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\title{Counterexamples to F. Morel's conjecture on~$\boldsymbol{\pi}_0^{\mathbb{A}^1}$}

\author{\firstname{Joseph} \lastname{Ayoub}}
\address{University of Zurich / LAGA - Universit\'e Sorbonne Paris Nord}
\email{joseph.ayoub@math.uzh.ch}
\urladdr{user.math.uzh.ch/ayoub/}

%~\keywords{$\mathbb{A}^1$-algebraic topology, F.~Morel's conjecture on $\bfpi_0^{\mathbb{A}^1}$}

\begin{abstract}
We exhibit counterexamples to F.~Morel's conjecture on the $\mathbb{A}^1$-invariance of the sheaves of connected components of $\mathbb{A}^1$-local spaces.
\end{abstract}

\begin{document}
\maketitle

For a scheme $S$, we denote by $\Spc(S)$ the $\infty$-category $\Shv_{\Nis}(\Sm_S)$ of Nisnevich sheaves on smooth $S$-schemes. An object of $\Spc(S)$ is called an $S$-space. The Morel--Voevodsky $\infty$-category $\MV(S)$ is the full sub-$\infty$-category of $\Spc(S)$ consisting of $\mathbb{A}^1$-local $S$-spaces. Recall that an $S$-space $\mathcal{X}$ is $\mathbb{A}^1$-local if, for every $U\in \Sm_S$, the map $\pr_1^*:\mathcal{X}(U) \to \mathcal{X}(U\times \mathbb{A}^1)$ is an equivalence (in the $\infty$-category of spaces). The obvious inclusion admits a left adjoint $\Lder_{\mathbb{A}^1}:\Spc(S) \to \MV(S)$.

\begin{nota}\label{not:pi0-A1-of-a-space-}
Let $\bfpi_0$ be the $0$-th truncation functor in the $\infty$-topos $\Spc(S)$ and, for $i\geq 1$, let $\bfpi_i$ be the composition of $\bfpi_0$ with the $i$-th loop space functor. For an $S$-space $\mathcal{X}$, we set $\bfpi_0^{\mathbb{A}^1}(\mathcal{X})=\bfpi_0(\Lder_{\mathbb{A}^1}(\mathcal{X}))$ and, if $\mathcal{X}$ is pointed, we set $\bfpi_i^{\mathbb{A}^1}(\mathcal{X})=\bfpi_i(\Lder_{\mathbb{A}^1}(\mathcal{X}))$.
\end{nota}

Now, assume that $S$ is the spectrum of a perfect field $k$. In his monograph~\cite{A1-alg-top}, F.~Morel proved that the sheaves $\bfpi_i^{\mathbb{A}^1}(\mathcal{X})$ are $\mathbb{A}^1$-invariant in the strongest possible sense for every pointed $k$-space $\mathcal{X}$ and every integer $i\geq 1$. (See~\cite[Definition~1.7 \& Theorem~1.9]{A1-alg-top} for a precise statement.) The case $i=0$ was left open and, in~\cite[Conjecture~1.12]{A1-alg-top}, F.~Morel expressed the hope that $\bfpi_0^{\mathbb{A}^1}(\mathcal{X})$ is also $\mathbb{A}^1$-invariant for every $k$-space $\mathcal{X}$.

We will exhibit counterexamples to F.~Morel's conjecture. Interestingly, our counterexample is based on an old counterexample to a different conjecture of F.~Morel, namely his $\mathbb{A}^1$-connectivity conjecture over a general base, which we disproved in~\cite{contre-A1-connex}.

\begin{defi}\label{dfn:rationally-discrete}
Let $X$ be a smooth $k$-scheme. We say that $X$ is $\mathbb{A}^1$-discrete if, for any extension $K/k$, every $k$-morphism $\mathbb{A}^1_K \to X$ factors as the structural projection $\mathbb{A}^1_K \to \Spec(K)$ followed by a $K$-point $\Spec(K) \to X$ of the scheme $X$.
\end{defi}

We have the following well known fact.

\begin{lemm}\label{lemma:proper-and-rat-discrete}
Let $X$ be a smooth $k$-scheme. Assume that $X$ is proper and $\mathbb{A}^1$-discrete. Then, for a dense open immersion $j:V\to U$ of smooth $k$-schemes, composition with $j$ gives a bijection
\[
\hom(U,X) \simeq \hom(V,X).
\]
\end{lemm}

\begin{proof}
See~\cite[Corollary~1.44]{Debarre-HDAG}.
\end{proof}

We now give a general construction of $\mathbb{A}^1$-local $k$-spaces.

\begin{enonce}{Construction}\label{cons:gen-const-local-spc}
Let $X$ be a smooth $k$-scheme and let $\mathcal{M}\in \MV(X)$ be an $\mathbb{A}^1$-local $X$-space. We denote by $\Phi_X(M)$ the presheaf on $\Sm_k$ given informally by
\[
U\in (\Sm_k)^{\op}\;\mapsto \;
\coprod_{s:U \to X} \Gamma(U;s^*\mathcal{M})
\]
where, for every morphism $s:U \to X$, we write $s^*:\MV(X)\to \MV(U)$ for the pullback of $\mathbb{A}^1$-local spaces. More precisely, the functor $\Phi_X$ is given by the following composition
\[
\MV(X) \overset{(\star)}{\simeq} \Sect^{\mathrm{cocart}}\left(\left.\int_{s:U \to X \in
((\Sm_k)_{/X})^{\op}} \MV(U)\right/((\Sm_k)_{/X})^{\op}\right)
\xrightarrow{\Gamma}
\Psh((\Sm_k)_{/X})
\xrightarrow{\ff_{X,\,\sharp}}
\Psh(\Sm_k)
\]
where $\ff_{X,\,\sharp}$ is the left Kan extension along the forgetful functor $\ff_X:(\Sm_k)_{/X} \to \Sm_k$ and $(\star)$ is the obvious equivalence between $\MV(X)$ and the $\infty$-category of cocartesian sections of the cocartesian fibration classified by $(U\to X)\mapsto \MV(U)$.
\end{enonce}

\begin{rema}\label{rmk:Phi-X-and-sharp-images}
Denote by $p:X \to \Spec(k)$ the structural projection. It can be shown that $\Lder_{\mathbb{A}^1}(\Phi_X(\mathcal{M}))$ is equivalent to $p_{\sharp}(\mathcal{M})$ where $p_{\sharp}:\MV(X) \to \MV(k)$ is the left adjoint to the pullback functor $p^*$. We will not prove this here since we do not need it.
\end{rema}


\begin{prop}\label{prop:criterion-A-1-localness}
Keep the notations as in Construction~\ref{cons:gen-const-local-spc}. Assume that $X$ is proper and $\mathbb{A}^1$-discrete. Then $\Phi_X(\mathcal{M})$ belongs to $\MV(k)$, i.e., it has Nisnevich descent and is $\mathbb{A}^1$-invariant.
\end{prop}

\begin{proof}
We check the Brown--Gersten property for $\Phi_X(\mathcal{M})$. Clearly, the space $\Phi_X(\mathcal{M})(\emptyset)$ is contractible since $\MV(\emptyset)$ is the final category. If $U=U_1\coprod U_2$, a map $s:U \to X$ is the union of two maps $s_1:U_1\to X$ and $s_2:U_2\to X$, and we have
\[
\Gamma(U;s^*\mathcal{M})=\Gamma(U_1,s_1^*\mathcal{M})
\times \Gamma(U_2;s_2^*\mathcal{M}).
\]
This yields an equivalence $\Phi_X(\mathcal{M})(U)\simeq \Phi_X(\mathcal{M})(U_1)\times
\Phi_X(\mathcal{M})(U_2)$. Consider now a Nisnevich square of smooth $k$-schemes:
\[
\xymatrix{U' \ar[r]^-{j'} \ar[d]^-{e'} & V'\ar[d]^-{e}\\
U \ar[r]^-j & V.\!}
\]
We need to show that
\[
\xymatrix{{\displaystyle \coprod_{s':U' \to X}} \Gamma(U';s'^*\mathcal{M})
& {\displaystyle\coprod_{t':V' \to X}} \Gamma(V';t'^*\mathcal{M}) \ar[l] \\
{\displaystyle \coprod_{s:U \to X}} \Gamma(U;s^*\mathcal{M})\ar[u] & {\displaystyle \coprod_{t:V \to X}}
\Gamma(V;t^*\mathcal{M})\ar[l] \ar[u]}
\]
is cartesian in the $\infty$-category of spaces. Using what we just said, we may assume that $V$ and $V'$ are connected, and that $j$ and $j'$ have dense images. By Lemma~\ref{lemma:proper-and-rat-discrete}, we have bijections $\hom(V,X)\simeq \hom(U,X)$ and $\hom(V',X)\simeq \hom(U',X)$. Thus, we may rewrite the above square as follows:
\[
\xymatrix{{\displaystyle \coprod_{t':V' \to X}} \Gamma(U';t'^*\mathcal{M})
& {\displaystyle\coprod_{t':V' \to X}} \Gamma(V';t'^*\mathcal{M}) \ar[l] \\
{\displaystyle \coprod_{t:V \to X}} \Gamma(U;t^*\mathcal{M})\ar[u] & {\displaystyle \coprod_{t:V \to X}}
\Gamma(V;t^*\mathcal{M}).\!\ar[l] \ar[u]}
\]
The obvious map $\hom(V,X) \to \hom(V',X)$ is injective and the vertical arrows in the above square factor through the summands
\[
\coprod_{t:V\to X}\Gamma(U';t^*\mathcal{M})
\qquad \text{and} \qquad
\coprod_{t:V\to X}\Gamma(V';t^*\mathcal{M})
\]
respectively. Thus, we are left to show that the square
\[
\xymatrix{\Gamma(U';t^*\mathcal{M})
& \Gamma(V';t^*\mathcal{M}) \ar[l] \\
\Gamma(U;t^*\mathcal{M})\ar[u] &
\Gamma(V;t^*\mathcal{M})\ar[l] \ar[u]}
\]
is cartesian for every $t:V \to X$. This is obvious, since $t^*\mathcal{M}$ belongs to $\MV(V)$ by design.


It remains to see that $\Phi_X(\mathcal{M})$ is $\mathbb{A}^1$-local. Using that $X$ is $\mathbb{A}^1$-discrete, we see that the map $\Phi_X(\mathcal{M})(U) \to \Phi_X(\mathcal{M})(U\times \mathbb{A}^1)$ is the coproduct over $s:U \to X$ of the maps
\[
\Gamma(U;s^*\mathcal{M}) \to \Gamma(U\times \mathbb{A}^1;s^*\mathcal{M}).
\]
These are equivalences since the $s^*\mathcal{M}$'s belong to $\MV(U)$ by design.
\end{proof}

Next, we describe the sheaves of connected components of the $\mathbb{A}^1$-local $k$-spaces we just constructed.


\begin{prop}\label{prop:pi-0-A-1-of-Phi-X-M}
Keep the notations as in Construction~\ref{cons:gen-const-local-spc}. Assume that $X$ is proper and $\mathbb{A}^1$-discrete. Then the sheaf $\bfpi^{\mathbb{A}^1}_0(\Phi_X(\mathcal{M}))$ is given by
\[
U\mapsto \coprod_{s:U \to X}\Gamma(U;\bfpi_0^{\mathbb{A}^1}(s^*\mathcal{M})).
\]
In particular, $\bfpi^{\mathbb{A}^1}_0(\Phi_X(\mathcal{M}))$ is $\mathbb{A}^1$-invariant if and only if $\bfpi^{\mathbb{A}^1}_0(s^*\mathcal{M})$ is $\mathbb{A}^1$-invariant for every morphism $s:U \to X$. (In particular, a necessary condition is that $\bfpi^{\mathbb{A}^1}_0(\mathcal{M})$ is $\mathbb{A}^1$-invariant.)
\end{prop}


\begin{proof}
Since $\Phi_X(\mathcal{M})$ is $\mathbb{A}^1$-local, we have $\bfpi_0^{\mathbb{A}^1}(\Phi_X(\mathcal{M}))=
\bfpi_0(\Phi_X(\mathcal{M}))$. Thus, it is the sheafification of the ordinary presheaf of sets
\[
U\mapsto \coprod_{s:U \to X}\pi_0\Gamma(U;s^*\mathcal{M}),
\]
which we denote by $F$. Let $G$ be the presheaf described in the statement. We will show that $G$ is a Nisnevich sheaf and that the obvious map $F\to G$ induces isomorphisms on stalks. This will prove the first statement.

The proof that $G$ is a Nisnevich sheaf is identical to the proof that $\Phi_X(\mathcal{M})$ has Nisnevich descent: we check that $G$ takes a Nisnevich square to a cartesian square of sets, and this boils down to the property that $\bfpi_0^{\mathbb{A}^1}(t^*\mathcal{M})$ is a Nisnevich sheaf for every $t:V \to X$, which is true by design. To prove that $F\to G$ induces an isomorphism on stalks, we fix a henselian essentially smooth $k$-scheme $W$. The map $F(W) \to G(W)$ is then the coproduct, over $r:W \to X$, of the maps $\pi_0\Gamma(W;r^*\mathcal{M}) \to \Gamma(W;\bfpi_0(r^*\mathcal{M}))$, which are obviously isomorphisms.

For the last statement, using that $X$ is $\mathbb{A}^1$-discrete, we see that $\bfpi^{\mathbb{A}^1}_0(\Phi_X(\mathcal{M}))$ is $\mathbb{A}^1$-invariant if and only if, for every $s:U\to X$, the map
\[
\Gamma(U;\bfpi_0^{\mathbb{A}^1}(s^*\mathcal{M}))
\to \Gamma(U\times \mathbb{A}^1;\bfpi_0^{\mathbb{A}^1}(s^*\mathcal{M}))
\]
is an equivalence. Applying this property for a composite $q\circ s:V \to X$ with $q:V \to U$ a smooth morphism, we deduce immediately that the previous condition is equivalent to asking that $\bfpi_0^{\mathbb{A}^1}(s^*\mathcal{M})$ is $\mathbb{A}^1$-invariant for every $s:U \to X$.
\end{proof}

\goodbreak
It is now clear how to produce counterexamples to F.~Morel's conjecture.

\begin{enonce}{Construction}\label{cons:final-constr-}
Let $X$ be a smooth, proper and $\mathbb{A}^1$-discrete $k$-scheme. (For example, $X$ can be an abelian variety or a product of curves of genera $\geq 1$.) Let $\mathcal{M}\in \MV(X)$ be an $\mathbb{A}^1$-local $X$-space such that $\bfpi_0^{\mathbb{A}^1}(\mathcal{M})$ is not $\mathbb{A}^1$-invariant. Then Proposition~\ref{prop:pi-0-A-1-of-Phi-X-M} insures that $\bfpi_0^{\mathbb{A}^1}(\Phi_X(M))$ is also not $\mathbb{A}^1$-invariant. An explicit example of such an $\mathcal{M}$ can be obtained as follows, assuming that $X$ has dimension $\geq 3$. Let $Y\subset X$ be a closed integral surface and $o\in Y(k)$ a rational point admitting a Zariski neighbourhood $N\subset Y$ which is also an \'etale neighbourhood of the singular point of the projective surface $S\subset \mathbb{P}^3$ defined by the equation $w(x^3-y^2z)+F(x,y,z)=0$, where $F$ is a general homogeneous polynomial of degree~$4$. This is the surface used in~\cite{contre-A1-connex} to produce a counterexample to Morel's connectivity conjecture. In particular, we have a complex of abelian groups $\mathcal{K}^{M,\,!}_{S,1}$ on $\Sm_S$, concentrated in homological degrees $0$ and $1$ and sending an irreducible $T\in \Sm_S$ to the two-term Gersten complex
\[
k(T)^{\times} \to \coprod_{x\in T^{(1)}}\Z.
\]
We write also $\mathcal{K}^{M,\,!}_{S,1}$ for the associated Eilenberg--Mac Lane space which is an object of $\MV(S)$. Letting $i:N \to X$ be the obvious inclusion and $e:N \to S$ the \'etale neighbourhood of the singular point of $S$, we set $\mathcal{M}=i_*e^*\mathcal{K}^{M,\,!}_{S,\,1}$. As was shown in~\cite{contre-A1-connex}, the sheaf $\bfpi_0^{\mathbb{A}^1}(\mathcal{M})$ restricted to a neighbourhood of $o$ in $X$ is not $\mathbb{A}^1$-invariant.
\end{enonce}



\subsection*{Acknowledgement} This note was written while the author was visiting the Hausdorff Research Institute for Mathematics in Bonn for the trimester program ``Spectral Methods in Algebra, Geometry, and Topology''. He thanks the organisers for the invitation and the institute for the excellent working conditions.



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