ScalingStacks

For a more intrinsic approach to this section, cf [De, §1.1].

Let ${\mathcal{A}}$ be an additive category and $n\geq 0$. We denote by $\operatorname{Comp}\nolimits({\mathcal{A}})$ the category of complexes of objects of ${\mathcal{A}}$. The category $n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$ of $n$-fold complexes is defined inductively by $n\operatorname{\!-Comp}\nolimits({\mathcal{A}})=\operatorname{Comp}\nolimits((n-1)\operatorname{\!-Comp}\nolimits({\mathcal{A}}))$ and $0\operatorname{\!-Comp}\nolimits({\mathcal{A}})={\mathcal{A}}$. Its objects are families $(X,d_{1},\ldots,d_{n})$ where $X$ is an object of ${\mathcal{A}}$ graded by ${\mathbf{Z}}^{n}=\bigoplus_{i=1}^{n}{\mathbf{Z}}e_{i}^{*}$, $d_{i}$ is a graded map of degree $e_{i}$ and $d_{i}^{2}=[d_{i},d_{j}]=0$ for all $i,j$.

Given $X$ an $n$-complex and $i\in\{1,\ldots,n\}$, we define $Y=X[e_{i}]$ as the $n$-complex given by $Y^{b}=X^{e_{i}+b}$ and differentials $\partial_{i}^{b}=(-1)^{\delta_{ij}}d_{j}^{e_{i}+b}$.

Let $f:\{1,\ldots,n\}\to\{1,\ldots,m\}$ be a map. It induces a map $\sigma:{\mathbf{Z}}^{n}\to{\mathbf{Z}}^{m}$ and gives by duality a map ${\mathbf{Z}}^{m}\to{\mathbf{Z}}^{n}$. This provides a functor from ${\mathbf{Z}}^{n}$-graded objects to ${\mathbf{Z}}^{m}$-graded objects of ${\mathcal{A}}$. Let $X$ be an $n$-complex. We have a corresponding ${\mathbf{Z}}^{m}$-graded object $X^{\prime}$. We define a structure of $m$-complex by

where $b=\sum_{i}b_{i}e_{i}$.

Note that when $f$ is an injection, the sum above has only positive signs. When $f$ is a bijection, then $\operatorname{Tot}\nolimits^{f}$ is a self-equivalence of $n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$. We write $\operatorname{Tot}\nolimits=\operatorname{Tot}\nolimits^{f}$ when $m=1$.

We have defined an additive functor

Let $g:\{1,\ldots,m\}\to\{1,\ldots,p\}$ be a map and $\tau:{\mathbf{Z}}^{m}\to{\mathbf{Z}}^{p}$ the associated morphism. Let $X\in n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$. The ${\mathbf{Z}}^{p}$-graded objects underlying $\operatorname{Tot}\nolimits^{gf}(X)$ and $\operatorname{Tot}\nolimits^{g}(\operatorname{Tot}\nolimits^{f}(X))$ have their component of degree $a$ equal to $\bigoplus_{c\in(\tau\sigma)^{-1}(a)}X^{c}$. We define an isomorphism between these $p$-complexes by multiplication by $(-1)^{\varepsilon(c)}$ on $X^{c}$, where

This gives an isomorphism of functors

Let $k$ be a commutative ring and $A$, $B$ and $C$ be three $k$-algebras. Let $X\in n\operatorname{\!-Comp}\nolimits((A\otimes B)\operatorname{\!-Mod}\nolimits)$ and $Y\in m\operatorname{\!-Comp}\nolimits((B\otimes C)\operatorname{\!-Mod}\nolimits)$. Then $X\otimes_{B}Y$ defines an object of $(n+m)\operatorname{\!-Comp}\nolimits((A\otimes C)\operatorname{\!-Mod}\nolimits)$. We have $(X\otimes_{B}Y)^{(a_{1},\ldots,a_{n+m})}=X^{(a_{1},\ldots,a_{n})}\otimes_{B}Y^{(a_{n+1},\ldots,a_{n+m})}$.