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109.59 Sheaves on the category of Noetherian schemes

Let $S$ be a locally Noetherian scheme. As in Artin's Axioms, Section 97.25 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 $B$.

Lemma 109.59.1. 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.

Proof. See discussion above. $\square$


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