I'm definitely not an expert of $K$-theory but I need to know a couple of results in order to complete a computation. I tried o find these things on the standard books but without success. I'm sorry if this question may seem dumb for the experts in the sector. I'll be happy also if you show me some exact references.
- For any field $F$, we have that $K_1(F)=F^\times$. Now suppose that I want to calculate $K_1$ of a ring $A=\prod^{\infty}_{n} F_n$ where $F_n$ is a field. Is there any relation between $K_1(A)$ and $\prod F^\times_n$?
- Suppose that I have a complex of modules $$M^\bullet:\quad\dots\to M^1\to M^2\to M^3\to\dots$$ Is there any notion of $K_1(M^\bullet)$? In particular $K_1(M^\bullet)$ should be a complex where each term is $K_1(M^i)$ but I don't know if it exists and moreover how the boundary maps are defined.
Your ring $A = \prod F_i$ is unit von Neumann regular, which means that for every $x\in A$ there exists a unit $y\in A$ such that $xyx = x$. Menal and Moncasi [1, Theorem 1.6] proved that for such a ring, $K_1(A)$ is isomorphic to $A^\times/V(A)$ where $V(A)$ is the subgroup generated by the set $ \{(ab + 1)(ba + 1)^{-1} : (ab + 1)\in A^\times \}$. For commutative rings $V(A)$ is trivial. Therefore: $$K_1(A) \cong \prod_i F_i^\times.$$ As for your second question, it doesn't make sense. The $K$-theory functors are in general applied to some sort of category: f.g. projective modules, exact categories, monoidal categories, Waldhausen categories, etc. So, it doesn't make sense to apply it to a single module in a meaningful and interesting way. Though, you could attempt to apply it to some category associated to a given module, but I'm not sure that is along the lines of what you were looking for.
[1] Menal, Pere; Moncasi, Jaume. $K_{1}$ of von Neumann regular rings. J. Pure Appl. Algebra 33 (1984), no. 3, 295--312.