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Exact Sequences - Diagrams that 'commute' - Example

I am reading Dummit and Foote Section 10.5 on Exact Sequences.

I am trying to understand Example 1 as given at the bottom of page 381 and continued at the top of page 382 - please see attachment for the diagram and explanantion of the example.

The example, as you can no doubt see, requires an understanding of the nature of the quotient module

To make this quotient more tangible, in this example take m = 6, n = 3 so k = 2.

Then we are trying to understand the nature of the quotient module

Now consider the nature of

We have = { ... ... -18, -12, -6, 0 , 6, 12, 18, 24, ... ... }

and = {... ... -17, -11, -5, 1, 7, 13, 19, 25, ... }

and so on

But what is ? and indeed, further, what is ?

Can someone please help clarify this matter?

Peter

Re: Exact Sequences - Diagrams that 'commute' - Example

I am an amateur but i find your question interesting. I thought Z/6Z = {0 , 1 , 2 , 3 , 4 , 5} ?

That would make 3Z/6Z = {0 , 3} ?

So (Z/6Z)/(3Z/6Z) = {0, 1, 2, 3, 4, 5}/{0, 3} = {1 , 2 , 4 , 5} <---- really not sure about this but it appears the parenthesis force the division in the center to become a SUBTRACTION of sorts ... {0, 1, 2 , 3, 4 , 5} - {0 , 3} = {1 , 2 , 4 , 5} where the elements 0 and 3 have been subtracted from Z/6Z.

I apologize if this is way off.

P.S. My Google Docs Viewer can't decode your pdf attachment.

:D

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Re: Exact Sequences - Diagrams that 'commute' - Example

Hi Bernhard,

I hope the following attachment helps.

Attachment 29020