# 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.