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

$$d_{i}^{\prime a}=\sum_{\begin{subarray}{c}b\in\sigma^{-1}(a)\\ j\in f^{-1}(i)\end{subarray}}(-1)^{\sum_{k\in f^{-1}(i),k<j}b_{k}}d_{j}^{b}$$

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

$$\operatorname{Tot}\nolimits^{f}:n\operatorname{\!-Comp}\nolimits({\mathcal{A}})\to m\operatorname{\!-Comp}\nolimits({\mathcal{A}}).$$

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

$$\varepsilon(c)=\sum_{\begin{subarray}{c}l<l^{\prime}\\ f(l)>f(l^{\prime})\\ gf(l)=gf(l^{\prime})\end{subarray}}c_{l}c_{l^{\prime}}.$$

This gives an isomorphism of functors

$$\operatorname{Tot}\nolimits^{gf}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Tot}\nolimits^{g}\circ\operatorname{Tot}\nolimits^{f}.$$

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 $(X\otimes_{B}Y)^{(a_{1},\ldots,a_{n+m})}=X^{(a_{1},\ldots,a_{n})}\otimes_{B}Y^{(a_{n+1},\ldots,a_{n+m})}$.

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)

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

Let $k$ be a commutative ring and $A$ a $k$-algebra. The $k$-linear functor $H^{0}:\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\to A\operatorname{\!-Mod}\nolimits$ restricted to the full subcategory of complexes $M$ with $H^{i}(M)=0$ for $i\not=0$ is an equivalence. Let $C=C_{A}:A\operatorname{\!-Mod}\nolimits\to\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)$ be an inverse, composed with the inclusion functor. By construction the resolution functor $C$ is fully faithful. It induces a functor, still denoted by $C$,

$$C:\operatorname{Ho}\nolimits^{b}(A\operatorname{\!-Mod}\nolimits)\to\operatorname{Ho}\nolimits^{b}\bigl(\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\bigr).$$

We view $\operatorname{Ho}\nolimits^{b}\bigl(\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\bigr)$ as a triangulated category with the canonical structure on $\operatorname{Ho}\nolimits^{b}({\mathcal{C}})$, where ${\mathcal{C}}$ is the additive category $\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)$. The functor $C$ is a fully faithful triangulated functor.

Assume $A$ is projective as a $k$-module. Let $X=C_{A^{\mathrm{en}}}(A)$ be a projective resolution of $A$ as an $A^{\mathrm{en}}$-module. The functor

$$-\otimes_{A^{\operatorname{opp}\nolimits}}X:\mathrm{Comp}^{b}(A\operatorname{\!-Mod}\nolimits)\to 2\operatorname{\!-Comp}\nolimits(A\operatorname{\!-Proj}\nolimits)$$

composed with the canonical functor $2\operatorname{\!-Comp}\nolimits(A\operatorname{\!-Proj}\nolimits)\to\operatorname{Ho}\nolimits\bigl(\operatorname{Ho}\nolimits(A\operatorname{\!-Proj}\nolimits)\bigr)$ is isomorphic to $C$: we have $M\otimes_{A^{\operatorname{opp}\nolimits}}X\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{A}(M)$ for $M\in\operatorname{Ho}\nolimits^{b}(A\operatorname{\!-Mod}\nolimits)$.

Let $i\in{\mathbf{Z}}$. The functor $H^{i}:\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\to A\operatorname{\!-Mod}\nolimits$ induces a functor

$$H^{i}_{d_{2}}:\operatorname{Ho}\nolimits^{b}\bigl(\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\bigr)\to\operatorname{Ho}\nolimits^{b}(A\operatorname{\!-Mod}\nolimits).$$

It extends the functor $H^{i}_{d_{2}}:2\operatorname{\!-Comp}\nolimits(A\operatorname{\!-Mod}\nolimits)\to\mathrm{Comp}(A\operatorname{\!-Mod}\nolimits)$.

Let $B,B^{\prime}$ be two $k$-algebras, projective as $k$-modules. Let $L\in\operatorname{Ho}\nolimits^{b}\bigl((A\otimes B^{\operatorname{opp}\nolimits})\operatorname{\!-Mod}\nolimits\bigr)$ and $M\in\operatorname{Ho}\nolimits^{b}((B\otimes B^{\prime{\operatorname{opp}\nolimits}})\operatorname{\!-Mod}\nolimits)$. Assume the components of $M$ are projective right $B^{\prime}$-modules. We deduce from (1) an isomorphism

(2)

$$\operatorname{Tot}\nolimits^{13\to 1,2\to 2}\bigl(C_{A\otimes B^{\operatorname{opp}\nolimits}}(L)\otimes_{B}M\bigr)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{A\otimes B^{\prime{\operatorname{opp}\nolimits}}}\bigl(\operatorname{Tot}\nolimits(L\otimes_{B}M)\bigr).$$

Note that when $A$ is coherent, then $A\operatorname{\!-Mod}\nolimits$ can be replaced by the abelian category $A\operatorname{\!-mod}\nolimits$ and $A\operatorname{\!-Proj}\nolimits$ by $A\operatorname{\!-proj}\nolimits$. If $A$ is graded, we can replace these categories by the corresponding categories of graded modules.