Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $\Omega_{X/Y} = \Delta^{-1} \mathscr{I}/\mathscr{I}^2$ where $\mathscr{I}$ is the sheaf representing the immersion $X\to X\times_Y X$ (it's the kernel of $\mathscr{O}_{X\times_Y X} \to \Delta_* \mathscr{O}_X$.

Algebraically, this works out fine, due to the theory of abstract Kaehler derivation defined on algebras. Is there a way to actually see the motivation behind this? Also, what's the analog in higher infinitesimal approximation (instead of just 1st order one given by the differentials)? What's the insight behind the relationship between higher infinitesimal and higher diagonal?

Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $\Omega_{X/Y} = \Delta^{-1} \mathscr{I}/\mathscr{I}^2$ where $\mathscr{I}$ is the sheaf representing the immersion $X\to X\times_Y X$ (it's the kernel of $\mathscr{O}_{X\times_Y X} \to \Delta_* \mathscr{O}_X$.

Algebraically, this works out fine, due to the theory of abstract Kahler derivation defined on algebras. Is there a way to actually see the motivation behind this?

Moreover, what's the analog in higher infinitesimal approximation (instead of just 1st order one given by the differentials)? What's the (say, "analytic") insight behind the relationship between higher infinitesimal and higher diagonal?

Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $\Omega_{X/Y} = \Delta^{-1} \mathscr{I}/\mathscr{I}^2$ where $\mathscr{I}$ is the sheaf representing the immersion $X\to X\times_Y X$ (it's the kernel of $\mathscr{O}_{X\times_Y X} \to \Delta_* \mathscr{O}_X$.

Algebraically, this works out fine, due to the theory of abstract Kaehler derivation defined on algebras. Is there a way to actually see the motivation behind this?

Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $\Omega_{X/Y} = \Delta^{-1} \mathscr{I}/\mathscr{I}^2$ where $\mathscr{I}$ is the sheaf representing the immersion $X\to X\times_Y X$ (it's the kernel of $\mathscr{O}_{X\times_Y X} \to \Delta_* \mathscr{O}_X$.

Algebraically, this works out fine, due to the theory of abstract Kaehler derivation defined on algebras. Is there a way to actually see the motivation behind this? Also, what's the analog in higher infinitesimal approximation (instead of just 1st order one given by the differentials)? What's the insight behind the relationship between higher infinitesimal and higher diagonal?