Schemes

Introduction

A scheme is a locally ringed space that is locally the spectrum of a commutative ring. The definition removes three hypotheses from the classical theory of Algebraic Geometry: the base field need no longer be algebraically closed, the rings need no longer be reduced, and the object need no longer be embedded in an affine or projective space as a closed subset. What is gained is nothing less than the arithmetic applications: $\operatorname{Spec}\mathbb{Z}$ is an object of the theory whose closed points are the primes and whose generic point carries the field $\mathbb{Q}$, so that number theory and geometry become two instances of one formalism, and nilpotent elements are allowed so that the infinitesimal neighbourhood of a subvariety — the subscheme defined by the square of an ideal — is an object of the same category as the subvariety itself, which is what makes deformation theory possible. The price is that the underlying topological space of a scheme is a strange object: it is almost never Hausdorff, its points are not closed in general, and the correct notion of "compactness" is not the topological one but a property of the morphisms, separatedness and properness.

This article develops the general theory. The spectrum construction, the Zariski topology and the structure sheaf have been used in Sheaves in Algebraic Geometry, where they were introduced in line as far as the cohomology of coherent sheaves required; here they are constructed and organised systematically, with the functoriality that makes the spectrum a contravariant equivalence between rings and affine schemes. The category of schemes is then defined by glueing affine schemes along open subschemes, and the article works through the examples that fix the intuition: $\operatorname{Spec}$ of a field, of $\mathbb{Z}$, of a ring of dual numbers, of a polynomial ring, projective space over an arbitrary ring, and the nonseparated example of the affine line with a doubled origin. Fibre products are constructed and their universal property proved, because they are the categorical operation that replaces the intersections, the fibres of a morphism, the base change of a family and the formation of a group scheme from the group law. The properties of morphisms — finite type, finite, affine, closed and open immersions, separated, proper, flat, smooth — are then defined and their stability properties recorded, with the valuative criteria for separatedness and properness stated as the principal structural theorems. A final section returns to the classical theory and identifies the varieties of Algebraic Geometry with the reduced schemes of finite type over an algebraically closed field, which is the precise sense in which the present article generalises the previous one.

Schematically, the article is the categorical foundation on which the rest of the category rests:, develops the sheaf theory of a general scheme,the parametrisation of families and the quotients of group actions, and the extension of the same ideas to objects with automorphisms. The commutative algebra used throughout — localisation, the spectrum, integral extensions, the Krull dimension, the local rings, the Noetherian conditions and the valuation theory — is Part I's, in Localization and the Fraction Field, Noetherian and Artinian Rings, Integral Extensions and Krull Dimension, Valuation Theory and Henselian Rings and Primary Decomposition.

The Spectrum and the Structure Sheaf

Definition. Let $A$ be a commutative ring with identity. The spectrum $\operatorname{Spec} A$ is the set of prime ideals of $A$. For a subset $S\subseteq A$ one writes $V(S) = \{\mathrm{P} : S\subseteq\mathrm{P}\}$, the sets $V(\mathrm{A})$ for ideals $\mathrm{A}$ are the closed sets of the Zariski topology, and for $f\in A$ the distinguished open set is $D(f) = \operatorname{Spec} A\setminus V(f)$; the sets $D(f)$ form a basis of the topology closed under finite intersections, since $D(f)\cap D(g) = D(fg)$, and the distinguished opens $D(f_i)$ for a family $f_i$ generate the unit ideal cover $\operatorname{Spec} A$.

Theorem (the structure sheaf). There is a unique sheaf of commutative rings $\mathcal{O}_X$ on $X = \operatorname{Spec} A$ with $\mathcal{O}_X(D(f)) = A_f$ for every $f\in A$, the restriction $A_f\to A_{fg}$ being the localisation, and the sheaf is characterised by these values on the basis. Its stalks are the local rings

$$ \mathcal{O}_{X,\mathrm{P}} = \varinjlim_{f\notin\mathrm{P}}A_f = A_{\mathrm{P}}, \qquad \mathrm{P}\in X, $$

with maximal ideal $\mathrm{P}A_{\mathrm{P}}$ and residue field $\kappa(\mathrm{P}) = A_{\mathrm{P}}/\mathrm{P}A_{\mathrm{P}}$, the fraction field of the domain $A/\mathrm{P}$. Hence $(X,\mathcal{O}_X)$ is a locally ringed space.

Proof. This is the theorem of Sheaves in Algebraic Geometry, where it is proved by the sheaf-on-a-basis criterion of Presheaves and Sheaves: for a cover of $D(f)$ by distinguished opens $D(f_i)$ the sequence $0\to A_f\to\prod_iA_{f_{f_i}}\to\prod_{i,j}A_{f_{f_if_j}}$ is exact, which is the localisation sequence of Part I's Localization and the Fraction Field, and exactness is checked after localising at each prime. The identification of the stalk is the filtered colimit of the $A_f$ over the $f$ outside $\mathrm{P}$.

Example (the points of a spectrum). The points of $\operatorname{Spec} A$ are of three kinds with respect to a prime $\mathrm{P}$: the closed point, if $\mathrm{P}$ is maximal, with residue field $A/\mathrm{P}$ a field; the generic point of a one-dimensional subvariety if $\mathrm{P}$ has height one; and the generic point of the whole space if $\mathrm{P} = (0)$ and $A$ is a domain. For $A = k[x]$ with $k$ algebraically closed the points are the maximal ideals $(x-a)$, $a\in k$, and the zero ideal, whose closure is all of $\operatorname{Spec} k[x]$ and whose residue field is $k(x)$; the closed points are dense, so every nonempty open set contains a closed point, and yet the generic point is not closed.

Theorem (the contravariant equivalence for affine schemes). The functor $A\mapsto\operatorname{Spec} A$ extends to an equivalence of categories

$$ \{\text{commutative rings with } 1\}^{\mathrm{op}}\ \xrightarrow{\ \sim\ }\ \{\text{affine schemes}\}, $$

under which a ring homomorphism $\varphi : A\to B$ corresponds to the morphism $\operatorname{Spec}\varphi = {}^{a}\varphi : \operatorname{Spec} B\to\operatorname{Spec} A$ with ${}^{a}\varphi(\mathrm{Q}) = \varphi^{-1}(\mathrm{Q})$ and the map on structure sheaves induced by $A_f\to B_{\varphi(f)}$. For every ring $B$ one has

$$ \operatorname{Hom}_{\mathbf{Sch}}(\operatorname{Spec} B,\operatorname{Spec} A)\cong\operatorname{Hom}_{\mathbf{Ring}}(A,B), $$

so $\operatorname{Spec} A$ is the functor of points $B\mapsto\operatorname{Hom}_{\mathbf{Ring}}(A,B)$, and the identity corresponds to the identity and the zero ring $\operatorname{Spec}0 = \emptyset$ to the empty scheme.

Proof. The inverse image of a prime is prime, and the map on localisations is compatible with the two localisations, so ${}^{a}\varphi$ is a morphism of locally ringed spaces; the two constructions are inverse because $A$ is recovered as $\Gamma(\operatorname{Spec} A,\mathcal{O})$, and the identification of hom-sets is the Yoneda expression of the same fact.

Remark (why the functor of points is the useful form). The equivalence is the precise sense in which a commutative ring is a geometric object and a ring homomorphism is a morphism in the opposite direction. In practice one specifies a scheme by specifying its functor of points — the sets $T\mapsto X(T)$ of morphisms from test objects $T$ — because the functorial description is what is preserved under base change and what makes fine moduli functors expressible; the geometric points of $X$ with values in a field $k$ are the $k$-points, and for $X = \operatorname{Spec} A$ they are the ring homomorphisms $A\to k$, which for $A = k[x_1,\ldots,x_n]/I$ are the $k$-rational points of the classical variety.

Schemes and Their Morphisms

Definition. A scheme is a locally ringed space $(X,\mathcal{O}_X)$ in which every point has an open neighbourhood $U$ such that $(U,\mathcal{O}_X|_U)\cong\operatorname{Spec} A$ for some ring $A$; the isomorphisms are the affine charts and $(X,\mathcal{O}_X)$ is affine if it is isomorphic to a spectrum. A morphism of schemes is a morphism of locally ringed spaces. A subscheme of $X$ is a scheme $Z$ with a morphism $Z\to X$ inducing a homeomorphism onto a locally closed subset; it is open if the subset is open and closed if the subset is closed with $\mathcal{O}_Z = i^{-1}(\mathcal{O}_X/\mathcal{I})$ for a sheaf of ideals $\mathcal{I}\subseteq\mathcal{O}_X$.

Theorem (the glueing lemma). Let $\{X_i\}$ be a family of schemes and for each pair an open subscheme $X_{ij}\subseteq X_i$ with isomorphisms $\varphi_{ij} : X_{ij}\to X_{ji}$ satisfying $\varphi_{ii} = \mathrm{id}$, $\varphi_{ij}(X_{ij}\cap X_{ik}) = X_{ji}\cap X_{jk}$ and the cocycle condition $\varphi_{ik} = \varphi_{jk}\circ\varphi_{ij}$ on the triple intersections. Then there is a scheme $X$, unique up to unique isomorphism, with open subschemes $U_i\cong X_i$ covering $X$ and $U_i\cap U_j = X_{ij}$ under these identifications. Consequently the category of schemes is closed under glueing and, more generally, a sheaf of sets on the category of schemes for a suitable topology is the same as the data of its values on the affine schemes with the descent conditions.

Proof. On the disjoint union $\coprod_iX_i$ impose the equivalence relation generated by the $\varphi_{ij}$; the quotient has a topology and a structure sheaf defined by glueing the sheaves along the open subsets, the compatibility being exactly the cocycle condition, and the result is locally affine. The uniqueness is the universal property of the quotient.

Definition. Let $S$ be a scheme. A scheme over $S$, or an $S$-scheme, is a morphism $X\to S$, and a morphism of $S$-schemes is a commutative triangle over $S$; the category of $S$-schemes has fibre products, constructed below. For $S = \operatorname{Spec} A$ one speaks of $A$-schemes, and a morphism $X\to\operatorname{Spec} A$ is equivalently a sheaf of $A$-algebras $\mathcal{O}_X$; for $A = \mathbb{Z}$ every scheme is a $\mathbb{Z}$-scheme, since $\mathbb{Z}$ is the initial ring.

Definition. Let $f : X\to Y$ be a morphism of schemes.

$f$ is locally of finite type if for every affine open $\operatorname{Spec} B\subseteq Y$ and affine open $\operatorname{Spec} A\subseteq f^{-1}(\operatorname{Spec} B)$ the ring $A$ is a finitely generated $B$-algebra; $f$ is of finite type if in addition $X$ can be covered by finitely many such affines. It is of finite presentation if the $A$ are finitely presented $B$-algebras.
$f$ is finite if it is affine — that is, the inverse images of affine opens are affine — and the induced ring maps make $A$ a finitely generated $B$-module; $f$ is integral if the $A$ are integral over $B$.
$f$ is flat if for every point the local ring $\mathcal{O}_{X,x}$ is flat as a module over $\mathcal{O}_{Y,f(x)}$; it is smooth or étale if it is locally of finite presentation, flat and the fibres are smooth, respectively étale, over their residue fields.
$f$ is a closed immersion if it induces a homeomorphism onto a closed subset and the map $\mathcal{O}_Y\to f_*\mathcal{O}_X$ is surjective; an open immersion is an isomorphism onto an open subscheme. A locally closed immersion is the composite of a closed and an open immersion.

Example (the fundamental examples). (i) $\operatorname{Spec}\mathbb{Z}$ has points $(0)$ and $(p)$ for the primes, the latter closed with residue field $\mathbb{F}_p$; its structure sheaf has global sections $\mathbb{Z}$, so the whole arithmetic of the integers is the geometry of this one scheme. (ii) $\operatorname{Spec} k[x]/(x^2)$, the scheme of the dual numbers over $k$ — the algebra $k[\varepsilon]$ with $\varepsilon^2 = 0$ — is a scheme with a single point whose structure sheaf is not reduced: it is the infinitesimal thickening of a $k$-point, and a morphism from it to a scheme $X$ is a $k$-point of $X$ together with a tangent vector, which is why it encodes first-order deformations. (iii) $\operatorname{Spec} k[x_1,\ldots,x_n] = \mathbb{A}^n_k$ and $\operatorname{Proj} k[x_0,\ldots,x_n] = \mathbb{P}^n_k$, the affine and projective spaces; $\mathbb{P}^n_A = \operatorname{Proj} A[x_0,\ldots,x_n]$ for any ring $A$, and $\mathbb{P}^n_\mathbb{Z}$ is the universal projective space. (iv) The affine line with a doubled origin is obtained by glueing two copies of $\operatorname{Spec} k[t]$ along the common open subscheme $\operatorname{Spec} k[t,t^{-1}]$ by the identity, giving a scheme in which the two origins are not identified and which is not separated; it is the standard example showing that schemes need not be separated, and that the underlying Zariski topology alone does not detect the failure.

Theorem (fibre products). Let $f : X\to S$ and $g : Y\to S$ be morphisms of schemes.

The fibre product $X\times_SY$ exists in the category of schemes: a scheme with morphisms $p : X\times_SY\to Y$ and $q : X\times_SY\to X$ making the square commute and universal among such data, with $$\operatorname{Hom}_S(T,X\times_SY)\cong\operatorname{Hom}_S(T,X)\times_{\operatorname{Hom}_S(T,S)}\operatorname{Hom}_S(T,Y)$$ for every $S$-scheme $T$.
On affine charts it is computed by the tensor product: if $X = \operatorname{Spec} A$, $Y = \operatorname{Spec} B$, $S = \operatorname{Spec} C$, then $X\times_SY = \operatorname{Spec}(A\otimes_CB)$, and the construction glues over the charts.
The fibre of $f$ over a point $s\in S$, $X_s = X\times_S\operatorname{Spec}\kappa(s)$, is a scheme over the residue field, and the formation of fibres and of fibre products commutes with base change; the fibre product is the categorical form of the intersection of the two families over $S$.

Proof. (2) is the universal property of the tensor product: a $C$-algebra map $A\otimes_CB\to D$ is the same as a pair of $C$-algebra maps $A\to D$ and $B\to D$. For (1) one glues the affine fibre products of the charts. The tensor product of rings exists and the compatibility of the maps with the localisations makes the glueing and the cocycle conditions automatic, so the result is a scheme and the universal property is inherited. (3) is the definition and the associativity of the fibre product.

Remark (the fibre product is the intersection). For $X,Y\subseteq Z$ subschemes of a common scheme the fibre product $X\times_ZY$ is the scheme-theoretic intersection, and it remembers multiplicities: the intersection of a line with a conic in the projective plane over $k$ is a closed subscheme of length two, a reduced pair of points when the intersection is transverse and the doubled point of the tangent line when it is not. The classical set-theoretic intersection is the underlying reduced subscheme, and the additional structure is exactly the nilpotent information that the scheme-theoretic definition retains.

Separatedness and Properness

Definition. A morphism $f : X\to S$ is separated if the diagonal $\Delta : X\to X\times_SX$ is a closed immersion, and proper if it is separated, of finite type and universally closed: for every $S$-scheme $T$ the base change $X\times_ST\to T$ carries closed subsets to closed subsets. A scheme over a field $k$ is complete if the structure morphism $X\to\operatorname{Spec} k$ is proper. An open immersion is separated and closed immersions are proper; the composite and the base change of separated morphisms are separated, and similarly for proper morphisms.

Theorem (the valuative criteria). Let $f : X\to S$ be a morphism of schemes and let $R$ be a valuation ring with fraction field $K$, with $j : \operatorname{Spec} K\to\operatorname{Spec} R$ the inclusion of the generic point.

$f$ is separated if and only if for every valuation ring $R$ with fraction field $K$, and every pair of morphisms $\operatorname{Spec} K\to X$ and $\operatorname{Spec} R\to S$ whose restrictions to the generic point of $\operatorname{Spec} R$ make the square commute, there is at most one morphism $\operatorname{Spec} R\to X$ completing the square.
$f$ is proper if and only if it is separated, of finite type, and for every such diagram there is at least one such completing morphism, so that properness is the existence, and separatedness the uniqueness, of the specialisation of an $R$-valued point of $X$.
Consequently $\operatorname{Spec} A$ is separated over any base and $\mathbb{A}^n_k$ and $\mathbb{P}^n_k$ over $k$ are separated; $\mathbb{P}^n_k$ is proper, while $\mathbb{A}^n_k$ of positive dimension is not proper, since the map $\mathbb{A}^1_k\to\mathbb{A}^1_k\setminus\{0\}$ is not closed.

Proof. The criteria are the standard ones of the theory, quoted here; the "at most one" is the valuative criterion for separatedness and the "at least one" for properness, and both are proved by reducing to the affine local case and using the valuative criterion for integrality and the extension of morphisms from a generic fibre. The properness of projective space is the fundamental theorem of elimination theory in the form of Algebraic Geometry, and the failure of properness of the affine line is witnessed by the open immersion $\mathbb{A}^1_k\setminus\{0\}\to\mathbb{A}^1_k$ and the closedness requirement for its base change.

Remark (properness is algebraic compactness). The valuative criteria make precise the sense in which properness replaces compactness: a $K$-point of $X$ extends to a valuation ring $R$ if the point can be specialised to the closed point of $\operatorname{Spec} R$ keeping its "limit" inside $X$, and the existence of such an extension for all valuation rings is a completeness condition. The Zariski topology is quasicompact for every scheme of finite type over a field, so the topological notion cannot distinguish $\mathbb{A}^1$ from $\mathbb{P}^1$; properness can, and it is the notion that appears in the theorems of the subject — for instance in the finiteness theorem for the cohomology of a coherent sheaf under a proper morphism, which is stated in Sheaf Cohomology.

Classical Varieties as Schemes

Theorem (the comparison with the classical theory). Let $k$ be an algebraically closed field.

A closed subset $X\subseteq\mathbb{A}^n_k$ that is reduced and irreducible — an affine variety of Algebraic Geometry — has a structure of a reduced scheme of finite type over $k$, unique, with $\mathcal{O}_X(U)$ the regular functions on $U$, and the assignment is compatible with the Nullstellensatz dictionary: the points of the scheme are the prime ideals of the coordinate ring, the closed points are the points of the variety, and the closed subschemes correspond to the ideals, the radical ideals giving the reduced ones.
The assignment extends to all quasiprojective varieties with their morphisms and gives an equivalence between the category of varieties over $k$ that admit an immersion into a projective space and the category of reduced separated schemes of finite type over $k$ that are quasiprojective. A reduced separated scheme of finite type over $k$ is always covered by finitely many affine charts, each the spectrum of a finitely generated reduced $k$-algebra and so a classical affine variety, but it need not be quasiprojective; the classical quasiprojective theory therefore corresponds to a proper subcategory, and the general theory is needed for the nonquasiprojective objects.
The set of points of a scheme of finite type over $k$ is the set of irreducible closed subsets of the classical variety, the closed points corresponding to the maximal ideals; consequently the generic point of an irreducible variety carries the function field, and every nonempty open set contains a closed point.

Proof. (1) The regular functions on a Zariski open set $U$ are the localisations of $k[X]$ and form the structure sheaf of $\operatorname{Spec} k[X]$, which is reduced because $X$ is reduced and finitely generated because $k[x_1,\ldots,x_n]$ is Noetherian; the dictionary is the Nullstellensatz. (2) the equivalence is the full faithfulness of the functor on affine pieces together with the glueing lemma applied to the affine charts of a quasiprojective variety; conversely a scheme obtained by glueing the affine charts of a variety is the corresponding variety, so the functor is essentially surjective onto the quasiprojective finite-type separated schemes, and the abstract ones not admitting an immersion into projective space exhibit the strictness of the essential image. (3) the identification of the points with the irreducible closed subsets is the definition of the Zariski topology of the spectrum.

Remark (what the generalisation adds). Three things are added by the scheme-theoretic formulation, and each is a theorem generator rather than a reformulation. First, non-reduced structure: the ideal- and nilpotent-thickening information lets one talk about the infinitesimal neighbourhood of a subscheme, the fibre of a morphism with multiplicity, and the tangent space as the scheme of morphisms from $\operatorname{Spec} k[x]/(x^2)$, the algebra of dual numbers, and this is the basis of deformation theory. Second, non-algebraically-closed and non-reduced base: $\operatorname{Spec}\mathbb{Z}$, $\operatorname{Spec}\mathbb{F}_p$, and schemes over a general ring $A$ make the theory arithmetic, and reduction modulo a prime becomes the passage to the fibre over a closed point of the base. Third, nonseparated and nonquasiprojective objects: the category is closed under glueing and fibre products, so quotients and moduli functors have a natural home, which is not covered here.

Dimension and Local Properties

Definition. Let $X$ be a scheme of finite type over a field $k$ or, more generally, a locally Noetherian scheme. The dimension of $X$ at a point $x$ is the Krull dimension of the local ring $\mathcal{O}_{X,x}$, and $\dim X$ is the supremum over the points, finite for a scheme of finite type over a field. The codimension of a closed subscheme $Z\subseteq X$ in an irreducible $X$ is $\dim X - \dim Z$, and a scheme is equidimensional of dimension $d$ if every component has dimension $d$.

Theorem (dimension). Let $X$ be a scheme of finite type over a field $k$, irreducible of dimension $d$.

$\dim X = \operatorname{tr.deg}_kk(X)$, the transcendence degree of the function field of the generic point, and the dimension is the maximum length of a chain of irreducible closed subschemes.
The points of codimension one are exactly those whose local ring is a discrete valuation ring; the regular local rings of dimension one are the discrete valuation rings of Part I's Valuation Theory and Henselian Rings.
For an effective divisor $Z\subseteq X$ on a scheme $X$ that is regular at a codimension-one point $z$, the local ring $\mathcal{O}_{X,z}$ is a discrete valuation ring and the local equation of $Z$ at $z$ is a uniformiser; the multiplicity of $Z$ at $z$ is the length of $\mathcal{O}_{X,z}$ modulo the ideal of $Z$, and the degree of $Z$ is the sum of these multiplicities over the points of a fibre of a proper morphism, which is the scheme-theoretic form of the count of intersections.

Proof. (1) is the dimension theorem for finitely generated algebras over a field, Part I's Integral Extensions and Krull Dimension, applied to the coordinate ring of an affine chart and transported to the scheme; the maximum length of a chain of irreducible closed subsets is the definition of the Krull dimension of the local ring at the generic point. (2) a Noetherian local ring of dimension one is regular exactly when it is a discrete valuation ring, and conversely; the points of codimension one have local rings of dimension one. (3) is the local description of a divisor as the zero set of one equation, Part I's Primary Decomposition for the multiplicity.

Definition. A scheme $X$ is reduced if all its local rings are reduced, equivalently if $\mathcal{O}_X$ has no nilpotent sections; normal if the local rings are integrally closed domains; regular or smooth if the local rings are regular local rings; Cohen–Macaulay if the local rings are. For a scheme of finite type over a field the implications regular $\Rightarrow$ normal $\Rightarrow$ reduced hold, and none reverses: the cuspidal cubic of Algebraic Geometry is reduced and not normal, and for a curve normal and regular coincide, so the standard example of a normal but not regular point is of dimension two, the vertex of the quadric cone $\operatorname{Spec} k[x,y,z]/(xy-z^2)$.

Theorem (the local structure over a field). Let $X$ be a scheme of finite type over $k$ and $x\in X$ a closed point. Then $\mathcal{O}_{X,x}$ is regular if and only if the maximal ideal satisfies $\dim_k\mathrm{M}_x/\mathrm{M}_x^2 = \dim\mathcal{O}_{X,x}$; the tangent space at $x$ is the dual of $\mathrm{M}_x/\mathrm{M}_x^2$, it is the fibre of the cotangent sheaf $\Omega^1_{X/k}$, and its dimension is the embedding dimension of $\mathcal{O}_{X,x}$. If $X$ is smooth at $x$ then $\hat{\mathcal{O}}_{X,x}$ is a formal power series ring in $\dim X$ variables over $k$, and over a perfect field the smooth locus is a nonempty dense open subset.

Proof. The tangent space computation is the one of Algebraic Geometry, now with the cotangent sheaf in place of the Jacobian matrix of a particular embedding: the regularity criterion is the equality of the embedding dimension with the Krull dimension, and the cotangent sheaf is locally free of rank $d$ exactly on the smooth locus, which is open and nonempty. The completions are the formal description of the local ring and are quoted as standard.

Remark (sheaves and cohomology on a general scheme). The quasi-coherent sheaves on a scheme are locally given by modules over the coordinate rings, and on an affine scheme the global sections functor is an exact equivalence with the category of modules, as in Sheaves in Algebraic Geometry; the coherent sheaves on a Noetherian scheme and the statements of the finiteness theorem for a proper morphism, the vanishing theorem and the duality belong to Sheaf Cohomology. The present article's contribution to that theory is the categorical one: the structure sheaf is a sheaf of rings on the topological space, the direct and inverse images of sheaves of modules are the functors induced by scheme morphisms, and the six operations of Grothendieck are the derived-category formalisation of these, as recorded in Derived Functors and Sheaf Cohomology. If $X$ is a scheme over $\mathbb{C}$ the associated complex-analytic space carries the same coherent-sheaf cohomology, which is the theorem of GAGA; its analytic part lies outside the present Part.

Summary

A scheme is a locally ringed space that is locally the spectrum of a commutative ring. The spectrum of a ring $A$ is the set of its primes with the Zariski topology, and its structure sheaf is characterised by $\mathcal{O}_X(D(f)) = A_f$, with stalks the local rings $A_{\mathrm{P}}$ and residue fields $\kappa(\mathrm{P})$; the construction is a contravariant equivalence between commutative rings and affine schemes, equivalently a representation of $\operatorname{Spec} A$ as the functor of points $B\mapsto\operatorname{Hom}_{\mathbf{Ring}}(A,B)$. Schemes are obtained from the affine ones by glueing, which the glueing lemma assures is always possible under the cocycle condition, and the category has fibre products, computed on affine charts by tensor products over the base ring and providing the scheme-theoretic intersection and the fibre of a morphism; over a fixed base $S$ the category of $S$-schemes is the setting for the notions of localised algebra — finite type and finite presentation, finite and integral, flat, smooth and étale, immersions — and the structural properties of separatedness and properness are characterised by the valuative criteria, which give properness as the algebraic replacement for compactness, in sharp contrast with the quasicompact Zariski topology of every scheme of finite type.

The classical varieties of Algebraic Geometry are the reduced separated schemes of finite type over an algebraically closed field, so the general theory contains the previous article and adds three genuinely new features: nilpotent structure, which encodes infinitesimal information and makes deformation theory possible; arbitrary base rings, which make the theory arithmetic, with $\operatorname{Spec}\mathbb{Z}$ as the fundamental example; and nonseparated and nonquasiprojective objects, which allow the glueings and the fibre products that the theory of moduli requires. The dimension of a scheme of finite type over a field is the transcendence degree of the function field of its generic point and the Krull dimension of the local rings, the codimension-one points are the discrete valuation rings, and the local properties — reduced, normal, regular — organise the singularities that the Jacobian criterion of the previous article detects.

Summary of Notation

Symbol	Meaning
$\operatorname{Spec} A$, $V(\mathrm{A})$, $D(f)$	spectrum; closed set of an ideal; distinguished open
$\mathcal{O}_X$, $\mathcal{O}_{X,\mathrm{P}} = A_{\mathrm{P}}$	structure sheaf; stalk; local ring
$\kappa(\mathrm{P})$	residue field $A_{\mathrm{P}}/\mathrm{P}A_{\mathrm{P}}$
$(X,\mathcal{O}_X)$, morphism	scheme: locally affine locally ringed space; morphism of locally ringed spaces
$X\to S$, $S$-scheme, $X(T)$	scheme over a base; functor of points
$X\times_SY$, $X_s$	fibre product; fibre over $s\in S$, $X\times_S\operatorname{Spec}\kappa(s)$
finite type, finite, flat, smooth, étale	local conditions on $X\to Y$ via the coordinate rings
closed/open immersion	locally closed subscheme with surjective map to a closed set / isomorphism onto an open subscheme
separated, proper	closed diagonal; separated, finite type, universally closed
$\dim X = \operatorname{tr.deg}_kk(X)$	dimension; codimension-one points are the discrete valuation rings
reduced, normal, regular	local rings reduced; integrally closed; regular
$\Omega^1_{X/k}$, $\mathrm{M}_x/\mathrm{M}_x^2$	cotangent sheaf; cotangent space, dual to the tangent space