The Stacks project

28.14 The singular locus

Here is the definition.

Definition 28.14.1. Let $X$ be a locally Noetherian scheme. The regular locus $\text{Reg}(X)$ of $X$ is the set of $x \in X$ such that $\mathcal{O}_{X, x}$ is a regular local ring. The singular locus $\text{Sing}(X)$ is the complement $X \setminus \text{Reg}(X)$, i.e., the set of points $x \in X$ such that $\mathcal{O}_{X, x}$ is not a regular local ring.

The regular locus of a locally Noetherian scheme is stable under generalizations, see the discussion preceding Algebra, Definition 10.110.7. However, for general locally Noetherian schemes the regular locus need not be open. In More on Algebra, Section 15.47 the reader can find some criteria for when this is the case. We will discuss this further in Morphisms, Section 29.19.

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