nLab
scheme

Contents

Idea

A scheme is a space that locally looks like a particularly simple ringed space: an affine scheme. This can be formalised either within the category of locally ringed spaces or within the category of presheaves of sets on the category of affine schemes AffAff.

The notion of scheme originated in algebraic geometry where it is, since Grothendieck’s revolution of that subject, a central notion.

However, the idea that

A scheme is a ringed space that is locally isomorphic to an affine space.

is much more general and need not be restricted to the locality in Zariski topology and to a notion of affine spaces that are formal duals of rings. If one takes another subcanonical Grothendieck topology τ\tau on AffAff then one talks about τ\tau-locally affine spaces. More generally one can take another “category of local models” LocLoc replacing AffAff and suitable topology and consider sheaves on it, as locally affine space in this generalized sense. The category LocLoc can sometimes be represented by ringed spaces of special type and the gluing can be sometimes made in a genuine (classical, not Grothendieck) topology, within the category of ringed spaces.

For instance a smooth manifold is a ringed space locally isomorphic to a “smooth affine space” n\mathbb{R}^n, with its standard smooth structure.

The standard concept of scheme in algebraic geometry is therefore usefully understood as a special case of generalized schemes that naturally appear for instance also in differential geometry, in synthetic differential geometry and many other topics.

Definition

Throughout this article, “ring” will mean “commutative ring with unit”.

As locally ringed spaces

A scheme is a locally ringed space (X,𝒪 X)(X, \mathcal{O}_X) with an open cover (as locally ringed spaces), by affine schemes: the spectra SpecA=(SpecA,𝒪 SpecA)Spec A = (|Spec A|, \mathcal{O}_{Spec A}) of unital commutative rings.

A morphism f:XYf : X \to Y of schemes is a morphism of the underlying locally ringed spaces. This means it is a morphism of ringed spaces such that for each point xXx \in X the induced map of local rings

(𝒪 Y) f(x)(𝒪 X) x (\mathcal{O}_Y)_{f(x)} \to (\mathcal{O}_X)_x

is local (in that it carries the maximal ideal to the maximal ideal). See functor of points.

As sheaves on CRing opCRing^{op}

Definition

(k-ring, k-functor,affine k-scheme)

For a ring kk the category of kk-rings, denoted by M k,M_k, is defined to be the category of commutative associative kk-algebras with unit. This is equivalently the under category kCRingk\downarrow CRing of pairs (R,f:kR)(R,f:k\to R) where RR is a commutative ring and ff is a ring homomorphism.

The category of kk-functors, denoted by coPsh(M k)co Psh (M_k), is defined to be the category of covariant functors M kSetM_k\to Set.

The forgetful functor O k:RRO_k:R\to R sending a kk-ring to its underlying set is called affine line.

For the full and faithful contravariant functor

Sp k:{M k coPsh(M k) A M k(A,)Sp_k:\begin{cases} M_k&\to& co Psh(M_k) \\ A&\to& M_k(A,-) \end{cases}

Sp kASp_k A (and every isomorphic functor) is called an affine kk-scheme. Sp kSp_k restricts to an equivalence between the categories of kk-rings and the category AffSch kAff Sch_k of affine kk-schemes. We think of this category as of M k opM_k^{op}. The functor Sp kSp_k commutes with limits and scalar extension (see below). Consequently AffSch kAff Sch_k is closed under limits and base change.

The affine line O k=M k(k[t],)O_k=M_k(k[t],-) is an affine kk-scheme.

A function on a kk-scheme XX is defined to be an object fO(X):=coPsh(M k)(X,O k)f\in O(X):=co Psh (M_k)(X,O_k). O(X)O(X) is a kk-ring by component-wise addition and -multiplication.

Remark

The category of kk-functors has limits.

The terminal object is e:R{}e:R\mapsto\{\varnothing\}. Products and pullbacks are computed component-wise.

Remark

For ϕ:kk\phi:k\to k' the ”base change” functor () kk:coPsh(M k)coPsh(M k)(-)\otimes_k k':co Psh(M_k)\to co Psh(M_{k'}) induced by ()ϕ:M kM k(-)\circ \phi:M_k\to M_{k'} given by postcompositions with ϕ\phi is called scalar extension.

Now we come to the definition of not necessarily affine k-schemes

For a kk-functor XcoPsh(M k)X\in coPsh(M_k) and EO(X)=M k(X,O k)E\subseteq O(X)=M_k(X,O_k) a set of functions on XX, Definition in k-ring?, we define

V(E)(R):={xX(R)fE,f(x)=0}V(E)(R):=\{x\in X(R) | f\in E, f(x)=0\}

and

D(E)(R):={xX(R)fE,thef(x)generate the unit ideal ofR}D(E)(R):=\{x\in X(R)|f\in E, \text{the} f(x) \text{generate the unit ideal of} R\}

For a transformation u:YXu:Y\to X of kk-functors and ZXZ\subseteq X a subfunctor we define

u 1(Z)(U):={yY(R)u(Y)Z(R)}u^{-1}(Z)(U):=\{y\in Y(R)|u(Y)\in Z(R)\}

A subfunctor YXY\subseteq X is called open subfunctor resp. closed subfunctor if for every transformation u:TXu:T\to X we have u 1(Y)u^{-1}(Y) is of the form V(E)V(E) resp. D(E)D(E).

Definition

A kk-functor XX is called a kk-scheme if the following two conditions hold:

  1. (XX is a sheaf for the Zariski Grothendieck topology? on M k opM_k^{op}) For all kk-rings and all families (f i) i(f_i)_i such that R= iRf iR=\coprod_i R f_i we have: if for all x iR[f i 1]x_i\in R[f_i^{-1}] such that the images of x ix_i and x jx_j coincide in X(R[f i 1f j 1])X(R[f_i^{-1} f_j^{-1}]) there is a unique xX(R)x\in X(R) mapping to the x ix_i.

  2. (XX has a cover of Zariski open immersions of affine kk-schemes) The exists a small family (U i) i(U_i)_i of open affine subfunctors of XX such that for all fields KM kK\in M_k we have that X(K)= iU i(K)X(K)=\coprod_i U_i(K).

Remark

The category of kk-schemes is closed under finite limits, forming open- and closed subfunctors, and scalar extension. As a subcategory of the category of Zariski sheaves, it is also closed under taking small coproducts.

Translation between the two approaches

The fundamental theorem on morphisms of schemes asserts that there is a fully faithful functor from the category SchSch of schemes to the category Aff:=Psh(CRing op)Aff:=Psh(C Ring^{op}) of presheaves on the opposite category of commutative rings given by

(X,𝒪 X)Sch((Spec(),𝒪 Spec()),(X,𝒪 X))(X,\mathcal{O}_X)\mapsto Sch((|Spec (-)|,\mathcal{O}_{Spec(-)}),(X,\mathcal{O}_X))

This identifies schemes with those presheaves on CRing op{}^{op} that

  1. are sheaves with respect to the Zariski Grothendieck topology on CRing opCRing^{op};
  2. have a cover by Zariski-open immersions of affine schemes in the category of presheaves over AffAff.

The standard reference for the functor-of-points approach to schemes is Demazure-Gabriel.

Remark

Different authors take different approaches to the underlying set-theoretic issues. The astute reader will have noticed that we consider the category of all functors CRingSetCRing \to Set – which is not a locally small category. Nor is the category of sheaves on the site CRing opCRing^{op} with its Zariski topology actually a Grothendieck topos. With due regard to such set-theoretic issues, this approach seems to be conceptually the simplest. Or, one could keep one’s options open: for some suitable small category of rings left to one’s discretion, for example the category of finitely presented rings, one can consider schemes locally modelled on that category.

Demazure-Gabriel steer a middle course involving universes: assuming two universes UU and VV with UV\mathbb{N} \in U \in V, one has a category of “small rings” (belonging to UU) and a category of sets (belonging to VV) and one considers functors MSetM \to Set from small rings (called “models”) to (not necessarily small) sets. They remark that the device of using universes is really just a convenience that could mostly be dispensed with: one could work within the standard Bernays-Gödel framework by assuming that the models are inclusive enough to hold various standard commutative algebra constructions (e.g., quotients, localizations, completions) while still remaining a small category. However, since they wish to avail themselves of direct limits in the category of models, they choose to work with universes instead.

Generalizations

In algebraic geometry this is a basic object of study, since the revolution of Grothendieck. There are generalizations like relative schemes (which are just objects in a slice category Sch/SSch/S), relative noncommutative schemes in noncommutative algebraic geometry introduced by A. Rosenberg in terms of categories and covers defined using pairs of adjoint functors, the generalized schemes of Nikolai Durov, the algebraic stacks of Deligne-Mumford and Artin, the dg-schemes of Kapranov, the derived schemes of Jacob Lurie, the higher algebraic stacks of Toën–Vezzosi, almost schemes (Ofer Gabber and Lorenzo Ramero), formal schemes (Cartier–Grothendieck), locally affine spaces in the fpqc, fppf or étale topology (Grothendieck), algebraic spaces, etc.

Underlying topological space vs. underlying locale

Jacob Lurie argues that underlying locale point of view is better than underlying topological space point of view, see schemes as locally affine structured (∞,1)-toposes.

Simplicial schemes

References

Terminology: EGA says prescheme, for what we call algebraic scheme, and says scheme for what we call separated scheme.

Standard monographs

  • Robin Hartshorne, Algebraic geometry, Springer

  • Qing Liu, Algebraic geometry and arithmetic curves, 592 pp. Oxford Univ. Press 2002

  • D. Eisenbud, J. Harris, The geometry of schemes, Springer Grad. Texts in Math.

  • David Mumford, Red book of varieties and schemes

  • Amnon Neeman, Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345

  • William Fulton, Intersection theory, Springer 1984

  • Ulrich Görtz, Torsten Wedhorn, Algebraic geometry I. Schemes with examples and exercises, Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010. viii+615 pp. Springerlink book

  • M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 (functor of points approach, mainly)

Other references

MathOverflow: arbitrary-products-of-schemes-dont-exist, model-of-a-scheme-regular-over-the-generic-point, categorical-construction-of-the-category-of-schemes, when-is-an-algebraic-space-a-scheme, is-an-algebraic-space-group-always-a-scheme, connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov

Revised on November 22, 2013 03:22:12 by Urs Schreiber (77.251.114.72)