Subsection 3.3 (0PN2) · ScalingStacks

3.3 skew-commutative cubes

We next define skew-commutative $\mathcal{I}$ -cubes over an additive category $\mathcal{B}.$ A skew-commutative $\mathcal{I}$ -cube is almost the same as a commutative $\mathcal{I}$ -cube but now we require that for every square facet of the cube the associated diagram of objects and morphisms of $\mathcal{B}$ anticommutes.

0PKZ

Definition 2 Let $\mathcal{I}$ be a finite set and $\mathcal{B}$ an additive category. A skew-commutative $\mathcal{I}$ -cube $V$ over $\mathcal{B}$ is a collection of objects $V(\mathcal{L})\in Ob(\mathcal{B})$ for $\mathcal{L}\subset\mathcal{I}$ and morphisms

\xi^{V}_{a}(\mathcal{L}):V(\mathcal{L})\longrightarrow V(\mathcal{L}a).

such that for each triple $(\mathcal{L},a,b),$ where $\mathcal{L}$ is a subset of $\mathcal{I}$ and $a,b,a\not=b$ are two elements of $\mathcal{I}$ that do not lie in $\mathcal{L},$ there is an equality

\xi^{V}_{b}(\mathcal{L}a)\xi^{V}_{a}(\mathcal{L})+\xi^{V}_{a}(\mathcal{L}b)\xi^{V}_{b}(\mathcal{L})=0.

We will call a skew-commutative $\mathcal{I}$ -cube over $\mathcal{B}$ a skew $\mathcal{I}$ -cube or, without specifying $\mathcal{I}$ , a skew cube.

Given $\mathcal{I}$ -cubes or skew $\mathcal{I}$ -cubes $V$ and $W$ over $R{\mbox{-mod}_{0}}$ , their tensor product is defined to be an $\mathcal{I}$ -cube (if $V$ and $W$ are both cubes or both skew cubes) or a skew $\mathcal{I}$ -cube (if one of $V,W$ is a cube and the other is a skew cube), denoted $V\otimes W,$ given by

	$\displaystyle(V\otimes W)(\mathcal{L})=V(\mathcal{L})\otimes W(\mathcal{L}),\hskip 21.68121pt\mathcal{L}\subset\mathcal{I},$
	$\displaystyle\xi_{a}^{V\otimes W}(\mathcal{L})=\xi_{a}^{V}(\mathcal{L})\otimes\xi_{a}^{W}(\mathcal{L}),\hskip 21.68121pt(\mathcal{L},a)\in r(\mathcal{I}),$

where, recall, the tensor products are taken over $R.$

For a finite set $\mathcal{L}$ denote by $o(\mathcal{L})$ the set of complete orderings or elements of $\mathcal{L}.$ For $x,y\in o(\mathcal{L})$ let $p(x,y)$ be the parity function, $p(x,y)=0$ if $y$ can be obtained by from $x$ via an even number of transpositions of two neighboring elements in the ordering, otherwise, $p(x,y)=1.$ To a finite set $\mathcal{L}$ associate a graded $R$ -module $E(\mathcal{L})$ defined as the quotient of the graded $R$ -module, freely generated by elements $x$ for all $x\in o(\mathcal{L}),$ by relations $x=(-1)^{p(x,y)}y$ for all pairs $x,y\in o(\mathcal{L}).$ Module $E(\mathcal{L})$ is a free graded $R$ -module of rank $1.$ For $a\not\in\mathcal{L}$ there is a canonical isomorphism of graded $R$ -modules $E(\mathcal{L})\longrightarrow E(\mathcal{L}a)$ induced by the map $o(L)\to o(La)$ that takes $x\in o(L)$ to $xa\in o(La).$ Moreover, for $a,b,a\not=b,$ the diagram below anticommutes

\begin{CD}E(\mathcal{L})@>{}>{}>E(\mathcal{L}a)\\ @V{}V{}V@V{}V{}V\\ E(\mathcal{L}b)@>{}>{}>E(\mathcal{L}ab)\end{CD}

(32)

Denote by $E_{\mathcal{I}}$ the skew $\mathcal{I}$ -cube with $E_{\mathcal{I}}(\mathcal{L})=E(\mathcal{L})$ for $\mathcal{L}\subset\mathcal{I}$ and the structure map $E_{\mathcal{I}}(\mathcal{L})\to E_{\mathcal{I}}(\mathcal{L}a)$ being canonical isomorphism $E(\mathcal{L})\to E(\mathcal{L}a).$

We will use $E_{\mathcal{I}}$ to pass from $\mathcal{I}$ -cubes over $R{\mbox{-mod}_{0}}$ to skew $\mathcal{I}$ -cubes over $R{\mbox{-mod}_{0}}$ by tensoring an $\mathcal{I}$ -cube with $E_{\mathcal{I}}.$

Mikhail Khovanov

Original source: arXiv:math/9908171v2