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Let $A$ be a ring and $E$ a module. If $\mathrm{spec} A$ is connected, then so is $\mathrm{spec} S^\bullet E$. If this is not true in general, then what are some minimal conditions that make it true?

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If I'm not mistaken the geometric fibers are nonempty affine spaces (hence connected, even if E is the zero module), and the augmentation gives you a zero section everywhere. I think that means the answer is yes.

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