# Comparison of model structures on $sMod_{B}$

Let $B$ be a commutative ring with unity.

By Theorem 4 from Chapter 2 Section 4 of 'Homotopical Algebra' we can impose a model structure on $sMod_{B}$ by declaring a map of simplicial $B$-modules $f:M \rightarrow N$ to be a fibration (resp. a weak equivalence) if $\underline{Hom}(P,M) \rightarrow \underline{Hom}(P,N)$ is a fibration (resp. a weak equivalence) for all projective $B$-modules $P$. Call this model structure MS1

Alternatively, using the forgetful/free adjunction between $sMod_{B}$ and $sSet$, by Theorem 5.1 of Goerss-Jardine, we have another model structure on $sMod_{B}$. Call this model structure MS2.

Now since $B$ is a projective $B$-module, and $\underline{Hom}(B,M) \cong M$ as simplicial sets, it follows that $f:M \rightarrow N$ a fibration/weak equivalence in MS1 implies $f:M \rightarrow N$ is a fibration/weak equivalence in MS2.

Do these model structures coincide? If not, can we exhibit an example highlighting their nonequivalence?

Yes, MS1=MS2. In MS2, fibrations and weak equivalences are created by $hom(B,-)$. But since fibrations and weak equivalences of simplicial sets are preserved under (arbitrary) products, then you can equivalently create these by the functors $hom(B^{\oplus S},-)$ for all sets $S$. Then, any projective module $P$ is a retract of a free module $B^{\oplus S}$, and thereafter the corepresented functor $hom(P,-)$ is a retract of $hom(B^{\oplus S},-)$. Since fibrations and weak equivalences of simplicial sets are also stable under retracts, it follows that any fibration or weak equivalence in MS2 is necessarily also such in MS1.