ScalingStacks

Let $k$ be a commutative ring. We write $\otimes$ for $\otimes_{k}$. Let $A$ be a $k$-algebra. We denote by $A^{\operatorname{opp}\nolimits}$ the opposite algebra to $A$ and we put $A^{{\mathrm{en}}}=A\otimes A^{\operatorname{opp}\nolimits}$.

We denote by $A\operatorname{\!-Mod}\nolimits$ the category of $A$-modules, by $A\operatorname{\!-mod}\nolimits$ the category of finitely generated $A$-modules, by $A\operatorname{\!-Proj}\nolimits$ the category of projective $A$-modules and by $A\operatorname{\!-proj}\nolimits$ the category of finitely generated projective $A$-modules. Assume $A$ is graded. We denote by $A\operatorname{\!-modgr}\nolimits$ (resp. $A\operatorname{\!-projgr}\nolimits$) the category of finitely generated (resp. and projective) graded $A$-modules.

Given $M$ a graded $k$-module and $n\in{\mathbf{Z}}$, we denote by $M\langle n\rangle$ the graded $k$-module given by $M\langle n\rangle_{i}=M_{n+i}$.

Given ${\mathcal{A}}$ an additive category, we denote by $\operatorname{Comp}\nolimits({\mathcal{A}})$ (resp. $\operatorname{Ho}\nolimits({\mathcal{A}})$) the category (resp. the homotopy category) of complexes of objects of ${\mathcal{A}}$. If ${\mathcal{A}}$ is an abelian category, we denote by $D({\mathcal{A}})$ its derived category.

Given ${\mathcal{C}}$ a category, we denote by $\{1\}$ the self equivalence of ${\mathcal{C}}^{\mathbf{Z}}$ given by $(M\{1\})_{i}=M_{i+1}$.

Let ${\mathcal{T}}$ be a triangulated category equipped with an automorphism $M\mapsto M\langle 1\rangle$. We denote by $q$ the automorphism of $K_{0}({\mathcal{T}})$ given by $[M]\mapsto[M\langle 1\rangle]$. This endows $K_{0}({\mathcal{T}})$ with a structure of ${\mathbf{Z}}[q,q^{-1}]$-module.