# Questions tagged [diagram-chasing]

For questions about proofs using equivalent map compositions in commutative diagrams in homological algebra, or in category theory in general.

52 questions
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### Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
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### Looking for student's guide to diagram chasing

I'm teaching myself some category theory, and I find that I'm very slow with diagram chasing. It takes me some times a very long time to decide whether adding an arrow to a diagram preserves the ...
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### Commutative hexagonal diagram of Abelian groups; proving a certain equality

I'm trying to prove the following lemma by diagram chasing, but I've had no success, so I decided to ask for help here. Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be Abelian groups, and let $a_{1}$, ...
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### How would one solve Weibel 1.3.1 in a general Abelian category?

Working through Weibel's Introduction to Homological Algebra, I am frequently unsure when it is acceptable to prove results using diagram-chasing and elements, and when Weibel has in mind a category-...
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### A question about the functoriality of the module of derivations on the category of algebras

Assume that all rings are commutative with identity. Let $k$ be a fixed ring with $k$-algebras $\varphi: k \rightarrow R$ and $\tau: k \rightarrow A$. Let $\tau' : R \rightarrow B$ define an $R$-...
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### How to verify commutativity of a diagram?

Let $C$ be a category, and let all objects $X_i, Y_j$ belong to $ob(C)$, and morphisms $f_{ij}, h_i, g_{ij}$ be morphisms between them in $C$. Let us have a diagram then: How do we verify it's ...
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### Proof that the dual of a coalgebra is an algebra via commutative diagrams

We know that the algebra and coalgebra axioms are given via following commutative diagrams (algebra $A$ and coalgebra $C$ are over a field $\mathbb{K}$): I am now trying to show that the dual of a ...
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### The Existence of a Natural Isomorphism Confirming the Existence of a Limit, as a Commutativity Diagram

I am reading Category Theory for Programmers and I am having some trouble in Part 2, Chapter 2: Limits and Colimits. The author writes: Now that we have two functors, we can talk about natural ...
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### Diagram chasing, and more

1) Assume that $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \rightarrow 0$ and $0 \rightarrow C_1 \rightarrow C_2 \rightarrow D \rightarrow 0$ are exact, $i=1,2$. Show, using a diagram chase, ...
Let $A,B,C,D$ be R-modules. Suppose there is a commutative diagram as the following. $A\to B$ is the injective, $A\to C$ and $B\to D$ are surjective. Is $C\to D$ injective?
I am trying to prove that $E'$ is an extension of $Q$ by $N'$ \begin{array} 00 &\longrightarrow & N & \overset{i_1} \longrightarrow & E & \overset{\pi_1} \longrightarrow& Q &...