Timeline for Smoothness of homomorphisms between graded algebras
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2022 at 12:30 | comment | added | Johan | Let $X \to Y \to S$ be the morphisms on spectra. The augmentations of $S$ and $T$ gives compatible sections $S \to X$ and $S \to Y$. Let $s \in S$ be a closed point and $x \in X$ and $y \in Y$ the images. Smoothness of $X \to Y$ says that $\mathcal{O}_{X, x}^\wedge \cong \mathcal{O}_{Y, y}^\wedge[[t_1, \ldots, t_r]]$. I think that this means that the conormal sheaf of $S \to X$ is a locally free module of rank $r$ direct sum the conormal sheaf of $S \to Y$ in a neighborhood of $s$. Statements like this can be found in many references, try eg Fulton's book on intersection theory. Good luck. | |
| Aug 3, 2022 at 3:27 | comment | added | Kim | @KonstantinosKanakoglou Here means smooth as $A$-algebra homomorphism | |
| Aug 2, 2022 at 17:16 | comment | added | Konstantinos Kanakoglou | What do you mean "smooth" here ? | |
| Aug 2, 2022 at 15:24 | history | asked | Kim | CC BY-SA 4.0 |