# Free Abelian Groups ... Aluffi Proposition 5.6

Gold Member
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:

In 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## ... does anyone have an idea what Aluffi is doing ... ... ??? ... ... he does a similar thing when explaining free modules ... ... am I missing something ... ... ??? ... ...

======================================================

To give Physics Forum 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:

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fresh_42
Mentor
What do you think about the following definitions:
$$j = \bigoplus_{a \in A}^{} j_a \text{ and } j_a = \bigoplus_{b \in A}^{} \delta_{ab} \cdot a$$

Math Amateur
mathwonk
Homework Helper
2020 Award
Unless you are really having fun reading that book, I would suggest it is not going to do you a lot good. If you want to learn some algebra, you might try Michael Artin's book, Algebra,.

fresh_42
Mentor
Unless you are really having fun reading that book, I would suggest it is not going to do you a lot good. If you want to learn some algebra, you might try Michael Artin's book, Algebra,.
I agree. It seems Paolo Aluffi has a very categorial approach. This might have its right when (co-)homology theory is the goal. However, I've seen physical concepts in which the latter is needed. But I agree it's a rather dusty ground to start with algebra. I took my first steps with these books:
https://www.amazon.com/s/ref=nb_sb_...van+der+waer,stripbooks,384&tag=pfamazon01-20

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Gold Member
I agree. It seems Paolo Aluffi has a very categorial approach. This might have its right when (co-)homology theory is the goal. However, I've seen physical concepts in which the latter is needed. But I agree it's a rather dusty ground to start with algebra. I took my first steps with these books:
https://www.amazon.com/s/ref=nb_sb_...van+der+waer,stripbooks,384&tag=pfamazon01-20

Thanks mathwonk and fresh_42 ... but note that I wish to understand a category theory approach ... seems an interesting theoretical basis and language for algebra ... and possibly category theory has an interesting way to help in algebraic topology as well ... it is challenging abstract though ...

Peter

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Gold Member
What do you think about the following definitions:
$$j = \bigoplus_{a \in A}^{} j_a \text{ and } j_a = \bigoplus_{b \in A}^{} \delta_{ab} \cdot a$$

Fresh_42 ... can you expand a bit on how to interpret your definitions ...

Peter

Gold Member
Hi Fresh_42 ... just some vague thoughts on your definitions ... what they may mean, that is ...

Assume (in order to get an idea of the meaning of the definitions) that ##A## is an ordered and countable set ...

... then we have ##j \ : \ A \longrightarrow \mathbb{Z}^{ \oplus A}## can be written as follows:

##j = \bigoplus_{a \in A} \ j_a##

##= ( \ j_{a_1} \ , \ j_{a_2} \ , \ j_{a_3}, \ ... \ ... \ ...\ ... )##

... so we can write ... ...

##j(a_1) = ( \ j_{a_1} (a_1) \ , \ j_{a_2} (a_1) \ , \ j_{a_3} (a_1) , \ ... \ ... \ ...\ ... )##

... BUT ... how do we interpret ## j_{a_1} (a_1) \ , \ j_{a_2} (a_1) \ , \ j_{a_3} (a_1) , \ ... \ ... ## etc ...

I am assuming that ## j_{a_1} (a_1) \ = 1 \ , \ j_{a_2} (a_1) \ = 0 \ , \ j_{a_3} (a_1) \ = 0 \ , \ ... \ ## etc

But ... I am not sure how that comes out of your definition/formula for ##j_a## ... ...

Can you please comment and clarify ...

Peter

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mathwonk
Homework Helper
2020 Award
please forgive me, but to be blunt, before learning a fancy way to express something, it helps to actually understand the topic. and you do not. so in my opinion you are handicapping yourself by pursuing this less than useful presentation of the material. long ago i read briefly paolo's book and wondered just who in the world he aimed it at. i would never recommend this book to any young person, not just you.

your questions make it obvious that you are not understanding anything in this book. this is not a criticism of you by the way. so i suggest an experiment, try one of the other books we recommend and see for yourelf if they do not speak more clearly to you.

fresh_42
Mentor
I should probably better have written ##j_a## without the plus because it's only a function ##j_a : A → ℤ##. Your dots behind the ##ℤ## above misleaded me together with Aluffi's tendency to denote different things with the same letter.

##j_a## is defined exactly as the Kronecker-delta(-function): ##j_a(b) = \delta_{ab}##, i.e. ##1## if ##a=b## and ##0## elsewhere. It is a set function which picks a special ##a## out of ##A## and shouts "Got it!" by returning a ##1##.

##j## is then the cartesian product of all ##j_a##, i.e. ##j = \oplus_{a \in A} j_a##, a function ##j: A → \{0,1\}^A ⊆ ℤ^{A}##. It is also a set function only that here you can pick any element of ##A##.
If you want to define it as the general function ##j :A → F^{ab}(A)## then it would be ##j = \oplus_{a \in A}j_a \cdot a## and
$$j(b) = \oplus_{a \in A}j_a(b) \cdot a = \oplus_{a \in A}\delta_{ab} \cdot a = 1 \cdot b = b \in F^{ab}(A)$$

Unfortunately Aluffi uses both versions and denotes them equally by ##j##.
In 5.4. it's the natural embedding ##j : A → F^{ab}(A)## where you have to multiply the ##j_a## with an ##a \in A## for ##1 \in ℤ## isn't an element of ##F^{ab}(A)## and there is no order either,
while in Prop. 5.6. he omits these multiplications by ##a## and ends up in ##ℤ^A## instead. Regarding that he exactly wants to prove that ##F^{ab}(A) ≅ ℤ^{\oplus A}## it's rather sloppy.

Of course you could as well define the multiplication by an ##a \in A## as part of ##j_a## and omit it in the definition of ##j##. But in this case ##j_a## would be the natural embedding ##j_a : \{a\} → F^{ab}(A)## and not a mapping in ##ℤ##.

The crucial part of the story in the book, however, is another. Although ##A## can have arbitrary many elements and the cartesian product will be respectively large, the elements of ##F^{ab}(A)## are all products ##a_1^{n_1} \dots a_N^{n_N}## of (arbitrary but) finite length ##N##.
This is the difference in Aluffi's notation between ##ℤ^A## and ##ℤ^{\oplus A}## (see above with ##H## in the role of ##ℤ##).

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Math Amateur
fresh_42
Mentor
long ago i read briefly paolo's book and wondered just who in the world he aimed it at
This is something that makes me wonder, too. For a representation like this, full of categorial context, Aluffi is rather sloppy in his notations. A fact that doesn't work very well on category theory. And I seriously doubt that there are more than five lines needed to explain a free (Abelian) group in an algebra textbook.

Gold Member
please forgive me, but to be blunt, before learning a fancy way to express something, it helps to actually understand the topic. and you do not. so in my opinion you are handicapping yourself by pursuing this less than useful presentation of the material. long ago i read briefly paolo's book and wondered just who in the world he aimed it at. i would never recommend this book to any young person, not just you.

your questions make it obvious that you are not understanding anything in this book. this is not a criticism of you by the way. so i suggest an experiment, try one of the other books we recommend and see for yourelf if they do not speak more clearly to you.

Gold Member
Hi mathwonk ... look I know you are essentially trying to help ... but I think when you write:

" ... your questions make it obvious that you are not understanding anything in this book. ... "

I think you are being somewhat negative, de-motivating and overly harsh in your judgement ...

Peter

Last edited:
Gold Member
I should probably better have written ##j_a## without the plus because it's only a function ##j_a : A → ℤ##. Your dots behind the ##ℤ## above misleaded me together with Aluffi's tendency to denote different things with the same letter.

##j_a## is defined exactly as the Kronecker-delta(-function): ##j_a(b) = \delta_{ab}##, i.e. ##1## if ##a=b## and ##0## elsewhere. It is a set function which picks a special ##a## out of ##A## and shouts "Got it!" by returning a ##1##.

##j## is then the cartesian product of all ##j_a##, i.e. ##j = \oplus_{a \in A} j_a##, a function ##j: A → \{0,1\}^A ⊆ ℤ^{A}##. It is also a set function only that here you can pick any element of ##A##.
If you want to define it as the general function ##j :A → F^{ab}(A)## then it would be ##j = \oplus_{a \in A}j_a \cdot a## and
$$j(b) = \oplus_{a \in A}j_a(b) \cdot a = \oplus_{a \in A}\delta_{ab} \cdot a = 1 \cdot b = b \in F^{ab}(A)$$

Unfortunately Aluffi uses both versions and denotes them equally by ##j##.
In 5.4. it's the natural embedding ##j : A → F^{ab}(A)## where you have to multiply the ##j_a## with an ##a \in A## for ##1 \in ℤ## isn't an element of ##F^{ab}(A)## and there is no order either,
while in Prop. 5.6. he omits these multiplications by ##a## and ends up in ##ℤ^A## instead. Regarding that he exactly wants to prove that ##F^{ab}(A) ≅ ℤ^{\oplus A}## it's rather sloppy.

Of course you could as well define the multiplication by an ##a \in A## as part of ##j_a## and omit it in the definition of ##j##. But in this case ##j_a## would be the natural embedding ##j_a : \{a\} → F^{ab}(A)## and not a mapping in ##ℤ##.

The crucial part of the story in the book, however, is another. Although ##A## can have arbitrary many elements and the cartesian product will be respectively large, the elements of ##F^{ab}(A)## are all products ##a_1^{n_1} \dots a_N^{n_N}## of (arbitrary but) finite length ##N##.
This is the difference in Aluffi's notation between ##ℤ^A## and ##ℤ^{\oplus A}## (see above with ##H## in the role of ##ℤ##).

Thanks Fresh_42 ... most helpful ... appreciate the help ...

Peter

fresh_42
Mentor
Thanks Fresh_42 ... most helpful ... appreciate the help ...

Peter
You're welcome!

Gold Member
Hi Fresh_42, mathwonk,

What do you think of Dummit and Foote: Abstract Algebra as an alternative to Aluffi ...

Peter

fresh_42
Mentor
A note on the diagram in 5.4.

The free abelian group ##F^{ab}(A)## is basically the set of all words over an alphabet ##A## plus commutativity and formal inverse. (The neutral element can be seen as empty word.)
It has no additional structure other than neutral element, (formal) inverse, associativity and commutativity. No rules like ##a^2=1## which reflections have, or ##a^3=1## for rotations of 120°.
##j## is the embedding of the alphabet in it, the one-letter-words.
Now you can have any abelian group ##G## with an additional structure (e.g. rotations, reflections) and a mapping ##f : A → G##.
Then there is a group homomorphism ##φ: F^{ab}(A) → G## which puts the structure upon ##F^{ab}(A)## and gets ##G##.

Math Amateur
Gold Member
A note on the diagram in 5.4.

The free abelian group ##F^{ab}(A)## is basically the set of all words over an alphabet ##A## plus commutativity and formal inverse. (The neutral element can be seen as empty word.)
It has no additional structure other than neutral element, (formal) inverse, associativity and commutativity. No rules like ##a^2=1## which reflections have, or ##a^3=1## for rotations of 120°.
##j## is the embedding of the alphabet in it, the one-letter-words.
Now you can have any abelian group ##G## with an additional structure (e.g. rotations, reflections) and a mapping ##f : A → G##.
Then there is a group homomorphism ##φ: F^{ab}(A) → G## which puts the structure upon ##F^{ab}(A)## and gets ##G##.

Thanks fresh_42 ... yes ... do understand that ... but appreciate the further help ...

Peter

fresh_42
Mentor
Hi Fresh_42, mathwonk,

What do you think of Dummit and Foote: Abstract Algebra as an alternative to Aluffi ...

Peter
I don't know it. Perhaps I got driven to van der Waerden because I knew the girl who typed the German version . But I never regretted it and I still use it to look things up which I have forgotten in detail. But it's more about groups, fields and numbers and practically nothing homological.

Math Amateur
Gold Member
I don't know it. Perhaps I got driven to van der Waerden because I knew the girl who typed the German version . But I never regretted it and I still use it to look things up which I have forgotten in detail. But it's more about groups, fields and numbers and practically nothing homological.
Thanks anyway fresh_42 ... I did check van der Waerden at Amazon ... seems it is a classic ... but may be a bit dated ..

Peter

fresh_42
Mentor
Thanks anyway fresh_42 ... I did check van der Waerden at Amazon ... seems it is a classic ... but may be a bit dated ..

Peter
It arose from lectures by Artin and Emmy Noether. What a pedigree!

Gold Member
It arose from lectures by Artin and Emmy Noether. What a pedigree!
Indeed fresh_42 ... could not improve on that!

Peter

fresh_42
Mentor
I just like to add a thought on Aluffi's approach on algebra.

I've just read lavinia's post #8 here: https://www.physicsforums.com/threa...-forms-manifolds-algebra.868945/#post-5474164 which I liked very much. Next I read https://en.wikipedia.org/wiki/Differential_form which in my own language is by far more categorial and emphasizes the role of cohomology groups (and thus explaining better what closed or exact forms are, why they are called so, resp.).

Both display how modern physics uses seemingly abstract concepts which are mathematically pure homology theory.

Math Amateur
mathwonk
Homework Helper
2020 Award
well forgive the negativity, of course that does not help. rather you are understanding what paolo has written, but his treatment focuses on how things are expressed rather than what they mean, so i fear you are not understanding the topic, which is not your fault. It is fundamental that the categorical approach which he takes, emphasizes that to understand a type of object such as a module, one should understand maps between them, i.e. how they are compared. So one should understand an abelian group G by understanding Hom(G,H) for all abelian groups H, i.e. we want to know how to define maps out of our group G.

a basic construction is the direct sum of abelian groups. So we want to know how to define maps out of a direct sum of abelian groups {Gj}. The basic result is that this is an abelian group Sum(Gj) such that to define a map out of Sum(Gj) all you have to do is to define a map out of each individual summand Gj, and that will give a unique map out of Sum(Gj).

Now a free abelian group on a set A is simply a direct sum of #(A) copies of the integers, so given an abelian group H, a map from the free abelian group on A to H, is given by one map out of each copy of Z, for each element of A, and that means simply one element chosen from H for each element in the family A. So a map from the free abelian group on the set A, to H, corresponds to a choice of one element of H for each element of A, i.e. to a function from A to H.

OK, if you got that out of paolo's presentation, then I stand down, and you are doing fine.

I myself think paolo has made this needlessly complicated and abstract. so i am trying to divert you to a book where it is already explained better. forgive me if this if not helpful, god bless.

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Math Amateur
Gold Member
well forgive the negativity, of course that does not help. rather you are understanding what paolo has written, but his treatment focuses on how things are expressed rather than what they mean, so i fear you are not understanding the topic, which is not your fault. It is fundamental that the categorical approach which he takes, emphasizes that to understand a type of object such as a module, one should understand maps between them, i.e. how they are compared. So one should understand an abelian group G by understanding Hom(G,H) for all abelian groups H, i.e. we want to know how to define maps out of our group G.

a basic construction is the direct sum of abelian groups. So we want to know how to define maps out of a direct sum of abelian groups {Gj}. The basic result is that this is an abelian group Sum(Gj) such that to define a map out of Sum(Gj) all you have to do is to define a map out of each individual summand Gj, and that will give a unique map out of Sum(Gj).

Now a free abelian group on a set A is simply a direct sum of #(A) copies of the integers, so given an abelian group H, a map from the free abelian group on A to H, is given by one map out of each copy of Z, for each element of A, and that means simply one element chosen from H for each element in the family A. So a map from the free abelian group on the set A, to H, corresponds to a choice of one element of H for each element of A, i.e. to a function from A to H.

OK, if you got that out of paolo's presentation, then I stand down, and you are doing fine.

I myself think paolo has made this needlessly complicated and abstract. so i am trying to divert you to a book where it is already explained better. forgive me if this if not helpful, god bless.

Hi mathwonk ...

Thank you for such a considered and also considerate post ... most helpful ...

Yes, I did follow most of that but struggled with both the mechanics of Aluffi's approach and the notation ... BUT ... I think what you say has a lot of wisdom to it ... so I am moving to another text ...

I am an obsessive math book collector so I have a number of options ... indeed I will definitely use Michael Artin's book ...

Mind you, I am trying to get an understanding of the theory of modules ... and within that topic an understanding of the algebra of tensors ... so I thought that Dummit and Foote's book "Abstract Algebra" might well be appropriate ... what do you think?

Peter

Gold Member
I just like to add a thought on Aluffi's approach on algebra.

I've just read lavinia's post #8 here: https://www.physicsforums.com/threa...-forms-manifolds-algebra.868945/#post-5474164 which I liked very much. Next I read https://en.wikipedia.org/wiki/Differential_form which in my own language is by far more categorial and emphasizes the role of cohomology groups (and thus explaining better what closed or exact forms are, why they are called so, resp.).

Both display how modern physics uses seemingly abstract concepts which are mathematically pure homology theory.

Thanks for your post, fresh_42 ... yes, checked that ... BUT ... I think that you still feel I should move on from Aluffi's text ... is that right?

By the way, I thought I'd share some excellent and helpful posts I got on the topic of free abelian groups ... at the Math Help Boards forum on Linear and Abstract Algebra ...

See: http://mathhelpboards.com/linear-ab...an-groups-aluffi-proposition-5-6-a-18586.html