ScalingStacks

Assume ${\mathcal{A}}$ is an abelian category. Let $X\in n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$. Let $r\in\{1,\ldots,n\}$ and $Y=\ker d_{r}/\operatorname{im}\nolimits d_{r}$. This is an $n$-complex with $d_{r,Y}=0$. Let $i\in{\mathbf{Z}}$. Consider the map $f:{\mathbf{Z}}^{n-1}\to{\mathbf{Z}}^{n},\ (a_{1},\ldots,a_{n-1})\mapsto(a_{1},\ldots,a_{r-1},i,a_{r},\ldots,a_{n-1})$. We put $H^{i}_{d_{r}}(X)=\bigoplus_{a\in{\mathbf{Z}}^{n-1}}Y^{f(a)}$. This defines a functor

$$H^{i}_{r}:n\operatorname{\!-Comp}\nolimits({\mathcal{A}})\to(n-1)\operatorname{\!-Comp}\nolimits({\mathcal{A}}).$$

Let $g:\{1,\ldots,n\}\to\{1,\ldots,m\}$ be a map and let $r\in\{1,\ldots,m\}$ such that $f^{-1}(r)=\{s\}$ for some $s\in\{1,\ldots,n\}$. Define maps

$$\alpha:\{1,\ldots,n-1\}\to\{1,\ldots,n\},\ i\mapsto\begin{cases}i&\text{ if }i<s\\ i+1&\text{ if }i\geq s\end{cases}$$

and

$$\beta:\{1,\ldots,m\}\to\{1,\ldots,m-1\},\ i\mapsto\begin{cases}i&\text{ if }i<r\\ i-1&\text{ if }i\geq r\end{cases}$$

Then, we have a canonical isomorphism

(1)

$$H^{i}_{d_{r}}(\operatorname{Tot}\nolimits^{f}(M))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Tot}\nolimits^{\beta f\alpha}(H^{i}_{d_{s}}(M)).$$