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\begin{document}
\title{Corrigendum to ``The alternative operad is not Koszul''}
\author{Askar Dzhumadil'daev}
\address{Kazakh-British Technical University, Almaty, Kazakhstan}
\email{dzhuma@hotmail.com}
\author{Pasha Zusmanovich}
\address{Tallinn University of Technology, Tallinn, Estonia}
\email[as of April 4, 2021]{pasha.zusmanovich@gmail.com}
\date{first written September 9, 2012; last minor revision April 4, 2021}
\thanks{Experiment. Math. \textbf{21} (2012), no.4, 418}
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In \cite[\S 4]{alternative}, we formulated a conjecture that in characteristic
$3$, the dimension of the $n$th homogeneous component of the dual alternative
operad, i.e. an operad governed by two identities -- associativity and
\begin{equation}\label{eq}\tag{*}
xyz + yxz + zxy + xzy + yzx + zyx = 0
\end{equation}
(or, what is the same, dimension of the multilinear
component of the corresponding free algebra), is equal to $2^n - n$.
In fact, this was proved earlier by Lopatin
(see \cite[\S 7, Remark 2]{lopatin}): he computes the corresponding dimension
for the variety of associative algebras satisfying the identity $x^3 = 0$,
what for multilinear components is equivalent to the corresponding dimensions
of its full linearization (\ref{eq}). Lopatin's proof consists of computer
calculations for small values of $n$ (as we did in \cite{alternative}), and an
argument based on the composition (=diamond) lemma reducing the general case
to the cases of small $n$'s.
Thanks are due to Ivan Kaygorodov for bringing this fact to our attention,
and to Artem Lopatin for explaining some points of \cite{lopatin}.
Recently, a more general result was established by Dotsenko in \cite{dotsenko}.
Dotsenko's proof utilizes a generalization of composition lemma for operads,
and does not depend on any computer calculations.
\begin{thebibliography}{DZ}
\bibitem[D]{dotsenko} V. Dotsenko,
\emph{Dual alternative algebras in characteristic three};
\textsf{arXiv:1111.2289v2}.
\bibitem[DZ]{alternative} A. Dzhumadil'daev and P. Zusmanovich,
\emph{The alternative operad is not Koszul},
Experiment. Math. \textbf{20} (2011), 138--144; \textsf{arXiv:0906.1272}.
\bibitem[L]{lopatin} A.A. Lopatin,
\emph{Relatively free algebras with the identity $x^3 = 0$},
Comm. Algebra \textbf{33} (2005), 3583--3605; \textsf{arXiv:math/0606519}.
\end{thebibliography}
\end{document}
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