nLab
scheme

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 $\mathrm{Aff}$.

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 $\tau$ on $\mathrm{Aff}$ then one talks about $\tau$-locally affine spaces. More generally one can take another “category of local models” $\mathrm{Loc}$ replacing $\mathrm{Aff}$ and suitable topology and consider sheaves on it, as locally affine space in this generalized sense. The category $\mathrm{Loc}$ 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” ${ℝ}^n$, 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

As locally ringed spaces

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

A morphism $f:X\to Y$ of schemes is a morphism of the underlying ringed spaces, such that for each point $x\in X$ the induced map of local rings

$$({𝒪}_Y{)}_{f(x)}\to ({𝒪}_X{)}_x$$(\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.

Zoran: This is superfluous: the definition of morphisms in the category of LOCALLY ringed spaces already asks for this condition, one does not need to introduce that as late as when introducing schemes.

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.

Functor of points approach: as sheaves on ${\mathrm{CRing}}^{\mathrm{op}}$

The fundamental theorem on morphisms of schemes asserts that there is a fully faithful functor from the category of schemes to the category of presheaves on $\mathrm{Aff}={\mathrm{CRing}}^{\mathrm{op}}$ which sends $(X,{𝒪}_X)$ to the functor

$$h_X{\mid }_{\mathrm{Aff}}:A\mapsto {\mathrm{Hom}}_{\mathrm{Schemes}}(\mathrm{Spec}A,(X,{𝒪}_X)).$$h_X|_{Aff} : A \mapsto Hom_{Schemes}(Spec A , (X, \mathcal{O}_X)) \,.

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

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

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

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 $\mathrm{Sch}/S$), 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.

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)
• EGA, FGA explained

Other references

• Ravi Vakil’s Berkeley course notes

• Paul Goerss, Topological Algebraic Geometry - A Workshop – at the beginning one fins a quick introduction in the light of its higher categorical generalizations

• Wikipedia: scheme (mathematics).

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