Noetherian scheme

In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets $\operatorname {Spec} A_{i}$ , where each $A_{i}$ is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and compact. As with Noetherian rings, the concept is named after Emmy Noether.

It can be shown that, in a locally Noetherian scheme, if $\operatorname {Spec} A$ is an open affine subset, then A is a Noetherian ring; in particular, $\operatorname {Spec} A$ is a Noetherian scheme if and only if A is a Noetherian ring. For a locally Noetherian scheme X, the local rings ${\mathcal {O}}_{X,x}$ are also Noetherian rings.

A Noetherian scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring.

The definitions extend to formal schemes.

Properties and Noetherian hypotheses

Having a (locally) Noetherian hypothesis for a statement about schemes generally makes a lot of problems more accessible because they sufficiently rigidify many of its properties.

Dévissage

One of the most important structure theorems about Noetherian rings and Noetherian schemes is the dévissage theorem. This makes it possible to decompose arguments about coherent sheaves into inductive arguments. Given a short exact sequence of coherent sheaves

$0\to {\mathcal {E}}'\to {\mathcal {E}}\to {\mathcal {E}}''\to 0,$

proving one of the sheaves has some property is equivalent to proving the other two have the property. In particular, given a fixed coherent sheaf ${\mathcal {F}}$ and a sub-coherent sheaf ${\mathcal {F}}'$ , showing ${\mathcal {F}}$ has some property can be reduced to looking at ${\mathcal {F}}'$ and ${\mathcal {F}}/{\mathcal {F}}'$ . Since this process can only be non-trivially applied only a finite number of times, this makes many induction arguments possible.

Number of irreducible components

Every Noetherian scheme can only have finitely many components.^[1]

Morphisms from Noetherian schemes are quasi-compact

Every morphism from a Noetherian scheme $X\to S$ is quasi-compact.^[2]

Homological properties

There are many nice homological properties of Noetherian schemes.^[3]

Čech and sheaf cohomology

Čech cohomology and sheaf cohomology agree on an affine open cover. This makes it possible to compute the sheaf cohomology of $\mathbb {P} _{S}^{n}$ using Čech cohomology for the standard open cover.

Compatibility of colimits with cohomology

Given a direct system $\{{\mathcal {F}}_{\alpha },\phi _{\alpha \beta }\}_{\alpha \in \Lambda }$ of sheaves of abelian groups on a Noetherian scheme, there is a canonical isomorphism

$\varinjlim H^{i}(X,{\mathcal {F}}_{\alpha })\to H^{i}(X,\varinjlim {\mathcal {F}}_{\alpha })$

meaning the functors

$H^{i}(X,-):{\text{Ab}}(X)\to {\text{Ab}}$

preserve direct limits and coproducts.

Derived direct image

Given a locally finite type morphism $f:X\to S$ to a Noetherian scheme $S$ and a complex of sheaves ${\mathcal {E}}^{\bullet }\in D_{Coh}^{b}(X)$ with bounded coherent cohomology such that the sheaves $H^{i}({\mathcal {E}}^{\bullet })$ have proper support over $S$ , then the derived pushforward $\mathbf {R} f_{*}({\mathcal {E}}^{\bullet })$ has bounded coherent cohomology over $S$ , meaning it is an object in $D_{Coh}^{b}(S)$ .^[4]

Examples

Most schemes of interest are Noetherian schemes.

Locally of finite type over a Noetherian base

Another class of examples of Noetherian schemes^[5] are families of schemes $X\to S$ where the base $S$ is Noetherian and $X$ is of finite type over $S$ . This includes many examples, such as the connected components of a Hilbert scheme, i.e. with a fixed Hilbert polynomial. This is important because it implies many moduli spaces encountered in the wild are Noetherian, such as the Moduli of algebraic curves and Moduli of stable vector bundles. Also, this property can be used to show many schemes considered in algebraic geometry are in fact Noetherian.

Quasi-projective varieties

In particular, quasi-projective varieties are Noetherian schemes. This class includes algebraic curves, elliptic curves, abelian varieties, calabi-yau schemes, shimura varieties, K3 surfaces, and cubic surfaces. Basically all of the objects from classical algebraic geometry fit into this class of examples.

Infinitesimal deformations of Noetherian schemes

In particular, infinitesimal deformations of Noetherian schemes are again Noetherian. For example, given a curve $C/{\text{Spec}}(\mathbb {F} _{q})$ , any deformation ${\mathcal {C}}/{\text{Spec}}(\mathbb {F} _{q}[\varepsilon ]/(\varepsilon ^{n}))$ is also a Noetherian scheme. A tower of such deformations can be used to construct formal Noetherian schemes.

Non-examples

Schemes over Adelic bases

One of the natural rings which are non-Noetherian are the Ring of adeles $\mathbb {A} _{K}$ for an algebraic number field $K$ . In order to deal with such rings, a topology is considered, giving topological rings. There is a notion of algebraic geometry over such rings developed by Weil and Alexander Grothendieck.^[6]

Rings of integers over infinite extensions

Given an infinite Galois field extension $K/L$ , such as $\mathbb {Q} (\zeta _{\infty })/\mathbb {Q}$ (by adjoining all roots of unity), the ring of integers ${\mathcal {O}}_{K}$ is a Non-noetherian ring which is dimension $1$ . This breaks the intuition that finite dimensional schemes are necessarily Noetherian. Also, this example provides motivation for why studying schemes over a non-Noetherian base; that is, schemes ${\text{Sch}}/{\text{Spec}}({\mathcal {O}}_{E})$ , can be an interesting and fruitful subject.

One special case^[7]^{pg 93} of such an extension is taking the maximal unramified extension $K^{ur}/K$ and considering the ring of integers ${\mathcal {O}}_{K^{ur}}$ . The induced morphism

${\text{Spec}}({\mathcal {O}}_{K^{ur}})\to {\text{Spec}}({\mathcal {O}}_{K})$

forms the universal covering of ${\text{Spec}}({\mathcal {O}}_{K})$ .

Polynomial ring with infinitely many generators

Another example of a non-Noetherian finite-dimensional scheme (in fact zero-dimensional) is given by the following quotient of a polynomial ring with infinitely many generators.

${\frac {\mathbb {Q} [x_{1},x_{2},x_{3},\ldots ]}{(x_{1},x_{2}^{2},x_{3}^{3},\ldots )}}$

References

^ "Lemma 28.5.7 (0BA8)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.
^ "Lemma 28.5.8 (01P0)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.
^ "Cohomology of Sheaves" (PDF).
^ "Lemma 36.10.3 (08E2)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.
^ "Lemma 29.15.6 (01T6)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.
^ Conrad, Brian. "Weil and Grothendieck Approaches to Adelic Points" (PDF). Archived (PDF) from the original on 21 July 2018.
^ Neukirch, Jürgen (1999). "1.13". Algebraic Number Theory. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-03983-0. OCLC 851391469.

Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: Springer-Verlag. ISBN 978-0-387-90244-9. MR 0463157. Zbl 0367.14001.
Harder, Günter. "Cohomology of Arithmetic Groups" (PDF). Archived from the original (PDF) on 2020-07-24.
Danilov, V.I. (2001) [1994], "Noetherian scheme", Encyclopedia of Mathematics, EMS Press

[1] "Lemma 28.5.7 (0BA8)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.

[2] "Lemma 28.5.8 (01P0)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.

[3] "Cohomology of Sheaves" (PDF).

[4] "Lemma 36.10.3 (08E2)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.

[5] "Lemma 29.15.6 (01T6)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-07-24.

[6] Conrad, Brian. "Weil and Grothendieck Approaches to Adelic Points" (PDF). Archived (PDF) from the original on 21 July 2018.

[7] Neukirch, Jürgen (1999). "1.13". Algebraic Number Theory. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-03983-0. OCLC 851391469.

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