The nonsingular variety is a manifold and irreduciblity

Question

For the claim that a nonsingular variety is a smooth manifold, do we need to require the nonsingular variety to be irreducible? I am thinking that each irreducible component is a smooth manifold and different components might have different dimensions, which might lead to a problem of the dimension of the manifold.

Answer 1

If your definition of a manifold does not require that all connected components have the same dimension, you need not worry about irreducibilty. In a smooth variety, irreducible components are connected components, so one may just deal with each component separately.