The general linear group scheme and its coordinate ring

Define a $k$-algebra homomorphism $\bar{\mathrm{\Delta }}\text{\hspace{0.17em}⁣}:B\to A{\otimes }_{k}A$ by $\bar{\mathrm{\Delta }}(x_{ij})={\sum }_l{x}_{il}\otimes x_{lj}$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums). With $X^{(1)}=(x_{ij}\otimes 1)$ and $X^{(2)}=(1\otimes x_{ij})$ one has $\bar{\mathrm{\Delta }}(X)=X^{(1)}{X}^{(2)}$, so [F1] gives $\bar{\mathrm{\Delta }}(d)=\det (X^{(1)})\det (X^{(2)})=(d\otimes 1)(1\otimes d)=d\otimes d$, a unit of $A{\otimes }_{k}A$; the identities $\det (X^{(1)})=d\otimes 1$ and $\det (X^{(2)})=1\otimes d$ are the Leibniz formula (For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix) applied termwise to the ring homomorphisms $B\to A{\otimes }_{k}A$, $x_{ij}\mapsto x_{ij}\otimes 1$ and $x_{ij}\mapsto 1\otimes x_{ij}$ (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$). By [F3] there is a unique $k$-algebra homomorphism $\mathrm{\Delta }\text{\hspace{0.17em}⁣}:A\to A{\otimes }_{k}A$ with $\mathrm{\Delta }\lambda =\bar{\mathrm{\Delta }}$.

Define $\bar{\epsilon }\text{\hspace{0.17em}⁣}:B\to k$ by $\bar{\epsilon }(x_{ij})={\delta }_{ij}$; then $\bar{\epsilon }(d)=\det (I_n)=1$ is a unit, so by [F3] there is a unique $k$-algebra homomorphism $\epsilon \text{\hspace{0.17em}⁣}:A\to k$ with $\epsilon \lambda =\bar{\epsilon }$.

Let $M=(m_{ij})$ be the matrix over $A$ with entries $m_{ij}=d^{-1}(\mathrm{adj}X{)}_{ij}$, and define the $k$-algebra homomorphism $\bar{S}\text{\hspace{0.17em}⁣}:B\to A$ by $\bar{S}(x_{ij})=m_{ij}$. Since $X\mathrm{adj}(X)=\mathrm{adj}(X)X=d{I}_n$ over $A$ by [F1], multiplying by $d^{-1}$ gives $XM=I_n=MX$; multiplicativity of the determinant gives $\bar{S}(d)\cdot d=\det (M)\det (X)=\det (MX)=\det (I_n)=1$, so $\bar{S}(d)=d^{-1}$ is a unit and [F3] yields a unique $k$-algebra homomorphism $S\text{\hspace{0.17em}⁣}:A\to A$ with $S\lambda =\bar{S}$.

Coassociativity holds on the generators: $(\mathrm{\Delta }\otimes \mathrm{id})\mathrm{\Delta }(x_{ij})={\sum }_{l,k}{x}_{ik}\otimes x_{kl}\otimes x_{lj}=(\mathrm{id}\otimes \mathrm{\Delta })\mathrm{\Delta }(x_{ij})$ by associativity of matrix multiplication in [F1]. Both sides are $k$-algebra homomorphisms $A\to A{\otimes }_{k}A{\otimes }_{k}A$ agreeing on all $x_{ij}$, hence on $d$ and on $d^{-1}$, so they agree on $A$ by [F3].

The counit identities hold on the generators: $(\epsilon \otimes \mathrm{id})\mathrm{\Delta }(x_{ij})={\sum }_l{\delta }_{il}{x}_{lj}=x_{ij}={\sum }_l{x}_{il}{\delta }_{lj}=(\mathrm{id}\otimes \epsilon )\mathrm{\Delta }(x_{ij})$, with the canonical identification $k{\otimes }_{k}A\cong A\cong A{\otimes }_{k}k$; agreement on generators and on $d^{-1}$ as in step 2.1 extends this to $A$.

The antipode identities hold on the generators: $m_A(S\otimes \mathrm{id})\mathrm{\Delta }(x_{ij})={\sum }_{l}S(x_{il})x_{lj}=(MX{)}_{ij}={\delta }_{ij}=\epsilon (x_{ij})$ and $m_A(\mathrm{id}\otimes S)\mathrm{\Delta }(x_{ij})=(XM{)}_{ij}={\delta }_{ij}$, so $m_A(S\otimes \mathrm{id})\mathrm{\Delta }=u_A\epsilon =m_A(\mathrm{id}\otimes S)\mathrm{\Delta }$ on generators, both sides being $k$-algebra homomorphisms $A\to A$; agreement on the generators extends the identity to $A$ as in step 2.1.

By [F2] the ring maps $\mathrm{\Delta },\epsilon ,S$ are comorphisms of morphisms $m\text{\hspace{0.17em}⁣}:{\mathrm{GL}}_n{\times }_k{\mathrm{GL}}_n\to {\mathrm{GL}}_n$, $e\text{\hspace{0.17em}⁣}:\mathrm{Spec}k\to {\mathrm{GL}}_n$ and $i\text{\hspace{0.17em}⁣}:{\mathrm{GL}}_n\to {\mathrm{GL}}_n$, where ${\mathrm{GL}}_n=\mathrm{Spec}A$: the identifications $\mathrm{Spec}(A{\otimes }_{k}A)\cong {\mathrm{GL}}_n{\times }_k{\mathrm{GL}}_n$ and $\mathrm{Spec}(A{\otimes }_{k}A{\otimes }_{k}A)\cong {\mathrm{GL}}_n{\times }_k{\mathrm{GL}}_n{\times }_k{\mathrm{GL}}_n$ hold. The comorphisms of $m\circ (m\times \mathrm{id})$ and $m\circ (\mathrm{id}\times m)$ are $(\mathrm{\Delta }\otimes \mathrm{id})\mathrm{\Delta }$ and $(\mathrm{id}\otimes \mathrm{\Delta })\mathrm{\Delta }$, equal by step 2.1, so the two composites agree since $\mathrm{Spec}$ is a contravariant equivalence; the identities $m\circ (e\times \mathrm{id})=\mathrm{id}=m\circ (\mathrm{id}\times e)$ and $m\circ (i,\mathrm{id})=e\circ p=m\circ (\mathrm{id},i)$ follow in the same way from steps 3.1 and 3.2. Thus ${\mathrm{GL}}_n$ is a $k$-group scheme, and it is of finite type because $A$ is the finitely generated $k$-algebra $k[x_{ij},d^{-1}]$ and [F4] applies.

For every commutative unital $k$-algebra $R$ there is a natural bijection between $k$-algebra homomorphisms $A\to R$ and $n\times n$ matrices $N$ over $R$ with unit determinant: a map restricts to $B\to R$ giving $N=(N_{ij})$ with $\det (N)=\phi (d)$ a unit, and conversely a matrix with unit determinant gives $B\to R$, $x_{ij}\mapsto N_{ij}$, which sends $d$ to a unit and factors uniquely through $A$ by [F3]; by [F1] the unit-determinant matrices are exactly the invertible ones. Under this bijection the group law induced by $m$ is matrix multiplication, $(\phi \psi )(x_{ij})={\sum }_l\phi (x_{il})\psi (x_{lj})$, the identity is $I_n$, and $i$ induces matrix inversion, so ${\mathrm{GL}}_n(R)$ is the group of invertible matrices over $R$.

If $V$ is a $k$-vector space with basis $e_1,\dots ,e_n$, then [F5] identifies ${\mathrm{Aut}}_R(V{\otimes }_{k}R)$ with the invertible $n\times n$ matrices over $R$ naturally in $R$, and step 5.1 identifies the latter with ${\mathrm{GL}}_n(R)$; hence the functor $R\mapsto {\mathrm{Aut}}_R(V{\otimes }_{k}R)$ is naturally identified with ${\mathrm{GL}}_n$.

For $n=1$ the constructions specialize: $d=x_{11}$, $\mathrm{adj}(X)=1$, so with $t=x_{11}$ one has $A=k[t,t^{-1}]$, $\mathrm{\Delta }(t)=t\otimes t$, $\epsilon (t)=1$ and $S(t)=t^{-1}$, and step 5.1 identifies the points with $R^{\times }$; this is ${\mathbf{G}}_m$. For $V=0$, the singleton functor $R\mapsto {\mathrm{Aut}}_R(0)$ is represented by $\mathrm{Spec}k$, whose identity, multiplication and inverse are the unique possible maps; it is of finite type since its one-point space is quasi-compact. This supplies ${\mathrm{GL}}_0$ without a determinant formula. The coordinate and point constructions are choice-free; [F4] uses AC for the finite-type assertion when $n\ge 1$.