Double complex

In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a \mathbb{Z} -grading, the objects in the bicomplex have a \mathbb{Z}\times\mathbb{Z} -grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category \mathcal{A} . A bicomplex is a sequence of objects C_{p,q} \in \text{Ob}(\mathcal{A}) with two differentials, the horizontal differential d^h: C_{p,q} \to C_{p+1,q} and the vertical differential d^v:C_{p,q} \to C_{p,q+1} which have the compatibility relation d_h\circ d_v = d_v\circ d_h Hence a double complex is a commutative diagram of the form \begin{matrix} & & \vdots & & \vdots & & \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_{p,q+1} & \to & C_{p+1,q+1} & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_{p,q} & \to & C_{p+1,q} & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ & & \vdots & & \vdots & & \\ \end{matrix} where the rows and columns form chain complexes. Some authors instead require that the squares anticommute. That is d_h\circ d_v + d_v\circ d_h = 0. This eases the definition of Total Complexes. By setting f_{p,q} = (-1)^p d^v_{p,q} \colon C_{p,q} \to C_{p,q-1} , we can switch between having commutativity and anticommutativity.…

In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a \mathbb{Z} -grading, the objects in the bicomplex have a \mathbb{Z}\times\mathbb{Z} -grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category \mathcal{A} . A bicomplex is a sequence of objects C_{p,q} \in \text{Ob}(\mathcal{A}) with two differentials, the horizontal differential d^h: C_{p,q} \to C_{p+1,q} and the vertical differential d^v:C_{p,q} \to C_{p,q+1} which have the compatibility relation d_h\circ d_v = d_v\circ d_h Hence a double complex is a commutative diagram of the form \begin{matrix} & & \vdots & & \vdots & & \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_{p,q+1} & \to & C_{p+1,q+1} & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_{p,q} & \to & C_{p+1,q} & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ & & \vdots & & \vdots & & \\ \end{matrix} where the rows and columns form chain complexes. Some authors instead require that the squares anticommute. That is d_h\circ d_v + d_v\circ d_h = 0. This eases the definition of Total Complexes. By setting f_{p,q} = (-1)^p d^v_{p,q} \colon C_{p,q} \to C_{p,q-1} , we can switch between having commutativity and anticommutativity. If the commutative definition is used, this alternating sign will have to show up in the definition of Total Complexes.

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