Suppose $X$ is a commutative $R$-algebra with $R$ a commutative ring. Is it the case that there will always be a Projective resolution of $X$ as an $R$-module that is a graded commutative DGA?
Tate showed this for $R$-algebras of the form $R/I$ for $I$ an ideal of $R$. I am wondering if there is a standard counterexample or if this result is known/unknown. It seems like one of those facts that someone in commutative algebra might know off the top of their head.
Thanks for your time.