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?



Is the total space of a module connected?
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.


