The two-square lemma
type: Item
instance of: scholarly article
title: The two-square lemma
zbMATH Open document ID: 0699.18003
DOI: 10.5565/PUBLMAT_33189_10
published in: Publicacions Matemàtiques
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41072
review text: An essential step in the Snake Lemma in Homological Algebra is the construction of a connecting homomorphism. Generalizing this aspect of the Snake Lemma (in an Abelian category), the authors here give an interesting new result which they call the Two Square Lemma: Consider the commutative diagram \[ \begin{matrix} A_ {pa} && @>\psi>> && B && @>\varphi>> && C \\ &&& \overset\tau\swarrow &&&& \overset\sigma\nearrow \\ \text{\llap{\(\alpha\)}}\downarrow && Q && \text{\llap{\(\beta\)}}\downarrow && P_{pb} && \downarrow\text{\rlap{\(\gamma\)}} \\ & \overset{\tau'}\nearrow &&&& \overset{\sigma'}\swarrow \\ A' && @>>\psi'> && B' && @>>\varphi'> && C' \end{matrix} \] with exact rows and Q the pushout of \(\psi\) and \(\alpha\) and P the pullback of \(\gamma\) and \(\phi '\). Then (i) there exists a unique \(\theta\) : \(Q\to B'\) such that \(\theta \tau =\beta\), \(\theta \tau '=\psi '\); (ii) there exists a unique \(\rho\) : \(B\to P\) such that \(\sigma \rho =\phi\), \(\sigma '\rho =\beta\); (iii) there exists a unique \(\eta\) : \(Q\to P\) such that \(\eta \tau =\rho\), \(\sigma '\eta =\theta\), \(\sigma \eta \tau '=0.\) Moreover, \(\eta\) is monic if \(\psi '\) is monic, and \(\eta\) is epic if \(\phi\) is epic. As an easy application of this lemma they then obtain the Snake Lemma.
Mathematics Subject Classification ID: 18G99
zbMATH DE Number: 4146225
zbMATH Keywords: Snake Lemma
zbMATH Keywords: connecting homomorphism
author: Temple H. Fay
author: K. A. Hardie
author: Peter Hilton
reviewed by: Lekh Raj Vermani
MaRDI profile type: Publication
Wikidata QID: Q125026004
OpenAlex ID: W2084320142
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description: scientific article; zbMATH DE number 4146225