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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.

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$X$ is of the form $R[x_1,\dots]/\text{some ideal}$. There is a free resolution of $X$ as an $R[x_1,\dots]$-algebra which is a DGCA. This is also a free $R$-resolution, no? –  Mariano Suárez-Alvarez Sep 12 '11 at 18:11
Hi Mariano, thanks, this is exactly the sad news i was hoping to not to get, oh well. –  Sean Tilson Sep 13 '11 at 4:33

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