Questions tagged [diagram-chasing]
For questions about proofs using equivalent map compositions in commutative diagrams in homological algebra, or in category theory in general.
176 questions
2
votes
1
answer
109
views
Diagram chasing mod $S$
$$\require{AMScd}
\begin{CD}
A @>{\phi}>> B @>{\psi}>> C\\
@V{\alpha}VV @V{\beta}VV @V{\gamma}VV\\
A^\prime @>{\mathsf{id}}>> B^\prime @>{x \mapsto xA^\prime}>> {B^\...
- 103
2
votes
1
answer
92
views
How can we show this "Small Wall-Kervaire Braid Lemma"?
Consider the following diagram of abelian groups:
Assume that the $\color{blue}{\text{blue braid}}$ and the $\color{green}{\text{green braid}}$ are long exact sequences and that the $\color{red}{\...
- 5,287
10
votes
1
answer
704
views
Where does this attempted proof of the "magic diagram" being Cartesian fail?
I'm trying to prove the Diagonal-Base-Change-Diagram (i.e. "magic diagram") from Vakil's book, and I can't find a mistake in my proof attempt, even though it's clearly incorrect:
We need to ...
- 101
2
votes
0
answers
246
views
Equivalence of infinite descending normal towers
Let $R = \mathbb{Z}\langle\!\langle X_1,\ldots,X_\mu\rangle\!\rangle$ be the Magnus Ring of formal power series on the non-commuting indeterminates $X_1,\ldots,X_\mu$ with integer coefficients. Let $X$...
- 12.5k
3
votes
1
answer
134
views
How to prove the 5-lemma using subtraction of members?
I'm asking specifically about exercise 8.3.2 of the 2nd edition of Mac Lane, which reads "In the five-lemma, prove $f_3$ epi using members (not comembers)."
The context for this exercise is ...
- 373
1
vote
0
answers
55
views
Replacing objects with certain kernels in an element-free proof of the snake lemma
I am having difficulty with a part of Jonathan Wise’s oft-cited element-free proof of the snake lemma in an abelian category. At a high-level, he takes the snake-lemma starting-diagram, then applies a ...
- 3,097
0
votes
0
answers
328
views
A generalization of the 🐍 Snake Lemma; why hasn't this appeared naturally in HA development?
Define reverse homology to be the situation where $\operatorname{im}g \supset \ker f$, as is the case when we take $C_n = \Bbb{Z}/p^n$, then $\ker \pi_n$ is $p^{n-1}C_n$. And $\text{im } \pi_{n+1} = ...
- 22.9k
0
votes
1
answer
62
views
How can we paste together some finite long exact sequences (LES's) at the top page 14 of Weibel (in the SES of complexes to infinite LES proof)?
The Snake Lemma starting diagram at the bottom of page 13 in this proof is:
Here is a link to the above diagram.
Now at the top of page 14, it states "The kernel of the left vertical is $H_n(A)$ ...
- 22.9k
0
votes
2
answers
68
views
Page 13 of Weibel's Intro to Hom. Alg. - How do we get from one diagram to the other (question in the SES to Long Exact Sequence proof)
The goal theorem the book is trying to explain is:
Theorem 1.3.1. Let $0\to A_{\bullet} \xrightarrow{f} B_{\bullet} \xrightarrow{g} C_{\bullet} \to 0$ be a short exact sequence (SES) of chain ...
- 22.9k
6
votes
1
answer
229
views
Stuck on exercise from Goldblatt's "Topoi"
On page 78 of R. Goldblatt's "Topoi: The Categorial Analysis of Logic", the author introduces the notion of an element of a category thusly: if a category $\mathcal C$ has a terminal object $...
- 91
1
vote
1
answer
91
views
Commutative Diagram Problem for Exact Sequences of a Length-2 Module
Let $A$ be a finite-dimensional algebra over a field $k$.
Let $M$ be an indecomposable module of length $2$ over $A$ such that $\operatorname{soc}(M) \ncong \operatorname{top}(M)$. Here, $\...
- 1,475
1
vote
1
answer
124
views
Proving that an arrow in a diagram is an isomorphism.
The following commutative diagram is from Sheaves in Geometry and Logic by Mac Lane and Moerdijk, Proposition II.2.2. We know that $\beta, \gamma$ are isomorphisms, $a$ is the equalizer of $b,c$; $d$ ...
- 679
1
vote
1
answer
65
views
Commutativity of a diagram involving braids
I have this diagram and I want to prove its commutativity
Let me explain what it means.
$\beta$ is a braid with $n$ strings, that is represented in $\mathbb{D} \times [0,1]$. Its uper ends are $\beta(...
- 65
0
votes
1
answer
145
views
Mac Lane Chapter 7 Section 2 Exercise 1
Let $\mathcal{C}$ be a monodical category, with the monodical product written $\otimes$, the associator denoted $\alpha$, and the left/right unitors denoted $\iota^\ell,\iota^r$ respectively. Mac ...
- 1,873
1
vote
0
answers
42
views
Chasing diagrams of chain complexes which are pointwise projective
For context this question arose in a simple K-theory computation, where I wish to show that the inclusion of the category of bounded degreewise projective chain complexes into the category of bounded ...
- 1,797