I Questions about non existence of GCDs vs (coimages, cokernels)

[elias001]

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho.

In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states

"Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels).

Also in the reference url page above, the authors present two theorems as facts along with their proofs and two other ones as i) and ii) along with their proofs, finally culminating in presenting fact 3 as theorem along with its proof. Fact 3 states:

"Fact 3: Let $p<n$ be a prime number, and suppose that there exist integers $a,b$ such that $p^2=a^2+nb^2$ and $P$ does not divide at least one of $a,b.$ Let $z:=a+b\sqrt{n}\;\;i\in R_n.$ Then $\text{gcd}(p^2, pz)$ does not exist."

My questions are as follows:

1) For Fact 3, does anyone knows of a book reference source? I am not surprised the proof is presented as proof by contradiction.

2) For the case of non existence of coimages or cokernels, how does one go about showing its non existence. I forgot, one of them applies to either the case of groups, rings, or modules. I know for the case of kernel in the case of topological space, it only exists for pointed topological spaces, rather ones endowed with a metric. I have never seen how to show kernel does not exists for general topological spaces in any category theory text.

Thank you in advance.

 

[fresh_42]

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho.

In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states

"Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels).

Also in the reference url page above, the authors present two theorems as facts along with their proofs and two other ones as i) and ii) along with their proofs, finally culminating in presenting fact 3 as theorem along with its proof. Fact 3 states:

"Fact 3: Let $p<n$ be a prime number, and suppose that there exist integers $a,b$ such that $p^2=a^2+nb^2$ and $P$ does not divide at least one of $a,b.$ Let $z:=a+b\sqrt{n}\;\;i\in R_n.$ Then $\text{gcd}(p^2, pz)$ does not exist."

My questions are as follows:

1) For Fact 3, does anyone knows of a book reference source? I am not surprised the proof is presented as proof by contradiction.

What are $P\, , \,i\, , \,R_n\,$?

I guess, the idea is that $p^2=z(a-b\sqrt{n})$ so $p\,|\,p^2 \wedge p\,|\,pz$ and $z\,|\,p^2\wedge z\,|\,pz.$ Hence, we have two common divisors $p$ and $z,$ but cannot tell which one is larger, if all happens in complex numbers.

2) For the case of non existence of coimages or cokernels, how does one go about showing its non existence. I forgot, one of them applies to either the case of groups, rings, or modules. I know for the case of kernel in the case of topological space, it only exists for pointed topological spaces, rather ones endowed with a metric. I have never seen how to show kernel does not exists for general topological spaces in any category theory text.

You cannot show non-existence in these categories since kernels and cokernels exist there.

Generally, you have to show that a certain universal mapping problem isn't solvable. That is that a morphism doesn't factor over the, say, kernel. You can do this by showing that no subset satisfies this condition, or that it cannot be uniquely defined. I don't know any, and my book doesn't mention a category in which there are no kernels. I think FLD (fields) is a place to look at. GRP, MOD, and RNG do have kernels and cokernels, images, and coimages. $\operatorname{ker}$ has to be a covariant functor, so non-existence requires the non-existence of such a functor. I would guess that a proof assumes one and shows that it cannot be well-defined by displaying two different solutions.

 

[elias001]

@fresh_42 Fact 3 should read assembly

"Fact 3: Let $p<n$ be a prime number, and suppose that there exist integers $a,b$ such that $p^2=a^2+nb^2$ and $p$ does not divide at least one of $a,b.$ Let $z:=a+bi\sqrt{n}\in R_n.$ Then $\text{gcd}(p^2, pz)$ does not exist."

I misread the LaTex rendering on my PC.

Basically i wanted to ask to show the non existence of kernel, cokernel in one of the math categories, is it the same logic as how Fact 3 is proved.

 

[fresh_42]

@fresh_42 Fact 3 should read assembly

"Fact 3: Let $p<n$ be a prime number, and suppose that there exist integers $a,b$ such that $p^2=a^2+nb^2$ and $p$ does not divide at least one of $a,b.$ Let $z:=a+bi\sqrt{n}\in R_n.$ Then $\text{gcd}(p^2, pz)$ does not exist."

Then my argument holds. $p$ and $z$ both divide $p^2$ and $pz,$ but there is no order for complex numbers that tells us which one is greater.

Basically i wanted to ask to show the non existence of kernel, cokernel in one of the math categories, is it the same logic as how Fact 3 is proved.

The $\operatorname{gcd}$ did not exist since the $\operatorname{g}$ in it isn't defined. We do have common divisors, we just can't tell which one is bigger.

A category without kernels means we cannot solve a universal mapping problem. To decide why it is unsolvable depends on the case. The analogy would be that the problem cannot be stated.

If such an example uses the absence of an initial (or terminal) object in such a category, then in a way it is analogous. We would not be able to set up the universal mapping problem. However, if we do were able to phrase the problem, and it would be simply unsolvable, e.g., by the lack of well-definition, then it would be a different case. So it depends on the case.

Your question asks whether non-existence always has the same reason. And the answer to that question is no. There are usually many options that can prevent existence: you need the description itself, and you need a list of properties. So we can fail to describe it, or we can fail to prove one of the properties. The $\operatorname{gcd}$ example fails on the description level since we cannot define greater.

A missing kernel can be due to the lack of an initial object in which case it is analogous, or due to the lack of well-definition, in which case it is not analogous.

 

[elias001]

@fresh_42 when you said

'A category without kernels means we cannot solve a universal mapping problem. To decide why it is unsolvable depends on the case. The analogy would be that the problem cannot be stated.

If such an example uses the absence of an initial (or terminal) object in such a category, then in a way it is analogous. We would not be able to set up the universal mapping problem. However, if we do were able to phrase the problem, and it would be simply unsolvable, e.g., by the lack of well-definition, then it would be a different case. So it depends on the case.'

Did you meant well definedness?

Also, are there theorems in category theory that characterizes under what condition does say a kernel or cokernel does or does not exists in a particular category?

 

[fresh_42]

@fresh_42 when you said
Did you meant well definedness?

Yes.

Also, are there theorems in category theory that characterizes under what condition does say a kernel or cokernel does or does not exists in a particular category?

I don't know. I only had a brief look at it. There are tons of properties for a category (additive, abelian, large, normal, punctured, concrete, and whatever), so I suspect that some of them imply the existence of kernels.

I think that the existence of kernels is equivalent to the existence of nullobjects, i.e., objects that are initial and terminal. My suspicion is that this is equivalent to being punctured, but I am not sure.

The usual candidates have kernels etc., the category of fields doesn't.

 

[elias001]

@fresh_42 oh kk. Thank you

 

[martinbn]

There many examples! One is the category of vector bundles over some space.

 

[elias001]

@martinbn I am sure there are many more examples. In Saunders Mac Lane's text, he has a chart that list for each particular math category, what its kernel, cokernel, coimage image are. I think he lists all four, it might just be cokernel, kernel and coimage. Anyways, what I like to know is how to show if say whether one can talk about the cokernel for the category of groups, and if not, how does one go about showing it does not exists.

In Arbib and Manes' text, in the topic of limit and colimits. He asks about showing that the same methods for the constructions for the colimits of diagrams for category of Sets does not work in the case for category of posets. I am not sure if a similar idea for the proof is the same for the cases of kernel, cokernel, coimage, and image.

 

[martinbn]

@martinbn I am sure there are many more examples. In Saunders Mac Lane's text, he has a chart that list for each particular math category, what its kernel, cokernel, coimage image are. I think he lists all four, it might just be cokernel, kernel and coimage. Anyways, what I like to know is how to show if say whether one can talk about the cokernel for the category of groups, and if not, how does one go about showing it does not exists.

In Arbib and Manes' text, in the topic of limit and colimits. He asks about showing that the same methods for the constructions for the colimits of diagrams for category of Sets does not work in the case for category of posets. I am not sure if a similar idea for the proof is the same for the cases of kernel, cokernel, coimage, and image.

I don't understand what you are trying to achieve and what you are asking! To show that, say the kernel, doesn't exist in a given category, it is enough to find an example of a morphism that doesn't have a kernel. And a quick web search gives you plenty of examples. But you are not satisfied! So what are you asking?

 

[elias001]

@martinbn what does it mean in mathematical notations for a morphism to not to have a kernel, coimage or cokernel? Also, does it require a proof?

I think in our interactions in my past post, i might have given you the impression of being interested in algebraic geometry. I want to clarify that I just want to know my commutative algebra well either at the level of Atiyah & MacDonald and also grobner bases. I have not made up my mind how to go about learning algebraic geometry. I know i can learn it through Hartshorne or learn it through the algebraic function where i weasel my way in via complex analysis and riemann surfaces. Actually, I rather be able to generate examples in whatever topics that I am learning.l as well as knowing how to do computations in the subjects. I hope I won't ever have to experience what I felt when learning about localisation. That topic felt like the fifth level of math hell for me.

 

[martinbn]

@martinbn what does it mean in mathematical notations for a morphism to not to have a kernel, coimage or cokernel? Also, does it require a proof?

Mathematical notations do not change the meaning. So the answer to your questions is that the meaning is the same whether you use mathematical notations or plain sentences. Did you find any examples?

I think in our interactions in my past post, i might have given you the impression of being interested in algebraic geometry. I want to clarify that I just want to know my commutative algebra well either at the level of Atiyah & MacDonald and also grobner bases. I have not made up my mind how to go about learning algebraic geometry. I know i can learn it through Hartshorne or learn it through the algebraic function where i weasel my way in via complex analysis and riemann surfaces. Actually, I rather be able to generate examples in whatever topics that I am learning.l as well as knowing how to do computations in the subjects. I hope I won't ever have to experience what I felt when learning about localisation. That topic felt like the fifth level of math hell for me.

 

[elias001]

@martinbn when i got into that localisation topic rabbit hole, I almost had a heart attack since all i am seeing were definitions, theorems and proofs with very little examples. A friend who is a grad student who works in algebraic combinatorics told me that I better get used to it with very little or no examples given if I want to do anything algebra related. I even asked a friend who was a student of Hanna Neumann who also said the exact same thing. I was trying to see if I can find examples for very simple cases, like single variables polynomials or integer mod p type examples. After that, if I can move onto more complicated ones.