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Mikhail Borovoi
Mikhail Borovoi
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Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ be a presheaf of sets on $C$ (i.e., an object of $\widehat{C}$) and let $a$ be the associated sheaf functor. Now consider the presheaf $\underline{\rm Aut}(\mathcal P)$ given by $\underline{\rm Aut}(\mathcal P)(T)={\rm Aut}(\mathcal P\times h_{T})$ for all $T\to S$ in $C$, where $h\,\colon C\to \widehat{C}$ is the usual functor (note that $h_{T}$ is, in fact, a sheaf). My question is:

Question. Is it true that there exists a canonical isomorphism of sheaves $$ a(\underline{\rm Aut}(\mathcal P))=\underline{\rm Aut}(a(\mathcal P))? $$

is it true that there exists a canonical isomorphism of sheaves $$ a(\underline{\rm Aut}(\mathcal P))=\underline{\rm Aut}(a(\mathcal P))? $$ ThisThis seems plausible, but I haven't been able to find a proof, so either I'm getting old :-) or my guess is false.

Note that if $\mathcal P$ is a sheaf, then my guess is true as it is well-known [SGA 3, IV, Corollary 4.5.13] that, in this case, the presheaf $\underline{\rm Aut}(\mathcal P)$ is, in fact, a sheaf.

Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ be a presheaf of sets on $C$ (i.e., an object of $\widehat{C}$) and let $a$ be the associated sheaf functor. Now consider the presheaf $\underline{\rm Aut}(\mathcal P)$ given by $\underline{\rm Aut}(\mathcal P)(T)={\rm Aut}(\mathcal P\times h_{T})$ for all $T\to S$ in $C$, where $h\,\colon C\to \widehat{C}$ is the usual functor (note that $h_{T}$ is, in fact, a sheaf). My question is:

is it true that there exists a canonical isomorphism of sheaves $$ a(\underline{\rm Aut}(\mathcal P))=\underline{\rm Aut}(a(\mathcal P))? $$ This seems plausible, but I haven't been able to find a proof, so either I'm getting old :-) or my guess is false.

Note that if $\mathcal P$ is a sheaf, then my guess is true as it is well-known [SGA 3, IV, Corollary 4.5.13] that, in this case, the presheaf $\underline{\rm Aut}(\mathcal P)$ is, in fact, a sheaf.

Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ be a presheaf of sets on $C$ (i.e., an object of $\widehat{C}$) and let $a$ be the associated sheaf functor. Now consider the presheaf $\underline{\rm Aut}(\mathcal P)$ given by $\underline{\rm Aut}(\mathcal P)(T)={\rm Aut}(\mathcal P\times h_{T})$ for all $T\to S$ in $C$, where $h\,\colon C\to \widehat{C}$ is the usual functor (note that $h_{T}$ is, in fact, a sheaf).

Question. Is it true that there exists a canonical isomorphism of sheaves $$ a(\underline{\rm Aut}(\mathcal P))=\underline{\rm Aut}(a(\mathcal P))? $$

This seems plausible, but I haven't been able to find a proof, so either I'm getting old :-) or my guess is false.

Note that if $\mathcal P$ is a sheaf, then my guess is true as it is well-known [SGA 3, IV, Corollary 4.5.13] that, in this case, the presheaf $\underline{\rm Aut}(\mathcal P)$ is, in fact, a sheaf.

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Cristian D. Gonzalez-Aviles
Cristian D. Gonzalez-Aviles
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Sheaf associated to presheaf Aut

Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ be a presheaf of sets on $C$ (i.e., an object of $\widehat{C}$) and let $a$ be the associated sheaf functor. Now consider the presheaf $\underline{\rm Aut}(\mathcal P)$ given by $\underline{\rm Aut}(\mathcal P)(T)={\rm Aut}(\mathcal P\times h_{T})$ for all $T\to S$ in $C$, where $h\,\colon C\to \widehat{C}$ is the usual functor (note that $h_{T}$ is, in fact, a sheaf). My question is:

is it true that there exists a canonical isomorphism of sheaves $$ a(\underline{\rm Aut}(\mathcal P))=\underline{\rm Aut}(a(\mathcal P))? $$ This seems plausible, but I haven't been able to find a proof, so either I'm getting old :-) or my guess is false.

Note that if $\mathcal P$ is a sheaf, then my guess is true as it is well-known [SGA 3, IV, Corollary 4.5.13] that, in this case, the presheaf $\underline{\rm Aut}(\mathcal P)$ is, in fact, a sheaf.