Given $a_1,\dots,a_n\in A$, with $A$ a suitable ring, my algebra teacher defined the Koszul complex associated to $a_1,\dots,a_n$ with coefficients in $A$ in this way: $$K(a_1,\dots,a_n;A):=\bigoplus_{i=0}^n\bigwedge^iA^n,$$ with differential $\partial$ such that $\partial(e_i)=a_i$. With the rule $$\partial(uv)=\partial(u)v+(-1)^iu\partial(v)$$if $u\in K(a_1,\dots,a_n;A)_i$, the Koszul complex is a DG-algebra. But if we consider the restriction of $\partial$ to $K(a_1,\dots,a_n;A)_i$ we obtain differentials $\partial_i$ for a chain complex.
I have to prove that the following isomorphism holds $$K(a_1,\dots,a_n;A)\simeq K(a_1;A)\otimes\dots\otimes K(a_n;A).$$
This is clear "algebraically", but I need to find an isomorphism of complexes. Could you please help me?
Have a nice day.
Asdrubale