Let $A$ be a simplicial commutative ring, then it is known that the ideal of elements of degree $\ge1$ in the associated CDGA has a "DG divided power structure," which induces a divided power structure on the ideal of elements of degree $\ge1$ in the graded-commutative ring $\pi_*A$. I am trying to interpret this as saying something like "the infinitesimal thickening $\operatorname{Spec}\pi_0A\to\operatorname{Spec}N(A)$ of spectral schemes has the structure of a PD-thickening," where $N$ is the forgetful from simplicial commutative rings to $E_\infty$-rings.
This suggests the following question: let $f:A\to B$ be a morphism of connective $E_\infty$-rings in spectra or $D(\mathbb{Z})$ inducing a surjection on $\pi_0$. Is there a suitable notion of "divided power structure on $\operatorname{fib}(f)$?" I want there to be an automatic PD-structure on $\operatorname{fib}(N(A)\to\pi_0(A))$ for $A$ a simplicial commutative ring. As well, the blunt notion "$\pi_*\operatorname{fib}(f)\to\pi_*A$ is a quasi-ideal and is given a PD-structure" (it is not clear to me if the first condition is automatic) seems "not coherent enough," though I don't know any formal list of "properties I want" that would exclude this.
It seems to me any kind of animation procedure would just spit out something in the strictly commutative rather than the $E_\infty$-setting, and divided powers are too nonlinear to be tractable operadically, so I don't see what tools are available here.
I am doubtful about this, since I don't even know what structure on $\operatorname{fib}(f)$ should correspond to its being the fiber of a $\pi_0$-surjection of connective $E_\infty$-rings (though see the discussion here: there is a notion of Smith ideal for $A_\infty$-rings, and the paper of David White linked there has a notion which may be suitable.) (Correction: Smith ideals for the $E_\infty$-setting are handled by Theorem A in the paper of White and Yau linked in the same answer.) But I am posting this in the hopes someone knows something like this.
