In this section, the authors present some fundamental differential calculus concepts that are particularly useful in algebraic geometry. They introduce the notion of differential invariants for smooth morphisms, focusing on the normal invariants of an immersion. The key points include: 1. **Definition of Normal Invariants**: For a morphism \( f = (\psi, 0) : Y \to X \) where \( \psi^*(\mathcal{O}_X) \to \mathcal{O}_Y \) is surjective, the \( n \)-th normal invariant of \( f \) is defined as the quotient \( \psi^*(\mathcal{O}_X) / \mathcal{I}_f^{n+1} \). The \( n \)-th infinitesimal neighborhood \( Y^{(n)} \) is the space \( (Y, \psi^*(\mathcal{O}_X) / \mathcal{I}_f^{n+1}) \). 2. ** associated graded module**: The associated graded module \( \mathcal{I}_f_\bullet(f) = \bigoplus_{n \geq 0} (\mathcal{I}_f^n / \mathcal{I}_f^{n+1}) \) is introduced, and the conormal module \( \mathcal{I}_f / \mathcal{I}_f^2 \) is defined. 3. **Projective System of Rings**: The \( \mathcal{O}_{Y^{(n)}} \) form a projective system of rings over \( Y \), and the \( Y^{(n)} \) form an inductive system of anchored spaces with the same underlying space \( Y \). 4. **Examples**: - **Local Case**: If \( X \) is a local ringed space and \( Y \) is a single point, the \( \mathcal{O}_{Y(n)} \) are identified with \( \mathcal{O}_{x} / \mathfrak{m}_x^{n+1} \). - **Closed Subset**: If \( Y \) is a closed subset of an open subset \( U \) of \( X \), the \( \mathcal{O}_{Y(n)} \) are identified with \( \psi_0^*(\mathcal{O}_U / \mathcal{I}^{n+1}) \). 5. **Proposition**: - For an immersion \( f = (\psi, \theta) : Y \to X \), the graded \( \mathcal{O}_Y \)-algebra \( \mathcal{G}_{r_*}(f) \) is quasi-coherent. This section provides a foundational understanding of the differential invariants and their applications in algebraic geometry, particularly in the context of immersions and their infinitesimal neighborhoods.In this section, the authors present some fundamental differential calculus concepts that are particularly useful in algebraic geometry. They introduce the notion of differential invariants for smooth morphisms, focusing on the normal invariants of an immersion. The key points include: 1. **Definition of Normal Invariants**: For a morphism \( f = (\psi, 0) : Y \to X \) where \( \psi^*(\mathcal{O}_X) \to \mathcal{O}_Y \) is surjective, the \( n \)-th normal invariant of \( f \) is defined as the quotient \( \psi^*(\mathcal{O}_X) / \mathcal{I}_f^{n+1} \). The \( n \)-th infinitesimal neighborhood \( Y^{(n)} \) is the space \( (Y, \psi^*(\mathcal{O}_X) / \mathcal{I}_f^{n+1}) \). 2. ** associated graded module**: The associated graded module \( \mathcal{I}_f_\bullet(f) = \bigoplus_{n \geq 0} (\mathcal{I}_f^n / \mathcal{I}_f^{n+1}) \) is introduced, and the conormal module \( \mathcal{I}_f / \mathcal{I}_f^2 \) is defined. 3. **Projective System of Rings**: The \( \mathcal{O}_{Y^{(n)}} \) form a projective system of rings over \( Y \), and the \( Y^{(n)} \) form an inductive system of anchored spaces with the same underlying space \( Y \). 4. **Examples**: - **Local Case**: If \( X \) is a local ringed space and \( Y \) is a single point, the \( \mathcal{O}_{Y(n)} \) are identified with \( \mathcal{O}_{x} / \mathfrak{m}_x^{n+1} \). - **Closed Subset**: If \( Y \) is a closed subset of an open subset \( U \) of \( X \), the \( \mathcal{O}_{Y(n)} \) are identified with \( \psi_0^*(\mathcal{O}_U / \mathcal{I}^{n+1}) \). 5. **Proposition**: - For an immersion \( f = (\psi, \theta) : Y \to X \), the graded \( \mathcal{O}_Y \)-algebra \( \mathcal{G}_{r_*}(f) \) is quasi-coherent. This section provides a foundational understanding of the differential invariants and their applications in algebraic geometry, particularly in the context of immersions and their infinitesimal neighborhoods.