ScalingStacks

We define a ${\mathbf{Z}}$-monoid $G$ to be a monoid $G$ endowed with an action of the group ${\mathbf{Z}}$, denoted by $g\mapsto g+n$ for $g\in G$ and $n\in{\mathbf{Z}}$, and such that $(g+n)(g^{\prime}+n^{\prime})=gg^{\prime}+n+n^{\prime}$. Note that $e_{G}+{\mathbf{Z}}$ is a central submonoid of $G$, where $e_{G}$ denotes the unit of $G$. So, the data above is equivalent to the data of a morphism of monoids ${\mathbf{Z}}\to Z(G)$. This is itself determined by the image of $1$, a central invertible element $\upsilon$ of $G$.

We define a differential $G$-graded $k$-module to be a $G$-graded $k$-module $M$ together with a differential module structure such that $d(M_{g})\subset M_{g+1}$ (cf [LiOzTh1, §2.5]).

Given $g\in G$, we define $M\langle g\rangle$ to be the differential $G$-graded $k$-module given by $(M\langle g\rangle)_{h}=M_{hg}$. Similarly, we define $\langle g\rangle M$ by $(\langle g\rangle M)_{h}=M_{gh}$.

We define similarly the notion of differential $G$-graded algebra, of differential $G$-graded category, etc.

When $G={\mathbf{Z}}$ and $\upsilon=1$, we recover the usual notion of differential graded $k$-module, etc.

Let $G_{1}$ and $G_{2}$ be two ${\mathbf{Z}}$-monoids. We define $G_{1}\times_{{\mathbf{Z}}}G_{2}$ as the quotient of $G_{1}\times G_{2}$ by the equivalence relation $(g_{1},g_{2}+n)\sim(g_{1}+n,g_{2})$ for $g_{1},g_{2}\in G$ and $n\in{\mathbf{Z}}$. Denote by $p:G_{1}\times G_{2}\to G_{1}\times_{{\mathbf{Z}}}G_{2}$ the quotient map, a morphism of monoids. There is a structure of ${\mathbf{Z}}$-monoid on $G_{1}\times_{{\mathbf{Z}}}G_{2}$ given by $p(g_{1},g_{2})+1=p(g_{1}+1,g_{2})=p(g_{1},g_{2}+1)$.

Let $M_{i}$ be a differential $G_{i}$-graded $k$-module for $i\in\{1,2\}$. We define a structure of differential $(G_{1}\times_{{\mathbf{Z}}}G_{2})$-module on the differential module $M_{1}\otimes M_{2}$ by setting $(M_{1}\otimes M_{2})_{g}=\bigoplus_{(g_{1},g_{2})\in p^{-1}(g)}(M_{1})_{g_{1}}\otimes(M_{2})_{g_{2}}$.