Noetherian scheme

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

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

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 ${\displaystyle {\mathcal {F}}}$ and a sub-coherent sheaf ${\displaystyle {\mathcal {F}}'}$, showing ${\displaystyle {\mathcal {F}}}$ has some property can be reduced to looking at ${\displaystyle {\mathcal {F}}'}$ and ${\displaystyle {\mathcal {F}}/{\mathcal {F}}'}$. Since this process can only be applied a finite number of times in a non-trivial manner, this makes many induction arguments possible.

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

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

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

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

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

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

meaning the functors

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

preserve direct limits and coproducts.

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

Another class of examples of Noetherian schemes[5] are families of schemes ${\displaystyle X\to S}$ where the base ${\displaystyle S}$ is Noetherian and ${\displaystyle X}$ is of finite type over ${\displaystyle S}$. 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 encoutered 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.

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.

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

One of the natural rings which are non-Noetherian are the Ring of adeles ${\displaystyle \mathbb {A} _{K}}$ for an algebraic number field ${\displaystyle K}$. 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]

Given an infinite Galois field extension ${\displaystyle K/L}$, such as ${\displaystyle \mathbb {Q} (\zeta _{\infty })/\mathbb {Q} }$ (by adjoining all roots of unity), the ring of integers ${\displaystyle {\mathcal {O}}_{K}}$ is a Non-noetherian ring which is dimension ${\displaystyle 1}$. 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 ${\displaystyle {\text{Sch}}/{\text{Spec}}({\mathcal {O}}_{E})}$, can be an interesting and fruitful subject.

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.

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