# Proving the multiplicative group scheme is representable by punctured affine line given an internal description

I’m having trouble with what may be a simple exercise in the internal language of a topos.

Let E\mathcal E be a topos of sheaves over kk-Algop\mathsf{Alg}^\text{op} with some Grothendieck topology. Let R=aySpeck[x]R=\mathbf{ay} \operatorname{Spec} k[x] be its chosen (commutative unitary) ring object. Suppose we cut out the following subobject using the internal language of E\mathcal E: R×=[[r∈R∣∃s∈R:rs=1]].R^\times =[[r\in R\mid \exists s\in R:rs=1]].

How can I prove this subsheaf R×↣RR^\times \rightarrowtail R is representable by the punctured affine line k[x,x^{-1}]=k[x,y]/ \left\langle xy-1 \right\ranglek[x,x^{-1}]=k[x,y]/ \left\langle xy-1 \right\rangle?

Added. I think the point is R^\times (\mathbf{ay} \operatorname{Spec} A)\cong A^\timesR^\times (\mathbf{ay} \operatorname{Spec} A)\cong A^\times, and I’m sure there’s a very simple reason why this is true, but I don’t know it…

=================

=================

=================