Reduced scheme

Remarks

Equivalently, every stalk $\mathcal{O}_{X,x}$ is a reduced ring. An affine scheme $\operatorname{Spec} A$ is reduced if and only if $A$ is reduced. Example: $\operatorname{Spec} k[x, y]/(xy)$ for a field $k$ is reduced, since $k[x,y]/(xy)$ has no nonzero nilpotents. Indeed, a power of a polynomial lies in $(xy)$ only if the polynomial itself is divisible by both $x$ and $y$. Non-example: $\operatorname{Spec} k[x]/(x^2)$ is not reduced, since the class of $x$ is nonzero but $x^2 = 0$. Every scheme $X$ has a reduced scheme $X_{\mathrm{red}}$ with the same underlying space, obtained by dividing out the nilpotents.