Consider the infinity category of simplicial rings $CAlg=Fun^{\prod}(Poly^{op},Spc)$, and the under-category over $R$: $CAlg_{A/}$ is equivalent to $CAlg_{A}=Fun^{\prod}(Poly^{op}_{R},Spc)$ by 25.1.4.3 and 25.1.4.4 in [SAG by Lurie][1]. And there is a final object $*$ in $CAlg$, which is the 'zero ring', the functor valued in the final object $*\in Set\hookrightarrow Spc$. As the property of under category, we can see that this zero ring $*$ is again a final object in $CAlg_{A}$.

I can see there is much development about the spectrum of the over-category $CAlg_{/A}$, for example, in [25.3.3 in SAG][1] or [7.3.4 in HA][2]. For the over-category $CAlg_{/A}$, the final object is the $id_{A}$ and the pointed objects of $(CAlg_{/A})_{*}=(CAlg_{/A})_{A/}$ is equivalent to $CAlg(Mod(A))_{/A}$ by [3.4.17 in HA][2], and this is exactly the 'augmented algebra in $Mod(A)$' as defined in [definition 7.3.4.3 HA][2], which is in the shape like the following triangle:[![][3]][3]

And we can show that the spectrum of this infinity category of augmented algebra is exactly the stable infinity category $Mod(A)$. Thus we get the spectrum of the over-category.

I can't help thinking about the dual case: 

$\textbf{Question:}$
$\textbf{What is the spectrum object of the infinity category of simplicial commutative rings over $A$?}$


Note that we used the fact that $Sp(C)\cong Sp(C_{*})$ above, where $C=CAlg_{/A}$. So I also imitate this in the case when $C=CAlg_{A/}$, if we consider the pointed objects in $CAlg_{A}$, that is, a morphism from final object $*$ to $A$. Note that a discrete ring admitting a pointed morphism is exactly the zero ring, however, in the derived setting, the condition for the existence of a morphism(natural transformation) between simplicial rings(functors) boils down to the homotopy commutativity in $\textbf{Spc}$, unwinding these diagrams leads to the conclusion: the pointed objects in $CAlg_{A}$ are $A$ rings with $\pi_{0}$ trivial, that is, the path connected $A$ rings. This is indeed a interesting fact, but it doesn't help solve the problem.

Any ideas or thoughts on the spectrum object of $CAlg_{A}$ is very welcome.


 $\textbf{Remark}$: As suggested by [comments by Maxime Ramiz][4], indeed, if $\pi_{0}(A)$ is 0, then the whole simplicial ring is 0, because we have $\pi_{k}(A)$ as $\pi_{0}(A)$ module, which is zero because the only module over zero ring is zero ring itself. But indeed it's a very strange thing that stablization of over-category $CAlg_{A}$ is only a singleton?


  [1]: https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf
  [2]: https://people.math.harvard.edu/~lurie/papers/HA.pdf
  [3]: https://i.sstatic.net/7AGV8mde.png
  [4]: https://mathoverflow.net/questions/477697/the-spectrum-object-in-the-infty-category-calg-r#comment1242683_477697