Subsection 2.2 (0PMW) · ScalingStacks

2.2 The algebra $A$

Let $A$ be a free graded $R$ -module of rank $2$ spanned by $\mathbf{1}$ and $X$ with

{\mathrm{deg}}(\mathbf{1})=1,\hskip 21.68121pt{\mathrm{deg}}(X)=-1

(11)

We equip $A$ with a commutative algebra structure with the unit $\mathbf{1}$ and multiplication

\mathbf{1}X=X\mathbf{1}=X,X^{2}=0.

(12)

We denote by $\iota$ the unit map $R\to A$ which sends $1$ to $\mathbf{1}.$ This map is a graded map of graded $R$ -modules and it increases the degree by $1.$

We equip $A$ with a coalgebra structure with a coassociative cocommutative comultiplication

		$\displaystyle\Delta(\mathbf{1})$	$\displaystyle=\mathbf{1}\otimes X+X\otimes\mathbf{1}+cX\otimes X$		(13)
		$\displaystyle\Delta(X)$	$\displaystyle=X\otimes X$		(14)

and a counit

\epsilon(\mathbf{1})=-c,\hskip 36.135pt\epsilon(X)=1.

(15)

$A,$ equipped with these structures, is not a Hopf algebra. Instead, the identity

\Delta\circ m=(m\otimes\mbox{Id})\circ(\mbox{Id}\otimes\Delta)

(16)

holds.

Grading ${\mathrm{deg}},$ given by (7),(11), induces a grading, also denoted ${\mathrm{deg}},$ on tensor powers of $A$ by

{\mathrm{deg}}(a_{1}\otimes\dots\otimes a_{n})={\mathrm{deg}}(a_{1})+\dots+{\mathrm{deg}}(a_{n})\mbox{ for }a_{1},\dots,a_{n}\in A

(17)

Here and further on all tensor products are taken over the ring $R$ unless specified otherwise.

We next describe the effect of the structure maps $\iota,m,\epsilon,\Delta$ on the gradings. We say that a map $f$ between two graded $R$ -modules $V=\oplus V_{n}$ and $W=\oplus W_{n}$ has degree $k$ if $f(x)\in W_{n+k}$ whenever $x\in V_{n}.$