nLab regular scheme

Idea

A regular scheme $(X,\mathcal{O}_X)$ is an algebraic scheme such that for every $x\in X$ there exists an affine open neighbourhood $U\subset X$ of $x$ such that the ring $\mathcal{O}_X(U)$ is regular (many authors also require Noetherian). Equivalently, stalk $\mathcal{O}_{X,x}$ of the structure sheaf $\mathcal{O}_X$ at any point $x\in X$ (as a locally ringed space) is a regular local ring.

References

• stacks project: 28.9 Regular schemes
• eom: regular scheme
• wikipedia: regular scheme

On pro-algebraic resolutions of regular schemes over algebraically complete fields:

• Adrian Clough, Pro-algebraic resolutions of regular schemes, MSc thesis, ETC Zürich (2014) [pdf]

category: algebraic geometry