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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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How does the fundamental group of $\mathbb G_{m,S}$ depend on the base scheme

Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$. How to compute $\pi_1^{et}(X)$? Note. I am only interested in the part not coming trivially from the finite etale cov …
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