Regular in codimension one

A locally Noetherian scheme $X$ is regular in codimension one, or satisfies $(R_1)$, if every local ring ${\mathcal{O}}_{X,x}$ of Krull dimension at most one is a regular local ring. This includes both the generic points and the codimension-one points of its irreducible components. Together with the depth condition $(S_2)$, it is equivalent to normality by Serre criterion for normality.^[ref]

For an integral scheme, the local ring at its generic point is already its function field, so only the codimension-one points need checking; their local rings must be discrete valuation rings. For a general scheme, omitting the generic-point test changes the condition. The nonreduced double point has no codimension-one points at all, but its zero-dimensional local ring $k[\epsilon ]/({\epsilon }^2)$ is not a regular local ring. Thus it fails $(R_1)$ despite satisfying the empty list of exactly-codimension-one tests.