# Short Exact Sequences: Splitting

• jgens
In summary, the authors of Dummit and Foote define a short exact sequence of R-modules to be said to split if there exists an R-module complement to the homomorphism \psi:A \rightarrow B. However, they do not give a clear definition of what is meant by "an R-module complement to". After some research, it seems that this means there exists another R-module C' such that B is the internal direct sum of \psi(A) and C'. The same concept applies for groups as well, with "subgroup complement" and "internal semi-direct product" replacing "R-module complement" and "internal direct sum" respectively.
jgens
Gold Member
In Dummit and Foote, a short exact sequence of R-modules $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ ($\psi:A \rightarrow B$ and $\phi:B \rightarrow C$) is said to split if there is an R-module complement to $\psi(A)$ in $B$. The authors are not really clear on what the phrase "an R-module complement to" means, so I was hoping someone here could help clear my confusion up. My best guess is that they mean "there exists some R-module $C'$ such that $B$ is the internal direct sum of $\psi(A)$ and $C'$."

In Dummit and Foote, a short exact sequence of groups $1 \rightarrow A \rightarrow B \rightarrow C \rightarrow 1$ ($\psi:A \rightarrow B$ and $\phi:B \rightarrow C$) is said to split if there is a subgroup complement to $\psi(A)$ in $B$. Again the authors are not really clear on what the phrase "a subgroup complement to" means, so I was hoping someone here could help clear my confusion up. My best guess is that they mean "there exists some group $C'$ such that $B$ is the internal semi-direct product of $\psi(A)$ and $C'$."

Are my interpretations correct?

Last edited:
The relevant definitions are on page 180 and page 383 (which is not so clear, but you get the idea).

micromass said:
The relevant definitions are on page 180 and page 383 (which is not so clear, but you get the idea).

The definition on page 180 is clear and it appears that my interpretation was correct for groups. Since R-modules are an abelian category, I am pretty sure I got the right interpretation for modules too. Thanks!

## 1. What is a short exact sequence?

A short exact sequence is a sequence of mathematical objects (usually groups, rings, or modules) connected by homomorphisms in a specific order, such that the image of one object equals the kernel of the next. This means that there are no missing or extra elements in the sequence, and it provides a precise way to study the structure and relationships between these objects.

## 2. What does it mean for a short exact sequence to be "splitting"?

A short exact sequence is considered "splitting" if it can be broken up into two smaller sequences that are easier to work with. This is similar to factoring a number into smaller prime numbers, where the prime numbers are simpler and easier to understand than the original number.

## 3. How do you determine if a short exact sequence is splitting?

There are several ways to determine if a short exact sequence is splitting. One method is to check if there exists a homomorphism from the last object in the sequence to the first object, such that composing it with the previous homomorphism gives the identity map. Another method is to use the concept of "splitting maps" which are homomorphisms that can be used to split the sequence into smaller ones.

## 4. What are some applications of splitting short exact sequences?

Splitting short exact sequences have many applications in mathematics, physics, and other scientific fields. They are often used to study the structure of groups, rings, and modules, and can provide insight into the behavior of more complex mathematical objects. They can also be used to simplify difficult problems and make them more manageable.

## 5. Are there any limitations to splitting short exact sequences?

While splitting short exact sequences can be very useful, not all sequences can be split. Some sequences may be too complex or have special properties that do not allow for splitting. Additionally, splitting a sequence may not always provide a complete understanding of the original object, as some information may be lost in the process.

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