MHB Simple question of interpretation - D&F Ch 10 - Proposition 30

[Math Amateur]

I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.

I am studying Proposition 30 (D&F, page 389)

I need some help in order to interpret one of the statements in Proposition 30.

Proposition 30 reads as follows:

View attachment 2506

Statement (3) of Proposition 30 begins as follows:

"(3) If P is a quotient of the R-module N ... "

I am uncertain regarding the exact meaning of this statement ... I suspect it means that there exists a sub-module module Q of N such that P = N/Q ...

BUT ... firstly, this seems a vague thing to assert and secondly, I am most uncertain of this interpretation ...

Can someone please clarify the matter for me?

Peter

 

[Deveno]

Actually, it is saying:

If $P \cong M/L$ then $M \cong L \oplus P$.

 

[Math Amateur]

Actually, it is saying:

If $P \cong M/L$ then $M \cong L \oplus P$.

Thanks Deveno ... but you are saying the L of the short exact sequence is the relevant sub-module ... how does one arrive at this conclusion from the mere statement that "P is a quotient of M"?

Can you help?

Peter
EDIT - apologies for the typo in my original post in this thread - I typed N when I should have typed M so the text that read:

""(3) If P is a quotient of the R-module N ... "

I am uncertain regarding the exact meaning of this statement ... I suspect it means that there exists a sub-module module Q of N such that P = N/Q ... "

obviously should have read:

""(3) If P is a quotient of the R-module M ... "

I am uncertain regarding the exact meaning of this statement ... I suspect it means that there exists a sub-module module Q of N such that P = M/Q ..."

again ... apologies ...

 

[Deveno]

By exactness we have that the kernel of the map $M \to P$ is the image of the map $L \to M$.

Also, by exactness, we have that the kernel of the map $P \to 0$ is the image of the map $M \to P$.

Since all of $P$ maps to $0$, the image of the map $M \to P$ is all of $P$, that is, the map $M \to P$ is surjective.

By the fundamental isomorphism theorem, $P$ is thus a quotient of $M$ by the image of $L$.

Also by exactness, we have that the kernel of the map $L \to M$ is the image of the map $0 \to L$.

Since that image is clearly just $0_L$, the map $L \to M$ is injective.

It is somewhat of an abuse of notation to write $M/L$, as $L$ may not be, strictly speaking, a submodule of $M$, but there is an isomorphic copy of $L$ which is.

*************

So what Proposition 30 is saying is:

The functor $\text{Hom}_R(P,-)$ is exact (that is: preserves short exact sequences) iff every short exact sequence with $P$ as the 3rd module splits. This is a condition on $P$.

Earlier, you should have already proven that the functor $\text{Hom}_R(D,-)$ is "left-exact" (preserves the first 3 arrows of a short exact sequence) no matter what module $D$ is.

Thus Proposition 30 is giving you several equivalent criteria for a moduule to be PROJECTIVE.

The name "projective" is suggested by:

$0 \to L \to L\oplus P \to P \to 0$, where the map $L \oplus P \to P$ is a "projection onto $P$".

In the special case when $R$ is a field (such as the real numbers), this has its usual GEOMETRICAL interpretation as "projection onto a subspace". Although one need not study vector spaces BEFORE studying modules, the analytic geometry underlying the vector spaces $\Bbb R^2$ and $\Bbb R^3$ can often be comprehended INTUITIVELY.

Short exact sequences impart a lot of information in 9 symbols. They become particularly useful in investigating algebraic structures of topological objects.

There is another notion in modules, which preserves "right exactness", this is the functor:

$M \otimes_R -$

and we say that a module $M$ is flat if the functor $M \otimes_R -$ is exact (and not just "right exact"). There is a certain sense in which the hom-functor and tensor functor are "inverses" to each other (more correctly, they are categorical adjoints).

As you progress further into the theory, you will start to encounter many proofs containing the phrases: "by exactness..." (which may, in some cases, be merely left or right). D&F are approaching this "from a low level" easing you gradually into the mode of thinking more about $R$-linear maps, and less about the modules that produce these maps.

This should not be all that surprising: maps are "where the action is". For example, when studying calculus, the whole point of assuming the backdrop of the real numbers, is so the NOTION of continuous function can be investigated, as many phenomena which we would like to model by this "vary continually". The question: "What is the limit of the real number 3?" is not very interesting.

 

[Math Amateur]

By exactness we have that the kernel of the map $M \to P$ is the image of the map $L \to M$.

Also, by exactness, we have that the kernel of the map $P \to 0$ is the image of the mgap $M \to P$.

Since all of $P$ maps to $0$, the image of the map $M \to P$ is all of $P$, that is, the map $M \to P$ is surjective.

By the fundamental isomorphism theorem, $P$ is thus a quotient of $M$ by the image of $L$.

Also by exactness, we have that the kernel of the map $L \to M$ is the image of the map $0 \to L$.

Since that image is clearly just $0_L$, the map $L \to M$ is injective.

It is somewhat of an abuse of notation to write $M/L$, as $L$ may not be, strictly speaking, a submodule of $M$, but there is an isomorphic copy of $L$ which is.

*************

So what Proposition 30 is saying is:

The functor $\text{Hom}_R(P,-)$ is exact (that is: preserves short exact sequences) iff every short exact sequence with $P$ as the 3rd module splits. This is a condition on $P$.

Earlier, you should have already proven that the functor $\text{Hom}_R(D,-)$ is "left-exact" (preserves the first 3 arrows of a short exact sequence) no matter what module $D$ is.

Thus Proposition 30 is giving you several equivalent criteria for a moduule to be PROJECTIVE.

The name "projective" is suggested by:

$0 \to L \to L\oplus P \to P \to 0$, where the map $L \oplus P \to P$ is a "projection onto $P$".

In the special case when $R$ is a field (such as the real numbers), this has its usual GEOMETRICAL interpretation as "projection onto a subspace". Although one need not study vector spaces BEFORE studying modules, the analytic geometry underlying the vector spaces $\Bbb R^2$ and $\Bbb R^3$ can often be comprehended INTUITIVELY.

Short exact sequences impart a lot of information in 9 symbols. They become particularly useful in investigating algebraic structures of topological objects.

There is another notion in modules, which preserves "right exactness", this is the functor:

$M \otimes_R -$

and we say that a module $M$ is flat if the functor $M \otimes_R -$ is exact (and not just "right exact"). There is a certain sense in which the hom-functor and tensor functor are "inverses" to each other (more correctly, they are categorical adjoints).

As you progress further into the theory, you will start to encounter many proofs containing the phrases: "by exactness..." (which may, in some cases, be merely left or right). D&F are approaching this "from a low level" easing you gradually into the mode of thinking more about $R$-linear maps, and less about the modules that produce these maps.

This should not be all that surprising: maps are "where the action is". For example, when studying calculus, the whole point of assuming the backdrop of the real numbers, is so the NOTION of continuous function can be investigated, as many phenomena which we would like to model by this "vary continually". The question: "What is the limit of the real number 3?" is not very interesting.

Thanks for a VERY informative post ... Still reflecting on your post ...

Based on your advice I am going to get more familiarity with category theory ... probably via Steve Awodey's book on the topic ... May also look at "Conceptual Mathematics" by F. William Lawvere and Stephen Schanuel ...

Peter