. Tate made systematic use of commutative di erential graded algebras over a commutative noetherian ring R and showed that there always exists a free resolution X of the residue class ring R = M which is a commutative di erential graded algebra over R in (1). I want to introduce a generalization of the result in a suitable way to a case when M is not necessarily an ideal of R. It turns out that homotopy Lie theory, so called L∞-homotopy theory, plays an essential role. 

(1) Tate, John; Homology of Noetherian rings and local rings. Illinois J. Math. 1 (1957), 14-27.