ScalingStacks

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 $C$,

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

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

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

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.