The group schemes Ga, Gm, and GLn

For ${\mathbf{G}}_a$, the comorphisms of multiplication, identity and inverse send $x$ respectively to $x\otimes 1+1\otimes x$, $0$, and $-x$. They are algebra maps and hence morphisms by [F2]. Evaluation on $R$ identifies its points with $R$ and its operations with addition, zero, and negation. For ${\mathbf{G}}_m$, the corresponding formulas are $t\mapsto t\otimes t$, $t\mapsto 1$, and $t\mapsto t^{-1}$. Each image of $t$ is a unit, so these maps are defined on the Laurent algebra. Evaluation identifies its points with $R^{\times }$ and its operations with multiplication, one and inversion. These formulas satisfy the group-object identities as ring identities and hence as scheme morphisms by [F1]–[F2].

For $X=(x_{ij})$, define multiplication by $x_{ij}\mapsto {\sum }_l{x}_{il}\otimes x_{lj}$. Its determinant is $(d\otimes 1)(1\otimes d)$ by [F3], a unit, so the formula extends to the localized coordinate ring. The identity has $x_{ij}\mapsto {\delta }_{ij}$, with determinant one. Define inversion by the entries of $d^{-1}\mathrm{adj}(X)$; they belong to the same localized algebra. Its determinant is a unit, since the adjugate identity gives $X{X}^{-1}=I$ and determinant multiplicativity gives $\det (X^{-1})=d^{-1}$. Thus inversion also gives a morphism. The points of the localized spectrum are exactly matrices with unit determinant, equivalently invertible matrices by [F3].

Matrix associativity, the identity matrix, and the two inverse identities in [F3] verify all group identities on ${\mathrm{GL}}_n(R)$, for every $R$. They also verify the scheme identities: each domain in those identities is affine by [F2]; testing its coordinate algebra with its universal point tests the morphisms themselves. All three displayed coordinate algebras are finitely generated over $k$ (write the determinant inverse as a generator subject to $zd-1=0$), so the schemes are finite type. They are therefore group schemes by [F1]. No field-valued-point or smoothness argument substitutes for these formulas.