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Let $A$ be a finitely generated $\mathbb{C}$-algebra. Let $S^{\bullet}=\bigoplus_{i\geq 0} S^i$ and $T^{\bullet}=\bigoplus_{i\geq 0} T^i$ be two graded $A$-algebras such that $S^0=T^0=A$ and $S^1, T^1$ are finitely generated $A$-module. Moreover, $S^{\bullet}$ and $T^{\bullet}$ are both generated by degree one elements as $A$-algebra.

Now given a smooth injective graded homomorphism between graded $A$-algebras $f: S^{\bullet}\hookrightarrow T^{\bullet}$. Is it always true that the induced homomorphism $g: \mathrm{Sym}(S^1)\hookrightarrow \mathrm{Sym}(T^1)$ is also smooth?

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    $\begingroup$ What do you mean "smooth" here ? $\endgroup$
    – Konstantinos Kanakoglou
    Commented Aug 2, 2022 at 17:16
  • $\begingroup$ @KonstantinosKanakoglou Here means smooth as $A$-algebra homomorphism $\endgroup$
    – Kim
    Commented Aug 3, 2022 at 3:27
  • $\begingroup$ 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. $\endgroup$
    – Johan
    Commented Aug 3, 2022 at 12:30

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