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Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes_{A}B, f_{\ast})$ which is an equivalence. In this situation, $(-\otimes_{A}B, f_{\ast})$ is an equivalence precisely if the unit map

$$\eta_{M} : M \rightarrow f_{\ast}(M \otimes_{A}B)$$

is a weak equivalence for all cofibrant $A$-modules $M$. I am having trouble seeing why this map is a weak equivalence. My guess is use the sequence of $A$-modules $\ker(f) \rightarrow A \rightarrow B$ to compute the homotopy groups of the sequence $\ker(\eta_{M}) \rightarrow M \rightarrow f_{\ast}(M\otimes_{A}B)$.

Is this the right idea, or is there some easier way to demonstrate the pair is a Quillen equivalence?

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The counit map is cocontinuous in M, so using the fact that any cofibrant object is a retract of a transfinite composition of cobase changes of generating cofibrations of A-modules, combined with the left properness of the model category of A-modules and the fact that the left adjoint sends generating cofibrations to monomorphisms and the right adjoint preserves monomorphisms, the problem boils down to showing the claim for the case when M is the domain or codomain of a generating cofibration. In this case, this amounts to M=A[n] or M=(A[n−1]←A[n]), and in both cases the claim is trivial.

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  • $\begingroup$ I don't know how you are using the preservation of monomorphisms. Could you elaborate on this? I have been trying to use the cube diagram obtained from the unit map, along with Hirschorn Prop. 17.9.2, to conclude the original $\eta_{M}$ is a weak equivalence. $\endgroup$
    – Liam Keenan
    Commented Jan 15, 2019 at 5:55
  • $\begingroup$ The intended use of monomorphisms is that monomorphisms in this case are h-cofibrations, which under left properness implies that a cobase change along such a map is also a homotopy cobase change, in particular, it preserves weak equivalences. Thus, in the resulting cube diagram three parallel maps are weak equivalences, hence so is the fourth one. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 15, 2019 at 15:13
  • $\begingroup$ How is it that tensoring with B preserves monomorphisms? $\endgroup$
    – Liam Keenan
    Commented Jan 22, 2019 at 22:09
  • $\begingroup$ @LPK: I adjusted the post to reflect the statement that I actually use. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 23, 2019 at 2:47
  • $\begingroup$ Thanks very much. I appreciate it. $\endgroup$
    – Liam Keenan
    Commented Jan 23, 2019 at 2:51

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