ScalingStacks

Denote by $\mbox{Kom}(\mathcal{B})$ the category of complexes of an abelian category $\mathcal{B}.$ An object $N$ of $\mbox{Kom}(\mathcal{B})$ is a collection of objects $N^{i}\in\mathcal{B},i\in\mathbb{Z}$ together with morphisms $d^{i}:N^{i}\longrightarrow N^{i+1},i\in\mathbb{Z}$ such that $d^{i+1}d^{i}=0.$ A morphism $f:M\to N$ of complexes is a collection of morphisms $f^{i}:M^{i}\to N^{i}$ such that $f^{i+1}d^{i}=d^{i}f^{i},i\in\mathbb{Z}.$ A morphism $f:M\to N$ is called a quasi-isomorphism if the induced map of the cohomology groups $H^{i}(f):H^{i}(M)\to H^{i}(N)$ is an isomorphism for all $i\in\mathbb{Z}.$

For $n\in\mathbb{Z}$ denote by $[n]$ the automorphism of $\mbox{Kom}(\mathcal{B})$ that is defined on objects by $N[n]^{i}=N^{i+n},d[n]^{i}=(-1)^{n}d^{i+n}$ and continued to morphisms in the obvious way.

The cone of a morphism $f:M\to N$ of complexes is a complex $C(f)$ with

The automorphism $\{n\}$ of shifting the grading down by $n,$ introduced in Section 2.1, can be naturally extended to an automorphism of the category $\mbox{Kom}(R{\mbox{-mod}_{0}})$ of complexes of graded $R$-modules. This automorphism of $\mbox{Kom}(R{\mbox{-mod}_{0}})$ will also be denoted $\{n\}.$

To a complex $M$ of graded $R$-modules we associate a graded $R$-module ${\mathop{\oplus}\limits_{i\in\mathbb{Z}}}M^{i}.$ Each $M^{i}$ is a graded $R$-module, $M^{i}={\mathop{\oplus}\limits_{j\in\mathbb{Z}}}M^{i}_{j}$ and thus ${\mathop{\oplus}\limits_{i\in\mathbb{Z}}}M^{i}$ is a bigraded $R$-module when we extend our usual grading of $R$ to a bigrading with $c\in R$ having degree $(0,2).$

From this viewpoint the differential $d_{M}$ of a complex $M$ is a homogeneous map of degree $(1,0)$ of bigraded $R$-modules.