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let $S=\operatorname{Spec}(A)$ be an affine scheme. For which ring $A$, not field is it known that $H^1(S,\mathcal{O}_S^{*})$ is trivial?

If $X\to S$ is a finite map and $H^1(S,\mathcal{O}_S^{*})$ is trivial, is it true that also $H^1(X,\mathcal{O}_X^{*})$ is trivial?

Thanks

1 Answers 1

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Sample answers to your first question: If $S$ is Spec of a local ring, or of a UFD, then $H^1(S, \mathcal O_S^{\times})$ is trivial.

The answer to your second question is no: $X = $Spec $\mathbb C[x,y]/(y^2 - x^3 +x) \to S =$ Spec $\mathbb C[x]$ gives a counterexample of a geometric nature, and $X =$Spec $\mathbb Z[\sqrt{-5}] \to S =$ Spec $\mathbb Z$ gives a counterexample of an arithmetic nature.

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    @MattE: Dear Matt, this was indeed useful. Thank you!2014-06-09