Each exact sequence can be arised by short exact sequences

In summary, the conversation discusses the exactness of sequences in a commutative ring with unit. It is stated that if two sequences are exact, then a third sequence formed by combining them is also exact. This leads to the question of how to prove this and a hint is requested. The idea of using induction on the length of a sequence is suggested, with the base case being a sequence of length 2 and the inductive step being a sequence of length n+1. A specific exercise is mentioned as a reference for further understanding.
  • #1
mathmari
Gold Member
MHB
5,049
7
Hey! :eek:

Let $R$ be a commutative ring with unit.

We have that if the sequences $0\rightarrow A\rightarrow B\overset{f}{\rightarrow}C\rightarrow 0$ and $0\rightarrow C\overset{g}{\rightarrow}D\rightarrow E\rightarrow 0$ are exact, then the sequence $0\rightarrow B\overset{gf}{\rightarrow} D\rightarrow E\rightarrow 0$ is exact.

So, each exact sequence can be arised by short exact sequences as above, right? (Wondering)

But how could we prove this? Could you give me a hint? (Wondering)
 
Physics news on Phys.org
  • #2
Do we show that maybe using induction on the length of a sequence?

Base case: We consider the sequence $0\rightarrow A\rightarrow B\rightarrow 0$. This is exact.

Inductive hypothesis: We assume that each exact sequnez of length $n$ is made by short exact sequences.

Inductive step: We consider a sequence of length $n+1$. The first $n$ are made by short exact sequences, because of the inductive hypothesis, right? How could we continue? (Wondering)
 
  • #3
See exercise 2.6 on p.65 of Rotman - An Introduction to Homological Algebra 2nd edition 2009.
 

FAQ: Each exact sequence can be arised by short exact sequences

What is the meaning of "each exact sequence can be arised by short exact sequences"?

This statement is a fundamental concept in mathematics and refers to the idea that any exact sequence can be decomposed into a series of shorter exact sequences. An exact sequence is a sequence of objects and morphisms (or functions) between them, where the image of one morphism is equal to the kernel of the next. This means that the composition of two consecutive morphisms is always equal to zero. By breaking down the original exact sequence into smaller pieces, we can better understand and analyze the relationships between the objects and the morphisms.

What is an exact sequence?

An exact sequence is a sequence of mathematical objects, such as groups, vector spaces, or modules, along with morphisms (or functions) between them. The key property of an exact sequence is that the image of one morphism is equal to the kernel of the next. This means that the composition of two consecutive morphisms is always equal to zero. Exact sequences are important in many areas of mathematics, including algebra, topology, and homology.

Why is it useful to decompose an exact sequence into shorter exact sequences?

Breaking down an exact sequence into smaller pieces allows us to better understand and analyze the relationships between the objects and the morphisms. It also allows us to apply different mathematical techniques and tools to each individual piece, which can make it easier to solve problems or prove theorems. Additionally, decomposing an exact sequence can provide insights and connections between seemingly different objects and concepts.

What is the role of short exact sequences in this concept?

Short exact sequences are the building blocks of the decomposition process. They are exact sequences that consist of only three objects and two morphisms, where the image of the first morphism is equal to the kernel of the second. These sequences are important because they can be easily manipulated and combined to create longer exact sequences. By repeatedly applying this process, we can break down any exact sequence into a series of short exact sequences.

How does the idea of "each exact sequence can be arised by short exact sequences" relate to other mathematical concepts?

This concept is closely related to the concept of exactness and the use of exact sequences in homological algebra. It also has connections to other areas of mathematics, such as category theory and algebraic topology, where exact sequences are used to study the relationships between different objects and structures. Additionally, the decomposition of exact sequences into short exact sequences is a common technique in many mathematical proofs and constructions.

Similar threads

Short Exact Sequences 0→A→B→C→0: Explained
Replies
2
Views
2K
MHB Pointwise and uniform convergence
Replies
5
Views
2K
Short Exact Sequences: Splitting
Replies
2
Views
3K
MHB Subsets of permutation group: Properties
Replies
18
Views
1K
MHB Show equivalence of propositions
Replies
15
Views
2K
MHB Sub-Category of Short Exact Sequences - Bland pages 85-86
Replies
1
Views
1K
Short Exact Sequence: Explaining C = B/A
Replies
1
Views
1K
MHB Is There a Typo in the Refinement of Schanuel's Lemma in Passman's Book?
Replies
4
Views
2K
MHB A little problem on exact sequences.
Replies
7
Views
2K
I Split Short Exact Sequences .... Bland, Proposition 3.2.6 ....
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top