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Yang
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What is the Isomorphism subspace of the mapping space in an infiniteinfinity category

When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this establishes a correspondence between $GL_{n}$ tensor on $X$ and locally free sheaf of rank $n$. What I want to do is to extend this correspondence to the derived case when $X$ is a derived scheme.

$\textbf{Question 1:}$ What is the proper replacement of the isomorphism set of two objects in the category of sheaf to the 'space of equivalence' $Isom_{QCoh(S)}(O_{S}^{n},E)$ between two objects in the infiniteinfinity category of quasi-coherent sheaf on $S$? (Here I take the infiniteinfinity category $QCoh(S)$ instead of the infiniteinfinity category $Mod(O_{S})$ because I never see the latter thing being used in the context of derived algebraic geometry) I think this is some subspace of the mapping space $Map_{QCoh(S)}(O_{S}^{n},E)$. But how do we describe this subspace explicitly?

$\textbf{Question 2:}$ Especially, if we have the appropriate definition of the isomorphism space, then for the 'automorphism space' $Aut(O_{S}^{n})\cong Isom(O_{S}^{n},O_{S}^{n})$, is it equivalent to the discrete space $GL_{n}(S)$ (it is discrete because of this post )? Or this equivalence may not be true here?

What is the Isomorphism subspace of the mapping space in an infinite category

When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this establishes a correspondence between $GL_{n}$ tensor on $X$ and locally free sheaf of rank $n$. What I want to do is to extend this correspondence to the derived case when $X$ is a derived scheme.

$\textbf{Question 1:}$ What is the proper replacement of the isomorphism set of two objects in the category of sheaf to the 'space of equivalence' $Isom_{QCoh(S)}(O_{S}^{n},E)$ between two objects in the infinite category of quasi-coherent sheaf on $S$? (Here I take the infinite category $QCoh(S)$ instead of the infinite category $Mod(O_{S})$ because I never see the latter thing being used in the context of derived algebraic geometry) I think this is some subspace of the mapping space $Map_{QCoh(S)}(O_{S}^{n},E)$. But how do we describe this subspace explicitly?

$\textbf{Question 2:}$ Especially, if we have the appropriate definition of the isomorphism space, then for the 'automorphism space' $Aut(O_{S}^{n})\cong Isom(O_{S}^{n},O_{S}^{n})$, is it equivalent to the discrete space $GL_{n}(S)$ (it is discrete because of this post )? Or this equivalence may not be true here?

What is the Isomorphism subspace of the mapping space in an infinity category

When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this establishes a correspondence between $GL_{n}$ tensor on $X$ and locally free sheaf of rank $n$. What I want to do is to extend this correspondence to the derived case when $X$ is a derived scheme.

$\textbf{Question 1:}$ What is the proper replacement of the isomorphism set of two objects in the category of sheaf to the 'space of equivalence' $Isom_{QCoh(S)}(O_{S}^{n},E)$ between two objects in the infinity category of quasi-coherent sheaf on $S$? (Here I take the infinity category $QCoh(S)$ instead of the infinity category $Mod(O_{S})$ because I never see the latter thing being used in the context of derived algebraic geometry) I think this is some subspace of the mapping space $Map_{QCoh(S)}(O_{S}^{n},E)$. But how do we describe this subspace explicitly?

$\textbf{Question 2:}$ Especially, if we have the appropriate definition of the isomorphism space, then for the 'automorphism space' $Aut(O_{S}^{n})\cong Isom(O_{S}^{n},O_{S}^{n})$, is it equivalent to the discrete space $GL_{n}(S)$ (it is discrete because of this post )? Or this equivalence may not be true here?

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Yang
Yang
  • 618
  • 1
  • 1
  • 11

What is the Isomorphism subspace of the mapping space in an infinite category

When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this establishes a correspondence between $GL_{n}$ tensor on $X$ and locally free sheaf of rank $n$. What I want to do is to extend this correspondence to the derived case when $X$ is a derived scheme.

$\textbf{Question 1:}$ What is the proper replacement of the isomorphism set of two objects in the category of sheaf to the 'space of equivalence' $Isom_{QCoh(S)}(O_{S}^{n},E)$ between two objects in the infinite category of quasi-coherent sheaf on $S$? (Here I take the infinite category $QCoh(S)$ instead of the infinite category $Mod(O_{S})$ because I never see the latter thing being used in the context of derived algebraic geometry) I think this is some subspace of the mapping space $Map_{QCoh(S)}(O_{S}^{n},E)$. But how do we describe this subspace explicitly?

$\textbf{Question 2:}$ Especially, if we have the appropriate definition of the isomorphism space, then for the 'automorphism space' $Aut(O_{S}^{n})\cong Isom(O_{S}^{n},O_{S}^{n})$, is it equivalent to the discrete space $GL_{n}(S)$ (it is discrete because of this post )? Or this equivalence may not be true here?