Reduced scheme

A scheme is reduced when every local ring ${\mathcal{O}}_{X,x}$ is a reduced ring, equivalently when its structure sheaf has no nonzero nilpotent element. The affine scheme $\mathrm{Spec}A$ is reduced exactly when $A$ is reduced.

A Noetherian scheme with no embedded associated primes is reduced if it is reduced at all its generic points. Locally, a nonzero nilpotent ideal would have an associated prime, necessarily a minimal prime. Its localization there would be nonzero, contradicting generic reducedness. This is useful with the unmixedness of a complete intersection.

An integral scheme is a nonempty scheme that is both reduced and irreducible. An affine scheme $\mathrm{Spec}A$ is integral exactly when $A$ is an integral domain.

On an integral scheme $X$, every nonempty open $U$ contains its generic point $\eta$. Taking a germ gives an injective ring homomorphism from its regular functions to the function field ${\mathcal{O}}_{X,\eta }$. Injectivity follows on each nonempty affine open subscheme from the injection of an integral domain into its field of fractions, and then from the sheaf gluing axiom. This lets regular functions on different open subsets be compared inside one field.