In a 1957 paper ([Link](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-1/issue-1/Homology-of-Noetherian-rings-and-local-rings/10.1215/ijm/1255378502.full)), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R then there is a graded commutative DGA $X$ over $R$ with $H\_i X=0$ except $H\_0 X= R/I$ (I guess R should be noetherian). Further, $X$ is a free $R$ module in each degree. Is it known if there is a similar result for any other classes of commutative $R$ algebra?


If such an extension of the result is false, I would very much appreciate a counterexample.


Thanks for your time,
Sean


**EDIT:** (by other classes I mean commutative R algebras (Not DGAs) that are not of the form $R/I$). Also, I am happy with counter examples where the DGA is not level-wise free but instead projective or flat (although it shouldn't end up mattering for my purposes).


I am not concerned with relaxing the noetherian hypothesis but in resolving a different commutative $R$-algebra (not in the sense of Tate).


An extension of the above result would be equivalent to saying that there are no level wise free/projective/flat $E\_\infty$ algebras with homology a given commutative $R$-algebra $A$ that are not strictly commutative (in the graded sense).


I worry that I am making things more confusing, sorry if that is the case, and thank you for the answers so far.



Extension of Tate's result regarding Tor
I am not sure what the protocol is on answering ones own question so I am making it community wiki.


What I was most curious about is the following:
When does the projective resolution of a commutative algebra as a module over a commutative ring have the structure of a graded commutative (skew in Tate's paper) DGA?


As Mariano pointed out, when I asked a different version on MSE, this is always the case.
Tate showed that this is true for algebras of the form $R/I$ where $I$ is an ideal of $R$ (the commutative ring we are working over). As Mariano points out, every finitely generated (I think this can probably be dropped) commutative $R$ algebra is a quotient of $R[x\_1, x\_2, ... x\_n]$. So there is a graded commutative DGA that is a free resolution of $X$ as an $R[x\_1, x\_2, ... x\_n]$-module by Tate's result. You simply forget from $R[x\_1, x\_2, ... x\_n]$-modules to $R$-modules and you still have a graded commutative DGA that is a free resolution of $X$ as an $R$-module.


Anyway this is essentially what I was looking for.


(Mariano, if you want to post something like this, I will un-accept this answer and accept yours.)


