Cech and sheaf cohomology

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.

Compatibility of colimits with cohomology

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.

Derived direct image

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]

Schemes over Adelic bases

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]

Rings of integers over infinite extensions

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.

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

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

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

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.

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