Let CDGA denote the category of non-negatively graded, commutative, differential graded (degree -1) algebras over $\mathbb{Q}$. In Kathryn Hess's "Rational homotopy theory: a brief introduction", she says the following about these:
Proposition 1.16: Any morphism $f: A \rightarrow B$ in CDGA can be factored as the inclusion $A \rightarrow A \otimes \Lambda W$ followed by a quasi-isomorphism $A \otimes \Lambda W \rightarrow B$, where $\Lambda W$ is the CDGA generated by some graded vector space $W$, and the differential on $A \otimes \Lambda W$ makes it Sullivan.
Then she says, "In particular, if $H^0 (A)= \mathbb{Q}, H^1(A)=0$ and $H^*(B)$ is finite type (finite dimension in every degree), we have $W$ is finite type and $W=W^{\geq 2}$."
I'm not understanding how she is getting this. For example, let $A$ be $\mathbb{Q}$ concentrated in dimension 0 and have our map be the unit of $B$. Let $B$ be any CDGA with nontrivial first cohomology. The proposition says that $\Lambda W$ has the same cohomology as $B$ (since this is just a cofibrant replacement), but if $W$ has nothing in degree $1$ then neither does $\Lambda W$, so it certainly doesn't have the same cohomology as $B$.
It seems at the very least we would want $B$ to be 1-connected. Is this assessment correct?