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- Jun 22, 2012

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

I need help with some of the conclusions to Example 2, D&F Section 10.5, pages 379-380 - see attached. However, note that the question is essentially about isomorphisms. However, I would like someone to confirm the correctness of the exact sequences context.

To establish the context of the example as I interpret it ...

On pages 378-379 D&F, as I see it, establish the following:

\(\displaystyle A \stackrel{\psi}{\longrightarrow} B \stackrel{\phi}{\longrightarrow} C \)

where \(\displaystyle \psi \) is an injective homomorphism

and \(\displaystyle \phi \) is a surjective homomorphism and \(\displaystyle \text{ ker } \phi = \psi (A) \)

(1) \(\displaystyle A \cong \psi (A) \) because the homomorphism \(\displaystyle \psi \ : \ A \to \psi (A) \) is both injective and surjective

and

(2) by the First Isomorphism Theorem \(\displaystyle C \cong B/ \text{ker } \phi \text{ or } C \cong B/ \psi (A) \)

so that C is isomorphic to the quotient of B by an isomorphic copy of A

or, in other words B is an extension of C by A.

Now Example 2 is a special case of Example 1 - see bottom of page 379 and top of page 380 on the attachment.

In Example 2 D&F give two short exact sequences (see attachment).

\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong ( \mathbb{Z} \oplus ( \mathbb{Z} / n \mathbb{Z}) ) / \text{ ker } \phi \)

that is,

\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong ( \mathbb{Z} \oplus ( \mathbb{Z} / n \mathbb{Z}) ) / i ( \mathbb{Z} ) \) ... ... ... (3)

\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong \mathbb{Z}) ) / \text{ ker } \phi \)

that is

\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong \mathbb{Z}) ) / n ( \mathbb{Z} ) \) ... ... ... (4)

Now it seems (intuitively) that given (3) and (4) above we should have

\(\displaystyle ( \mathbb{Z} \oplus ( \mathbb{Z} / n \mathbb{Z}) ) / i ( \mathbb{Z} ) \cong \mathbb{Z} / n ( \mathbb{Z} ) \) ... ... ... (5)

since they are both isomorphic to \(\displaystyle \mathbb{Z} / n \mathbb{Z} \)

BUT

D&F say that this is explicitly not the case ... indeed, they write:

"Note that the modules in the middle of the previous two exact sequences are not isomorphic even though the respective "A" and "C" terms are isomorphic. Then there are at least two "essentially different" or "inequivalent" ways of extending \(\displaystyle \mathbb{Z} / n \mathbb{Z} \) by \(\displaystyle \mathbb{Z} \).

Can someone please clarify this situation for me.

Peter

I need help with some of the conclusions to Example 2, D&F Section 10.5, pages 379-380 - see attached. However, note that the question is essentially about isomorphisms. However, I would like someone to confirm the correctness of the exact sequences context.

To establish the context of the example as I interpret it ...

On pages 378-379 D&F, as I see it, establish the following:

**If we have**

\(\displaystyle A \stackrel{\psi}{\longrightarrow} B \stackrel{\phi}{\longrightarrow} C \)

where \(\displaystyle \psi \) is an injective homomorphism

and \(\displaystyle \phi \) is a surjective homomorphism and \(\displaystyle \text{ ker } \phi = \psi (A) \)

**then**(1) \(\displaystyle A \cong \psi (A) \) because the homomorphism \(\displaystyle \psi \ : \ A \to \psi (A) \) is both injective and surjective

and

(2) by the First Isomorphism Theorem \(\displaystyle C \cong B/ \text{ker } \phi \text{ or } C \cong B/ \psi (A) \)

so that C is isomorphic to the quotient of B by an isomorphic copy of A

or, in other words B is an extension of C by A.

**Can someone please confirm that my interpretation of the situation regarding extensions of C by A as outlined by D&F is correct. I would definitely feel more confident if someone was to confirm the above as OK!**Now Example 2 is a special case of Example 1 - see bottom of page 379 and top of page 380 on the attachment.

In Example 2 D&F give two short exact sequences (see attachment).

**The first exact sequence**, I believe, establishes the following isomorphism:\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong ( \mathbb{Z} \oplus ( \mathbb{Z} / n \mathbb{Z}) ) / \text{ ker } \phi \)

that is,

\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong ( \mathbb{Z} \oplus ( \mathbb{Z} / n \mathbb{Z}) ) / i ( \mathbb{Z} ) \) ... ... ... (3)

**The second exact sequence**, I believe, establishes the following isomorphism\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong \mathbb{Z}) ) / \text{ ker } \phi \)

that is

\(\displaystyle \mathbb{Z} / n \mathbb{Z} \cong \mathbb{Z}) ) / n ( \mathbb{Z} ) \) ... ... ... (4)

Now it seems (intuitively) that given (3) and (4) above we should have

\(\displaystyle ( \mathbb{Z} \oplus ( \mathbb{Z} / n \mathbb{Z}) ) / i ( \mathbb{Z} ) \cong \mathbb{Z} / n ( \mathbb{Z} ) \) ... ... ... (5)

since they are both isomorphic to \(\displaystyle \mathbb{Z} / n \mathbb{Z} \)

BUT

D&F say that this is explicitly not the case ... indeed, they write:

"Note that the modules in the middle of the previous two exact sequences are not isomorphic even though the respective "A" and "C" terms are isomorphic. Then there are at least two "essentially different" or "inequivalent" ways of extending \(\displaystyle \mathbb{Z} / n \mathbb{Z} \) by \(\displaystyle \mathbb{Z} \).

Can someone please clarify this situation for me.

Peter

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