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 $\mathrm{Loc}$ 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 $\mathrm{Aff}={\mathrm{CommRing}}^{\mathrm{op}}$ locally representable in Zariski topology on $\mathrm{Aff}$. 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

$$X(-):\mathrm{CRing}\to \mathrm{Set}$$X(-) : CRing \to Set

$$A\mapsto {\mathrm{Hom}}_{\mathrm{CRing}}(\mathrm{Spec}A,X)$$A \mapsto Hom_{CRing}(Spec A, X)

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

In general various generalizations which do not have exactness properties of Zariski or etale 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

This is the definition in the context of

• Jacob Lurie, Structured Spaces.

Idea

The notion of structured (∞,1)-topos is in particular a categorification of the notion of ringed space. In fact it is more general also in that it allows a choice of local model spaces: for $𝒢$ an (∞,1)-category of local models that is a geometry (for structured (∞,1)-toposes), a $𝒢$-structured (∞,1)-topos is a generalization of a space with a structure sheaf of functions that may take values in the objects of $𝒢$.

Therefore the ordinary statement “a scheme is a Zariski-locally affine space” may therefore be generalized

A $𝒢$-scheme is a locally affine structured (∞,1)-topos.

Definition in the variant of Lurie

Let $𝒢$ be a geometry (for structured (∞,1)-toposes). Write ${𝒢}_0$ for the underlying discrete geometry. The identity functor

$$p:𝒢\to {𝒢}_0$$p : \mathcal{G} \to \mathcal{G}_0

is then a morphism of geometries.

Recall the notation $\mathrm{LTop}(𝒢)$ for the (∞,1)-category of $𝒢$-structured (∞,1)-toposes and geometric morphisms between them.

affine $𝒢$-schemes

Theorem ( StSp 2.1.1 )

There is a pair of adjoint (∞,1)-functors

$$p^{*}:\mathrm{LTop}(𝒢)\overleftarrow{\to }\mathrm{LTop}({𝒢}_0):{\mathrm{Spec}}_{{𝒢}_0}^{𝒢}$$p^* : LTop(\mathcal{G}) \stackrel{\leftarrow}{\to} LTop(\mathcal{G}_0) : \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}}

with ${\mathrm{Spec}}_{{𝒢}_0}^{𝒢}$ left adjoint to the canonical functor $p^{*}$ given by precomposition with $p$.

Remark ( StSp p. 38 )

There is a canonical morphism

$$\mathrm{can}:\mathrm{Pro}(𝒢{)}^{\mathrm{op}}\to \mathrm{LTop}({𝒢}_0)$$can : Pro(\mathcal{G})^{op} \to LTop(\mathcal{G}_0)

Definition ( affine $𝒢$-scheme, StSp 2.3.9)

Write ${\mathrm{Spec}}^{𝒢}$ for the (∞,1)-functor

$${\mathrm{Spec}}^{𝒢}:\mathrm{Pro}(𝒢{)}^{\mathrm{op}}\stackrel{\mathrm{can}}{\to }\mathrm{LTop}({𝒢}_0)\stackrel{{\mathrm{Spec}}_{{𝒢}_0}^{𝒢}}{\to }\mathrm{LTop}(𝒢).$$\mathbf{Spec}^{\mathcal{G}} : Pro(\mathcal{G})^{op} \stackrel{can}{\to} LTop(\mathcal{G}_0) \stackrel{ \mathbf{Spec}_{\mathcal{G}_0}^{\mathcal{G}} }{\to} LTop(\mathcal{G}) \,.

A $𝒢$-structured (∞,1)-topos in the image of this functor is an affine $𝒢$-scheme.

$𝒢$-schemes

Definition (geometric scheme, StSp 2.3.9)

Let $𝒢$ be a geometry (for structured (∞,1)-toposes).

A $𝒢$-structured (∞,1)-topos $(𝒳,{𝒪}_{𝒳})$ is a $𝒢$-scheme if

• there exists a collection $\{U_i\in 𝒳\}$

such that

• the $\{U_i\}$ cover $𝒳$ in that the canonical morphism ${\coprod }_i{U}_i\to *$ (with $*$ the terminal object of $𝒳$) is an effective epimorphism;

• for every $U_i$ there exists an equivalence

  $$(𝒳/U_i,{𝒪}_{𝒳}{\mid }_{U_i})\simeq {\mathrm{Spec}}^{𝒢}{A}_i$$(\mathcal{X}/{U_i}, \mathcal{O}_{\mathcal{X}}|_{U_i}) \simeq \mathbf{Spec}^{\mathcal{G}} A_i

  of structured $(\mathrm{\infty }\mathrm{,1})$-toposes for some $A_i\in \mathrm{Proj}(𝒢)$ (in the (∞,1)-category of pro-objects of $𝒢$).

Definition (pregeometric scheme, StSp, 3.4.6)

For $𝒯$ a pregeometry, a $𝒯$-structured (infinity,1)-topos $(𝒳,{𝒪}_{𝒳})$ is a $𝒯$-scheme if it is a $𝒢$-scheme for the geometric envelope $𝒢$ of $𝒯$.

This means that for $f:𝒯\to 𝒢$ the geometric envelope and for $𝒪{\prime }_{𝒳}$ the $𝒢$-structure on $𝒳$ such that ${𝒪}_{𝒳}\simeq 𝒪{\prime }_{𝒳}\circ f$, we have that $(𝒳,𝒪{\prime }_{𝒳})$ is a $𝒢$-scheme.

smooth $𝒢$-schemes

Let ${\rm T}$ be a pregeometry (for structured (∞,1)-toposes) and let ${\rm T}\hookrightarrow 𝒢$ be an inclusion into an enveloping geometry (for structured (∞,1)-toposes).

We think of the objects of ${\rm T}$ as the smooth test spaces – for instance the cartesian products of some affine line $R$ with itsef – and of the objects of $𝒢$ as affine test spaces that may have singular points where they are not smooth.

The idea is that a smooth $𝒢$-scheme is a $𝒢$-structured space that is locally not only equivalent to objects in $𝒢$, but even to the very nice – “smooth” – objects in $\mathrm{𝒯𝒶𝓊}$.

Definition ( smooth $𝒢$-scheme, StSp 3.5.6)

With an envelope ${\rm T}\hookrightarrow 𝒢$ fixed, a $𝒢$-scheme is called smooth if there the affine schemes ${\mathrm{Spec}}^{𝒢}{A}_i$ appearing in its definition may be chosen with $A_i$ in the image of the includion $\tau \hookrightarrow 𝒢$.

Examples

ordinary schemes

See the discussion at derived scheme for how ordinary schemes are special cases of generalized schemes.

ordinary Deligne-Mumford stacks

See the discussion at derived Deligne-Mumford stack for how ordinary Deligne-Mumford stacks are special cases of derived Deligne-Mumford stacks.

derived schemes

Definition (derived scheme, Structured Spaces, 4.2.8)

Let $k$ be a commutative ring. Recall the pregoemtry ${𝒯}_{\mathrm{Zar}}(k)$.

A derived scheme over $k$ is a ${𝒯}_{\mathrm{Zar}}(k)$-scheme.

derived Deligne-Mumford stacks

Definition (derived Deligne-Mumford stack, Structured Spaces, 4.3.19)

Let $k$ be a commutative ring. Recall the pregeometry ${𝒯}_{\mathrm{et}}(k)$

A derived Deligne-Mumford stack over $k$ is a ${𝒯}_{\mathrm{et}}(k)$-scheme.

Zoran: there is a serious collision here with the terminology in algebraic geometry, including in HAG and DAG of Toen et al. I mean if one just changes from Zariski to etale topology, one passes roughly from algebraic schemes to algebraic spaces. One needs to go further from algebraic spaces to algebraic stacks. I do not see these distinctions here. The terminology in HAG and DAG is perfectly in accord and refines classical terminology.

derived schemes with $E_{\mathrm{\infty }}$-ring valued structure sheaves

The above derived schemes have structure sheaves with values in simplicial commutative rings. There is also a notion of derived scheme whose structure sheaf takes values in E-infinity rings. The theory of these is to be described in full detail in

• Jacob Lurie, Spectral Schemes.

An indication of some details is in

• Paul Goerss, Topological Algebraic Geometry - A Workshop

derived smooth manifolds

…A ${𝒯}_{\mathrm{diff}}$-scheme…

See

• derived smooth manifold.

References

The theory of $𝒢$-schemes is due to Jacob Lurie.

Generalized schemes are definition 2.3.9 of

• Jacob Lurie, Structured Spaces

The definition of affine $𝒢$-schemes (absolute spectra) is in section 2.2.

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 $𝒪$-modules is not abelian in general. See also the separate entry generalized scheme after Durov.

Brave new schemes

Brave new algebraic geometry is a case of homotopic algebraic geometry where the local models are ringed spectra in the sense of stable homotopy theory. So one can talk about brave new schemes and brave new stacks…

Other generalized schemes

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

One should note that 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; Berkovits spectra are most often used) in so-called G-topology.