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On ncatlab, in explaining the basics of the Salamander lemma, they state the following.

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For $A_{\square}$, they write the denominator as $\operatorname{im}(\partial^{hor}_{in}) \oplus \operatorname{im}(\partial^{vert}_{in})$. Why are they using the direct sum here? Those two modules aren't disjoint, are they?

Eric Auld
  • 28,127
  • Probably not relevant anymore, but anyways … This was indeed a mistake and has been corrected. The direct sum has been replaced by the sum of submodules of $\operatorname{ker}(\partial^{\operatorname{diag}}{\operatorname{out}})$ (, i.e. the join in the lattice of submodules of $\operatorname{ker}(\partial^{\operatorname{diag}}{\operatorname{out}})$ ordered by inclusion). – Max Demirdilek May 03 '23 at 06:45

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