Example 8.3.3. Every associative \(R\)-algebra \(A\) defines an associative DGA \(A[0]\) concentrated in degree \(0\). If \(A\) is commutative, then \(A[0]\) is a commutative DGA. As an example with non-zero differential, let \(a\in R\) and consider the Koszul DGA \[ K_R(a):=(\Lambda _R(e),d),\qquad |e|=1,\quad d(e)=a. \] Here \(\Lambda _R(e)\) denotes the exterior algebra on one generator, so this is a commutative DGA in homological degrees \(1\) and \(0\).
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