Glossary of scheme theory

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This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory. See also list of algebraic geometry topics.

1 Points
2 Local properties
3 Properties of the underlying space
4 Properties of scheme morphisms

[edit] Points

A scheme S is a locally ringed space, so a fortiori a topological space, but the meaning of point of S are threefold:

a point P of the underlying topological space;
a T-valued point of S is a morphism from T to S, for any scheme T;
a geometric point, where S is defined over (is equipped with a morphism to) Spec(K), where K is a field, is a morphism from Spec(K*) to S where K* is an algebraic closure of K.

Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds, would be the ordinary-sense points. The points P of the underlying space include analogues of the generic points (in the sense of Zariski, not that of André Weil), which specialise to ordinary-sense points. The T-valued points are thought of, via Yoneda's lemma, as a way of identifying S with the representable functor h_S it sets up. Historically there was a process by which projective geometry added more points (e.g. complex points, line at infinity) to simplify the geometry by refining the basic objects. The T-valued points were a massive further step.

As part of the predominating Grothendieck approach, there are three corresponding notions of fiber of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a geometric fiber of a morphism S′ → S is thought of as

S′×_SSpec(K*).

This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.

[edit] Local properties

A property P of (commutative) rings is local in nature, for the Zariski topology if it remains stable under finite localization. Such a property automatically gives a property of schemes which is local in nature. We can require that the scheme is covered by affine open sets whose rings of coordinates have the property P in question.

For example, we can speak of locally noetherian schemes, namely those which are covered by the spectra of Noetherian rings; and we say that a scheme is noetherian when it is covered by 'finitely' many spectra of noetherian rings. Unfortunately, while it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false. Much of algebraic geometry is concerned only about noetherian (or, at any rate, locally noetherian) schemes, but non-noetherian and even non-locally noetherian schemes do turn up.

Another example of a local property is for a scheme to be reduced: this means that none of its rings of sections has any nilpotent element other than zero, or that it is covered by the spectra of reduced rings (viz. rings having no nonzero nilpotent elements).

Here are some local properties of rings:

Reduced
Noetherian
Artinian
Catenary
Universally catenary
Regular
Cohen-Macaulay
Krull dimension ≤ d
Global dimension ≤ d

[edit] Properties of the underlying space

Naturally, since a scheme has an underlying topological space, one can also apply to schemes whatever properties apply to topological spaces. For example, one might say of a scheme that it is connected which simply means that the underlying topological space is connected. Things aren't always that simple, however. A scheme is rarely a Hausdorff space. The word separated is used in a way (separated morphism) not familiar from topological separation axioms as conventionally stated.

A scheme X is said to be irreducible when (as a topological space) it is not the union of two closed subsets except if one is equal to X. (Any noetherian scheme can be written uniquely as the union of finitely many irreducible non-empty closed subsets, called its irreducible components.) A scheme that is both reduced and irreducible is called integral; this is equivalent to saying that the scheme is connected and that it covered by the spectra of integral domains (and a ring is an integral domain if and only if its spectrum, an affine scheme, is integral).

Of course, these are only a small subset of what adjectives can be applied to the word "scheme".

[edit] Properties of scheme morphisms

One of Grothendieck's fundamental ideas is to emphasize relative notions, i.e. conditions on morphisms rather than conditions on schemes themselves. The category of schemes has a final object, the spectrum of the ring Z of integers; so that any scheme S is over Spec(Z), and in a unique way.

For the following definitions, we take as standard notation

f:Y → X

to be a morphism of schemes. Parallel to the properties of schemes above, the following properties of morphisms are also of local nature, i.e. if there is an open covering of X by some open subschemes $U i$ , such that the restriction of f to $f - 1 (U i)$ has the property, then f has it, as well.

[edit] Open and closed immersions

A morphism f is an open immersion if locally on the target it is of the form of an inclusion of an open subset.

A closed immersion morphism is one defined by the vanishing of a global ideal of O_X-algebras, i.e. closed immersions correspond locally to morphisms of rings $A \rightarrow A/I$ , where I is the ideal of the closed subscheme Y. Equivalently, a morphism f: Y → X of schemes is a closed immersion if and only if f induces a homeomorphism from sp(Y), the underlying topological space of Y, onto a closed subset of sp(X), and if furthermore the induced morphism f^#: O_X → f_∗O_Y is surjective.

Note, that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: $S p e c A / I$ may be homeomorphic to $S p e c A / I'$ , without I = I' . When specifying a closed subset of a scheme without mentioning the scheme structure, mostly the so-called reduced scheme-structure is meant, i.e. (locally) $A / I$ should have no nilpotent elements, which uniquely determines the closed subscheme.

[edit] Affine and projective morphisms

A morphism is called affine, if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by the global Spec construction for sheaves of O_X-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles.

Projective morphisms are defined similarly, but in practice they turn out to be more important than affine morphisms: f is called projective, if it factors as a closed immersion followed by the projection of a projective space $\mathbb P^n_Y := \mathbb P^n \times Y$ to Y. Again, one may say, that f is projective if it is given by the global Proj construction on graded commutative O_X-Algebras.

[edit] Separated and proper morphism

A separated morphism is a morphism f such that the fiber product of Y with itself along f has its diagonal as a closed subscheme — in other words, the diagonal map is a closed immersion.

As a consequence, a scheme X is separated when the diagonal of X within the scheme product of X with itself, is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated, if the unique morphism $X \rightarrow Spec \mathbb Z$ is separated. Any affine scheme is separated.

This compares with the criterion (closed diagonal) for a topological space to be Hausdorff; with the usual topological space product used.

While the separatedness is of rather technical nature, properness has deep geometrically meaning.

A morphism f is proper if it is separated, universally closed (i.e. such that fiber products with it preserve closed immersions), and of finite type.

A projective morphism is proper; but the converse is not in general true. See also complete variety. A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.

See proper morphism.

[edit] Finite morphism

A morphism f is finite if, locally on X, it is represented by a finitely generated extension of commutative rings. (This extension is then automatically integral). Finite morphisms are proper. See finite morphism.

[edit] Flat morphism

A morphism f is flat if it gives rise to a flat map on stalks. When viewing a morphism as a family of schemes parametrized by the poins of Y, the geometric meaning of flatness could roughly be described by saying, that the fibers $f - 1 (y)$ do not vary too wildly.

[edit] Unramified and étale morphisms

For a point y in Y, consider the corresponding morphism of local rings

$f^\# \colon \mathcal{O}_{X, f(y)} \to \mathcal{O}_{Y, y}.$

Let m be the maximal ideal of O_X,f(y), and let

$n = f^\#(m) \mathcal{O}_{Y,y}$

be the ideal generated by the image of m in O_Y,y. The morphism f is unramified if for all y in Y, n is a maximal ideal of O_Y,y and the induced map

$\mathcal{O}_{X,f(y)}/m \to \mathcal{O}_{Y,y}/n$

is a finite, separable field extension.

A morphism f is étale if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties X and Y over a field, étale morphisms are precisely those inducing an isomorphism of tangent spaces $df: T_x X \rightarrow T_{f(x)} Y$ , which coincides with the usual notion of étale map in differential geometry.

Étale morphisms form a very important class of morphisms; they are used to build the so-called étale topology and consequently the étale cohomology, which is nowadays one of the cornerstones of algebraic geometry.

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Categories: Scheme theory | Glossaries