ScalingStacks

Let $V$ be a skew $\mathcal{I}$-cube over an abelian category $\mathcal{B}.$ To $V$ we associate a complex $\overline{C}(V)=(\overline{C}^{i}(V),d^{i}),i\in\mathbb{Z}$ of objects of $\mathcal{B}$ by

The differential $d^{i}:\overline{C}^{i}(V)\to\overline{C}^{i+1}(V)$ is given on an element $x\in V(\mathcal{L}),|\mathcal{L}|=i$ by

Examples:

1. 1.

   If $|\mathcal{I}|=1,\mathcal{I}=\{a\},$

   $$\overline{C}^{i}(V)=\left\{\begin{array}[]{ll}V(\emptyset)&\mbox{ if $i=0$ }\\ V(a)&\mbox{ if $i=1$ }\\ 0&\mbox{ otherwise }\end{array}\right.$$

   The differential $d^{0}=\xi_{a}^{V}(\emptyset)$ and $d^{i}=0$ if $i\not=0,$ so $\overline{C}(V)$ is the complex

   $$\cdots\longrightarrow 0\longrightarrow V(\emptyset)\stackrel{{\scriptstyle\xi^{V}_{a}(\emptyset)}}{{\longrightarrow}}V(\mathcal{I})\longrightarrow 0\longrightarrow\cdots$$

   (35)

2. 2.

   If $\mathcal{I}$ contains two elements, say, $\mathcal{I}=\{a,b\},$ then

   $$\overline{C}^{i}(V)=\left\{\begin{array}[]{ll}V(\emptyset)&\mbox{ if $i=0$ }\\ V(a)\oplus V(b)&\mbox{ if $i=1$ }\\ V(ab)&\mbox{ if $i=2$ }\\ 0&\mbox{ otherwise }\end{array}\right.$$

   and differentials

   		$\displaystyle d^{0}:$	$\displaystyle V(\emptyset)\longrightarrow V(a)\oplus V(b)$
   		$\displaystyle d^{0}=$	$\displaystyle\xi_{b}^{V}(\emptyset)+\xi_{a}^{V}(\emptyset)$
   		$\displaystyle d^{1}:$	$\displaystyle V(b)\oplus V(a)\longrightarrow V(ab)$
   		$\displaystyle d^{1}=$	$\displaystyle(\xi_{a}^{V}(b),\xi_{b}^{V}(a))$

Proof: The complex $\overline{C}(V)$ is isomorphic to the cone of the identity map of the complex $\overline{C}(V_{a}(\ast 1))[-1]$ and, therefore, acyclic. $\square$

Every map of $\mathcal{I}$-cubes $\phi:V\longrightarrow W$ over $R{\mbox{-mod}_{0}}$ induces a map of complexes

If $\phi$ is an isomorphism of commutative cubes, $\overline{C}(\phi)$ is an isomorphism of complexes.

Proof: Immediate from Proposition 4. $\square$

The following proposition and its corollary are obvious.