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shuhalo
shuhalo
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Standard homology result on double complexesbicomplexes

Suppose you have got a double complexbicomplex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows $(A_{r\bullet},d_{r\bullet})$ for $r > 0$ and the columns $(A_{\bullet s},d_{\bullet s})$ for $s > 0$ are exact.

Then a standard result says: The complexes the top-most row $(A_{0 \bullet},d_{0\bullet})$ and the left-most column $(A_{\bullet 0},d_{\bullet 0})$ are isomorphic on homology.

I am not aware of a citable reference for this standard result. Can help me out with that?

Standard homology result on double complexes

Suppose you have got a double complex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows $(A_{r\bullet},d_{r\bullet})$ for $r > 0$ and the columns $(A_{\bullet s},d_{\bullet s})$ for $s > 0$ are exact.

Then a standard result says: The complexes the top-most row $(A_{0 \bullet},d_{0\bullet})$ and the left-most column $(A_{\bullet 0},d_{\bullet 0})$ are isomorphic on homology.

I am not aware of a citable reference for this standard result. Can help me out with that?

Standard homology result on bicomplexes

Suppose you have got a bicomplex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows $(A_{r\bullet},d_{r\bullet})$ for $r > 0$ and the columns $(A_{\bullet s},d_{\bullet s})$ for $s > 0$ are exact.

Then a standard result says: The complexes the top-most row $(A_{0 \bullet},d_{0\bullet})$ and the left-most column $(A_{\bullet 0},d_{\bullet 0})$ are isomorphic on homology.

I am not aware of a citable reference for this standard result. Can help me out with that?

Source Link
shuhalo
shuhalo
  • 5.3k
  • 29
  • 43

Standard homology result on double complexes

Suppose you have got a double complex in an abelian category with objects $(A_{rs},d_{rs})$ such that $A_{rs} = 0$ for $r < 0$ or $s < 0$. Suppose furthermore that the rows $(A_{r\bullet},d_{r\bullet})$ for $r > 0$ and the columns $(A_{\bullet s},d_{\bullet s})$ for $s > 0$ are exact.

Then a standard result says: The complexes the top-most row $(A_{0 \bullet},d_{0\bullet})$ and the left-most column $(A_{\bullet 0},d_{\bullet 0})$ are isomorphic on homology.

I am not aware of a citable reference for this standard result. Can help me out with that?