Definition Let $A$ be a commutative ring. An ideal $I \triangleleft A$ is called quasiregular if $I/I^2$ is flat over $A/I$ and there is a canonical isomorphism of algebras $$ \Lambda_A I/I^2\xrightarrow{\cong} \operatorname{Tor}^A(A/I,A/I). $$
There is the following result due to Quillen (Theorem 6.12 in "On the (co)homology of commutative rings"):
Theorem Let $I_\bullet$ be a simplicial in a simplicial commutative ring $A_\bullet$ such that $H_0(I_\bullet)=0$ and $I_k$ is quasiregular in $A_k$ for all $k.$ Then $I_\bullet^r$ is $r-1$-connected (as a simplicial abelian group).
This result seems to be well-known. In particular, Andrè references it in Section XIII of "Homologie des algebres commutatives". However, there a condition similar to quasiregularity is formulated in non-homological terms.
Unfortunately, due to the format of Quillen's paper (proceedings of a conference), there is no detailed proof of this result. I think I more or less understand the idea of proof (we need to use quasiregularity condition to reduce the general case to the case of symmetric powers). But it would be nice to have a detailed account in the literature. Maybe someone here knows where to find it? Or maybe it is an immediate corollary of Andrè's results? I have to admit I did not study his book well enough.
