In a homework, I have to work with tensor products of chain complexes and have to show the following:
Let $\mathbb{K}$ be a principal ideal domain. For two chain complexes $A_{\bf{.}}$, $B_{\bf{.}}$ over $\mathbb{K}$ with boundary operators $\partial^{A}$, $\partial^{B}$, we define $C_{\bf{.}}$ by setting $C_n := \sum_{k+l=n} A_k \otimes B_l$ and $\partial^{A \otimes B} (a \otimes b) := \partial^{A} (a) \otimes b + (-1)^k a \otimes \partial^{B} b $ for two generators $a \in A_k , b \in B_l $.
Show that this is a chain complex.
I have problems computing the following :
$\partial^{A \otimes B} (a \otimes b + c \otimes d) $
$\partial^{A \otimes B} (a \otimes b) \circ \partial^{A \otimes B} (c \otimes d) $
, because if one has $(a \otimes b + c \otimes d = e \otimes f) $i can't write down simple e,f.
Since $\partial^{A+B}$ is a homomorphism, it must be linear. When we define $\partial^{A+B}$ only on elements of the form $a \otimes b$, we get the full homomorphism $\partial^{A+B}$ by extending it linearly to arbitrary elements of $A_k \otimes B_l$. So this means that $\partial^{A+B}(a \otimes b + c \otimes d) = \partial^{A+B}(a \otimes b) + \partial^{A+B}(c \otimes d)$.
See if you can show it is a chain complex now.