Is There a Limit to the Number of Elements in a Coproduct/Direct Sum?

[Sumanta]

Hi ,

I was trying to understand why or where would the problem arise in the definition of the direct sum for the coproduct/direct sum for the set (1, 1, 1, ...) infinite number of times. I was trying to reason out as follows. I posted this on the set theory but I am not sure how to tag it to algebra hence I decided to copy paste the same.

(1, 1, 1, ...)\rightarrow Y

\uparrow

f_{i}\rightarrow YPls note that the Y is the same as I cannot write the angular arrow.

Now if I say that f_{i} acting on (e_{i}) maps it to ( 0, 0, ...1 at the ith coordinate , 0, ...) then what is the place where I am making a mistake. The problem as I see is that either the map from the set (1, 1, 1, ...) to Y is either not unique or map from f_{i}\rightarrow Y does not give the same value as the other path. I am really not sure which is the one and why.

Now Y can be any kind of general space. So I first thought if Y was just the {0} or the whole space R^{\infty}. But I see very clearly that {0} choice does not give a counter example. Not sure how to handle the second case. Even thought with Y= N ( countable ) but still could not get to any kind of solution.
Thx

 

[Sumanta]

Hi,

I thought about the problem and I think this could be thought of as a way of thinking.

So what we need is the following.

Given data:

f_{i}: X_{i}\stackrel{injective}{\rightarrow} P
g_{i}: X_{i} \stackrel{homomorphism}{\rightarrow} X

Now define in someway

g: P \stackrel{homomorphism}{\rightarrow} X



P is the coproduct we are trying to define. Pls note that we need to define g given g_{i}. g is actually to be defined so that it completes the commutative diagram and then only P is the coproduct.

The most obvious way of defining g on each of the basis elements is g := \sum g_{i} where i ranges over the index set. Now all our group, ring homomorphism operations are defined over finite sums or finite products. So there is no way we could make sense of an infinite sum in the definition of g. Thus we have to restrict it to finite sum. Hence if we have to restrict it to finite sum then only a finite number of elements could be non zero and the rest are all 0s.

This is my understanding and if somebody could point out that there is some gaps it would be very helpful.

Thx