Just after posting this I think I've worked it out. If we supposed we have such that , then choose . Then . Therefore . By a similar argument we have .
Therefore in the direct limit, is the same no matter which we pick.
Hello, I just have a quick question about direct limits. Let be a directed system of -modules, where is a directed set. I want to show that
is an -module, where if there exists such that .
Now in my notes we define addition as follows:
The sum of and is given by choosing an element with and in the -module .
But my question is, surely this thing is not well defined because there might be some , where ! So this addition would be in a different -module. Does this limit module only make sense if in our directed set we make a specific choice of element for every pair in ?
Thanks for any help.
Just after posting this I think I've worked it out. If we supposed we have such that , then choose . Then . Therefore . By a similar argument we have .
Therefore in the direct limit, is the same no matter which we pick.