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I am reading Paolo Aluffi's book: Algebra: Chapter 0 ... ...
I am currently focussed on Section 5.4 Free Abelian Groups ... ...
I need help with an aspect of Aluffi's preamble to introduce Proposition 5.6 ...
Proposition 5.6 and its preamble reads as follows:
View attachment 5582In the above text from Aluffi's book we find the following:
" ... ... For $$H = \mathbb{Z}$$ there is a natural function $$j \ : \ A \longrightarrow \mathbb{Z}^{ \oplus A }$$ , obtained by sending $$a \in A$$ to the function $$j_a \ : \ A \longrightarrow \mathbb{Z}$$ ... ... "
My problem is in (precisely and rigorously) understanding the claim that $$j$$ sends $$a \in A$$ to the function $$j_a \ : \ A \longrightarrow \mathbb{Z}$$ ... ...
The function $$j_a$$ is actually a set of ordered pairs no two of which have the same member ... BUT ... $$j$$ does not (exactly anyway) seem to send $$a \in A$$ to this function ...If we pretend for a moment that $$A$$ is a countable ordered set ... then we can say that what $$j$$ seems to do is send $$a \in A$$ to the image-set of $$j_a$$, namely
$$( \ ... \ ... \ ,0,0, 0, \ ... \ ... \ 0,1,0 \ ... \ ... \ ... \ \ ,0,0, 0, \ ... \ ... ) $$
where the 1 is in the a'th position ...
So then $$j$$ seems to send $$a \in A$$ to the image-set of $$j_a$$ and not to $$j_a$$ itself ... ... (... ... not sure how to put this argument for the case where the set A is uncountable and not ordered ... ... )
Given my analysis ... how do we justify or make sense of Aluffi's claim that " there is a natural function $$j \ : \ A \longrightarrow \mathbb{Z}^{ \oplus A }$$ , obtained by sending $$a \in A$$ to the function $$j_a \ : \ A \longrightarrow \mathbb{Z}$$ "
Hope someone can critique my analysis and clarify the issue to which I refer ...
Help will be much appreciated ...
Peter*** EDIT ***
Just another concern over possibly missing something in fully understanding Aluffi's text above ... he introduces the general case with a general abelian group $$H$$ ... ... and then defines $$H^{ \oplus A}$$ ... ... but never uses $$H$$ ... he just puts it equal to $$\mathbb{Z}$$ ... if you are just going to put $$H = \mathbb{Z}$$ ... ... why bother with $$H$$ ... why introduce it ... just start with $$\mathbb{Z}$$ ... ... does anyone have an idea what Aluffi is doing ... ... ? ... ... he does a similar thing when explaining free modules ... ... am I missing something ... ... ? ... ...
======================================================
To give MHB members reading this post a sense of the approach and notation of Aluffi to free Abelian groups I am here providing Aluffi's introduction to free Abelian groups up to and including Proposition 5.6 ... as follows:View attachment 5583
View attachment 5584
View attachment 5585
I am currently focussed on Section 5.4 Free Abelian Groups ... ...
I need help with an aspect of Aluffi's preamble to introduce Proposition 5.6 ...
Proposition 5.6 and its preamble reads as follows:
View attachment 5582In the above text from Aluffi's book we find the following:
" ... ... For $$H = \mathbb{Z}$$ there is a natural function $$j \ : \ A \longrightarrow \mathbb{Z}^{ \oplus A }$$ , obtained by sending $$a \in A$$ to the function $$j_a \ : \ A \longrightarrow \mathbb{Z}$$ ... ... "
My problem is in (precisely and rigorously) understanding the claim that $$j$$ sends $$a \in A$$ to the function $$j_a \ : \ A \longrightarrow \mathbb{Z}$$ ... ...
The function $$j_a$$ is actually a set of ordered pairs no two of which have the same member ... BUT ... $$j$$ does not (exactly anyway) seem to send $$a \in A$$ to this function ...If we pretend for a moment that $$A$$ is a countable ordered set ... then we can say that what $$j$$ seems to do is send $$a \in A$$ to the image-set of $$j_a$$, namely
$$( \ ... \ ... \ ,0,0, 0, \ ... \ ... \ 0,1,0 \ ... \ ... \ ... \ \ ,0,0, 0, \ ... \ ... ) $$
where the 1 is in the a'th position ...
So then $$j$$ seems to send $$a \in A$$ to the image-set of $$j_a$$ and not to $$j_a$$ itself ... ... (... ... not sure how to put this argument for the case where the set A is uncountable and not ordered ... ... )
Given my analysis ... how do we justify or make sense of Aluffi's claim that " there is a natural function $$j \ : \ A \longrightarrow \mathbb{Z}^{ \oplus A }$$ , obtained by sending $$a \in A$$ to the function $$j_a \ : \ A \longrightarrow \mathbb{Z}$$ "
Hope someone can critique my analysis and clarify the issue to which I refer ...
Help will be much appreciated ...
Peter*** EDIT ***
Just another concern over possibly missing something in fully understanding Aluffi's text above ... he introduces the general case with a general abelian group $$H$$ ... ... and then defines $$H^{ \oplus A}$$ ... ... but never uses $$H$$ ... he just puts it equal to $$\mathbb{Z}$$ ... if you are just going to put $$H = \mathbb{Z}$$ ... ... why bother with $$H$$ ... why introduce it ... just start with $$\mathbb{Z}$$ ... ... does anyone have an idea what Aluffi is doing ... ... ? ... ... he does a similar thing when explaining free modules ... ... am I missing something ... ... ? ... ...
======================================================
To give MHB members reading this post a sense of the approach and notation of Aluffi to free Abelian groups I am here providing Aluffi's introduction to free Abelian groups up to and including Proposition 5.6 ... as follows:View attachment 5583
View attachment 5584
View attachment 5585
Last edited: