Lemma 110.59.1. With $S = \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$ the inclusion functor $(\textit{Noetherian}/S)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ does not define a morphism of sites.
110.59 Sheaves on the category of Noetherian schemes
Let $S$ be a locally Noetherian scheme. As in Artin's Axioms, Section 98.25 consider the inclusion functor
\[ u : (\textit{Noetherian}/S)_{fppf} \longrightarrow (\mathit{Sch}/S)_{fppf} \]
of the fppf site of locally Noetherian schemes over $S$ into a big fppf site of $S$. As explained in the section referenced, this functor is continuous. Hence we obtain an adjoint pair of functors
\[ u^ s : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\textit{Noetherian}/S)_{fppf}) \]
and
\[ u_ s : \mathop{\mathit{Sh}}\nolimits ((\textit{Noetherian}/S)_{fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}) \]
see Sites, Section 7.13. However, we claim that $u$ in general does not define a morphism of sites, see Sites, Definition 7.14.1. In other words, we claim that the functor $u_ s$ is not left exact in general.
Let $p$ be a prime number and set $S = \mathop{\mathrm{Spec}}(\mathbf{F}_ p)$. Consider the injective map of sheaves
\[ a : \mathcal{F} \longrightarrow \mathcal{G} \]
on $(\textit{Noetherian}/S)_{fppf}$ defined as follows: for $U$ a locally Noetherian scheme over $S$ we define
\[ \mathcal{G}(U) = \Gamma (U, \mathcal{O}_ U)^* = \mathop{\mathrm{Mor}}\nolimits _ S(U, \mathbf{G}_{m, S}) \]
and we take
\[ \mathcal{F}(U) = \{ f \in \mathcal{G}(U) \mid \text{fppf locally }f \text{ has arbitrary }p\text{-power roots}\} \]
A Noetherian $\mathbf{F}_ p$-algebra $A$ has a nilpotent nilradical $I \subset A$, the $p$-power roots of $1$ in $A$ are of the elements of the form $1 + a$, $a \in I$, and hence no-nontrivial $p$-power root of $1$ has arbitrary $p$-power roots. We conclude that $\mathcal{F}(U)$ is a $p$-torsion free abelian group for any locally Noetherian scheme $U$; some details omitted. It follows that $p : \mathcal{F} \to \mathcal{F}$ is an injective map of abelian sheaves on $(\textit{Noetherian}/S)_{fppf}$.
To get a contradiction, assume $u_ s$ is exact. Then $p : u_ s\mathcal{F} \to u_ s\mathcal{F}$ is injective too and we find that $(u_ s\mathcal{F})(V)$ is a $p$-torsion free abelian group for any $V$ over $S$. Since representable presheaves are sheaves on fppf sites, by Sites, Lemma 7.13.5, we see that $u_ s\mathcal{G}$ is represented by $\mathbf{G}_{m, S}$. Using that $u_ s\mathcal{F} \to u_ s\mathcal{G}$ is injective, we find a $p$-torsion free subgroup
\[ (u_ s\mathcal{F})(V) \subset \Gamma (V, \mathcal{O}_ V)^* \]
for every scheme $V$ over $S$ with the following property: for every morphism $V \to U$ of schemes over $S$ with $U$ locally Noetherian the subgroup
\[ \mathcal{F}(U) \subset \Gamma (U, \mathcal{O}_ U)^* \]
maps into the subgroup $(u_ s\mathcal{F})(V)$ by the restriction mapping $\Gamma (U, \mathcal{O}_ U)^* \to \Gamma (V, \mathcal{O}_ V)^*$.
The actual contradiction now is obtained as follows: let $k = \bigcup _{n \geq 0} \mathbf{F}_ p(t^{1/{p^ n}})$ and set
\[ B = k \otimes _{\mathbf{F}_ p(t)} k \]
and $V = \mathop{\mathrm{Spec}}(B)$. Since we have the two projection morphisms $V \to \mathop{\mathrm{Spec}}(k)$ corresponding to the two coprojections $k \to B$ and since $\mathop{\mathrm{Spec}}(k)$ is Noetherian, we conclude the subgroup
\[ (u_ s\mathcal{F})(V) \subset B^* \]
contains $k^* \otimes 1$ and $1 \otimes k^*$. This is a contradiction because
\[ (t^{1/p} \otimes 1) \cdot (1 \otimes t^{-1/p}) = t^{1/p} \otimes t^{-1/p} \]
is a nontrivial $p$-torsion unit of $B$.
Proof. See discussion above. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)