Let $A,B,C$ be three graded $R$- algebras. Can someone explicitely give me the algebra multiplication $(a\otimes b\otimes c) \cdot (a'\otimes b'\otimes c')$ where $a,b,...$ are of degree $|a|,|b|,...$?
I know the multiplication for the case of two algebras, but Im not sure how to generalise. Thank you!
Here's how I remember how to find the answer. The idea is that you are formally creating an algebra which $A$, $B$, and $C$ embed in and in which they (graded-)commute. In this picture, we identify $a\in A$ with $a\otimes 1\otimes 1$, $b\in B$ with $1\otimes b\otimes 1$, and $c\in C$ with $1\otimes 1\otimes c$. We then can think of $a\otimes b\otimes c$ as $abc$.
We can now figure out how to multiply $(a\otimes b\otimes c) \cdot (a'\otimes b'\otimes c')$ by just using the assumption that the elements of our different algebras are supposed to graded-commute. This gives: \begin{align*}abca'b'c'&=(-1)^{|a'||c|}aba'cb'c' \\ &=(-1)^{|a'||c|+|a'||b|}aa'bcb'c' \\ &=(-1)^{|a'||c|+|a'||b|+|b'||c|}aa'bb'cc'. \\ \end{align*}
That is, in tensor notation, $$(a\otimes b\otimes c) \cdot (a'\otimes b'\otimes c')=(-1)^{|a'||c|+|a'||b|+|b'||c|}aa'\otimes bb'\otimes cc'.$$
(How to turn this argument into a formal proof depends on exactly what your definition of $A\otimes B\otimes C$ as an algebra is.)