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\TopicEN{Number Theory}
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\title{Corrigendum to ``Symplectic and orthogonal $K$-groups of the integers''}
\alttitle{Corrigendum à « $K$-groupes symplectiques et orthogonaux de l’anneau des entiers »}

\author{\firstname{Marco} \lastname{Schlichting}}
\address{Marco Schlichting, Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK}
\email{m.schlichting@warwick.ac.uk}

\begin{abstract}
The action of the duality functor on the odd torsion of $K_n(\mathbb{Z})$ was stated incorrectly in~\cite{myCRAS}, in half of the cases, and lead to incorrect formulas for the odd primary torsion of $\pi_nB\mathrm{Sp}(\mathbb{Z})^+$ and $\pi_nBO_{\infty,\infty}(\mathbb{Z})^+$.
\end{abstract}


\begin{altabstract}
L’action du foncteur de dualité sur la torsion impaire de $K_n(\mathbb{Z})$ était énoncé incorrectement dans~\cite{myCRAS}, dans la moitié des cas, et a conduit à des formules incorrectes pour la torsion primaire impaire de $\pi_nB\mathrm{Sp}(\mathbb{Z})^+$ et $\pi_nBO_{\infty,\infty}(\mathbb{Z})^+$.
\end{altabstract}

\dateposted{2024-05-02}
\begin{document}
\maketitle

Let $R$ be a ring of integers in a number field and $\ell$ an odd prime. The formula~\cite[(2.3)]{myCRAS} for the $\ell$-primary torsion subgroups of $K_n(R)$ is false. The correct formulas extracted from~\cite[Proof of Theorem~70]{weibel:handbook} are for $i>1$
\[
K_{2i}(R)\{\ell\} = K_{2i}(R')\{\ell\} = H^2_{et}(R',(K_{2i+2})_{\ell})
\]
where the right hand term is the inverse limit $\lim_{\nu}H^2_{et}(R',(K_{2i+2})/{\ell^{\nu}})$. In particular, the duality acts by $(-1)^{i+1}$ on this group. In odd degree for $i>1$ we have
\[
K_{2i-1}(R)\{\ell\} = K_{2i-1}(R')\{\ell\} = H^0_{et}(R',(K_{2i})/{\ell^{\infty}})
\]
where the right hand term is the direct limit $\colim_{\nu}H^0_{et}(R',(K_{2i})/{\ell^{\nu}})$. In particular, the duality acts by $(-1)^{i}$ on this group. These formulas are not new; see~\cite[Theorem~70 and proof thereof]{weibel:handbook}; see also~\cite[Lemma~3.4.4]{9auth}. Together with~\cite[Lemma~2.1]{myCRAS} this yields now a correction of~\cite[Theorem~2.2]{myCRAS} in the cases $n\equiv 0,2\mod 4$.

\begin{theo}\label{thm1corrected}
Let $R$ be a ring of integers in a number field, and $\ell \in \Z$ an odd prime. Then for all $n\geq 1$ we have isomorphisms
\[
GW_{n}(R)\{\ell\} \cong K\Sp_{n}(R)\{\ell\} \cong KQ_{n}(R)\{\ell\} \cong
\begin{cases}
K_{n}(R)\{\ell\} & n\equiv 2,3\mod 4\\
0& n\equiv 0,1\mod 4.
\end{cases}
\]
\end{theo}

\cite[Section~3]{myCRAS} is unaffected by this error and thus, we obtain the following table of homotopy groups correcting~\cite[Th\'eor\`eme~0.1 and Theorem~1.1]{myCRAS} where the $2$-primary part comes from~\cite[4.7.2]{Karoubi:handbook} and~\cite[Theorem~3.3]{myCRAS} and the odd primary part from Theorem~\ref{thm1corrected}. See also~\cite[Theorem~3.2.1]{9auth}.

\begin{theo}\label{thm:Main}
The homotopy groups of the spaces $B\Sp(\Z)^+$ and $BO_{\infty,\infty}(\Z)^+$ for $n\geq 1$ are given in the following table
\[
\renewcommand\arraystretch{2}
\begin{array}{c|c|c|c|c|c|c|c|c}
n \mod 8& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\pi_nB\Sp(\Z)^+ & 0 & 0 &\Z \oplus K_n(\Z)_{odd} &K_n(\Z) &\Z/2 & \Z/2 & \Z \oplus K_n(\Z)_{odd} & K_n(\Z)\\
\hline
\pi_nBO_{\infty,\infty}(\Z)^+ &
\Z \oplus \Z/2 &(\Z/2)^{3} &{
\renewcommand\arraystretch{1}
\begin{matrix} (\Z/2)^2\\ \oplus \\ K_n(\Z)_{odd}
\end{matrix}}& {
\renewcommand\arraystretch{1}
\begin{matrix} \Z/8\\ \oplus \\ K_n(\Z)_{odd}
\end{matrix}}& \Z & 0 &K_n(\Z)_{odd} & K_n(\Z)
\end{array}
\]
\end{theo}
\medskip

Finally, this leads to a correction of~\cite[Remark~1.3]{myCRAS} and the following table for $n>0$. Denote by $B_k$ the $k$-th Bernoulli number~\cite[Example~24]{weibel:handbook} and let $d_n$ denote the denominator of $\frac{1}{n+1}B_{(n+1)/4}$ for $n=3$ mod~$4$. By~\cite[Introduction, Lemma~27]{weibel:handbook} we have $K_n(\Z) = \Z/2d_n$ for $n=3$ mod~$8$ and $K_n(\Z) = \Z/d_n$ for $n=7$ mod~$8$. Similarly, denote by $c_k$ the numerator of $B_k/4k$. Then $K_{4k-2}(\Z)$ is a finite group of order $c_k$ when $k$ is even and of order $2c_k$ when $k$ is odd. This group is conjectured to be cyclic. 
\[
\renewcommand\arraystretch{2}
\begin{array}{c|c|c|c|c|c|c|c|c}
n \mod 8& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\pi_nB\Sp(\Z)^+ &0 & 0 &\Z \oplus |\Z/c_k| &\Z/2d_n &\Z/2 & \Z/2 & \Z \oplus |\Z/c_k| & \Z/d_n\\
\hline
\pi_nBO_{\infty,\infty}(\Z)^+ &\Z \oplus \Z/2 &(\Z/2)^{3} &(\Z/2)^{2} \oplus |\Z/c_k| & \Z/d_n & \Z & 0 &|\Z/c_k| & \Z/d_n
\end{array}
\]
where $|\Z/m|$ denotes a finite group of order $n$ conjectured to be cyclic and $n=4k-2$.


Full proof of the claims in~\cite{myCRAS} are now available in~\cite{9auth}.

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