All job applications and no math makes Jack a dull boy. @_@

Say I have a symmetric monoidal model category in $M$ and that $A$ is a cofibrant commutative algebra. In this case, $A$ also determines a commutative algebra object of the underlying symmetric monoidal $\infty$-category $M^{cof}[W^{-1}]$.

When does the category of A-modules in M, with the model structure induced from M, model the $\infty$-category of $A$-modules in $M^{cof}[W^{-1}]$? Are there any references for this?

I'm willing to assume that things are as nice as possible.

@AaronMazel-Gee I like "exact triangle", and "exact square" for a pullback/pushout square of spectra

@PiotrPstrągowski 4.3.3.17 of HA proves it for bimodules over associative algebra objects, but i couldn't find the analogous statement for commutative algebra objects

@JonathanBeardsley I thought that Jack was only a nickname for John, not Jonathan =O

@skd I could have sworn this was in HA somewhere... I'm surprised that it's not!

it's just my lack of ability to search through HA, not jacob's lack of thoroughness

i'm sure it's in there somewhere

@skd @PiotrPstrągowski but piotr's statement is the special case when B=1