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Let $R$ be a Noetherian regular $k$-algebra (where $k$ any field of char = 0) of dimension greater than 0. Is it true that $H^{0}_{dR}(R \lvert k) = k$?.

In other words I am asking suppose a regular function $P \in R$ such that its derivative is zero i.e. $d(P) = 0$ in $\Omega^{1}_{R}$, then does $P$ necessarily belongs to $k$? where, $d : R \rightarrow \Omega^{1}_{R}$ is the Kahler derivation.

I know this holds for polynomial algebra and I thought it should be consistent with the fact in manifold that a smooth function which derivative is zero must be constant.

Let $R$ be a Noetherian regular $k$-algebra (where $k$ any field of char = 0) of dimension greater than 0. Is it true that $H^{0}_{dR}(R \lvert k) = k$?.

More generally we could ask, Is it true that $H^{0}_{dR}(R \lvert k)$ is finite $k$-algebra?

In other words I am asking suppose a regular function $P \in R$ such that its derivative is zero i.e. $d(P) = 0$ in $\Omega^{1}_{R}$, then does $P$ necessarily belongs to $k$? where, $d : R \rightarrow \Omega^{1}_{R}$ is the Kahler derivation.

I know this holds for polynomial algebra and I thought it should be consistent with the fact in manifold that a smooth function which derivative is zero must be constant.

Let $R$ be a Noetherian regular $k$-algebra (where $k$ any field of char = 0). Is it true that $H^{0}_{dR}(R \lvert k) = k$?.

In other words I am asking suppose a regular function $P \in R$ such that its derivative is zero i.e. $d(P) = 0$ in $\Omega^{1}_{R}$, then does $P$ necessarily belongs to $k$? where, $d : R \rightarrow \Omega^{1}_{R}$ is the Kahler derivation.

I know this holds for polynomial algebra and I thought it should be consistent with the fact in manifold that a smooth function which derivative is zero must be constant.

Let $R$ be a Noetherian regular $k$-algebra (where $k$ any field of char = 0) of dimension greater than 0. Is it true that $H^{0}_{dR}(R \lvert k) = k$?.

In other words I am asking suppose a regular function $P \in R$ such that its derivative is zero i.e. $d(P) = 0$ in $\Omega^{1}_{R}$, then does $P$ necessarily belongs to $k$? where, $d : R \rightarrow \Omega^{1}_{R}$ is the Kahler derivation.

I know this holds for polynomial algebra and I thought it should be consistent with the fact in manifold that a smooth function which derivative is zero must be constant.

Let $R$ be a Noetherian regular local $k$-algebra (where $k$ any field of char = 0). Is it true that $H^{0}_{dR}(R \lvert k) = k$?.

In other words I am asking suppose a regular function $P \in R$ such that its derivative is zero i.e. $d(P) = 0$ in $\Omega^{1}_{R}$, then does $P$ necessarily belongs to $k$? where, $d : R \rightarrow \Omega^{1}_{R}$ is the Kahler derivation.

I know this holds for polynomial algebra and I thought it should be consistent with the fact in manifold that a smooth function which derivative is zero must be constant.

Let $R$ be a Noetherian regular   $k$-algebra (where $k$ any field of char = 0). Is it true that $H^{0}_{dR}(R \lvert k) = k$?.

In other words I am asking suppose a regular function $P \in R$ such that its derivative is zero i.e. $d(P) = 0$ in $\Omega^{1}_{R}$, then does $P$ necessarily belongs to $k$? where, $d : R \rightarrow \Omega^{1}_{R}$ is the Kahler derivation.

I know this holds for polynomial algebra and I thought it should be consistent with the fact in manifold that a smooth function which derivative is zero must be constant.