/sci/ - Science & Math

>>12616348
You need to habe 1/p for every prime and then require it to be closed under multiplication and addtuon.

>>11963361
The homomorphism you described is one reason why people work with direct sums of abelian groups (or modules more generally) instead of direct products, as then you only have a finite number of non-identity coordinates in each element of your sum group/module. However, there could be some way to take the direct product of a single fixed group and then consider the system of all the finite products of the group with itself and include those into the infinite product. I must be honest, I don't remember if the infinite product is a direct limit of such finite products, but if it is, then you would obtain the homomorphism you so much desire (after defining the homomorphisms from the finite products in some way compatible with the inclusions). Then you could try to investigate the convergence of this induced homomorphism by looking at how the finite cases work. You should check that out if you feel like it.

>>11925727
Let's say you have the filtration [math]0 = F^MH \subset F{M-1}H \subset \cdots \subset F^1H \subset F^0H = H[/math]. Suppose [math]0 \to F^{n+1}H \to F^nH \to F^nH/F^{n+1}H \to 0[/math] is split exact for each [math]0\le n \le M-1[/math]. Then [math]F^nH \cong F^{n+1}H \oplus F^nH/F^{n+1}H[/math], so there is the isomorphism chain [math]H \cong F^1H\oplus H/F^1H \cong F^2H\oplus F^1H/F^2H \oplus H/F^1H \cong \cdots \cong F^MH\oplus F^{M-1}H/F^MH \oplus \cdots \oplus F^1H/F^2H \oplus H/F^1H \cong F^{M-1}H\oplus F^{M-2}H/F^{M-1}H \oplus \cdots \oplus F^1H/F^2H \oplus H/F^1H[/math]. There are probably more general cases, but this is the most obvious one.