Noetherian scheme
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| Noetherian scheme | |
|---|---|
| Name | Noetherian scheme |
| Field | Algebraic geometry |
| Introduced by | Emmy Noether |
| Introduced date | 1920s |
Noetherian scheme.
A Noetherian scheme is a scheme satisfying a finiteness condition modeled on the work of Emmy Noether and the development of Hilbert's basis theorem in the context of Alexander Grothendieck's scheme theory. It plays a central role in modern algebraic geometry as used in foundational texts by Jean-Pierre Serre, Robin Hartshorne, and Alexander Grothendieck and in applications to arithmetic geometry by Andrew Wiles and Pierre Deligne. Noetherian schemes provide a convenient setting for many finiteness and compactness arguments employed in the proof of the Riemann–Roch theorem and the study of moduli spaces such as the Moduli space of curves.
Definition and basic properties
A scheme X is called Noetherian when it admits a finite affine open cover by spectra of Noetherian rings, a notion that traces back to Emmy Noether and was reformulated in the language of schemes by Alexander Grothendieck in the context of the Séminaire de Géométrie Algébrique programs. Equivalently, X is Noetherian if and only if it is both quasi-compact and locally Noetherian, a property used throughout the work of Jean-Pierre Serre and Oscar Zariski. Key basic properties include the descending chain condition on closed subschemes, coherence of the structure sheaf in many cases studied by Robin Hartshorne, and compatibility with classical constructions in Hilbert scheme theory and Picard group computations.
Examples and non-examples
Standard examples include schemes of finite type over fields like André Weil's varieties, schemes of finite type over Spec Z used in arithmetic geometry by Alexander Grothendieck and Jean-Pierre Serre, and affine schemes Spec A for Noetherian rings A such as principal ideal domains studied by David Hilbert and Emmy Noether. Projective schemes arising from graded Noetherian rings appear in the work of Federigo Enriques and Guido Castelnuovo on algebraic surfaces. Non-examples arise from schemes constructed from non-Noetherian rings such as the ring of polynomials in infinitely many variables considered by Hilbert and pathological examples used by Nagata in counterexamples to finiteness, and certain formal schemes in the work of John Tate that fail global Noetherianity.
Noetherian conditions (locally, globally, and universally)
Local Noetherianity is defined by every point having an affine neighborhood Spec A with A Noetherian, a condition appearing in lectures by Alexander Grothendieck and in expositions by Robin Hartshorne. Global Noetherianity requires X to be quasi-compact and locally Noetherian, a hypothesis frequently assumed in the proofs of results by Jean-Pierre Serre and in the construction of Hilbert schemes by David Mumford. Universal Noetherianity, used in the study of base change along morphisms considered by Pierre Deligne and Nicholas Katz, requires that every scheme obtained by base change remains locally Noetherian, a strengthening relevant to work on étale cohomology and l-adic representations studied by Alexander Grothendieck and Pierre Deligne.
Consequences and key theorems
On Noetherian schemes one has finiteness of irreducible components, primary decomposition of coherent ideal sheaves akin to Noetherian ring theory developed by Emmy Noether, and the validity of the Krull intersection theorem in affine charts studied by Oscar Zariski. Important theorems relying on Noetherian hypotheses include the coherence results of Jean-Pierre Serre (Serre's finiteness theorem), the existence of dualizing complexes in work of Alexander Grothendieck and Robin Hartshorne, and proper mapping theorems employed by Gerd Faltings and Armand Borel in arithmetic contexts. Results on resolution of singularities by Heisuke Hironaka and improvements by Michael Artin often assume Noetherian ambient schemes.
Morphisms, operations, and stability properties
Morphisms between Noetherian schemes behave well under composition and base change in treatments by Alexander Grothendieck and Nicholas Katz, with finite type and proper morphisms preserving Noetherianity of sources or targets under standard hypotheses used by David Mumford and Gérard Laumon. Operations such as taking closed subschemes, finite unions, images under finite morphisms, and formation of projective bundles preserve Noetherian properties as detailed in expositions by Robin Hartshorne and Jean-Pierre Serre. Conversely, infinite fibered products or certain inductive limits described by Masayoshi Nagata can fail to be Noetherian, giving rise to counterexamples in the literature.
Cohomology and finiteness results
Cohomological finiteness theorems for coherent sheaves on Noetherian schemes are central to algebraic geometry: Serre's theorems on cohomology and cohomological dimension discussed by Jean-Pierre Serre and the coherence of higher direct images under proper morphisms established in Alexander Grothendieck's work form the backbone of many results used by Pierre Deligne, Gerd Faltings, and David Mumford. Theorems on vanishing, base change, and boundedness of cohomology groups for coherent sheaves on projective Noetherian schemes underlie the proofs of the Riemann–Roch theorem and the construction of moduli spaces handled in the work of Grothendieck and Mumford. Further refinements concerning étale and l-adic cohomology on Noetherian bases appear in the developments by Pierre Deligne and Alexander Grothendieck relating to the proof of the Weil conjectures.