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Definition
A locally Noetherian scheme whose local rings are regular.
A scheme is regular if it is locally Noetherian and each local ring is a regular local ring. Here locally Noetherian means that every point has an affine open neighborhood with Noetherian coordinate ring.
The local rings are the stalks of the structure sheaf of the underlying locally ringed space. Thus this definition concerns the scheme itself and does not require a chosen base field.
A scheme is a locally ringed space that has an open cover for which every restricted locally ringed space is an affine scheme.
In other words, a scheme is assembled by gluing prime spectra of commutative rings, while simultaneously gluing the algebraic functions carried by their structure sheaves.
A Noetherian local ring is regular if its maximal ideal can be generated by elements, where is its Krull dimension.
A locally ringed space is a pair consisting of a topological space and a sheaf of commutative rings such that every stalk is a local ring. The sheaf is called the structure sheaf.
The unique maximal ideal in distinguishes functions that vanish at from those locally invertible near . This extra condition makes points and local algebra interact correctly.