Graded algebras, graded modules, and the graded tensor product

Let $k$ be a field. A $\mathbb{Z}\mathrm{/}2$-graded $k$-vector space is a direct sum $A=A^0\oplus A^1$; a homogeneous element $a\in A^i$ has parity $\mathrm{\mid }a\mathrm{\mid }=i\in \{0,1\}$. A $\mathbb{Z}\mathrm{/}2$-graded unital associative $k$-algebra is such a space with $1\in A^0$ and $A^i{A}^j\subseteq A^{i+j(\mathrm{m}\mathrm{o}\mathrm{d}2)}$. A graded left module is a module $M=M^0\oplus M^1$ satisfying $A^i{M}^j\subseteq M^{i+j(\mathrm{m}\mathrm{o}\mathrm{d}2)}$. A graded submodule is an $A$-submodule $N=N^0\oplus N^1$; a graded-simple module is nonzero and has no proper nonzero graded submodule. An isomorphism of graded modules preserves both parity components.

For graded algebras $A,B$, their graded tensor product $A{\widehat{\otimes }}_{k}B$ has underlying vector space $A{\otimes }_{k}B$, parity $(A\widehat{\otimes }B{)}^r={⨁}_{i+j=r}{A}^i{\otimes }_k{B}^j$, and multiplication

$$(a\otimes b)(a^{\mathrm{\prime }}\otimes b^{\mathrm{\prime }})=(-1{)}^{\mathrm{\mid }b\mathrm{\mid }\mathrm{\mid }{a}^{\mathrm{\prime }}\mathrm{\mid }}(a{a}^{\mathrm{\prime }}\otimes b{b}^{\mathrm{\prime }})$$

for homogeneous factors, extended by the finite sum

$$\sum _{i,j,r,s\in \{0,1\}}(-1{)}^{jr}(a_i{a}_r^{\mathrm{\prime }}\otimes b_j{b}_s^{\mathrm{\prime }}),a=\sum _i{a}_i,b=\sum _j{b}_j,a^{\mathrm{\prime }}=\sum _r{a}_r^{\mathrm{\prime }},b^{\mathrm{\prime }}=\sum _s{b}_s^{\mathrm{\prime }}\mathrm{.}$$

Each parity projection is linear, so this formula is $k$-multilinear in the four inputs and therefore defines a multiplication on $(A{\otimes }_{k}B)\times (A{\otimes }_{k}B)$ by Finite iterated tensor products represent multilinear maps independently of parenthesization. The unit is $1_A\otimes 1_B$. For three homogeneous tensors, both associative bracketings contribute the sign exponent $\mathrm{\mid }b\mathrm{\mid }\mathrm{\mid }{a}^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }\mathrm{\mid }{a}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{b}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }{a}^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\mid }$, so the multiplication is associative.