\documentclass[CRMATH,Unicode,biblatex,XML]{cedram}

\TopicFR{Algèbre}
\TopicEN{Algebra}
\TopicFR{Géométrie algébrique}
\TopicEN{Algebraic geometry}


\addbibresource{CRMATH_Karpenko_20240177.bib}


\usepackage{mathtools}
\usepackage{amscd}
\usepackage{amssymb}


\renewcommand{\Im}{\mathop{\mathrm{Im}}}

\DeclareMathOperator{\CH}{CH}
\DeclareMathOperator{\BCH}{\bar{CH}}
\DeclareMathOperator{\Ch}{Ch}
\DeclareMathOperator{\BCh}{\bar{Ch}}
\DeclareMathOperator{\cd}{cd}
\DeclareMathOperator{\ed}{ed}

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\<}{\left<}
\renewcommand{\>}{\right>}
\renewcommand{\ll}{\<\mk\<}
\renewcommand{\phi}{\varphi}
\newcommand{\RatM}{\dashrightarrow}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
\let\save@mathaccent\mathaccent
\newcommand*\if@single[3]{%
 \setbox0\hbox{${\mathaccent"0362{#1}}^H$}%
 \setbox2\hbox{${\mathaccent"0362{\kern0pt#1}}^H$}%
 \ifdim\ht0=\ht2 #3\else #2\fi
 }
%The bar will be moved to the right by a half of \macc@kerna, which is computed by amsmath:
\newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
%If there's a superscript following the bar, then no negative kern may follow the bar;
%an additional {} makes sure that the superscript is high enough in this case:
\newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}}
%Use a separate algorithm for single symbols:
\newcommand*\wide@bar[2]{\if@single{#1}{\wide@bar@{#1}{#2}{1}}{\wide@bar@{#1}{#2}{2}}}
\newcommand*\wide@bar@[3]{%
 \begingroup
 \def\mathaccent##1##2{%
%Enable nesting of accents:
 \let\mathaccent\save@mathaccent
%If there's more than a single symbol, use the first character instead (see below):
 \if#32 \let\macc@nucleus\first@char \fi
%Determine the italic correction:
 \setbox\z@\hbox{$\macc@style{\macc@nucleus}_{}$}%
 \setbox\tw@\hbox{$\macc@style{\macc@nucleus}{}_{}$}%
 \dimen@\wd\tw@
 \advance\dimen@-\wd\z@
%Now \dimen@ is the italic correction of the symbol.
 \divide\dimen@ 3
 \@tempdima\wd\tw@
 \advance\@tempdima-\scriptspace
%Now \@tempdima is the width of the symbol.
 \divide\@tempdima 10
 \advance\dimen@-\@tempdima
%Now \dimen@ = (italic correction / 3) - (Breite / 10)
 \ifdim\dimen@>\z@ \dimen@0pt\fi
%The bar will be shortened in the case \dimen@<0 !
 \rel@kern{0.6}\kern-\dimen@
 \if#31
 \overline{\rel@kern{-0.6}\kern\dimen@\macc@nucleus\rel@kern{0.4}\kern\dimen@}%
 \advance\dimen@0.4\dimexpr\macc@kerna
%Place the combined final kern (-\dimen@) if it is >0 or if a superscript follows:
 \let\final@kern#2%
 \ifdim\dimen@<\z@ \let\final@kern1\fi
 \if\final@kern1 \kern-\dimen@\fi
 \else
 \overline{\rel@kern{-0.6}\kern\dimen@#1}%
 \fi
 }%
 \macc@depth\@ne
 \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
 \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
 \macc@set@skewchar\relax
 \let\mathaccentV\macc@nested@a
%The following initialises \macc@kerna and calls \mathaccent:
 \if#31
 \macc@nested@a\relax111{#1}%
 \else
%If the argument consists of more than one symbol, and if the first token is
%a letter, use that letter for the computations:
 \def\gobble@till@marker##1\endmarker{}%
 \futurelet\first@char\gobble@till@marker#1\endmarker
 \ifcat\noexpand\first@char A\else
 \def\first@char{}%
 \fi
 \macc@nested@a\relax111{\first@char}%
 \fi
 \endgroup
}
\makeatother

\let\oldbar\bar
\renewcommand*{\bar}[1]{\mathchoice{\widebar{#1}}{\widebar{#1}}{\widebar{#1}}{\oldbar{#1}}}

\let\oldtilde\tilde
\renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}}
%\let\tilde\widetilde

%\let\hat\widehat
\let\oldhat\hat
\renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}}
%\let\tilde\widetilde

\renewcommand*{\to}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}
\let\oldmapsto\mapsto
\renewcommand*{\mapsto}{\mathchoice{\longmapsto}{\oldmapsto}{\oldmapsto}{\oldmapsto}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}


\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\title
[Quadratic forms in $I^n$]
{Quadratic forms in $I^n$ 
of dimension $2^n+2^{n-1}$}

\alttitle{Formes quadratiques de dimension $2^n+2^{n-1}$ dans $I^n$}

\keywords{\kwd{Quadratic forms over fields}
\kwd{projective homogeneous varieties}
\kwd{Chow rings}}


\altkeywords{\kwd{Formes quadratiques sur des corps} \kwd{vari\'et\'es projectives homog\`enes} \kwd{anneaux de Chow.}}

\subjclass{11E04, 14C25}





\author
{\firstname{Curtis R.} \lastname{Harvey}}

\address{Mathematical \& Statistical Sciences,
University of Alberta, Edmonton, Canada}


\email
{rcharvey@ualberta.ca}


\author
{\firstname{Nikita A.} \lastname{Karpenko}\IsCorresp}


\address[1]{Mathematical \& Statistical Sciences, 
University of Alberta, Edmonton, Canada}


\email{karpenko@ualberta.ca}

\urladdr{www.ualberta.ca/~karpenko}




\thanks{This work has been supported by a Discovery Grant from the National Science and Engineering Research Council of Canada and completed when the second author was a visitor at the Universit\'e de Lorraine}

\begin{abstract}
For $n\geq3$,
confirming a weak version of a conjecture of Hoffmann,
we show that every anisotropic quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite extension of the base field
of degree not divisible by $4$.
The first new case is $n=4$, where we obtain a classification of the corresponding quadratic forms up to odd degree base field extensions
and get this way a strong upper bound on their essential $2$-dimension. As well,
we compute the reduced Chow group of the maximal orthogonal grassmannian of the quadratic form and
conclude that its canonical $2$-dimension is $2^n+2^{n-2}-2$.
\end{abstract}

\begin{altabstract}
Pour $n\geq3$, en confirmant une version faible d'une conjecture de Hoffmann, on montre que toute forme quadratique anisotrope
de dimension $2^n+2^{n-1}$ dans $I^n$ se d\'eploie sur une extension finie du corps de base d'un degr\'e qui n'est pas divisible par $4$.
Le premier nouveau cas est celui de $n=4$,
o\`u l'on obtient une classification des formes quadratiques correspondantes \`a une extension de degr\'e impair pr\`es ce qui donne une forte borne sup\'erieure pour leur $2$-dimension essentielle. De plus, on d\'etermine le groupe de Chow r\'eduit de la grassmannienne orthogonale maximale de la forme quadratique et on en d\'eduit que sa dimension $2$-canonique est \'egale \`a $2^n+2^{n-2}-2$.
\end{altabstract}

\begin{document}

\maketitle

Let $F$ be a field (of any characteristic) and let $I=I(F)$ be the Witt group of classes of
even-dimensional non-degenerate quadratic forms over $F$ defined as in~\cite[Section~8]{EKM}
(and denoted $I_q(F)$ there).
For $n\geq2$, we write $I^n=I^n(F)$ for the subgroup in $I(F)$ generated by the $n$-fold Pfister forms.
We refer to~\cite[9.B]{EKM} for other equivalent definitions of $I^n(F)$ (denoted $I_q^n(F)$ there).

Any element of $I$ is represented by an anisotropic quadratic form.
By the Arason--Pfister Hauptsatz,
the smallest possible dimension of a nonzero anisotropic quadratic form in $I^n$ is $2^n$\break
(see~\cite[Theorem~23.7(1)]{EKM} for the characteristic-free version).
The quadratic forms in $I^n$ of dimension $2^n$ are classified:
as a consequence of the Arason--Pfister Hauptsatz and~\cite[Corollary~23.4)]{EKM},
they are exactly the forms similar to $n$-fold Pfister forms.
In particular, any $2^n$-dimensional quadratic form in $I^n$ splits over a finite base field extension of degree dividing $2$.

The smallest possible dimension exceeding $2^n$ of an anisotropic quadratic form in $I^n$
is $2^n+2^{n-1}$.
For $n=3$ this has been shown in~\cite{MR0200270} (characteristic $\ne2$) and in~\cite{MR0491773} (arbitrary characteristic);
for $n=4$ in~\cite{MR1489898} (characteristic $\ne2$) and in~\cite[Theorem~4.2.11]{Faivre} (characteristic $2$);
for arbitrary $n$ and characteristic $\ne2$ a proof has been given in~\cite[Theorem~5.4]{MR2066515} and then
extended to characteristic $2$ in~\cite[Proposition~11.5]{primozic}.

For $n=2$, quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ are the well-understood Albert forms.
For $n\geq3$,
by a conjecture of Hoffmann (\cite[Conjecture~2]{MR1489898} for characteristic $\ne2$ and~\cite[Conjecture~4.3.1]{Faivre} for characteristic $2$),
quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ should be classified as products of an Albert
bilinear form (i.e., a $6$-dimensional symmetric bilinear form of determinant $-1$) by a Pfister form
(of foldness $n-2$).
In particular,
%for $n\geq3$ and in any characteristic,
such forms should split over a finite base field extension of degree dividing $2$ as well.

However, the two above conjectures are so far proved for $n=3$ only:
the proof for characteristic $\ne2$ of~\cite{MR0200270} is extended to characteristic $2$ in~\cite[Proposition~4.1.2]{Faivre}.
The main result of the present note is

\begin{theo}
\label{main}
For any $n\geq3$ and in any characteristic,
every quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite base field extension
of degree not divisible by $4$.
\end{theo}

\begin{proof}
Let $X$ be a connected component of the highest orthogonal grassmannian of a quadratic form $q$ in $I^n(F)$ of dimension $2^n+2^{n-1}$.
Theorem~\ref{main} means that the {\em index} $i(X)$ of the variety $X$, defined as the g.c.d. of the degrees of closed points on $X$,
divides $2$.
In other terms, taking into account Springer's Theorem~\cite[Corollary~18.5]{EKM}, $i(X)=2$ provided that $q$ is not split.

We write $\bar{X}$ for $X$ over an algebraic closure of $F$ and we write $\BCH(X)$ for the ring given by the image of the change of
field homomorphism $\CH(X)\to\CH(\bar{X})$ of the Chow rings.
Note that the kernel of the change of field homomorphism is the ideal of the elements of finite order.
For this reason, $\BCH(X)$ is sometimes called the {\em reduced Chow group} of $X$.

By~\cite[Theorem~86.12]{EKM}, the ring $\CH(\bar{X})$ is generated by certain homogeneous elements $e_1,\dots,e_l$ of codimensions
$1,\dots,l:=2^{n-1}+2^{n-2}-1$.
It is convenient to define $e_i:=0$ for $i>l$.
For any $i\geq1$, the element $e_i$ is characterized by the property that $(-1)^i2e_i$ is the $i$th Chern class of the tautological vector bundle on $\bar{X}$ (see~\cite[Proposition~87.13]{EKM});
in particular, $2e_i\in\BCH(X)$.

Since for any field extension $K/F$, the anisotropic part of the quadratic form $q_K$ over the field $K$ is
either $0$, or $2^n$, or $2^n+2^{n-1}$,
it follows by~\cite[Corollary~88.6]{EKM} (see also~\cite[Corollary~88.7]{EKM}) that $e_i\in\BCH(X)$ for all $i$ different from
$k:=2^{n-1}-1$ and $l$.
By~\cite[(86.15)]{EKM}, for any $i\geq1$, we have
$$
e_i^2-2e_{i-1}e_{i+1}+2e_{i-2}e_{i+2}-\dots+(-1)^{i-1}2e_1e_{2i-1}+(-1)^ie_{2i}=0\in\CH(\bar{X}).
$$
In particular,
%$e_i^2\in2\CH(\bar{X})$ for $i\geq d'$.
$$
2e_ke_l=2e_{k+1}e_{l-1}-2e_{k+2}e_{l-2}+\dots\pm2e_{m-1}e_{m+1}\pm e^2_m\in\BCH(X),
$$
where $m:=(k+l)/2$.
Therefore $2e\in\BCH(X)$, where
$e\in\CH(\bar{X})$ is the product $e_1\ldots e_l$ of all the generators.
Since $e$ is the class of a $0$-cycle of degree $1$ (see~\cite[Corollary~86.10]{EKM}),
the variety $X$ (over $F$) possesses a $0$-cycle of degree $2$.
\end{proof}


For $n=4$,
%in characteristic $\ne2$,
in view of~\cite[Proposition~4.1]{MR1489898} and~\cite[Proposition~4.3.2]{Faivre},
Theorem~\ref{main} provides a classification of
the corresponding quadratic forms ``up to odd degree extensions''
which yields a strong upper bound on their {\em essential $2$-dimension}.
We provide details right below, starting with the classification result:

\begin{theo}
\label{class}
For a field $F$
(of any characteristic),
%of characteristic $\ne2$,
let $q$ be a quadratic form in $I^4(F)$ of dimension $24=2^4+2^3$.
Then there exists a finite field extension $K/F$ of odd degree such that $q_K$
is isomorphic to the tensor product of an Albert bilinear form by a Pfister form.
\end{theo}

\begin{proof}
By Theorem~\ref{main}, we can find a finite field extension $K/F$ of odd degree
and a field extension $L/K$ of degree dividing $2$ such that $q_L$ is split.
The description of $q_K$ then follows from~\cite[Proposition~4.1]{MR1489898} (for characteristic $\ne2$)
and~\cite[Proposition~4.3.2]{Faivre} (for characteristic $2$).
\end{proof}

To formulate the result on the essential $2$-dimension, let us consider the functor $I^4_{24}$,
associating to every extension field $K$ of a fixed field $F$ the set of isomorphism classes of quadratic forms
in $I^4(K)$ of dimension $24$.
The essential $2$-dimension of an element in $I^4_{24}(K)$ as well as the essential $2$-dimension
$\ed_2 I^4_{24}$ of the functor $I^4_{24}$ are defined as in~\cite[Section~1]{MR3212863}.

\begin{coro}
\label{ess}
%For a base field of characteristic $\ne2$,
One has $\ed_2I^4_{24}\leq7$.
\end{coro}

\begin{proof}
Writing $F$ for the base field and taking $q\in I^4_{24}(K)$ for a field extension $K/F$,
we find by Theorem~\ref{class} an odd degree field extension $L/K$ such that
$q_L$ is isomorphic to the tensor product of the diagonal Albert bilinear form
$\<a_1,a_2,a_3,a_4,a_5,-a_1a_2a_3a_4a_5\>$ by the Pfister form $\ll b_1,b_2\right]\right]$
for some nonzero $a_1,\dots,a_5,b_1\in L$ and some $b_2\in L$, where in characteristic $\ne2$ the element $b_2$ is also
nonzero.
The subfield $F(a_1,\dots,a_5,b_1,b_2)\subset L$,
whose transcendence degree over $F$ is at most $7$, is then a {\em field of definition} of $q_L$.
It follows that $\ed_2q=\ed_2q_L\leq 7$ and so $\ed_2 I^4_{24}\leq 7$.
\end{proof}

Recall that the essential $2$-dimension is a $2$-local version of and constitutes a lower bound for the {\em essential dimension},
measuring, informally speaking, how many independent parameters are required to describe an isomorphism class of the corresponding type of objects;
in particular, $\ed_2I^4_{24}\leq\ed I^4_{24}$.
For $n=3$, since the description of the corresponding quadratic forms does not involve odd degree extensions,
similar to the proof of Corollary~\ref{ess} arguments show that $\ed_2I^3_{12}\leq\ed I^3_{12}\leq6$.
In fact, in characteristic $\ne2$,
$\ed_2I^3_{12}=\ed I^3_{12}=6$ by~\cite[Theorem~7.1]{MR3212863}:
the lower bound $6\leq\ed_2I^3_{12}$ is obtained by constructing a nontrivial degree $6$ cohomological invariant
with coefficients in $\Z/2\Z$ for $I^3_{12}$.

For $n\geq4$, assuming~\cite[Conjecture~2]{MR1489898}, one gets
$$
\ed_2I^n_{2^n+2^{n-1}}\leq\ed I^n_{2^n+2^{n-1}}\leq n+3.
$$

Finally, one has $\ed_2I^n_{2^n}=\ed I^n_{2^n}=n+1$ for any $n$.
Indeed, as already mentioned, any $q\in I^n_{2^n}$ is isomorphic to $b\cdot\ll b_1,b_2,\dots,b_n\right]\right]$
for some $n+1$ parameters $b,b_1,\dots,b_n$, ensuring that $n+1$ is an upper bound for $\ed I^n_{2^n}$.
On the other hand, associating in characteristic $\ne2$ to $q$ the symbol $(b,b_1,\dots,b_n)$
in the ($n+1)$st Galois cohomology group with coefficients in $\Z/2\Z$, one gets
a nontrivial degree $n+1$ cohomological invariant showing that $n+1$ is a lower bound for $\ed_2I^n_{2^n}$
(see~\cite[Theorem~3.4]{merk-newess}).
The non-triviality of the cohomological invariant is shown in~\cite[Section~3]{MR2029168}.
The characteristic $2$ case is treated similarly using cohomological invariants with values in
\'etale motivic cohomology groups (cf.~\cite[Section~3]{MR3941464}
and especially~\cite[Proof of Lemma~3.1]{MR3941464});
the non-triviality of the cohomological invariant follows from~\cite{MR1386649}.

\medskip
To conclude, let us return to the case of arbitrary $n\geq2$.
Let $X$ be the highest orthogonal grassmannian of an anisotropic quadratic form $q\in I^n$.
If $\dim q=2^n$, then $i(X)=2$ and therefore the ring $\BCH(X)$ contains $2\CH(\bar{X})$.
By~\cite[Corollary~88.6]{EKM}, $\BCH(X)$ also contains the elements $e_1,\dots,e_{2^{n-1}-2}$
-- the generators of the ring $\CH(\bar{X})$ with exception of the very last one $e_{2^{n-1}-1}$.
Since $i(X)\ne1$, we conclude that $\BCH(X)$ is exactly the subring in $\CH(\bar{X})$ generated by
$2\CH(\bar{X})$ and $e_1,\dots,e_{2^{n-1}-2}$ (cf.~\cite[Example~88.10]{EKM}).

Now let us assume that $\dim q=2^n+2^{n-1}$, where $n\geq3$.
Since $i(X)=2$ by Theorem~\ref{main}, we still have the inclusion
$\BCH(X)\supset2\CH(\bar{X})$.
Besides, it has been shown in the proof of Theorem~\ref{main} that
$\BCH(X)\ni e_i$ for all $i$ except $i=k:=2^{n-1}-1$ and $i=l:=2^{n-1}+2^{n-2}-1$.

\begin{theo}
\label{BCH}
For any $n\geq3$ and any anisotropic
quadratic form $q$ in $I^n$ of dimension $2^n+2^{n-1}$,
the ring $\BCH(X)$ of its highest grassmannian $X$ is
generated by $2\CH(\bar{X})$ and all $e_i$ with $i\not\in\{k,\;l\}$.
\end{theo}

\begin{proof}
Since $\BCH(X)\supset2\CH(\bar{X})$, it suffices to show that the ring $\BCh(X)$ is generated by
$e_i$ with $i\not\in\{k,\;l\}$,
where $\Ch(X):=\CH(X)/2\CH(X)$ and $\BCh(X):=\Im(\Ch(X)\to\Ch(\bar{X}))$.
By~\cite[Theorem~87.7]{EKM} (originally proved in~\cite{MR2148072}),
it suffices to show that neither $e_k$ nor $e_l$ is in $\BCh(X)$.

By~\cite[Corollary~82.3]{EKM} once again, the anisotropic part of $q$ over the function field of its quadric $Y$
has dimension $2^n$.
By~\cite[Corollary~88.7]{EKM}, we conclude that $e_k\not\in\BCh(X)$.

Finally, let us assume that $e_l\in\BCh(X)$ and seek for a contradiction.
By~\cite[Theorem~90.3]{EKM} (originally proved in~\cite{MR2148072}),
the {\em canonical $2$-dimension} of the variety $X$ equals $k$
and does not change when the base field is extended to the function field of $Y$.
It follows by~\cite[Theorem~3.2]{MR3404385} that a shift of the {\em upper Chow motive} $U(X)$ with coefficients $\Z/2\Z$
is a direct summand of the motive of $Y$.
On the other hand, by~\cite[Lemma~82.4]{EKM}, the complete motivic decomposition of the quadric $Y$ consists only of shifts of
the upper motive $U(Y)$.
Moreover, since the variety $X_{F(Y)}$ has no $0$-cycle of odd degree, the motives $U(Y)$ and $U(X)$ are not isomorphic,
see~\cite[Corollary~2.15]{upper}.
The contradiction obtained proves Theorem~\ref{BCH}.
\end{proof}

Regarding the motives of the varieties $X$ and $Y$ from the above proof, each of them decomposes in a finite direct sum of indecomposable motives; moreover, by~\cite[Corollary~2.6]{upper} , such a decomposition is unique in the usual sense.
The upper motive $U(X)$ (resp., $U(Y)$) is defined as the summand with nontrivial $\Ch^0$ (unique in any decomposition given).
By~\cite[Corollary~2.15]{upper}, the motives $U(X)$ and $U(Y)$ are isomorphic if and only if each of the two varieties $X_{F(Y)}$ and $Y_{F(X)}$ possesses a $0$-cycle of odd degree.

Let us also recall that {\em canonical dimension} $\cd(X)$ of a smooth projective variety $X$ is the minimum of dimension
of the image of a rational self-map $X\RatM X$, c.f.\!~\cite{canondim}.
See also~\cite[Definition~1.3]{canondim} for a definition using the essential dimension of a certain functor
related to $X$.
Canonical $2$-dimension, which appeared in the above proof, is its $2$-local version also providing a lower bound for it.

\begin{coro}
For any anisotropic $q$ as in Theorem~\ref{BCH}, the canonical $2$-dimension $\cd_2(X)$
of its highest grassmannian $X$ is equal to
$k+l=2^n+2^{n-2}-2$.
\end{coro}

\begin{proof}
As in the proof of Theorem~\ref{BCH}, $e_i \in \BCh(X)$ for $i\not\in\{k,\;l\}$.
Moreover, it follows from Theorem~\ref{BCH} (and has been shown in its proof explicitly) that
neither $e_k$ nor $e_l$ is in $\BCh(X)$.
Thus in terms of the $J$-invariant in~\cite[Chapter~88]{EKM}, we have $J(q) = \{k,l\}$ and so~\cite[Theorem~90.3]{EKM} tells us that
%the sum over the $J$-invariant is equal to
$\cd_2(X)=k+l$.
%, implying the result.
\end{proof}


\section*{Acknowledgements}
We thank Detlev Hoffmann for providing reference~\cite{Faivre}.
We also thank Alexander Merkurjev and Alexander Vishik for useful comments.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\printbibliography

\end{document}
