nLab generalized scheme

Idea

A general idea of Alexander Grothendieck was that to study a geometry more general than schemes instead of the gluing of affine schemes as ringed spaces, one glues the functors of points; hence a space is simply a sheaf of sets on some site $Loc$ of local models with a Grothendieck topology $\tau$ on it.

An algebraic scheme $X$ is a ringed space that is locally isomorphic to an affine scheme. Alternatively (see Gabriel-Demazure), it is a presheaf of sets on $Aff=CommRing^{op}$ locally representable in Zariski topology on $Aff$. The second approach Alexander Grothendieck calls functor of points approach.

To recall the equivalence between the two points of view, every scheme $X$ gives rise to a representable presheaf on the formal dual of commutative rings

$$X(-) : CRing \to Set$$

$$A \mapsto Hom_{Scheme}(Spec A, X)$$

and this is a sheaf with respect to Zariski Grothendieck topology on $Aff$. Sheaves in any other fixed subcanonical topology $\tau$ on $Aff$ are called $\tau$-locally affine spaces. The usual schemes are obtained for $\tau=Zariski$ and $Loc=Aff$. Algebraic spaces are another example. In other fields like analytic spaces, sheaves on other categories of local models $Loc$ instead of $Aff$ are considered in classical works.

In general various generalizations which do not have exactness properties of Zariski coverings or étale coverings, are usually among algebraic geometers called generalized spaces rather than (generalized) schemes; thus the terminology almost scheme is OK because though the local objects are more general the exactness properties are basically the same (similarly for derived schemes of Toen et al. noncommutative schemes of Rosenberg etc.).

There are various way to generalize the scope of the functor of points approach.

There are many generalizations of schemes, some are even called by their respective authors generalized schemes (e.g. Lurie, Durov). Deligne in Catégories Tannakiennes suggested algebraic geometry in arbitrary symmetric monoidal category. Aspects of toric geometry and the foundations of the geometry over a field of one element (Smirnov-Kapranov, Dietmar, Connes…) can be founded using structure sheaves of monoids, not rings. Another example is tropical geometry. Rings are sometimes noncommutative (e.g. D-schemes of Beilinson); the underlying topological space can be replaced by a site, locale, topos or a non-distributive lattice by localizations. Usual commutative unital rings suffice for manifolds, rigid analytic spaces, schemes, formal schemes and so on. The emphasis in Lurie is to categorify the space and to take the homotopy version of a ring, restating a formalism fitting the derived algebraic geometry, mainly of Simpson’s school.

Several different definitions by several authors exist.

Locally affine structured (∞,1)-toposes

In

• Jacob Lurie, Structured Spaces is considered the

  definition discussed at

• locally representable structured (infinity,1)-topos.

• E-∞ scheme

Generalized schemes of Durov

N. Durov replaces the commutative rings by commutative algebraic monads (aka generalized rings) in sets and defines spectra in that context, and glues them together. This way he defines what he calls generalized schemes: in a nutshell generalized schemes are schemes glued from affine spectra of generalized rings. The corresponding category of quasicoherent $\mathcal{O}$-modules is not abelian in general. See also the separate entry generalized scheme after Durov.

Other generalized schemes

O. Gabber considers replacing rings by almost rings, this results in the theory of almost schemes.

One should note that the Grothendieck school has occasionally studied ringed sites where ring is not required to be commutative and considered quasicoherent sheaves and cohomology in that context. D-schemes of Beilinson are an example where this formalism is useful.

Rosenberg considers generalized relative schemes as categories over an arbitrary base category with a relatively affine cover satisfying some exactness conditions. The scheme as a category is in fact abstracting the category of quasicoherent sheaves over some generalized scheme. Rosenberg calls the Zariski version of that formalism noncommutative scheme; some other versions of locally affine spaces can be also relativized.

Rigid analytic geometry is featuring locally affinoid spaces (affinoid spaces are spectra of Banach algebras over complete ultrametric fields which belong to a special class called affinoid algebras; Berkovich spectra are most often used) in so-called G-topology.