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Let $A$ be a noetherian commutative algebra over a perfect field $k$.

The algebra $A$ is said to be differentially smooth over $k$ if

(1) $\Omega^1_{A/k}$ is a projective $A$-module, and

(2) the canonical map $$\operatorname{Sym}_A(\Omega_{A/k}^1)\rightarrow \operatorname{Gr}_*(\operatorname{P}_{A/k})$$ is bijetive.

It is well known that if $A$ is a regular local ring, then $A$ is differentially smooth over $k$.

My question: Does it still hold if we only assume that $A$ is regular, namely, not necessarily a local ring?

Thank you.

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  • $\begingroup$ What are $P_{A/k}$ and $Gr$? $\endgroup$
    – Mattia Talpo
    Commented Dec 21, 2015 at 3:10
  • $\begingroup$ $P_{A/k}$ is the filtered $A$-algebra of principal parts and $Gr_*(P_{A/k})$ is its associated graded $A$-algebra. $\endgroup$
    – Diego Sulca
    Commented Dec 21, 2015 at 3:15
  • $\begingroup$ Yes, see EGA IV 17.12.4. $\endgroup$
    – abx
    Commented Dec 21, 2015 at 8:26
  • $\begingroup$ I am not assuming that $A$ is of finite type over $k$. Therefore, $A$ is not necessarily a smooth algebra over $k$. I am interested when $k$ is a prime field. $\endgroup$
    – Diego Sulca
    Commented Dec 21, 2015 at 13:08
  • $\begingroup$ If $f:k \to A$ is a homomorphism with $k$ a perfect field and $A$ regular, then $f$ is a regular homomorphism and then (2) holds (see e.g. M. Andre, Algèbres graduées associées et algèbres symétriques plates, Comment. Math. Helv., 49 (1974), pp. 277–301), but the flat $A$-module $\Omega_{A|k}$ is projective if and only if $f$ is formally smooth for the discrete topology (by the Jacobian criterion). $\endgroup$
    – Vinteuil
    Commented Dec 21, 2015 at 22:09

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