graded tensor product

If $A$ and $B$ are $\mathbb{Z}$-graded algebras, we define the graded tensor product (or super tensor product) $A{\otimes }_{su}B$ to be the ordinary tensor product as graded modules, but with multiplication - called the super product - defined by

$$(a\otimes b)(a^{\prime }\otimes b^{\prime })={(-1)}^{(\text{deg }b)(\text{deg }{a}^{\prime })}a{a}^{\prime }\otimes b{b}^{\prime }$$

where $a,a^{\prime },b,b^{\prime }$ are homogeneous. The super tensor product of $A$ and $B$ is itself a graded algebra, as we grade the super tensor product of $A$ and $B$ as follows:

$${(A{\otimes }_{su}B)}^n=\coprod _{p,q\text{ : }p+q=n}{A}^p\otimes B^q$$