ScalingStacks

Let $R=\mathbb{Z}[c]$ denote the ring of polynomials with integral coefficients. Introduce a $\mathbb{Z}$-grading on $R$ by

Denote by $R{\mbox{-mod}_{0}}$ the abelian category of graded $R$-modules. Denote the $i$-th graded component of an object $M$ of $R{\mbox{-mod}_{0}}$ by $M_{i}.$ Morphisms in the category $R{\mbox{-mod}_{0}}$ are grading-preservings homomorphisms of modules. For $n\in\mathbb{Z}$ denote by $\{n\}$ the automorphism of $R{\mbox{-mod}_{0}}$ given by shifting the grading down by $n.$ Thus for a graded $R$-module $N=\oplus_{i}N_{i},$ the shifted module $N\{n\}$ has graded components $N\{n\}_{i}=N_{i+n}.$

In this paper we will sometimes consider graded, rather than just grading-preserving, maps. A map $\alpha:M\to N$ of graded $R$-modules is called graded of degree $i$ if $\alpha(M_{j})\subset N_{i+j}$ for all $j\in\mathbb{Z}.$

Let $R{\mbox{-mod}}$ be the category of graded $R$-modules and graded maps between them. This category has the same objects as the category $R{\mbox{-mod}_{0}},$ but more morphisms. It is not an abelian category.

A graded map $\alpha$ is a morphism in the category $R{\mbox{-mod}_{0}}$ if and only if the degree of $\alpha$ is $0.$ At the end we will favor grading-preserving maps and when at some point we look at a graded map $\alpha:M\to N$ of degree $i,$ later we will make it grading-preserving by appropriately shifting the degree of one of the modules. For example, $\alpha$ gives rise to a grading-preserving map $M\to N\{i\},$ also denoted $\alpha.$

Let $M$ be a finitely-generated graded $R$-module. As an abelian group, $M$ is the direct sum of its graded components: $M={\mathop{\oplus}\limits_{j\in\mathbb{Z}}}M_{j},$ where each $M_{j}$ is a finitely-generated abelian group. Define the graded Euler characteristic $\widehat{\chi}(M)$ of $M$ by

Since

$\widehat{\chi}(M)$ is not, in general, a Laurent polynomial in $q$, but an element of the Laurent series ring. Moreover, for any $M$ as above, there are Laurent polynomials $a,b\in\mathbb{Z}[q,q^{-1}]$ such that