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

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

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:

 $(A\otimes_{su}B)^{n}=\coprod_{p,q\text{ : }p+q=n}A^{p}\otimes B^{q}$
Title graded tensor product GradedTensorProduct 2013-03-22 12:45:44 2013-03-22 12:45:44 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 16W55 TensorProduct super tensor product