathrm{Sq}^{3}$. The commutativity of (4.9) for
$\alpha=\mathrm{Sq}^{2}$ shows that $\mathrm{Sq}^{2}(y)\in
H^{1}(G,H^{5}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{2}^{\otimes
2}))\subset H^{6}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes 2})$.
In view of (4.7), one thus has

$\mathrm{Sq}^{2}(y)\in
F^{1}H^{6}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
2})=\mathrm{Ker}\big{(}H^{6}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
2})\to H^{6}_{\mathrm{\acute{e}t}}(\overline{X},\boldsymbol{\mu}_{2}^{\otimes
2})\big{)},$

where $F^{\bullet}H^{q}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
r})$ denotes the abutment filtration of the Hochschild–Serre spectral
sequence. Since $\omega\in
F^{1}H^{1}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/2)=\mathrm{Ker}\big{(}H^{1}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/2)\to
H^{1}_{\mathrm{\acute{e}t}}(\overline{X},\mathbb{Z}/2)\big{)}$, the
multiplicativity properties of Hochschild–Serre spectral sequences imply that
$\omega\cdot\mathrm{Sq}^{2}(y)\in
F^{2}H^{7}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes 2})$. This
group vanishes because $G$ has cohomological dimension $1$, so that
$\omega\cdot\mathrm{Sq}^{2}(y)=0$. It follows that
$(\mathrm{Sq}^{3}+\omega\cdot\mathrm{Sq}^{2})(y)\neq 0$. The class $y$ is not
algebraic because $\mathrm{Sq}^{3}+\omega\cdot\mathrm{Sq}^{2}$ kills
$H^{4}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes
2})_{\mathrm{alg}}$ by Proposition 3.12 (ii). ∎

###  4.5. A fourfold failing the integral Hodge conjecture over $\mathbb{R}$
but not over $\mathbb{C}$

In this paragraph, we work over $\mathbb{R}$. Let
$G:=\mathrm{Gal}(\mathbb{C}/\mathbb{R})\simeq\mathbb{Z}/2$ be generated by the
complex conjugation $\sigma$. Let $\mathbb{Z}(k)$ be the $G$-module whose
underlying abelian group is $\mathbb{Z}$, on which $\sigma$ acts by
$(-1)^{k}$. Let $\omega_{\mathbb{Z}/2}\in
H^{1}(G,\mathbb{Z}/2)=H^{1}_{G}(\mathrm{pt},\mathbb{Z}/2)\simeq\mathbb{Z}/2$
and ${\omega_{\mathbb{Z}}\in
H^{1}(G,\mathbb{Z}(1))=H^{1}_{G}(\mathrm{pt},\mathbb{Z}(1))\simeq\mathbb{Z}/2}$
be the generators. One has $\rho(\omega_{\mathbb{Z}})=\omega_{\mathbb{Z}/2}$.
We still denote by $\omega_{\mathbb{Z}/2}\in
H^{1}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$ and $\omega_{\mathbb{Z}}\in
H^{1}_{G}(X(\mathbb{C}),\mathbb{Z}(1))$ their pull-backs to the
$G$-equivariant Betti cohomology of any variety $X$ over $\mathbb{R}$. When
this causes no confusion, we denote any of these two classes by $\omega$. The
class $\omega\in H^{1}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$ corresponds to the one
defined in §2.2 through the isomorphism
$H^{*}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}/2)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{*}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$
between étale cohomology and equivariant Betti cohomology [Sch94, Corollary
15.3.1]. If $X$ is smooth, Krasnov ([Kra91, §2.1], see also [BW20a, §1.6.1])
has defined, for all $c\geq 0$, a cycle class map

(4.10) $\mathrm{cl}:\mathrm{CH}^{c}(X)\to
H^{2c}_{G}(X(\mathbb{C}),\mathbb{Z}(c)).$

The classes in the image of (4.10) are said to be algebraic.

Let $E$ be the real elliptic curve with set of complex points
$E(\mathbb{C})=\mathbb{C}/(\mathbb{Z}\oplus i\mathbb{Z})$, on which $\sigma\in
G$ acts through the usual complex conjugation of $\mathbb{C}$.

######  Proposition 4.13.

There exist $a\in H^{1}_{G}(E(\mathbb{C}),\mathbb{Z})$ and $b\in
H^{1}_{G}(E(\mathbb{C}),\mathbb{Z}(1))$ such that $a\cdot b\in
H^{2}_{G}(E(\mathbb{C}),\mathbb{Z}(1))$ is the cycle class of a real point
$P\in E(\mathbb{R})$.

###### Proof.

One has $H^{1}(E(\mathbb{C}),\mathbb{Z})\simeq\mathbb{Z}\oplus\mathbb{Z}(1)$
as $G$-modules, where the first factor is generated by the class
$a^{\prime}\in H^{1}(E(\mathbb{C}),\mathbb{Z})$ of the loop
$\mathbb{R}/\mathbb{Z}\hookrightarrow\mathbb{C}/(\mathbb{Z}\oplus
i\mathbb{Z})$ and the second factor by the class $b^{\prime}\in
H^{1}(E(\mathbb{C}),\mathbb{Z})$ of the loop
$i\mathbb{R}/i\mathbb{Z}\hookrightarrow\mathbb{C}/(\mathbb{Z}\oplus
i\mathbb{Z})$. We note that $a^{\prime}\cdot b^{\prime}\in
H^{2}(E(\mathbb{C}),\mathbb{Z})$ has degree $1$.

By work of Krasnov, the Hochschild–Serre spectral sequences

$E_{2}^{s,t}=H^{s}(G,H^{t}(E(\mathbb{C}),\mathbb{Z}(k))\implies
H^{s+t}_{G}(E(\mathbb{C}),\mathbb{Z}(k))$

degenerate (combine [Kra83, §5.1, Proposition 3.6, Corollary 3.4] and [Kra98,
Remark 1.2]). It follows that there exist $a\in
H^{1}_{G}(E(\mathbb{C}),\mathbb{Z})$ and $b\in
H^{1}_{G}(E(\mathbb{C}),\mathbb{Z}(1))$ lifting $a^{\prime}$ and $b^{\prime}$.
By Krasnov’s real Lefschetz $(1,1)$ theorem (see for instance [vH00, IV,
Theorem 4.1]), the class $a\cdot b\in H^{2}_{G}(E(\mathbb{C}),\mathbb{Z}(1))$
is the cycle class of a divisor $D$ on $E$. As $a^{\prime}\cdot b^{\prime}$
has degree $1$, the divisor $D$ has degree $1$. The Riemann–Roch theorem
applied on the elliptic curve $E$ now shows that $D$ is effective, hence
rationally equivalent to a point $P\in E(\mathbb{R})$. ∎

Let
$Q^{d}:=\\{\sum_{i=0}^{d+1}X_{i}^{2}=0\\}\subset\mathbb{P}^{d+1}_{\mathbb{R}}$
denote the $d$-dimensional anisotropic quadric.

######  Proposition 4.14.

If $n\geq 1$, then $\omega^{2^{n+1}-2}\in
H^{2^{n+1}-2}_{G}(Q^{2^{n}-1}(\mathbb{C}),\mathbb{Z}/2)$ is non-zero.

###### Proof.

The class $\omega^{2^{n+1}-2}\in
H^{2^{n+1}-2}_{G}(\mathbb{P}^{2^{n+1}-1}(\mathbb{C}),\mathbb{Z}/2)$ does not
vanish because it is non-zero in restriction to a real point of
$\mathbb{P}^{2^{n+1}-1}_{\mathbb{R}}$. It follows that the class
$\omega^{2^{n+1}-2}\in
H^{2^{n+1}-2}_{G}(Q^{2^{n+1}-2}(\mathbb{C}),\mathbb{Z}/2)$ is non-zero because
the restriction map

$H^{2^{n+1}-2}_{G}(\mathbb{P}^{2^{n+1}-1}(\mathbb{C}),\mathbb{Z}/2)\to
H^{2^{n+1}-2}_{G}(Q^{2^{n+1}-2}(\mathbb{C}),\mathbb{Z}/2)$

is injective by the weak Lefschetz theorem of [Del80, (4.1.6)] and comparison
with étale cohomology [Sch94, Corollary 15.3.1].

Pfister has shown the existence of a rational map
${Q^{2^{n+1}-2}\dashrightarrow Q^{2^{n}-1}}$ (see [Pfi65, Satz 5]). Resolving
its indeterminacies yields a smooth projective variety $X$ over $\mathbb{R}$,
a birational morphism $f:X\to Q^{2^{n+1}-2}$ and a morphism ${g:X\to
Q^{2^{n}-1}}$. One has
$f_{*}g^{*}\omega^{2^{n+1}-2}=f_{*}\omega^{2^{n+1}-2}=\omega^{2^{n+1}-2}$ in
$H^{2^{n+1}-2}_{G}(Q^{2^{n+1}-2}(\mathbb{C}),\mathbb{Z}/2)$ by the projection
formula. Since we have shown that this class does not vanish, we deduce that
$\omega^{2^{n+1}-2}\in
H^{2^{n+1}-2}_{G}(Q^{2^{n}-1}(\mathbb{C}),\mathbb{Z}/2)$ is non-zero, as
required. ∎

We now consider the smooth projective real fourfold
$X:=E\times_{\mathbb{R}}Q^{3}$.

######  Proposition 4.15.

There is a class
$x\in\mathrm{Ker}\big{(}H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))\to
H^{4}(X(\mathbb{C}),\mathbb{Z}(2))\big{)}$ such that
$\mathrm{Sq}^{3}(\rho(x))+\omega\cdot\mathrm{Sq}^{2}(\rho(x))+\omega^{3}\cdot\rho(x)\neq
0$ in $H^{7}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$.

###### Proof.

Let $a$, $b$ and $P$ be as in Proposition 4.13. Let $f:X\to E$ and $g:X\to
Q^{3}$ be the projections. Set $x:=f^{*}b\cdot\omega_{\mathbb{Z}}^{3}\in
H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(4))$. View it as an element of
$H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))$ using an isomorphism of $G$-modules
$\mathbb{Z}(4)\simeq\mathbb{Z}(2)$. The image of $x$ in
$H^{4}(X(\mathbb{C}),\mathbb{Z}(2))$ vanishes because so does the image of
$\omega_{\mathbb{Z}}$ in $H^{1}(X(\mathbb{C}),\mathbb{Z}(1))$.

As the elements $\omega$ and $f^{*}\rho(b)$ of
$H^{1}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$ lift to
$H^{1}_{G}(X(\mathbb{C}),\mathbb{Z}(1))$ and as $\mathrm{Sq}^{1}$ is the
Bockstein, one has $\mathrm{Sq}^{1}(\omega)=\omega^{2}$ and
$\mathrm{Sq}^{1}(f^{*}\rho(b))=f^{*}\rho(b)\cdot\omega$ (see for instance
[Ben19, §2.2]). Since $\rho(x)=f^{*}\rho(b)\cdot\omega^{3}$ in
$H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$, the Cartan formula implies that

$\mathrm{Sq}^{3}(\rho(x))+\omega\cdot\mathrm{Sq}^{2}(\rho(x))+\omega^{3}\cdot\rho(x)=f^{*}\rho(b)\cdot\omega^{6}$

in $H^{7}_{G}(X(\mathbb{C}),\mathbb{Z}/2)$. This class is non-zero because the
element

$g_{*}(f^{*}\rho(a)\cdot
f^{*}\rho(b)\cdot\omega^{6})=g_{*}(f^{*}\rho(\mathrm{cl}(P))\cdot\omega^{6})=\rho(g_{*}f^{*}\mathrm{cl}(P))\cdot\omega^{6}=\rho(\mathrm{cl}(X))\cdot\omega^{6}=\omega^{6}$

of $H^{6}_{G}(Q(\mathbb{C}),\mathbb{Z}/2)$ is non-zero by Proposition 4.14. ∎

We now reach the goal of this paragraph.

######  Theorem 4.16.

There exists a smooth projective variety $X$ of dimension $4$ over
$\mathbb{R}$ such that $X(\mathbb{R})=\varnothing$, and a non-algebraic class

$x\in\mathrm{Ker}\big{(}H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))\to
H^{4}(X(\mathbb{C}),\mathbb{Z}(2))\big{)}.$

###### Proof.

Let $X$ be as above and let $x\in H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))$ be
as in Proposition 4.15. One has $X(\mathbb{R})=\varnothing$ because $Q^{3}$ is
anisotropic. The inverse image $y\in
H^{4}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes 2})$ of $\rho(x)$
by the comparison isomorphism
$H^{4}_{\mathrm{\acute{e}t}}(X,\boldsymbol{\mu}_{2}^{\otimes 2})\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}/2(2))$
of [Sch94, Corollary 15.3.1] (which is $\mathcal{A}_{2}$-equivariant, see
§2.1) is not algebraic by Propositions 3.12 (ii) and 4.15. As a consequence,
neither is $x$. (This last assertion follows from the compatibility of cycle
class maps in étale and equivariant Betti cohomology. This result holds true,
but we know no published reference. Alternatively, one can use the fact that
the degree $2c$ classes that are algebraic are exactly those of coniveau $\geq
c$, both in étale and equivariant Betti cohomology, by purity (for which see
[SGA43, XIX, Théorèmes 3.2 et 3.4] and [BW20a, (1.21)]).) ∎

If $X$ is a smooth projective variety over $\mathbb{R}$, the real integral
Hodge conjecture for codimension $c$ cycles on $X$, defined in [BW20a,
Definition 2.2], is the statement that all the classes of
$H^{2c}_{G}(X(\mathbb{C}),\mathbb{Z}(c))$ whose images in
$H^{2c}(X(\mathbb{C}),\mathbb{Z})$ are Hodge, and that satisfy a topological
condition described in [BW20a, Definition 1.19] (and which is always satisfied
if $X(\mathbb{R})=\varnothing$), are algebraic.

######  Corollary 4.17.

There exists a smooth projective variety $X$ of dimension $4$ over
$\mathbb{R}$ such that $X$ fails the real integral Hodge conjecture but
$X_{\mathbb{C}}$ satisfies the usual (complex) integral Hodge conjecture.

###### Proof.

Let $X$ and $x$ be as in Theorem 4.16. The class $x\in
H^{4}_{G}(X(\mathbb{C}),\mathbb{Z}(2))$ is Hodge because it vanishes in
$H^{4}(X(\mathbb{C}),\mathbb{Z}(2))$. It satisfies the topological condition
appearing in the definition of the real integral Hodge conjecture because
${X(\mathbb{R})=\varnothing}$. As $x$ is not algebraic, the variety $X$ does
not satisfy the real integral Hodge conjecture.

Finally, as $H^{*}(Q^{3}(\mathbb{C}),\mathbb{Z})$ is algebraic and torsion-
free and $H^{2*}(E(\mathbb{C}),\mathbb{Z})$ is algebraic, the Künneth formula
implies that $H^{2*}(X(\mathbb{C}),\mathbb{Z})$ is algebraic, hence that
$X_{\mathbb{C}}$ satisfies the integral Hodge conjecture. ∎

######  Remarks 4.18.

(i) Three-dimensional counterexamples to the real integral Hodge conjecture
are constructed in [BW20a, Examples 4.7 and 4.8]. This is the smallest
dimension possible by [BW20a, Propositions 2.8 and 2.10].

(ii) The counterexamples to the real integral Hodge conjecture mentioned in
(i) are explained by a failure of the complex integral Hodge conjecture. A
$7$-dimensional counterexample to the real integral Hodge conjecture for which
the complex integral Hodge conjecture holds is described in [BW20a, Example
2.5]. The fourfold of Theorem 4.16 is the first counterexample of this kind in
dimension $\leq 6$. We do not know if some exist in dimension $3$ (a question
closely related to [BW20a, Question 4.9]).

(iii) Over the non-archimedean real closed field
$R:=\cup_{n}\mathbb{R}((t^{1/n}))$, there exist smooth projective threefolds
failing the (analogue in this context of the) real integral Hodge conjecture,
but for which the (analogue of the) complex integral Hodge conjecture holds,
see [BW20b, Examples 9.5, 9.6, 9.7, 9.10 and 9.22].

###  4.6. Applications to unramified cohomology

Let $X$ be a smooth variety over a field $k$. Let $\ell$ be a prime number
invertible in $k$, and let

$\mathrm{cl}:\mathrm{CH}^{2}(X)\to
H^{4}_{\mathrm{cont}}(X,\mathbb{Z}_{\ell}(2))$

be Jannsen’s cycle class map to continuous étale cohomology. As was discovered
by Colliot-Thélène and Voisin over the complex numbers [CTV12, Théorème 3.7],
and extended to arbitrary fields by Kahn [Kah12, Théorème 1.1], the torsion
subgroup $T$ of the cokernel of $\mathrm{cl}$ is a quotient of the third
unramified cohomology group $H^{3}_{\mathrm{nr}}(X,\mathbb{Q}/\mathbb{Z}(2))$.
It follows from the proofs of Theorems 4.2, 4.9, 4.12 and 4.16 that the smooth
projective varieties $X$ considered there satisfy $T\neq 0$. As a consequence,
one has $H^{3}_{\mathrm{nr}}(X,\mathbb{Q}/\mathbb{Z}(2))\neq 0$ for any of
these varieties.

##  5\. Algebraizability of cohomology classes of $\mathcal{C}^{\infty}$
manifolds

In this section, we study the algebraizability of cohomology classes of
compact $\mathcal{C}^{\infty}$ manifolds. We refer to §1.3 for context and for
the relevant definitions.

###  5.1. Steenrod operations and algebraizable classes

Our main results (Theorems 5.3 and 5.6 below) both rely on the fact that
Steenrod operations preserve algebraizable classes. This statement follows
from the next proposition, which is due to Akbulut and King [AK85, Theorem
6.6] at least when $X$ is projective. For the convenience of the reader, we
give a short proof based on the relative Wu theorem of Atiyah and Hirzebruch
[AH61, Satz 3.2]. Our argument is also closely related to [BH61, Proposition
5.19].

######  Proposition 5.1.

Let $X$ be a smooth variety over $\mathbb{R}$. Then
$H^{*}_{\mathrm{alg}}(X(\mathbb{R}),\mathbb{Z}/2)$ is stable by mod $2$
Steenrod operations.

###### Proof.

Let $g:Z\hookrightarrow X$ be an integral subvariety of codimension $c$ of $X$
and let $\nu:Y\to Z$ be a resolution of singularities of $Z$ (see [Hir64]).
Set $f:=g\circ\nu$ and denote by $f(\mathbb{R}):Y(\mathbb{R})\to
X(\mathbb{R})$ the map induced between real point sets. Let
$w(N_{f(\mathbb{R})})$ be the total Stiefel–Whitney class of the virtual
normal bundle ${N_{f(\mathbb{R})}:=f^{*}T_{X(\mathbb{R})}-T_{Y(\mathbb{R})}}$
of $f(\mathbb{R})$. The relative Wu theorem of Atiyah and Hirzebruch [AH61,
Satz 3.2] applied to the map $f(\mathbb{R})$ and to the class $1\in
H^{0}(Y(\mathbb{R}),\mathbb{Z}/2)$ (exactly as in [BW20a, (1.60) and Remark
1.6 (i)]) implies that

(5.1) $\mathrm{Sq}(f(\mathbb{R})_{*}1)=f(\mathbb{R})_{*}w(N_{f(\mathbb{R})}).$

One has $w(T_{X(\mathbb{R})})\in
H^{*}_{\mathrm{alg}}(X(\mathbb{R}),\mathbb{Z}/2)$ and $w(T_{Y(\mathbb{R})})\in
H^{*}_{\mathrm{alg}}(Y(\mathbb{R}),\mathbb{Z}/2)$ (see [BK98, Theorem 1.5]
when $X$ and $Y$ are projective; in general, one may combine [Kah87, Theorem
4] and [BW20a, Theorem 1.18]). As algebraic classes are stable by pull-backs,
proper pushforwards and cup-product (see [BK98, Theorems 1.3 and 1.4] under
projectivity hypotheses and [BW20a, §1.6.2] in general), the class (5.1) is
algebraic. Since $H^{*}_{\mathrm{alg}}(X(\mathbb{R}),\mathbb{Z}/2)$ is
generated by classes of the form
$\mathrm{cl}_{\mathbb{R}}(Z)=f(\mathbb{R})_{*}1$, the proof is now complete. ∎

Proposition 5.1 implies at once the following corollary.

######  Corollary 5.2.

Let $M$ be a compact $\mathcal{C}^{\infty}$ manifold. The subset of
$H^{*}(M,\mathbb{Z}/2)$ consisting of algebraizable classes is stable by mod
$2$ Steenrod operations.

###  5.2. Examples of non-algebraizable classes

Here is our first main result.

######  Theorem 5.3.

For all $c\geq 2$, there exists a compact $\mathcal{C}^{\infty}$ manifold $M$
and a class $x\in H^{c}(M,\mathbb{Z}/2)$ that is not algebraizable.

###### Proof.

Grant and Szücs have constructed in [GS13, Proof of Theorem 1.1 p. 334] a
compact $\mathcal{C}^{\infty}$ manifold $M$ and a class $x\in
H^{c}(M,\mathbb{Z}/2)$ such that $x^{2}$ is not in the image of the reduction
modulo $2$ map $H^{2c}(M,\mathbb{Z})\to H^{2c}(M,\mathbb{Z}/2)$ (if $c$ is
even), or such that $(\mathrm{Sq}^{2}\mathrm{Sq}^{1}x)^{2}$ is not in the
image of the reduction modulo $2$ map $H^{2c+6}(M,\mathbb{Z})\to
H^{2c+6}(M,\mathbb{Z}/2)$ (if $c$ is odd).

Akbulut and King have shown in [AK93, Theorem A (b)] that squares of
algebraizable classes are reductions modulo $2$ of integral classes. We deduce
that $x$ is not algebraizable (if $c$ is even) and that
$\mathrm{Sq}^{2}\mathrm{Sq}^{1}x$ is not algebraizable (if $c$ is odd).
Theorem 5.2 now shows that $x$ is not algebraizable even when $c$ is odd. ∎

######  Remark 5.4.

When $c$ is even, the proof of Proposition 5.3 is exactly the one given in
[Kuc08, Theorem 1.13].

###  5.3. Examples of algebraizable classes

Before stating our second main result, we recall the properties of Thom spaces
that will be used in its proof. Let $BO(n)$ be the classifying space for
$O(n)$, constructed as the increasing union of the Grassmannians
$\mathrm{Grass}(n,\mathbb{R}^{n+m})$ of $n$-planes in $\mathbb{R}^{n+m}$ (for
$m\geq 0$). Let $E(n)$ be the tautological rank $n$ topological real vector
bundle on $BO(n)$ ($BO(n)$ and $E(n)$ are denoted by $G_{n}$ and $\gamma^{n}$
in [MS74, p. 63]). One has

(5.2) $H^{*}(\mathrm{BO}(n),\mathbb{Z}/2)=\mathbb{Z}/2[w_{1},\dots,w_{n}]$

where the $w_{i}$ are the Stiefel–Whitney classes of $E(n)$ (see [MS74,
Theorem 7.1]).

Let $\mathrm{MO}(n)$ be the Thom space of $E(n)$ (defined in [MS74, p. 205]).
Denoting by $s_{n}\in H^{n}(\mathrm{MO}(n),\mathbb{Z}/2)$ the Thom class of
$E(n)$, the Thom isomorphism theorem [MS74, Theorem 10.2] shows that cup-
product by $s_{n}$ induces an isomorphism
$H^{*}(\mathrm{BO}(n),\mathbb{Z}/2)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\widetilde{H}^{*+n}(\mathrm{MO}(n),\mathbb{Z}/2)$.
By Thom’s definition of Stiefel–Whitney classes (see [MS74, p. 91]), one has

(5.3) $\mathrm{Sq}^{i}(s_{n})=w_{i}\cdot s_{n}.$

In particular, one computes that

(5.4) $s_{n}^{2}=\mathrm{Sq}^{n}(s_{n})=w_{n}\cdot s_{n}.$

Moreover, the Wu formula [MS74, Problem 8-B] shows that

(5.5) $\mathrm{Sq}^{i}(w_{j})=\sum_{t=0}^{i}\binom{j+t-i-1}{t}w_{i-t}w_{j+t},$

where it is understood that $w_{j}=0$ in $H^{*}(\mathrm{BO}(n),\mathbb{Z}/2)$
for $j>n$. These pieces of information and Cartan’s formula entirely determine
$H^{*}(\mathrm{MO}(n),\mathbb{Z}/2)$ as an algebra endowed with an action of
the mod $2$ Steenrod algebra.

Let $M$ be a compact $\mathcal{C}^{\infty}$ manifold and let $x\in
H^{n}(M,\mathbb{Z}/2)$ be a cohomology class. It follows from the
Thom–Pontrjagin construction [Tho54, Théorème II.1] that $x$ is Poincaré-dual
to the fundamental class of a $\mathcal{C}^{\infty}$ submanifold of
codimension $n$ in $M$ if and only if there exists a continuous map
$f:M\to\mathrm{MO}(n)$ such that $x=f^{*}s_{n}$.

We also include a proof of the following well-known lemma for lack of a
suitable reference.

######  Lemma 5.5.

Let $K$ be a finite $CW$-complex. For all $N\geq 0$, there exists a compact
$\mathcal{C}^{\infty}$ manifold $M$ and an $N$-equivalence $f:M\to K$.

###### Proof.

By [Hat02, Theorem 2C.5], we may assume that $K$ is a finite simplicial
complex. Viewing $K$ as a subcomplex of a simplex of high dimension, we may
embed it piecewise linearly in $\mathbb{R}^{d}$ for $d\gg 0$. Let
$\Theta\subset\mathbb{R}^{d}$ be a neighbourhood of $K$ that is a smooth
regular neighbourhood as in [Hir62, Theorem 1 (a) and (a’)], and set
$M:=\partial\Theta$. It follows from [HZ64, Corollary 1 p. 725] that the
inclusion $M\hookrightarrow\Theta\setminus K$ is a weak homotopy equivalence,
the inclusion $\Theta\setminus K\hookrightarrow\Theta$ is an $N$-equivalence
because $K$ has high codimension in $\Theta$ when $d\gg 0$, and the inclusion
$K\hookrightarrow\Theta$ is a homotopy equivalence by choice of $\Theta$.
These facts prove the lemma. ∎

We now reach the goal of this paragraph.

######  Theorem 5.6.

There exists a compact $\mathcal{C}^{\infty}$ manifold $M$ and an
algebraizable class $x\in H^{5}(M,\mathbb{Z}/2)$ that does not belong to the
subring $A(M)$ of $H^{*}(M,\mathbb{Z}/2)$ generated by Stiefel–Whitney classes
of vector bundles on $M$ and Poincaré-duals of fundamental classes of
$\mathcal{C}^{\infty}$ submanifolds of $M$.

###### Proof.

Let $K$ be the Thom space of the tautological rank $n$ vector bundle on
$\mathrm{Grass}(3,\mathbb{R}^{3+m})$. Comparing [MS74, Theorem 7.1 and Problem
7-B] and applying the Thom isomorphim theorem [MS74, Theorem 10.2] shows that
the restriction map $H^{*}(MO(3),\mathbb{Z}/2)\to H^{*}(K,\mathbb{Z}/2)$ is an
isomorphism in degree $\leq 18$ if $m\gg 0$. As $K$ is a finite CW-complex, it
follows from Lemma 5.5 that there exists a compact $\mathcal{C}^{\infty}$
manifold $M$ and a continuous map $f:M\to\mathrm{MO}(3)$ such that the pull-
back map $f^{*}:H^{*}(\mathrm{MO}(3),\mathbb{Z}/2)\to H^{*}(M,\mathbb{Z}/2)$
is an isomorphism in degree $\leq 18$.

By the Thom–Pontrjagin construction, the class $f^{*}s_{3}$ is Poincaré-dual
to the fundamental class of a $\mathcal{C}^{\infty}$ submanifold of $M$, hence
is algebraizable by [BT80, Corollary 4.5] or [AK81, Theorem 1.1]. It follows
Corollary 5.2 that the class $x:=f^{*}(w_{2}\cdot s_{3})\in
H^{5}(M,\mathbb{Z}/2)$, which is equal to $\mathrm{Sq}^{2}(f^{*}s_{3})$ by
(5.3), is algebraizable.

Assume for contradiction that $x\in A(M)$. As $H^{q}(M,\mathbb{Z}/2)=0$ for
$q\in\\{1,2\\}$ and $H^{5}(M,\mathbb{Z}/2)$ is generated by
$x=f^{*}(w_{2}\cdot s_{3})$ and by $y:=f^{*}((w_{1}^{2}+w_{2})\cdot s_{3})$,
we deduce that there exists $z\in\\{x,y\\}$ that is either the fifth
Stiefel–Whitney class of a vector bundle $E$ on $M$, or Poincaré-dual to the
fundamental class of a $\mathcal{C}^{\infty}$ submanifold of codimension $5$
in $M$. In the first case, one has

$\mathrm{Sq}^{1}(z)=\mathrm{Sq}^{1}(w_{5}(E))=w_{1}(E)\cdot w_{5}(E)=0$

in $H^{6}(M,\mathbb{Z}/2)$, where the second equality follows from (5.5) and
the third equality holds since $H^{1}(M,\mathbb{Z}/2)=0$. One then computes
(using the Cartan formula, (5.3) and (5.5)) that
$\mathrm{Sq}^{1}(x)=f^{*}(w_{3}\cdot s_{3})$ and
$\mathrm{Sq}^{1}(y)=f^{*}((w_{1}^{3}+w_{3})\cdot s_{3})$. This is a
contradiction since both classes are non-zero by the computation of
$H^{*}(\mathrm{MO}(3),\mathbb{Z}/2)$ recalled above and by the injectivity of
$f^{*}$ in degree $6$.

In the second case, the Thom–Pontrjagin construction shows the existence of a
continuous map $g:M\to\mathrm{MO}(5)$ with $z=g^{*}s_{5}$. Using the Cartan
formula, (5.3) and (5.5), one computes that

$\mathrm{Sq}^{2}\mathrm{Sq}^{1}(s_{5})\cdot
s_{5}^{2}=\mathrm{Sq}^{2}(w_{1}\cdot s_{5})\cdot s_{5}^{2}=w_{1}^{3}\cdot
s_{5}^{3}+w_{1}w_{2}\cdot
s_{5}^{3}=\mathrm{Sq}^{1}(s_{5})^{3}+\mathrm{Sq}^{1}(s_{5})\cdot\mathrm{Sq}^{2}(s_{5})\cdot
s_{5}.$

Pulling back this equality by the map $g$, we deduce that

(5.6) $\mathrm{Sq}^{2}\mathrm{Sq}^{1}(z)\cdot
z^{2}+\mathrm{Sq}^{1}(z)^{3}+\mathrm{Sq}^{1}(z)\cdot\mathrm{Sq}^{2}(z)\cdot
z=0$

in $H^{18}(M,\mathbb{Z}/2)$. Using the Cartan formula, (5.3), (5.4) and (5.5),
one shows that

$\displaystyle\mathrm{Sq}^{2}\mathrm{Sq}^{1}(x)$ $\displaystyle\cdot
x^{2}+\mathrm{Sq}^{1}(x)^{3}+\mathrm{Sq}^{1}(x)\cdot\mathrm{Sq}^{2}(x)\cdot x$
$\displaystyle=f^{*}(w_{1}^{2}w_{3}s_{3}\cdot(w_{2}s_{3})^{2}+(w_{3}s_{3})^{3}+w_{3}s_{3}\cdot(w_{1}^{2}w_{2}s_{3}+w_{1}w_{3}s_{3})\cdot
w_{2}s_{3})$ $\displaystyle=f^{*}((w_{1}w_{2}w_{3}^{4}+w_{3}^{5})\cdot
s_{3}),$

and a similar lengthier computation shows that

$\displaystyle\mathrm{Sq}^{2}\mathrm{Sq}^{1}(y)\cdot y^{2}$
$\displaystyle+\mathrm{Sq}^{1}(y)^{3}+\mathrm{Sq}^{1}(y)\cdot\mathrm{Sq}^{2}(y)\cdot
y$
$\displaystyle=f^{*}((w_{1}^{3}w_{2}^{3}w_{3}^{2}+w_{1}^{2}w_{2}^{2}w_{3}^{3}+w_{1}w_{2}w_{3}^{4}+w_{3}^{5})\cdot
s_{3}).$

As both these classes are non-zero by the computation of
$H^{*}(\mathrm{MO}(3),\mathbb{Z}/2)$ recalled above and by the injectivity of
$f^{*}$ in degree $18$, this contradicts (5.6). ∎

## References

  * [Ada74] J. F. Adams, _Stable homotopy and generalised homology_ , Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974.
  * [AH61] M. F. Atiyah and F. Hirzebruch, _Cohomologie-Operationen und charakteristische Klassen_ , Math. Z. 77 (1961), 149–187.
  * [AH62] by same author, _Analytic cycles on complex manifolds_ , Topology 1 (1962), 25–45.
  * [AK81] S. Akbulut and H. King, _A relative Nash theorem_ , Trans. Amer. Math. Soc. 267 (1981), no. 2, 465–481.