The square-root standard etale chart

The chart is standard 'etale. Take $P=T^2-a\in A[T]$, which is monic of degree $2$ with formal derivative $P^{\mathrm{\prime }}=2T$, and take $g=2T$ in the presentation $B=(A[T]\mathrm{/}(P){)}_{2T}$. In $B$ the image of $P^{\mathrm{\prime }}$ is the inverted element $2T$, hence a unit, so $B$ is standard 'etale over $A$ by [F1]; the presentation is a quotient of $A[T]$ by the principal ideal $(P)$ followed by a localisation, so $B$ is a finitely presented $A$-algebra, and therefore $\mathrm{Spec}B\to \mathrm{Spec}A$ is 'etale by [F1]. By [F1] again, $A[T]\mathrm{/}(P)$ is free over $A$ with basis $1,T$ before the localisation. This proves claim 1.

The degree-two locus. Let $\mathfrak{p}\in \mathrm{Spec}A$ with $2a\mathrm{\notin }\mathfrak{p}$. Then $2\mathrm{\notin }\mathfrak{p}$ and $a\mathrm{\notin }\mathfrak{p}$, so $2$ and $a$ are units of $A_{\mathfrak{p}}$; in the ring $A_{\mathfrak{p}}[T]\mathrm{/}(T^2-a)$ the relation $T\cdot (T{a}^{-1})=1$ makes $T$ a unit, hence $2T$ is a unit, and localising at a unit does not change the ring by [F2]. Therefore $B_{\mathfrak{p}}\cong A_{\mathfrak{p}}[T]\mathrm{/}(T^2-a)$, which is free of rank $2$ over $A_{\mathfrak{p}}$ with basis $1,T$ by [F1]. For the fibre, [F3] applied over $A_{\mathfrak{p}}$ gives $B{\otimes }_A\kappa (\mathfrak{p})\cong B_{\mathfrak{p}}\mathrm{/}\mathfrak{p}{A}_{\mathfrak{p}}{B}_{\mathfrak{p}}\cong \kappa (\mathfrak{p})[T]\mathrm{/}(T^2-\bar{a})$, where $\bar{a}\mathrm{\ne }0$ and $2\mathrm{\ne }0$ in the field $\kappa (\mathfrak{p})$; in this ring $T$ is a unit (with inverse $T{\bar{a}}^{-1}$), so the image of the derivative $2T$ of the monic polynomial $T^2-\bar{a}$ is a unit and [F1] makes $\kappa (\mathfrak{p})[T]\mathrm{/}(T^2-\bar{a})$ a standard 'etale, hence finite 'etale, $\kappa (\mathfrak{p})$-algebra of rank $2$. This proves claim 2.

The characteristic two boundary. If $2=0$ in $A$, then $2T=0$, so the localisation of $A[T]\mathrm{/}(T^2-a)$ at the powers of $0$ is the zero ring $B=0$ by [F2]; its spectrum is empty, and the structure morphism from the empty scheme to $\mathrm{Spec}A$ has no point at which 'etaleness could fail, so it is 'etale vacuously. In particular, for a field $k$ of characteristic two the chart $\mathrm{Spec}(k[T]\mathrm{/}(T^2-a){)}_{2T}$ is empty for every $a$.

The chart is supported over $D(2a)$. In $B$, the element $2T$ is a unit, so $4a=(2T{)}^2$ is a unit. Since $4a=2(2a)$, the factor $2a$ is a unit in $B$ as well. Hence $\mathrm{Spec}B\to \mathrm{Spec}A$ factors through $D(2a)$. On $D(2a)$, both $2$ and $a$ are units; the relation $T\cdot (T{a}^{-1})=1$ makes $T$ a unit, so localising at $2T$ changes nothing. Thus the restricted algebra is $A_{2a}[T]\mathrm{/}(T^2-a)$, finite free of rank $2$ and 'etale by the derivative calculation of step 1.2. This proves that the rank-two cover occurs exactly over $D(2a)$.

Conclusion and choice accounting. Claims 1, 2 and 3 follow from step 1.1, step 1.2, step 1.3 and step 2.1. The standard 'etale presentation, the free basis from monic division, the unit computations in a localisation and the fibre computation used above are all choice-free, and no Axiom of Choice is assumed or used in this example.