Bicomplex

binary complex, double complex

A graded module, i.e. one representable as the direct sum $ \sum A ^ {m,n} $ of its submodules $ A ^ {m,n} $, together with a pair of differentials

$$ d _ {1} : A ^ {m,n} \rightarrow A ^ {m+1,n} , $$

$$ d _ {2} : A ^ {m,n} \rightarrow A ^ {m,n+1} , $$

which satisfy the conditions

$$ d _ {1} \cdot d _ {1} = 0,\ \ d _ {2} \cdot d _ {2} = 0 ,\ \ d _ {2} d _ {1} +d _ {1} d _ {2} = 0 . $$

Instead of the direct sum, the set $ \{ A ^ {m,n} \} $ and the differentials

$$ d _ {1} : A ^ {m, n } \rightarrow A ^ {m - 1, n } , $$

$$ d _ {2} : A ^ {m, n } \rightarrow A ^ {m, n - 1 } , $$

satisfying the above conditions, may be considered.

References

[a1]	S. MacLane, "Homology" , Springer (1963)