Let $X = \text{Spec}(k[X_{1}, \ldots, X_{n}] / I)$ be an affine scheme. I have the two following questions:
- Suppose $Y \subset X$ is a locally closed subscheme of $X$ such that $Y$ is reduced. If we have the equality of Zariski tangent spaces $T_{y}Y = T_{y}X$ for all $y \in Y$, then $Y$ is open in $X$.
- Analogous to the first question, but replacing the condition $T_{y}Y = T_{y}X$ with the condition $T_{y}Y = T_{y}X_{red}$ for all $y \in Y$, where $X_{red}$ is the reduced subscheme of $X$.
I think that the proof of the first one is straightforward using the typical argument of dimension of $X$ as one do in the context of differentiable manifolds. Nevertheless I am not sure about how to proceed in the second case, is just because $X_{red}$ and $X$ have the same underlying topological space?