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This is again false. The geometric interpretation is as follows: given étale morphisms $Y \to X_1$ and $Y \to X_2$ of affine schemes, the image factorisation $$\Gamma(X_1,\mathcal O_{X_1}) \underset k\otimes \Gamma(X_2,\mathcal O_{X_2}) \twoheadrightarrow \Gamma(X,\mathcal O_X) \hookrightarrow \Gamma(Y,\mathcal O_Y)$$ corresponds geometrically to taking the (scheme-theoretic) image $Y \to X \hookrightarrow X_1 \times X_2$ of the product map $Y \to X_1 \times X_2$. If $Y$ is smooth, then so are the $X_i$, but $X$ need not be smooth. If $Y \to X$ is surjective, it cannot be étale [Tag 07NG].

We can turn this around to make a counterexample: start with a map $Y \to X_1 \times X_2$ of smooth schemes such that the image is singular but both projections $Y \to X_i$ are étale.

Example. This is the typical picture of a plane nodal curve: let $Y = \mathbf A^1 \setminus \bigl\{0,\pm\tfrac{\sqrt 3}{3}\bigr\}$ and $X_1 = X_2 = \mathbf A^1$, and consider the morphism $Y \to X_1 \times X_2$ given by $t \mapsto (t^2-1,t^3-t)$. The image lands in the nodal curve $\{y^2 = x^2(x+1)\}$, only missing the points $(-1,0)$ and $\bigl(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9}\bigr)$ where the tangent line is horizontal or vertical (hence the choice of $Y$). Thus the compositions $Y \to X_1 \times X_2 \to X_i$ are étale, but the image $$X = V\bigl(y^2-x^2(x+1)\bigr) \setminus \bigl\{(-1,0),\bigl(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9}\bigr)\bigr\}$$ of $Y \to X_1 \times X_2$ is singular at $(0,0)$.

(This parametrisation is explained on this page of Donu Arapura, which is literally the first result that my favourite search engine produces when I type 'nodal curve'.)

Translated back into algebra, this means $S = k\bigl[t,\tfrac{1}{t},\tfrac{1}{3t^2-1}\bigr]$, with $R_1 = k[t^2-1] \subseteq S$ and $R_2 = k[t^3-t] \subseteq S$.

This is again false. The geometric interpretation is as follows: write $Y = \operatorname{Spec} S$ and $X_i = \operatorname{Spec} R_i$. Given étale morphisms $f_1 \colon Y \to X_1$ and $f_2 \colon Y \to X_2$ of affine schemes, the image factorisation $$R_1 \underset k\otimes R_2 \twoheadrightarrow R \hookrightarrow S$$ corresponds geometrically to taking the (scheme-theoretic) image $Y \to X \hookrightarrow X_1 \times X_2$ of the product map $f \colon Y \to X_1 \times X_2$. If $Y$ is smooth, then so are the $X_i$, but $X$ need not be smooth. If $f$ is surjective, it cannot be flat when $X$ is singular [Tags 07NG and 00HQ].

We can turn this around to make a counterexample: start with a map $f \colon Y \to X_1 \times X_2$ of smooth schemes such that the image is singular but both projections $f_i \colon Y \to X_i$ are étale.

Example. This is the typical picture of a plane nodal curve: let $Y_0 = X_1 = X_2 = \mathbf A^1$, and consider the morphism $f \colon Y_0 \to X_1 \times X_2$ given by $t \mapsto (t^2-1,t^3-t)$. The scheme-theoretic image is the nodal curve $X_0 = V(y^2 - x^2(x+1))$, and the map $Y_0 \to X_0$ is surjective.

The projections $f_i \colon Y_0 \to X_i$ are not étale, but they become so after removing the points $0$ and $\pm\tfrac{\sqrt 3}{3}$ from $Y_0$. Let $Y \subseteq Y_0$ (resp. $X \subseteq X_0$) be the complement of these points (resp. their images in $X$). This gives the desired counterexample, since we did not remove the singular point $(0,0) = f(-1) = f(1)$ from $X$.

(This parametrisation is explained on this page of Donu Arapura, which is literally the first result that my favourite search engine produces when I type 'nodal curve'.)

Translated back into algebra, this means $S = k\bigl[t,\tfrac{1}{t},\tfrac{1}{3t^2-1}\bigr]$, with $R_1 = k[t^2-1] \subseteq S$ and $R_2 = k[t^3-t] \subseteq S$.

This is again false. The geometric interpretation is as follows: given étale morphisms $Y \to X_1$ and $Y \to X_2$ of affine schemes, the image factorisation $$\Gamma(X_1,\mathcal O_{X_1}) \underset k\otimes \Gamma(X_2,\mathcal O_{X_2}) \twoheadrightarrow \Gamma(X,\mathcal O_X) \hookrightarrow \Gamma(Y,\mathcal O_Y)$$ corresponds geometrically to taking the (scheme-theoretic) image $Y \to X \hookrightarrow X_1 \times X_2$ of the product map $Y \to X_1 \times X_2$. If $Y$ is smooth, then so are the $X_i$, but $X$ need not be smooth. If $Y \to X$ is surjective, it cannot be étale [Tag 07NG].

We can turn this around to make a counterexample: start with a map $Y \to X_1 \times X_2$ of smooth schemes such that the image is singular but both projections $Y \to X_i$ are étale.

Example. This is the typical picture of a plane nodal curve: let $Y = \mathbf A^1 \setminus \{0,\pm\tfrac{\sqrt 3}{3}\}$ and $X_1 = X_2 = \mathbf A^1$, and consider the morphism $Y \to X_1 \times X_2$ given by $t \mapsto (t^2-1,t^3-t)$. The image lands in the nodal curve $\{y^2 = x^2(x+1)\}$, only missing the points $(-1,0)$, $(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9})$ where the tangent line is horizontal or vertical (hence the choice of $Y$). Thus the compositions $Y \to X_1 \times X_2 \to X_i$ are étale, but the image $$X = V(y^2-x^2(x+1)) \setminus \{(-1,0),(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9})\}$$ is singular.

(This parametrisation is explained on this page of Donu Arapura, which is literally the first result that my favourite search engine produces when I type 'nodal curve'.)

Translated back into algebra, this means $S = k[t,\tfrac{1}{t},\tfrac{1}{3t^2-1}]$, with $R_1 = k[t^2-1] \subseteq S$ and $R_2 = k[t^3-t] \subseteq S$.

This is again false. The geometric interpretation is as follows: given étale morphisms $Y \to X_1$ and $Y \to X_2$ of affine schemes, the image factorisation $$\Gamma(X_1,\mathcal O_{X_1}) \underset k\otimes \Gamma(X_2,\mathcal O_{X_2}) \twoheadrightarrow \Gamma(X,\mathcal O_X) \hookrightarrow \Gamma(Y,\mathcal O_Y)$$ corresponds geometrically to taking the (scheme-theoretic) image $Y \to X \hookrightarrow X_1 \times X_2$ of the product map $Y \to X_1 \times X_2$. If $Y$ is smooth, then so are the $X_i$, but $X$ need not be smooth. If $Y \to X$ is surjective, it cannot be étale [Tag 07NG].

We can turn this around to make a counterexample: start with a map $Y \to X_1 \times X_2$ of smooth schemes such that the image is singular but both projections $Y \to X_i$ are étale.

Example. This is the typical picture of a plane nodal curve: let $Y = \mathbf A^1 \setminus \bigl\{0,\pm\tfrac{\sqrt 3}{3}\bigr\}$ and $X_1 = X_2 = \mathbf A^1$, and consider the morphism $Y \to X_1 \times X_2$ given by $t \mapsto (t^2-1,t^3-t)$. The image lands in the nodal curve $\{y^2 = x^2(x+1)\}$, only missing the points $(-1,0)$ and $\bigl(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9}\bigr)$ where the tangent line is horizontal or vertical (hence the choice of $Y$). Thus the compositions $Y \to X_1 \times X_2 \to X_i$ are étale, but the image $$X = V\bigl(y^2-x^2(x+1)\bigr) \setminus \bigl\{(-1,0),\bigl(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9}\bigr)\bigr\}$$ of $Y \to X_1 \times X_2$ is singular at $(0,0)$.

(This parametrisation is explained on this page of Donu Arapura, which is literally the first result that my favourite search engine produces when I type 'nodal curve'.)

Translated back into algebra, this means $S = k\bigl[t,\tfrac{1}{t},\tfrac{1}{3t^2-1}\bigr]$, with $R_1 = k[t^2-1] \subseteq S$ and $R_2 = k[t^3-t] \subseteq S$.

This is again false. The geometric interpretation is as follows: given étale morphisms $Y \to X_1$ and $Y \to X_2$ of affine schemes, the image factorisation $$\Gamma(X_1,\mathcal O_{X_1}) \underset k\otimes \Gamma(X_2,\mathcal O_{X_2}) \twoheadrightarrow \Gamma(X,\mathcal O_X) \hookrightarrow \Gamma(Y,\mathcal O_Y)$$ corresponds geometrically to taking the (scheme-theoretic) image $Y \to X \hookrightarrow X_1 \times X_2$ of the product map $Y \to X_1 \times X_2$. If $Y$ is smooth, then so are the $X_i$, but $X$ need not be smooth. If $Y \to X$ is surjective, it cannot be étale [Tag 07NG].

We can turn this around to make a counterexample: start with a map $Y \to X_1 \times X_2$ of smooth schemes such that the image is singular but both projections $Y \to X_i$ are étale.

Example. This is the typical picture of a plane nodal curve: let $Y = \mathbf A^1 \setminus \{0,\pm\tfrac{\sqrt 3}{3}\}$ and $X_1 = X_2 = \mathbf A^1$, and consider the morphism $Y \to X_1 \times X_2$ given by $t \mapsto (t^2-1,t^3-t)$. The image lands in the nodal curve $\{y^2 = x^2(x+1)\}$, only missing the points where the tangent line is horizontal or vertical (hence the choice of $Y$). Thus the compositions $Y \to X \to X_i$ are étale.

(This parametrisation is explained on this page of Donu Arapura, which is literally the first result that my favourite search engine produces when I type 'nodal curve'.)

Translated back into algebra, this means $S = k[t,\tfrac{1}{t},\tfrac{1}{3t^2-1}]$, with $R_1 = k[t^2-1] \subseteq S$ and $R_2 = k[t^3-t] \subseteq S$.

This is again false. The geometric interpretation is as follows: given étale morphisms $Y \to X_1$ and $Y \to X_2$ of affine schemes, the image factorisation $$\Gamma(X_1,\mathcal O_{X_1}) \underset k\otimes \Gamma(X_2,\mathcal O_{X_2}) \twoheadrightarrow \Gamma(X,\mathcal O_X) \hookrightarrow \Gamma(Y,\mathcal O_Y)$$ corresponds geometrically to taking the (scheme-theoretic) image $Y \to X \hookrightarrow X_1 \times X_2$ of the product map $Y \to X_1 \times X_2$. If $Y$ is smooth, then so are the $X_i$, but $X$ need not be smooth. If $Y \to X$ is surjective, it cannot be étale [Tag 07NG].

We can turn this around to make a counterexample: start with a map $Y \to X_1 \times X_2$ of smooth schemes such that the image is singular but both projections $Y \to X_i$ are étale.

Example. This is the typical picture of a plane nodal curve: let $Y = \mathbf A^1 \setminus \{0,\pm\tfrac{\sqrt 3}{3}\}$ and $X_1 = X_2 = \mathbf A^1$, and consider the morphism $Y \to X_1 \times X_2$ given by $t \mapsto (t^2-1,t^3-t)$. The image lands in the nodal curve $\{y^2 = x^2(x+1)\}$, only missing the points $(-1,0)$, $(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9})$ where the tangent line is horizontal or vertical (hence the choice of $Y$). Thus the compositions $Y \to X_1 \times X_2 \to X_i$ are étale, but the image $$X = V(y^2-x^2(x+1)) \setminus \{(-1,0),(-\tfrac{2}{3},\pm \tfrac{2\sqrt 3}{9})\}$$ is singular.

(This parametrisation is explained on this page of Donu Arapura, which is literally the first result that my favourite search engine produces when I type 'nodal curve'.)

Translated back into algebra, this means $S = k[t,\tfrac{1}{t},\tfrac{1}{3t^2-1}]$, with $R_1 = k[t^2-1] \subseteq S$ and $R_2 = k[t^3-t] \subseteq S$.