Classical schemes as derived schemes are discrete valued

Question

$\newcommand\Spc{\mathrm{Spc}}\newcommand\SCRing{\mathrm{SCRing}}\DeclareMathOperator\Map{Map}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ch{Ch}$Classical scheme can be considered as sheaf through 'functor of points', and when we consider 'sheaf valued in $\Spc$' on the infinity category $\SCRing$ with certain topos, we have the notion of derived stack. Every note I read says 'classical scheme can also be viewed as a derived stack.'

Let's start with the affine case:

Question: So does this mean the mapping space $\Map_{\SCRing}(A,S)$ is automatically discrete for any simplicial ring $A$ when $S$ is discrete? Any categorical explanation for this?

For instance, for the group scheme $\GL_{n}$ if viewed as a derived stack, what is its value $\GL_{n}(A)$ on the derived affine scheme $\Spec(A)$ with $A$ a simplicial ring?

Remark: here is my attempt on this through the Dold–Kan correspondence on the chain complex, if we regard the simplicial ring $A$ as a connective chain complex in $\Ch_{\geq 0}(A)$ and $S$ is concentrated in zero. Then the mapping space $\Map(A,S)$ (see remark 13.1 in Stable $\infty$-categories by Lurie) is constructed from the chain complex $[A,S]$ where $[A,S]_{n}=\prod_{m}\Hom(A_{m},S_{m+n})$, which is nonzero only when $m=-n$, so if $n$ is positive, and $m$ is negative, then $A_{m}=0$. Thus the only nontrivial term occurs in nonpositive part, and after composition with a truncated functor which is the right adjoint of the inclusion $\Ch_{\geq 0}\rightarrow \Ch$, the chain complex we get is concentrated in zero. Thus corresponds to a discrete simplicial ring by Dold–Kan correspondence.

Is this correct and is there any other way to see this?

Answer 1

There are a few different confusions going on here:

(1) If we take the statement "every classical scheme can be viewed as a derived stack" and specialize to affine schemes, this means that we send the affine scheme $\operatorname{Spec}S$ ($S$ discrete) to the 'functor of points' $A\mapsto\operatorname{Hom}_{\mathrm{SCRing}}(S,A)$ on simplicial commutative rings $A$. Note that this is the reverse of what you have written. It is not in general discrete. Nor is there any reason why the statement "every classical scheme can be viewed as a derived stack" should imply it is discrete. What this statement does mean (in its precise formulation that the functor from classical schemes to derived stacks is fully faithful) is that $\operatorname{Hom}_{\operatorname{Hom}(\mathrm{SCRing},\mathrm{Spc})}(\operatorname{Hom}_{\mathrm{SCRing}}(S,-),\operatorname{Hom}_{\mathrm{SCRing}}(S',-))$ is discrete for every pair of discrete rings $S$ and $S'$. One could indeed be tempted to believe that the only way this could be true is if $\operatorname{Hom}_{\mathrm{SCRing}}(S,A)$ were itself discrete for every simplicial commutative ring $A$, but this is in fact not the case. Geometrically speaking, the space of morphisms from a derived scheme to a classical scheme need not be discrete.

(2) Your statement that $\operatorname{Hom}_{\mathrm{SCRing}}(A,S)$ is discrete whenever $S$ is discrete is correct, and your justification is essentially correct. Alternatively, one can show this using the fact that the inclusion of discrete rings into space-valued rings is right adjoint to the $\pi_0$ functor. Geometrically, this corresponds to the statement that the space of maps from any classical scheme to any derived scheme is discrete. But this isn't really related to the functor of points of a classical scheme on derived rings.