A note on free differential graded algebra resolutions (Q1375801)

A standard way of constructing a DGA resolution of an algebra \(R\) yields a DGA \(U=T(E)\) which is a free tensor algebra over an appropriate module \(E\). The main result of this paper is that if \(U=T(E){\buildrel{\varepsilon}\over\to} R\) is a resolution, one can use it to construct a resolution \(R\otimes E\otimes R {\buildrel\sigma\over\to}\Omega_R\) of \(\Omega_R=\text{ker}(R\otimes R{\buildrel{\text{mult}}\over\longrightarrow} R)\) by free \(R\)-bimodules. In the context of connected graded algebras, \(\sigma\) is a resolution if and only if \(\varepsilon\) is. A given application is the construction of explicit free DGA resolutions of generalized Koszul algebras.