Questions tagged [diagram-chase]
Mathematical proofs based on chasing elements in commutative diagrams
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Is this an instance of the snake lemma?
I recently had need of the following fact (in the category of abelian groups, but I'm pretty sure it holds for all abelian categories): given a commutative diagram of the form
(quiver link), thus $k \...
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0
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Aggregation relationship in class diagram
How do you transform an aggregation relatinship between to classes in a class diagram (1 to many or many to many) into relational schema and is it needed/possible. Or is enough if we transform only ...
2
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1
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How to use $5$-lemma to prove that $F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M') $ is a (natural) isomorphism?
I am describing the question details, though the main question is short as below.
Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}_p$. Let $R$ be a finite $...
3
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1
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Show commutativity of a diagram involving multiplier $C^*$-algebras
Let me recall the following fact:
If $A$ is a $C^*$-algebra and $\pi: A \to \mathcal{B}(\mathcal{H})$ is a faithful non-degenerate representation, then we can explicitely realise the multiplier ...
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0
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Characterising exact sequence in terms of (quasi-)identities
First of all, hello everyone and thanks in advance of any kind of help.
I am currently working on automated proofs of diagram chases. To this end, I have to characterise the property of $A \overset{f}{...
2
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1
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Motivation for definitions of donor and receptor in Salamander Lemma?
$\newcommand{\im}{\operatorname{Im}}$Consider the following (subpart of) a double complex, using the same notation as in George Bergman's pre-print or in these lecture notes:
$$\require{AMScd}\begin{...
3
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1
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Cute/striking application(s) of snake lemma outside homological algebra
I already asked this question on MSE here https://math.stackexchange.com/questions/3254184/cute-striking-applications-of-snake-lemma-outside-homological-algebra, but still received no answer. I hope I ...
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Getting the most general form of Mayer-Vietoris from the Eilenberg-Steenrod axioms
I asked this question a while ago on MSE, got no answer, put a bounty on it, still got no answer, was advised to ask here instead, hesitated, forgot about the question for a while and now remembered ...
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Are these connecting homomorphisms commutative?
Are the connecting homomorphism induced by Kummer sequence and that of localization sequence commutative?
In other words, is the following statement true?
If it is true, then, how can one prove it?
...
14
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2
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Five-lemma for the end of long exact sequences of homotopy groups
Consider the commutative diagram below with exact rows (from the long exact sequence of homotopy groups) and $f_1,f_2,f_4,f_5$ bijective ($f_1,f_2$ homomorphisms). Does it follow that $f_3$ is also ...
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Is this isomorphism canonical?
Suppose $A\leq A',B$ and $C' \leq C$ are (finite dimensional) vector spaces.
Suppose that
$$ 0 \to A \to B \to C \to 0 $$
$$ 0 \to A' \to B \to C' \to 0 $$
are exact. Then using a dimension argument ...
6
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1
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What are the oldest illustrations of "Venn" diagrams?
Graphical representations of intersection of sets as logical combinations are much older than Venn.
Euler and Leibniz are often quoted and the current Wikipedia article also quotes Ramon Llull but I ...
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When does adding inverses of morphisms preserve commutativity of a diagram?
Here is the essence of a problem I have run in to: I have a finite poset D with a terminal object. If I formally invert all of the morphisms, and add these into my diagram, does the new diagram D' ...
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The sharp 3x3 lemma: a proof by universal properties?
I was reading this paper a while ago, and I couldn't figure out how to prove a lemma that was left as an exercise by only using universal properties and the definition of an abelian category.
I'll ...