double complex

{{Short description|Mathematical concept}}

{{redirect|Bicomplex|the type of number|Bicomplex number}}

In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a $\backslash mathbb\{Z\}$-grading, the objects in the bicomplex have a $\backslash mathbb\{Z\}\backslash times\backslash mathbb\{Z\}$-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category $\backslash mathcal\{A\}$. A bicomplex{{Cite web|title=Section 12.18 (0FNB): Double complexes and associated total complexes—The Stacks project|url=https://stacks.math.columbia.edu/tag/0FNB|access-date=2021-07-08|website=stacks.math.columbia.edu}} is a sequence of objects $C\_\{p,q\}\; \backslash in\; \backslash text\{Ob\}(\backslash mathcal\{A\})$ with two differentials, the horizontal differential

$d^h:\; C\_\{p,q\}\; \backslash to\; C\_\{p+1,q\}$

and the vertical differential

$d^v:C\_\{p,q\}\; \backslash to\; C\_\{p,q+1\}$

which have the compatibility relation

$d\_h\backslash circ\; d\_v\; =\; d\_v\backslash circ\; d\_h$

Hence a double complex is a commutative diagram of the form

$\backslash begin\{matrix\}$

& & \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{{Cite book|last=Weibel|first=Charles A.|url=https://www.worldcat.org/oclc/847527211|title=An introduction to homological algebra|date=1994|publisher=Cambridge University Press|isbn=978-1-139-64863-9|location=Cambridge [England]|oclc=847527211}} instead require that the squares anticommute. That is

$d\_h\backslash circ\; d\_v\; +\; d\_v\backslash circ\; d\_h\; =\; 0.$

This eases the definition of Total Complexes. By setting $f\_\{p,q\}\; =\; (-1)^p\; d^v\_\{p,q\}\; \backslash colon\; C\_\{p,q\}\; \backslash to\; C\_\{p,q-1\}$, 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.