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noetherian scheme

Contents

Definition

Definition

An algebraic scheme is locally noetherian if it can be covered by a family U αU_\alpha of open subsets of the form U α=SpecR αU_\alpha = Spec R_\alpha where R αR_\alpha are noetherian rings. A scheme is noetherian if it is locally noetherian and quasicompact.

Remark

If RR is a commutative noetherian ring, then every localization of the form R[f 1]R[f^{-1}] where fRf\in R is also noetherian. If a scheme XX is locally noetherian this implies that there is a base of topology of XX consisting of spectra of noetherian rings. In particular every affine subscheme in XX has a basis of topology of spectra of noetherian rings.

Properties

Proposition

An affine scheme is a spectrum of a noetherian ring precisely if it is a locally noetherian scheme.

Corollary

Every affine subscheme of a locally noetherian scheme is the spectrum of a noetherian ring. A scheme is noetherian if and only if it has a finite cover by spectra of noetherian rings.

References

Revised on January 11, 2014 11:14:11 by Urs Schreiber (89.204.154.8)