Affine Schemes
Schemes as Sets
We define a point of SpecR to be a prime — that is, a prime ideal — of
R To avoid confusion, we will sometimes write [p] for the point of SpecR corresponding to the primepofR We will adopt the usual convention that
Ritself is not a prime ideal Of course, the zero ideal (0) is a prime ifRis a domain.
The coordinate ring IfR of an ordinary affine variety V over an algebraically closed field connects to its spectrum, SpecR, which features points that represent the points of the affine variety through the maximal ideals of R Additionally, SpecR includes points corresponding to each irreducible subvariety of V Although the introduction of these new points, associated with subvarieties of positive dimension, may initially seem perplexing, they ultimately prove to be quite useful, serving as the "generic points" in classical algebraic geometry.
Exercise I-1 Find SpecR when R is (a) Z; (b) Z/(3); (c) Z/(6); (d)Z(3); (e)C[x]; (f)C[x]/(x 2 ).
In the context of the space SpecR, each element \( f \in R \) defines a function, denoted as \( f \) For a point \( x = [p] \in \text{SpecR} \), we represent the residue field at \( x \) as \( \kappa(x) \) or \( \kappa(p) \), which is the quotient field of the integral domain \( R/p \) The value of the function \( f \) at the point \( x \) is given by \( f(x) \in \kappa(x) \), which is derived from the image of \( f \) through the canonical maps.
Exercise I-2 What is the value of the “function” 15 at the point (7) ∈ SpecZ? At the point (5)?
Exercise I-3 (a) Consider the ring of polynomialsC[x], and letp(x) be a polynomial Show that ifα∈Cis a number, then (x−α) is a prime ofC[x], and there is a natural identification ofκ((x−α)) withCsuch that the value ofp(x) at the point (x−α)∈SpecC[x] is the number p(α).
In the context of an affine variety \( V \) over an algebraically closed field \( K \), the coordinate ring \( R \) plays a crucial role For a point \( x \in V \), the associated maximal ideal \( p \) leads to the conclusion that the residue field \( \kappa(x) \) equals \( K \) Consequently, the evaluation of a function \( f \) at the point \( x \) yields the standard value \( f(x) \).
In general, the “function” f has values in fields that vary from point to point Moreover, f is not necessarily determined by the values of this
In the context of a field K, the ring R=K[x]/(x^2) possesses a single prime ideal, specifically (x) Consequently, the nonzero element x within R generates a "function" that evaluates to 0 at every point in SpecR.
We define a regular function on SpecR to be simply an element of R.
So a regular function gives rise to a “function” on SpecR, but is not itself determined by the values of this “function”.
Schemes as Topological Spaces
By using regular functions, we make SpecR into a topological space; the topology is called theZariski topology The closed sets are defined as follows. For each subsetS ⊂R, let
The aim of this definition is to ensure that each function \( f \in R \) resembles a continuous function as closely as possible, despite the absence of topology in the fields \( \kappa(x) \) and their variability with \( x \) Each field includes a zero element, allowing us to identify the set of points in SpecR where \( f \) equals zero To maintain continuity, this set should be closed Consequently, since the intersection of closed sets is also closed, we arrive at the definition that \( V(S) \) represents the intersection of the loci where the elements of \( S \) vanish.
For a family of sets V(S) to qualify as the closed sets of a topology, it must be closed under arbitrary intersections This means that for any family of sets S, the intersection of the corresponding closed sets aV(S a) will always yield a subset of V aS a.
, as required It is worth noting also that, ifI is the ideal gener- ated byS, thenV(I) =V(S).
In the Zariski topology, an open set is defined as the complement of a set V(S) Open sets that correspond to single-element sets S are particularly significant, as they represent spectra of rings Consequently, these sets are given a specific name and notation For any element f in R, the distinguished or basic open subset of X = SpecR associated with f is established.
The points ofXf— that is, the prime ideals ofRthat do not containf— are in one-to-one correspondence with the prime ideals of the localization
In this book, we implicitly identify the points of SpecR f with X f by adjoining an inverse tof, establishing a correspondence that maps p ⊂ R to pR f ⊂ R f.
The distinguished open sets form a base for the Zariski topology in the sense that any open set is a union of distinguished ones:
Distinguished open sets are also closed under finite intersections; since a prime ideal contains a product if and only if it contains one of the factors, we have i=1, ,n
(SpecR) f i = (SpecR) g , where g is the productf1ã ã ãfn In particular, any distinguished open set that is a subset of the distinguished open set (SpecR)f has the form (SpecR)f g for suitableg.
SpecR is typically not a Hausdorff space due to the large size of its open sets, with the only closed points being those linked to maximal ideals of R The smallest closed set containing a point [p] is V(p), indicating that the closure of [p] includes all points [q] such that q ⊃ p A point [p] is closed if and only if p is maximal In the context where R represents the affine ring of an algebraic variety V over an algebraically closed field, the closed points of SpecR correspond directly to the points of V, with the closed points within the closure of [p] being precisely the points of V that lie in the subvariety defined by p.
Exercise I-4 (a) The points of SpecC[x] are the primes (x−a), for every a ∈ C, and the prime (0) Describe the topology Which points are closed? Are any of them open?
(b) Let K be a field and let R be the local ring K[x] (x) Describe the topological space SpecR (The answer is given later in this section.)
To fully define SpecR, it is essential to describe the structure sheaf, or the sheaf of regular functions on X Before delving into this, we will first outline some fundamental definitions of sheaf theory and prove an important proposition (Proposition I-12) that will be crucial for our later discussion.
An Interlude on Sheaf Theory
LetX be any topological space Apresheaf F onX assigns to each open set U in X a set, denoted F(U), and to every pair of nested open sets
12 I Basic Definitions satisfying the basic properties that res U, U = identity and res V, U ◦res W, V = res W, U for allU ⊂V ⊂W ⊂X.
The elements of F(U) are called thesections of F over U; elements of
A presheaf can be defined as a contravariant functor that maps open sets in a topological space X to sets, with morphisms corresponding to the inclusion of sets By altering the target category to abelian groups, we establish the concept of a presheaf of abelian groups, which can similarly be extended to rings, algebras, and other mathematical structures.
A key construction in algebraic geometry is the apresheaf of modules F over a presheaf of rings O on a space X This structure consists of a pair for each open set U of X, where O(U) represents a ring and F(U) denotes an O(U)-module Additionally, for any inclusion U ⊇ V, there exists a ring homomorphism α: O(U) → O(V) and a corresponding set map F(U) → F(V) This map is an O(U)-module homomorphism when F(V) is viewed as an O(U)-module via α.
A presheaf, which can be defined over sets, abelian groups, rings, or modules, qualifies as a sheaf when it meets an additional requirement known as the sheaf axiom This axiom stipulates that for every open covering \( U \) of an open set, the presheaf must satisfy specific conditions related to the elements in the covering.
U ⊂X and each collection of elements fa ∈F(Ua) for each a∈A having the property that for all a, b ∈ A the restrictions of fa and fb to
Ua∩Ub are equal, there is a unique elementf ∈F(U) whose restriction toUa isfa for alla.
The empty set ∅ is an open subset of Spec R and can be represented as the union of an empty family, leading to the conclusion that any sheaf has a unique section over the empty set Specifically, for a sheaf F of rings, F(∅) corresponds to the zero ring, where 0 equals 1 It is important to note that the zero ring has no prime ideals, making it the only ring with a unit that possesses this characteristic, provided one accepts the axiom of choice; thus, its spectrum is empty.
Exercise I-5 (a) LetX be the two-element set{0,1}, and makeX into a topological space by taking each of the four subsets to be open A sheaf onX is thus a collection of four sets with certain maps between them; describe the relations among these objects (Xis actually SpecR for some ringsR; can you find one?)
(b) Do the same in the case where the topology of X = {0,1} has as open sets only∅,{0}and{0,1} Again, this space may be realized as SpecR.
A presheaf \( F \) on a space \( X \) can be restricted to an open subset \( U \) of \( X \), resulting in a new presheaf \( F|_U \) This restriction is defined by setting \( F|_U(V) = F(V) \) for any open subset \( V \) of \( U \), while maintaining the same restriction maps as those of \( F \) Notably, if the original presheaf \( F \) is a sheaf, then the restricted presheaf \( F|_U \) is also a sheaf.
In the sequel we shall work exclusively with presheaves and sheaves of things that are at least abelian groups, so we will usually omit the phrase
In the context of presheaves of abelian groups, one can construct operations such as direct sums and tensor products on an open set basis For instance, if F and G are presheaves of abelian groups, their direct sum, denoted as F⊕G, can be defined accordingly.
This always produces a presheaf, and ifF andG are sheaves thenF ⊕G will be one as well Tensor product is not as well behaved: even ifF and
G are sheaves, the presheaf defined by
(F⊗G)(U) :=F(U)⊗G(U) may not be, and we define the sheafF ⊗G to be the sheafification of this presheaf, as described below.
The simplest sheaves on any topological space X are the sheaves of lo- cally constant functions with values in a setK— that is, sheavesK where
K(U) represents the collection of locally constant functions mapping from U to K When K is a group, it can be structured as a sheaf of groups through pointwise addition Likewise, if K is a ring, we can establish K as a sheaf of rings by defining multiplication in K(U) as pointwise multiplication Additionally, when K is endowed with a topology, we can create a sheaf of continuous functions that take values in K.
C, whereC(U) is the set of continuous functions fromU toK, again with pointwise addition IfX is a differentiable manifold, there are also sheaves of differentiable functions, vector fields, differential forms, and so on. Generally, ifπ:Y →X is any map of topological spaces, we may define the sheafI of sections ofπ; that is, for every open setU of X we define
In the context of continuous maps, I(U) represents the set of mappings σ: U → π − 1 U that satisfy the condition π ◦ σ = 1, where 1 denotes the identity on U Such mappings are referred to as sections of the projection π in a set-theoretical sense, and in a broader context, the elements of F(U) for any sheaf F are also termed sections by extension from this specific case.
Exercise I-6 (For readers familiar with vector bundles.) LetV be a vec- tor bundle on a topological spaceX.Check that the sheaf of sections ofV is a sheaf of modules over the sheaf of continuous functions onX (Sheaves of modules in general may in this way be seen as generalized vector bundles.)Another way to describe a sheaf is by its stalks For any presheafF and any pointx∈X, we define the stalk Fx ofF at xto be the direct limit
14 I Basic Definitions of the groupsF(U) over all open neighborhoodsU ofxinX— that is, by definition,
The disjoint union of F(U) for all open sets U that include the point x is considered, subject to the equivalence relation where σ ∼ τ if σ belongs to F(U) and τ belongs to F(V) This relation holds when there exists an open neighborhood W of x that is contained in the intersection U ∩ V, ensuring that the restrictions of σ and τ to W are identical, denoted as res U, W σ = res V, W τ.
For every element x in the set U, there exists a mapping from F(U) to Fx that associates sections with their equivalence class denoted as sx When F is a sheaf, a section s in F(U) is uniquely determined by its images in the stalks Fx for all x in U; specifically, s equals zero if and only if sx equals zero for every x in U This relationship is grounded in the sheaf axiom, which states that if sx equals zero for all x in U, then for each x, there exists a neighborhood Ux within U such that the restriction of s to U is also zero, leading to the conclusion that s is zero in F(U).
The concept of stalks relates closely to the geometric idea of rings of germs Specifically, for an analytic manifold \( X \) of dimension \( n \) and \( \mathcal{O}_{X} \) as the sheaf of analytic functions on \( X \), the stalk of \( \mathcal{O}_{X} \) at a point \( x \) represents the ring of germs of analytic functions at that point This means it consists of the ring of convergent power series in \( n \) variables.
Exercise I-7 Find the stalks of the sheaves you produced for Exercises I-5 and I-6.
Exercise I-8 Topologize the disjoint union F Fx by taking as a base for the open sets ofF all sets of the form
V(U, s) :={(x, sx) :x∈U}, whereU is an open set andsis a fixed section overU.
(a) Show that the natural map π : F → X is continuous, and that, for
U and s ∈F(U), the mapσ : x→sx from U to F is a continuous section ofπ overU (that is, it is continuous andπ◦σis the identity onU).
(b) Conversely, show that any continuous mapσ:U →F such thatπ◦σ is the identity onU arises in this way.
Hint.Takex∈U and a basic open setV(V, t) containingσ(x), where
The construction demonstrates that the sheaf of germs of sections of the map π: F → X is isomorphic to F, indicating that any sheaf can be represented as the sheaf of germs of sections from an appropriate map Historically, this definition of sheaves was utilized in early mathematical works The topological space F is referred to as the "espace étalé" of the sheaf, as its open sets correspond to the structure of the sheaf.
“stretched out flat” over open sets ofX.
A morphism ϕ: F → G of sheaves on a space X is defined simply to be a collection of mapsϕ(U) :F(U)→G(U) such that for every inclusion
? commutes (In categorical language, a morphism of sheaves is just a natural transformation of the corresponding functors from the category of open sets onX to the category of sets.)
Schemes as Schemes (Structure Sheaves)
We now return to the definition of the scheme X = Spec R and will finalize the construction by defining the structure sheaf OX as OSpec R Our goal is to generalize the relationship between Spec R and R, mirroring that of an affine variety and its coordinate ring Specifically, we aim for the ring of global sections of the structure sheaf OX to be equal to R.
We thus wish to extend the ring R of functions onX to a whole sheaf of rings This means that for each open setU ofX, we wish to give a ring
For every pair of open sets \( U \subset V \), we define a restriction homomorphism \( \text{res}_{V,U} : \mathcal{O}_X(V) \rightarrow \mathcal{O}_X(U) \) that adheres to the established axioms This process allows us to easily describe the corresponding rings.
OX(U) and the maps res V, U should be for distinguished open sets U and
If Xf ⊃ Xg, then a power of g is a multiple of f, indicating that the radical of (f) is the intersection of the primes containing f Consequently, the restriction map res X f, X g can be established as the localization map Rf → Rf g = Rg According to Proposition I-12, this is sufficient to define the structure sheaf O, provided we confirm that it adheres to the sheaf axiom concerning coverings of distinguished opens by distinguished opens Prior to this verification, Proposition I-18 presents a crucial lemma that outlines the coverings of affine schemes by distinguished open sets.
Lemma I-16 LetX = SpecR, and let{f a } be a collection of elements of
R The open sets X f a coverX if and only if the elementsf a generate the unit ideal In particular, X is quasicompact as a topological space.
Quasicompact spaces are defined by the property that every open cover has a finite subcover, but they are not necessarily Hausdorff, which is often the case for schemes This lack of Hausdorffness diminishes many benefits associated with compactness Unlike compact manifolds, the continuous image of one affine scheme in another may not be closed Therefore, in Section III.1, we will introduce a more suitable concept of "compactness" known as properness, which is equally significant in traditional geometric theories.
A cover \( X \) of a set is valid if no prime ideal of \( R \) contains all the elements \( f_a \), which occurs when the \( f_a \) generate the unit ideal This establishes the first statement For the second statement, it is important to note that every open cover can be refined to a form \( X \cup X_{f_a} \), where each \( f_a \) belongs to \( R \) Since the \( X_{f_a} \) covers \( X \), the elements \( f_a \) generate the unit ideal, allowing the element 1 to be expressed as a finite linear combination of the \( f_a \) By considering only the \( f_a \) used in this representation of 1, we confirm that the refined cover \( X \cup X_{f_a} \) possesses a finite subcover, thereby validating the original cover.
Exercise I-17 IfRis Noetherian, every subset of SpecRis quasicompact.
Proposition I-18 Let X = SpecR, and suppose that X f is covered by open sets X f a ⊂X f
(a) Ifg, h∈Rf become equal in eachRf a, they are equal.
(b) If for eachathere isg a ∈R f a such that for each pairaandbthe images ofga andgbinRf a f b are equal, then there is an elementg∈Rf whose image inRf a isga for alla.
If B represents the collection of distinguished open sets SpecRf of SpecR, and we define OX(SpecRf) as Rf, then OX functions as a B-sheaf According to Proposition I-12, OX can be uniquely extended to a sheaf on X.
Definition I-19 The sheaf OX defined in the proposition is called the structure sheaf ofX or thesheaf of regular functions on X.
Proof of Proposition I-18 We begin with the casef = 1, soRf =R and
In the initial analysis, if the functions g and h are equal for each f in a finite cover, then the difference g - h is annihilated by a power of each f According to Lemma I-16, we can consider a finite cover, which leads to the conclusion that g - h is annihilated by a power of the ideal generated by all f^N for some N Since this ideal includes a power of the ideal generated by all f, which is the unit ideal, we can deduce that g equals h in the ring R.
In part (b), we employ an argument similar to the classical partition of unity to combine the elements \( g_a \) into a single element \( g \in \mathbb{R} \) For large \( N \), the product \( f_a N g_a \in \mathbb{R} f_a \) represents the image of an element \( h_a \in \mathbb{R} \) By Lemma I-16, we can assume the covering \( \{X_{f_a}\} \) is finite, allowing us to use one \( N \) for all \( a \) Furthermore, since \( g_a \) and \( g_b \) converge in \( X_{f_a f_b} \), we conclude that \( f_b N h_a = (f_a f_b) N g_a = (f_a f_b) N g_b = f_a N h_b \) for sufficiently large \( N \) Thus, with the finite covering assumption, we can consolidate our findings.
N will do for allaandb By Lemma I-16 the elementsfa∈Rgenerate the unit ideal, and hence so do the elementsf a N , and we may write
In our analysis, we establish a partition of unity represented by a collection of elements \( a \in \mathbb{R} \) We propose that \( g \) is the element in \( \mathbb{R} \) we are looking for Specifically, for each \( b \), the relationship \( f(b) = f(b)N g \) leads to \( f(b)N e(a) = h(b) = f(b)N g(b) \) Consequently, \( g \) is determined to be equal to \( g(b) \) on the set \( X f(b) \), fulfilling our requirements.
Returning to the case of arbitrary f, setX =Xf,R =Rf, f a =f fa; thenX = SpecR andX f a =Xf a, so we can apply the case already proved to the primed data.
The proposition is still valid, and has essentially the same proof, if we replaceRf andRf a byMf andMf a for anyR-moduleM.
Exercise I-20 Describe the points and the sheaf of functions of each of the following schemes.
In many geometric theories, particularly in the realm of complex manifolds, there can be a scarcity of regular functions on a scheme Notably, when examining arbitrary schemes, we find that those analogous to compact manifolds may lack nonconstant regular functions entirely Consequently, partially defined functions on a scheme X, represented as elements OX(U) for some open dense subset U, become significantly important These functions are referred to as rational functions on X.
In the context of the SpecRwithRa domain, let X represent this domain and U denote Xf, where the elements of OX(Xf) = Rf are ratios of elements in R Notably, in many significant cases, every nonempty open set within X is dense, indicating that the behavior of rational functions is indicative of the overall properties of X.
Schemes in General
Subschemes
An open subset U of a scheme X, along with the restriction of the structure sheaf OX|U, forms a scheme itself, although this may not be immediately evident To demonstrate this, consider that any distinguished open set of an affine scheme remains an affine scheme; for instance, if X = Spec R and U = X_f, then (U, OX|U) corresponds to Spec R_f Since the distinguished open sets of X that lie within U collectively cover U, it follows that (U, OX|U) is indeed covered by affine schemes Consequently, an open subset of a scheme is termed an open subscheme of X, with its structure inherently understood.
The definition of a closed subscheme is more complicated; it is not enough to specify a closed subspace ofX,because the sheaf structure is not defined thereby.
Consider first an affine scheme X = SpecR For any idealI in the ring
R, we may make the closed subset V(I) ⊂ X into an affine scheme by identifying it withY = SpecR/I This makes sense because the primes of R/I are exactly the primes ofR that containI taken modulo I, and thus the topological space|SpecR/I|is canonically homeomorphic to the closed set V(I)⊂X We define a closed subscheme ofX to be a schemeY that is the spectrum of a quotient ring of R (so that the closed subschemes of
X by definition correspond one to one with the ideals in the ringR).
In the context of a given scheme X = SpecR, we can characterize the operations and relationships involving closed subschemes Specifically, a closed subscheme Y = SpecR/I is said to contain another closed subscheme Z = SpecR/J if Z is also a closed subscheme of Y.
J ⊃I This implies thatV(J)⊂V(I), but the converse is not true.
Exercise I-26 The schemes X1, X2, andX3 of Exercise I-20 may all be viewed as closed subschemes of SpecC[x] Show that
X1⊂X3 and X2⊂X3, but no other inclusions Xi ⊂ Xj hold, even though the underlying sets of X2 and X3 coincide and the underlying set of X1 is contained in the underlying set ofX2.
The union of closed subschemes SpecR/I and SpecR/J is defined as SpecR/(I∩J), while their intersection is represented as SpecR/(I+J) It is crucial to understand that the concepts of containment, intersection, and union do not adhere to all the typical properties found in set theory For instance, as illustrated on page 69, there exist closed subschemes X, Y, and Z of a scheme such that X∪Y equals X∪Z and X∩Y equals X∩Z, yet Y and Z are not identical.
We would now like to generalize the notion of closed subscheme to an arbitrary schemeX To do this, the first step must be to replace the ideal
To associate a closed subscheme \( Y \) of an affine scheme \( X = \text{Spec} R \) with a sheaf, we define the ideal sheaf \( J = J_{Y/X} \) as the sheaf of ideals of \( O_X \) on the distinguished open set \( V = X_f \) by \( J(X_f) = I R_f \) The structure sheaf \( O_Y \) of \( Y = \text{Spec} R/I \) can be identified with the pushforward \( j^* O_Y \), where \( j \) is the inclusion map from \( |Y| \) to \( |X| \), and is represented as the quotient sheaf \( O_X/J \) Additionally, the ideal sheaf \( J \) can be recovered as the kernel of the restriction map \( O_X \to j^* O_Y \).
It's important to note that not every sheaf of ideals in OX originates from ideals of R For instance, in the scenario where R equals K[x], as illustrated in Example I-22, we can establish a sheaf of ideals through a specific definition.
For a sheaf of idealsJ coming from an ideal of Rwe would have
In the context of algebraic geometry, the sheaf J(U) is defined as J(X)x = J(X)K(x), indicating that J does not originate from any ideal of the ring R When discussing closed subschemes, our focus is on sheaves of ideals that do derive from ideals of R To categorize these specific sheaves, the term "quasicoherent sheaves of ideals" is used, a designation that, while seemingly inadequate for such a fundamental concept, is well-established in the literature.
In the context of Noetherian rings, sheaves associated with finitely generated modules exhibit a property known as coherence Consequently, it is appropriate to describe sheaves derived from finitely generated modules as coherent, while those originating from arbitrary modules are classified as quasicoherent.
In the context of schemes, a sheaf of ideals J ⊂ OX is considered aquasicoherent if, for every open affine subset U of X, its restriction J|U functions as a quasicoherent sheaf of ideals on U.
Now we are ready to define a closed subscheme of an arbitrary scheme as something that looks locally like a closed subscheme of an affine scheme:
Definition I-27 IfX is an arbitrary scheme, aclosed subscheme Y ofX is a closed topological subspace|Y| ⊂ |X|together with a sheaf of ringsOY that is a quotient sheaf of the structure sheafOX by a quasicoherent sheaf of ideals J, such that the intersection of Y with any affine open subset
U ⊂X is the closed subscheme associated to the ideal J(U).
If V ⊂ X is any open set, we say that a regular function f ∈ OX(V) vanishes onY iff ∈J(V).
In fact, |Y| is uniquely determined by J, so closed subschemes of X are in one-to-one correspondence with the quasicoherent sheaves of ideals
Quasicoherence is defined in a broader context, specifically regarding sheaves A sheaf F on a space X is termed quasicoherent if it serves as a sheaf of OX-modules, meaning that for every affine set U, F(U) functions as an OX(U)-module.
U and distinguished open subset Uf ⊂U, theOX(Uf) =OX(U)f-module
F(Uf) is derived from F(U) by inverting the function f, resulting in an isomorphism between the restriction map F(U) and F(Uf) A sheaf F is termed coherent if all its modules F(U) are finitely generated While a more restrictive definition of coherence exists, it aligns with this interpretation when X is covered by a finite number of spectra of Noetherian rings, which is the primary focus Informally, qua-coherent sheaves can be understood as sheaves of modules whose restrictions to open affine sets are modules, with coherent sheaves being finitely generated on the corresponding rings In the realm of schemes, this concept serves as the appropriate analogue to modules over a ring, and for most applications, they can be regarded simply as modules.
Exercise I-28 To check that a sheaf of ideals (or any sheaf of modules) is quasicoherent (or for that matter coherent), it is enough to check the defining property on each setU of a fixed open affine cover ofX.
One of the most important closed subschemes of an affine scheme X is
Xred, thereduced scheme associated to X This may be defined by setting
In the context of ring theory, Xred is equivalent to SpecRred, where RredisR modulo its nilradical, which is defined as the ideal of nilpotent elements in R The nilradical of a ring R is characterized as the intersection of all prime ideals, specifically the minimal primes Consequently, the topological spaces |X| and |Xred| are identical.
Exercise I-29 Xredmay also be defined as the topological space|X|with structure sheaf OX red associating to every open subset U ⊂ X the ring
In the context of scheme theory, the nilradical is defined as a sheaf of ideals N ⊂ OX, where its value on any open set U corresponds to the nilradical of OX(U) This construction preserves localization, making N a quasicoherent sheaf of ideals The closed subscheme associated with X is referred to as the reduced scheme, denoted Xred, and a scheme X is considered reduced if it holds that X = Xred Additionally, irreducibility is an important property of schemes, which, despite its name, is not dependent on whether the scheme is reduced; a scheme X is deemed irreducible if its underlying set |X| cannot be expressed as the union of two properly contained closed sets.
Here are some easy but important remarks about reduced and irreducible schemes.
Exercise I-30 A scheme is irreducible if and only if every open subset is dense.
The Local Ring at a Point
The Noetherian property is essential in ring theory and extends to scheme theory, where a scheme \( X \) is classified as Noetherian if it has a finite cover by open affine subschemes, each represented by the spectrum of a Noetherian ring Importantly, this classification remains consistent regardless of the specific cover selected.
The germ of a scheme X at a point x ∈ X represents the intersection of all open subschemes that include the point, reflecting a natural conceptualization This idea is encapsulated in the local ring of X at x, which was defined previously.
The maximal ideal \( m_{X,x} \) of a local ring consists of all sections that vanish at the point \( x \) To compute this local ring and verify its properties, including the identification of the maximal ideal, one can start by substituting \( X \) with an affine open neighborhood around \( x \), leading to the simplification where \( X = \text{Spec} R \) and \( x = [p] \) Subsequently, the open subsets \( U \) in the direct limit can be restricted to the distinguished open sets \( \text{Spec} R_f \) where \( f(x) = 0 \), indicating that \( f \) belongs to the prime ideal \( p \).
The local ring of a scheme at a point, denoted as OX,x, is defined as the limit of the functions in the localization of R at p, while mX,x represents the limit of functions in the localization of pR This concept is fundamental to the theory of schemes, as it allows for the examination of the germ of X at a point x, which can be understood as Spec OX,x In the following chapter, we will explore various schemes of this nature and illustrate how to define different geometric notions using the local ring.
The dimension of a space X at a point x, denoted as dim(X, x), refers to the Krull dimension of the local ring OX,x, which is determined by the longest chain of prime ideals within OX,x This chain's length is measured by the number of strict inclusions The overall dimension of X, represented as dimX, is then defined as the supremum of these local dimensions across all points in X.
Exercise I-36 The underlying space of a zero-dimensional Noetherian scheme is finite.
The Zariski cotangent space at a point x on a scheme X is defined as the quotient of the maximal ideal mX,x by the square of the maximal ideal m2 X,x, and it is considered a vector space over the residue field κ(x), which is the quotient of the local ring OX,x by the maximal ideal mX,x The dual of this vector space is referred to as the Zariski tangent space at the point x.
To understand this definition, consider first a complex algebraic variety
X that is nonsingular In this setting the notion of the tangent space to
X at a point p is unambiguous: it may be taken as the vector space of derivations from the ring of germs of analytic functions at the point into
C If mX,p is the ideal of regular functions vanishing at p, then such a derivation induces aC-linear mapmX,p/m 2 X,p →C, and the tangent space may be identified in this way with Hom C (mX,p/m 2 X,p ,C) = (mX,p/m 2 X,p ) ∗ See Eisenbud [1995, Ch 16] It was Zariski’s insight that this latter vector space is the correct analogue of the tangent space for any point, smooth or singular, on any variety; Grothendieck subsequently carried the idea over to the context of schemes, as in the definition given above We shall return to this construction, from a new point of view, in Chapter VI.
Exercise I-37 If K is a field, the Zariski tangent space to the scheme SpecK[x 1 , , x n ] at [(x 1 , , x n )] isn-dimensional.
A variety X is considered nonsingular (or regular) at a point x ∈ X if the dimension of the Zariski tangent space at x equals the dimension of X at that point; otherwise, X is deemed singular at x In Noetherian varieties, X is nonsingular at x if and only if the local ring OX,x is a regular local ring This concept has historically played a crucial role in the algebraization of geometry, first introduced by Zariski in 1947, following Krull's earlier definition of regular local rings aimed at generalizing polynomial ring properties.
Exercise I-38 A zero-dimensional Noetherian scheme is nonsingular if and only if it is the union of reduced points.
Morphisms
In this article, we define morphisms of schemes, building on the classical theory where a regular map of affine varieties corresponds to a map of coordinate rings in the opposite direction This establishes an equivalence between regular maps of affine varieties and algebra homomorphisms of their coordinate algebras We generalize this concept by showing that maps between affine schemes correspond to maps of the respective rings While it may seem straightforward to define a morphism of schemes as being "locally a morphism of affine schemes," this approach can lead to complications regarding the independence of the definition from the choice of an affine cover Therefore, we present a more robust definition that avoids reliance on an affine cover, which, although initially appearing complex, proves to be practical and uniformly applicable to all "local ringed spaces," defined as topological spaces with a sheaf of rings whose stalks are local rings.
To grasp the motivation behind the definition of differentiable maps between manifolds, consider that a continuous map \( \psi: M \rightarrow N \) is differentiable if, for every differentiable function \( f \) defined on an open subset \( U \subset N \), the pullback \( \psi^* f := f \circ \psi \) results in a differentiable function on the preimage \( \psi^{-1} U \subset M \) This relationship can be succinctly articulated using sheaf theory, where any continuous map \( \psi: M \rightarrow N \) induces a corresponding map of sheaves on \( N \), represented as \( \psi^*: C(N) \rightarrow \psi^* C(M) \).
I.2 Schemes in General 29 sending a continuous function f ∈ C(N)(U) on an open subset U ⊂ N to the pullback f ◦ψ ∈ C(M)(ψ − 1 U) = (ψ ∗ C(M))(U) In these terms, a differentiable map ψ : M → N may be defined as a continuous map ψ:M →N such that the induced mapψ # carries the subsheafC ∞ (N)⊂
C(N) into the subsheaf ψ ∗ C ∞ (M) ⊂ ψ ∗ C(M) That is, we require that there be a commutative diagram
In the context of schemes, the structure sheaf OX of a scheme X differs from a predefined sheaf of functions on X, necessitating a specific approach for mapping schemes To establish a map of schemes, it is essential to define both a continuous map ψ #: X → Y on the underlying topological spaces and a corresponding pullback map ψ #: OX → ψ* OY.
To establish compatibility conditions for the morphism ψ # and the structure sheaf OY, it is essential to note that OY does not yield values in a fixed field but rather in a field κ(q) that varies depending on the point q ∈ Y This variability complicates the requirement for the value of f ∈ OY(U) at q ∈ U ⊂ Y to match the value of ψ # f ∈ ψ ∗ OX(U) at a corresponding point p ∈ ψ −1 U ⊂ X mapping to q, as these values belong to different fields Therefore, the only meaningful condition we can impose is that f vanishes at q if and only if ψ # f vanishes at p, which is the requirement we adopt in our definition.
Definition I-39 A morphism, or map, between schemes X and Y is a pair (ψ, ψ # ), where ψ : X → Y is a continuous map on the underlying topological spaces and ψ # :OY →ψ ∗ OX is a map of sheaves onY satisfying the condition that for any pointp∈X and any neighborhoodU ofq=ψ(p) inY a sectionf ∈OY(U) vanishes at qif and only if the sectionψ # f ofψ ∗ OX(U) =OX(ψ − 1 U) vanishes atp.
This last condition has a nice reformulation in terms of the local rings
OX,p andOY,q Any map of sheavesψ # :OY →ψ ∗ OX induces on passing to the limit a map
OY,q= lim−→q∈U⊂Y OY(U)→−→limq∈U⊂Y OX(ψ −1 U), and this last ring naturally maps to the limit
In the context of local rings, for any open subset \( V \) containing point \( p \), the notation \( O_{X,p} \) represents the local ring at \( p \) The induced map \( \psi^* : O_{Y,q} \to O_{X,p} \) establishes a relationship between sections, indicating that a section \( f \in O_{Y}(U) \) vanishes at \( q \) if and only if its image under \( \psi^* \), \( \psi^* f \), vanishes at \( p \) This implies that the mapping \( O_{Y,q} \to O_{X,p} \) sends the maximal ideal \( m_{Y,q} \) to \( m_{X,p} \), confirming that it functions as a local homomorphism of local rings.
A morphism of affine schemes, denoted as ψ: X = Spec S → Spec R = Y, corresponds to a ring homomorphism ϕ: R → S This relationship highlights a key result in algebraic geometry, further enhanced by an important advancement that characterizes mappings from any arbitrary scheme to an affine scheme.
Theorem I-40 For any schemeX and any ringR, the morphisms
(ψ, ψ # ) :X −→SpecR are in one-to-one correspondence with the homomorphisms of rings ϕ:R→OX(X) by the association ϕ=ψ # (SpecR) :R=OSpec R(SpecR)→ψ ∗ (OX)(SpecR) =OX(X).
Proof We describe the inverse association Set Y = SpecR, and let ϕ :
Let R → OX(X) be a mapping of commutative rings For any point p in |X|, the preimage of the maximal ideal through the composite R → OX(X) → OX,p results in a prime ideal, allowing the map ϕ to induce a continuous mapping of sets ψ: |X| → |Y| in the Zariski topology Furthermore, for each basic open set U = SpecRf ⊂ Y, we define the map ψ#: Rf = OY(U) → (ψ* OX)(U) as the composite.
By localizing the sheaf OX at ψ, we derive the expression Rf→OX(X)ϕ(f)→OX(ψ −1 U) According to Proposition I-12(ii), this suffices to establish a sheaf map Further localization reveals that if ψ(p) = q, then the map ψ # induces a local morphism of local rings Rq → OX,p, confirming that (ψ, ψ # ) constitutes a morphism of schemes It is evident that the induced map satisfies ψ # (Y) = ϕ, thereby demonstrating that this construction serves as the inverse of the original mapping.
Of course this result says in particular that all the information in the category of affine schemes is already in the category of commutative rings.
Corollary I-41 The category of affine schemes is equivalent to the cate- gory of commutative rings with identity, with arrows reversed, the so-called opposite category.
Exercise I-42 (a) Using this, show that there exists one and only one map from any scheme to SpecZ In the language of categories, this says that SpecZ is theterminal object of the category of schemes. (b) Show that the one-point set is the terminal object of the category of sets.
For example, each point [p] of X = SpecR corresponds to a scheme Specκ(p) that has a natural map to X defined by the composite map of rings
Of course, the inclusion makes [p] a closed subscheme if and only if p is a maximal ideal of R (in general, [p] is an infinite intersection of open subschemes of a closed subscheme).
In the context of morphisms of affine schemes, if ψ : Y → X is such that X = SpecR and Y = SpecT, and X is defined as a closed subscheme of itself by an ideal I in R, then the preimage ψ − 1(X) is defined as the closed subscheme of Y corresponding to the ideal ϕ(I)T in T When X is a closed point p in X, the preimage ψ − 1(p) is referred to as the fiber over X The underlying topological space of this preimage aligns with the set-theoretic preimage, while its scheme structure provides a nuanced understanding of the "correct multiplicity" for counting points in the preimage Two classical examples of this concept will be explored later, alongside a simpler example in Exercise II-2.
Exercise I-43 (a) Let ϕ : X → Y be the map of affine schemes illus- trated by
In the context of algebraic geometry, the structure X = SpecK[x, u]/(xu) represents the union of two intersecting lines at the point p = (x, u) Meanwhile, Y = SpecK[t] denotes a single line Notably, the mapping between these structures is an isomorphism on each line of X, which can be exemplified by a specific ring homomorphism.
The fiber over the point \( q = (t - a) \) with \( a = 0 \) is represented by the scheme \( \text{Spec}(K \times K) \), which consists of two distinct points In contrast, the fiber over \( q_0 \), which includes the double point \( p \), is isomorphic to \( \text{Spec}(K[x]/(x^2)) \) The algebra \( K[x]/(x^2) \) is two-dimensional as a vector space over \( K \), reflecting the local structure of the map at the point \( p \) Let \( \phi: X \to Y \) denote the map of affine schemes depicted in the illustration.
That is, X = SpecK[x, y, u, v]/((x, y)∩(u, v)) is the union of two planes in four-space meeting in a single point p = (x, y, u, v), while
Y = SpecK[s, t] is a plane, and the map is an isomorphism on each of the planes ofX; for example, it might be given by the map of rings
The fiber over the point \( q_{a,b} = (s-a, t-b) \) represents the scheme \( \text{Spec}(K \times K) \), which consists of two distinct points when either \( a \) or \( b \) is zero In contrast, the fiber over \( q_{0,0} \), which includes the "double point" \( p \), is isomorphic to a different structure.
The Gluing Construction
Using the notion of morphism, we can construct more complicated schemes (for example, nonaffine schemes) by identifying simpler schemes along open subsets This is a basic operation, called thegluing construction.
Suppose we are given a collection of schemes {Xα}I, and an open set
XαβinXαfor eachβ=αin I Suppose also that we are given a family of isomorphisms of schemes ψ αβ :X αβ →X βα for eachα=β inI, satisfying the conditionsψβα=ψ αβ −1 for allαandβ, ψ αβ (X αβ ∩X αγ ) =X βα ∩X βγ for allα, β, γ, and thecompatibility condition ψβγ◦ψαβ|(X αβ ∩X αγ )=ψαγ|(X αβ ∩X αγ ).
In this context, we can define a scheme X by seamlessly combining the schemes X α using the isomorphisms ψ αβ Specifically, there exists a unique scheme X that is covered by open subschemes isomorphic to the X α, ensuring that the identity maps on the intersections X α ∩ X β ⊂ X align with the isomorphisms ψ αβ.
This construction is instrumental in defining projective schemes based on affine ones and plays a significant role in the theory of toric varieties, as highlighted by Kempf et al [1973].
In various applications, it is often unnecessary to explicitly provide the maps ψαβ; instead, we are presented with a topological space |X| and a collection of open subsets |Xα| Each of these subsets is equipped with the structure of an affine scheme, which includes a structure sheaf OX α, facilitating a coherent framework for analysis.
The sheaves OX α and OX β, defined on the intersection Xα ∩ Xβ, can be naturally identified, especially when they are subsets of a common set This identification ensures that the conditions of Corollary I-14 are met, leading to a uniquely defined sheaf OX on X that extends all the OX α Consequently, the pair (|X|, OX) forms a scheme.
Probably the simplest example of this is the definition of affine space
In this article, we define affine n-space over an affine scheme \( X = \text{Spec} R \) as \( \text{Spec} R[x_1, \ldots, x_n] \), denoted as \( A^n X \) or \( A^n R \) The geometry of these affine spaces and their subschemes will be discussed in Chapter II Additionally, we observe that any morphism \( X \to Y \) of affine schemes induces a natural mapping \( A^n X \to A^n Y \) Consequently, we can apply the gluing construction: if \( S \) is an arbitrary scheme covered by affine schemes \( U_\alpha = \text{Spec} R_\alpha \), we define the affine space \( A^n S \) over \( S \) as the union of the affine spaces \( A^n U_\alpha \), with gluing maps derived from the identity maps on \( U_\alpha \cap U_\beta \).
We will see two other ways of defining affine spaceA n S over an arbitrary base Sin Exercises I-47 and I-54 below.
This exercise highlights the risks associated with the gluing construction, demonstrating that improper yet permissible gluing can lead to configurations that lack a basis in any geometric context.
Exercise I-44 PutY = SpecK[s] andZ= SpecK[t] LetU ⊂Y be the open set Ys and let V ⊂ Z be the open set Zt Let ψ : V → U be the isomorphism corresponding to the map
OY(U) =K[s, s − 1 ]→K[t, t − 1 ] =OZ(V) sendingstot, and letγbe the map sendingstot −1 LetX1be the scheme obtained by gluing together Y andZ along ψ, and let X2 be the scheme obtained by gluing alongγinstead.
X1 is not isomorphic to X2, as X2 represents the scheme associated with the projective line P1 K, which will be elaborated on in the subsequent section In contrast, X1 is characterized as an affine space featuring a doubled origin.
In Chapter III we will introduce a condition, called separatedness, that will preclude schemes such as thisX1.
Projective Space An important example of a scheme constructed by gluing isprojectiven-space over a ring R, denotedP n R It is made by gluing n+ 1 copies ofaffine space
A n R = SpecR[x1, , xn] overR An extensive treatment of projective schemes will begin in Chapter III Here we will use the idea only as an illustration of gluing.
The construction of projective space parallels the classical approach of defining it as a variety over a field While not strictly necessary, this method proves convenient We begin with the polynomial ring in n+1 variables, R[X0, , Xn], and proceed to form its localization.
Recall that the ringA has a naturalgrading, that is, a direct-sum decom- position (as an abelian group) into subgroupsA (n) , forn∈Z, such that
The relationship A(n) ⊂ A(m+n) indicates that A(n) is comprised of monomial rational fractions of degree n, with the degree 0 part, A(0), forming a subring of A To define an affine covering, we consider the rings as R-subalgebras of A(0), where each subring Ai consists of polynomials P/Xi with deg(P) being homogeneous elements of R[x0, , xn] It is evident that Ai is generated over R by n algebraically independent elements.
X0/Xi, , Xi/Xi, , Xn/Xi,
Relative Schemes
Fibered Products
The concept of the preimage of a set under a function is crucial in understanding the fibred product of schemes To lay the groundwork for this definition, we will first review the relevant context within the category of sets.
The fibered product of two setsX andY over a third setS— that is, of a diagram of maps of sets
?ϕ is by definition the set
The fibered product is sometimes called the pullback of X (or ofX →S) toY This construction generalizes several more elementary ones in a very useful way:
IfS is a point, it gives the usual direct product.
If X, Y are both subsets of S and ϕ, ψ are the inclusions, it gives the intersection.
IfY ⊂S andψis the inclusion, it gives the preimage of Y inX.
IfX =Y , it gives the set on which the mapsϕ, ψare equal, theequalizer of the maps.
Note that X ×S Y comes with natural projection maps to X and Y making the diagram
?ϕ commute Indeed, the setX×SY may be defined by the following universal property: among all setsZwith given maps toXandY making the diagram
?ϕ commute,X×SY with its projection maps is the unique “most efficient” choice in the sense that, given the diagram withZ above, there is a unique mapZ →X×SY making the diagram
In the realm of schemes, a fibered product is defined as a scheme that possesses a unique universal property, ensuring the existence of distinct projections to both X and Y This uniqueness is a fundamental characteristic of the fibered product in the context of schemes.
The concept of the fibered product allows us to define essential mathematical constructs such as products, intersections, preimages, and equalizers This raises an important inquiry regarding the existence of fibered products within the category of schemes Fortunately, fibered products do exist in this context, and we will now outline the construction involved.
First, we treat the affine case Recall that the category of affine schemes is opposite to the category of commutative rings, by Corollary I-41 Therefore, if we have schemes
X= SpecA, Y = SpecB, S= SpecR, whereX andY map toS(so thatAandBareR-algebras), we must define the fibered productX×SY to be
This is because the natural diagram
The tensor product 6ϕ serves as a fibred coproduct or fibred sum in the category of commutative rings, exhibiting a universal property that contrasts with that of the fibred product.
To check that this definition is reasonable, one may note that in the situation whereY is a closed subscheme ofS defined by an idealI, so that
B = R/I, we haveA⊗RB =A/IA Thus X×S Y = SpecA/IA is the same as the preimage of Y inX, as previously defined.
Exercise I-46 A few simple special cases are a great help when comput- ing fibered products Prove the following facts directly from the universal property of the tensor product of algebras:
(a) For anyR-algebraS we haveR⊗RS=S.
(b) If S, T areR-algebras andI⊂S is an ideal, then
R[x1, , xn]⊗RR[y1, , ym] =R[x1, , xn, y1, , ym]. Use these principles to solve the remainder of this exercise.
(d) Letm, nbe integers Compute the fibered product
(e) Compute the fibered product SpecC×Spec RSpecC.
(f) Show that for any polynomial rings R[x] and R[y] over a ringR, we have
In example (d), the underlying set of the fibered product corresponds to the fibered product of the underlying sets; however, this relationship does not hold in examples (e) and (f) Additionally, example (g) introduces the concept of ring homomorphisms.
R[x]→R[y]; x→y 2 Show that with respect to these maps we have
In general, we represent the space S using affine schemes SpecR ρ, while the preimages in spaces X and Y are covered by affine schemes SpecA ρα and SpecB ρβ, respectively This setup allows us to construct a coherent diagram that captures the relationships between these schemes.
?ϕ is covered by diagrams of the form
The fiber product depicted in the diagram is Spec(A ρα ⊗R ρB ρβ) By applying the gluing concept discussed earlier, we can verify that these schemes align on overlaps and combine to create the scheme X×SY, although the computation is omitted for brevity An alternative method will be outlined in section VI.2.1 Additionally, the concept of product offers a different perspective on affine space A n S over a scheme S.
Exercise I-47 Let S be any scheme Let A n Z = SpecZ[x1, , xn] be affine space over SpecZ, as defined above (this scheme will be discussed in detail in the next chapter) Show that affine space A n S over S may be described as a product:A n S =A n Z ×Spec ZS.
The fiber of a morphism ψ : Y → X over a point p in X, associated with a prime ideal p of R, is defined using the fibered product of Y and the one-point scheme Specκ(p) For affine schemes, such as Y = SpecT and X = SpecR, the fiber over p is given by ψ^(-1)(p) = Specκ(p) ×_X Y = Spec(R_p / p_p ⊗_R T) = Spec(R_p / p_p ⊗_R T / pT), representing the primes of T whose preimages in R correspond to p Additionally, the preimage or inverse image of a closed subscheme X of X under ψ is defined as the fibered product X ×_X Y.
In the context of algebraic geometry, the preimage ψ −1 X of a closed subscheme X is also a closed subscheme of Y By utilizing the OX-algebra structure on OY, the ideal sheaf corresponding to this preimage can be expressed as Iψ −1 X = IX ã OY.
A common application of the fibered product is analyzing how different varieties behave when a base field is extended, often referred to as a "base change." This concept plays a crucial role in the upcoming chapter, where we will explore various examples to illustrate its significance.
I.3 Relative Schemes 39 for the great flexibility and convenience of the theory of schemes in handling arithmetic questions.
In the context of schemes, the fibered product X×SY often differs from the fibered product of the sets of points of X and Y, as illustrated in Exercise I-46 This discrepancy is not a significant issue; rather, it highlights the complexity of functions f(x, y) compared to those of the form g(x)h(y) However, the definitions presented in Chapter VI offer a perspective that resolves this peculiar situation.
The Category of S-Schemes
In the context of schemes, we can establish an absolute product by utilizing the fibered product, where S is designated as a terminal object This means that for every scheme, there exists a unique morphism leading to S.
In Exercise I-42, it is established that the terminal object in the category of schemes is SpecZ Interestingly, the absolute product exhibits surprising characteristics, as demonstrated in Exercise I-46(d), where the product of nonempty sets can be empty when m and n are relatively prime Additionally, the dimension of an irreducible scheme can be defined as the Krull dimension of the coordinate ring from any of its affine open sets, highlighting more peculiarities in this context.
X×Y =X×Spec ZY of two schemes to have dimension equal to the sum of the dimensions ofX andY But in fact we have the result in the next exercise.
Exercise I-48 Show that ifX = SpecZ[x] andY = SpecZ[y], then dimX×Y = dimX+ dimY −dim SpecZ= dimX+ dimY −1.
To address certain oddities in the study of schemes, we can generalize our definitions to focus on K-schemes, which are schemes X over a specific field or ring K This approach involves using morphisms that respect the K-algebra structure on OX(X) In this framework, SpecK serves as the terminal object, and the absolute product is represented as the fibered product over SpecK When K is a field, the behavior of products in the category of K-schemes aligns more closely with elementary geometric intuition.
Exercise I-49 Let K be a field If X and Y are nonemptyK-schemes, then the productX ×Y =X×Spec KY in the category of K-schemes is nonempty.
In this context, the dimension of SpecK is zero, and it can be verified that for schemes derived from spectra of finitely generated K-algebras, the dimension of products exhibits additivity, which aligns with expected mathematical principles.
To enhance the understanding of families of schemes, we can broaden the concept further A K-algebra structure on OX(X) can be defined as a ring homomorphism from K to OX(X) According to Theorem I-40, this is equivalent to a mapping from X to SpecK By substituting SpecK with any arbitrary scheme, we can extend this framework.
S, we define a scheme over S, or S-scheme, to be a scheme X together with a morphism X → S We may think of a scheme over S informally as a family of schemes “parametrized by points of S” — for each point of
S we have the fiber over that point A morphism of schemes over S (or S-morphism) is a commutative diagram
If X and Y are schemes over S, then we write MorS(X, Y) for the set of S-morphisms Note that the fibered productX×SY of schemes overS is precisely the ordinary direct product in the category of schemes overS.
As usual, if S= SpecRis affine, we will use the terms “R-scheme” and
“the category of R-schemes” interchangeably with “S-scheme” and “the category ofS-schemes”.
Introducing the category of schemes over the complex numbers simplifies classical algebraic geometry rather than complicating it When working with schemes over C, the point SpecC has no nontrivial automorphisms, while the scheme SpecC[x]/(x² + 1), which consists of two points, has an automorphism group isomorphic to Z/2 In contrast, in the broader category of all schemes, SpecC has a large automorphism group, specifically the Galois group of C over Q, leading to a more complex structure for SpecC[x]/(x² + 1) Therefore, focusing on schemes over C streamlines the study of algebraic geometry.
Cremoves the (presumably unwanted) extra structure of the Galois group Gal(C/Q).
Exercise I-50 Find the automorphism groups of the schemesX 1 andX 3 of Exercise I-20 in the category of schemes overC.
Global Spec
In the context of an affine scheme, an affine S-scheme is defined as the spectrum of an R-algebra This article aims to extend this concept to explore similar constructs within the category of S-schemes for any given S.
In this article, we introduce the concept of an aquasicoherent sheaf of OS-algebras for any schemes Specifically, we define a sheaf F of OS-algebras that applies to any affine open subset U = SpecR within the scheme S, particularly focusing on distinguished open subsets.
In this article, we explore the relationship between quasicoherent sheaves of OS-algebras on a scheme S and their associated schemes Specifically, we define a scheme X = SpecF for any quasicoherent sheaf F of OS-algebras on S, along with a structure morphism X → S When S is an affine scheme, specifically S = SpecR, this simplifies to X = SpecF(S), with the structure morphism X → S derived from the R = OS(S)-algebra structure on F(S).
One approach to constructing the space SpecF is to utilize gluing via affine open subsets Uα = SpecRα, defining X as the union of schemes SpecF(Uα) with attaching maps derived from the restriction maps F(Uα) → F(Uα∩Uβ) While this method is feasible, it complicates the verification of the independence of the resulting space SpecF from the chosen cover and poses challenges in describing its points Therefore, we will present an alternative construction for clarity and ease of understanding.
We start with a definition: given a quasicoherent sheafF ofOS-algebras, we define a prime ideal sheaf in F to be a quasicoherent sheaf of ideals
I F, such that for each affine open subsetU ⊂S, the ideal I(U) ⊂
F(U) is either prime or the unit ideal (Observe that for any affine scheme
X, the points ofX are simply the prime ideal sheaves ofOX.) Now, we will defineX = SpecF in three stages, as we did the spectrum of a ring First, as a set,X is the set of prime ideal sheaves inF Second, as a topological space: for every openU ⊂S(not necessarily affine) and sectionσ∈F(U), letVU,σ ⊂Xbe the set of prime ideal sheavesP ⊂F such thatσ /∈P(U); take these as a basis for the topology Finally, we define the structure sheaf
OX on basis open sets by setting
In the context of the morphism f: X → S, we associate a prime ideal sheaf P ⊂ F with its inverse image in OS → F The pullback map on functions, denoted as f#: OS(U) → OX(f −1(U)) = F(U), corresponds directly to the structure map OS → F on the open set U.
Exercise I-51 Show that the points of an affine schemeX are in one-to- one correspondence with the set of prime ideal sheaves inOX.
Exercise I-52 Show that iff : Y →X is a morphism andP is a prime ideal sheaf of OY, thenf ∗ (P) is a prime ideal sheaf inf ∗ OY.
Exercise I-53 Show that iff : Y →X is a morphism, the map on sets corresponding tof sendsP ⊂OY to (f # ) −1 (f ∗ (P))⊂OX.
The simplest example of global Spec gives us yet another construction of affine space over an arbitrary schemeS:
Exercise I-54 LetS be any scheme Show that affine spaceA n S over S may be constructed as a global Spec: