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. Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... This article does not cite its references or sources. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... This is a list of algebraic geometry topics, by Wikipedia page. ...

Points

A scheme S is a locally ringed space, so a fortiori a topological space, but the meanings of point of S are threefold: In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

1. a point P of the underlying topological space;
2. a T-valued point of S is a morphism from T to S, for any scheme T;
3. 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 to S where 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. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... This article is about algebraic varieties. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... In mathematics, in the fields of general topology and particularly of algebraic geometry, a generic point P of a topological space X is a point such that every point Q of X is a specialization of P, in the sense of the specialization order (or pre-order). ... Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object. ... In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ... Projective geometry is a non-metrical form of geometry. ... In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ...

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 is thought of as Grothendiecks relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of objects explicitly depending on parameters, as the basic field of study, rather than a single such object. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...

.

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. In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on RZS then makes it a coproduct in the category of commutative rings. ...

Properties of schemes

Most important properties of schemes are local in nature, i.e. a scheme X has a certain property P if and only for any cover of X by open subschemes X_i, i.e. X= X_i, every X_i has the property P. It is usually the case that is enough to check one cover, not all possible ones. One also says that a certain property is Zariski-local, if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology. In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition but is only weakly related to their geometric properties; it is due to Oscar Zariski and took a place of particular importance in the field around... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...

Consider a scheme X and a cover by affine open subschemes Spec A_i. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings A_i. A property P is local in the above sense, iff the corresponding property of rings is stable under localization. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ...

For example, we can speak of locally noetherian schemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of a noetherian ring are still noetherian then means that the property of a scheme of being locally noetherian is local in the above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements), then so are its localizations. In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In ring theory, a ring R is said to be reduced if it has no non-zero nilpotent elements. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...

An example for a non-local property is separatedness (see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme.

The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let X = Spec A_i be a covering of a scheme by open affine subschemes. For definiteness, let k denote a field in the following. Most of the examples also work with the integers Z as a base, though, or even more general bases. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

notion	definition	example	non-example
related to scheme structure
irreducible	A scheme X is said irreducible when (as a topological space) it is not the union of two closed subsets except if one is equal to X. Using the correspondence of prime ideals and points in an affine scheme, this means X is irreducible iff the affine schemes Spec Ai all have exactly one minimal prime ideal. Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its irreducible components.	affine space, projective space	Spec k[x,y]/(xy) =
reduced	The Ai are reduced rings. Equivalently, none of its rings of sections (U any open subset of X) has any nonzero nilpotent element. Allowing non-reduced schemes is one of the major generalizations varieties to schemes.	varieties (by definition)	k[x]/(x2)
integral	A scheme that is both reduced and irreducible is called integral. Equivalently, a connected scheme that is covered by the spectra of integral domains. (Strictly speaking, this is not a local property, because the disjoint union of two integral schemes is not integral. However, for irreducible schemes, it is a local property).	Spec k[t]/f, f irreducible polynomial	Spec A ⊕ B. (A, B ≠ 0)
normal	An integral scheme is called normal, if the Ai are integrally closed domains.	regular schemes	singular curves
related to regularity
regular	The Ai are regular.	smooth varieties over a field	Spec k[x,y]/(x2+x3-y3)=
Cohen-Macaulay	All local rings are Cohen-Macaulay.	regular schemes, Spec k[x,y]/(xy)
related to "size"
locally noetherian	The Ai are Noetherian rings. If in addition a finite number of affine spectra covers X, the scheme is called noetherian. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false.	(Virtually everything in algebraic geometry).
dimension	The dimension, by definition the maximal length of a chain of irreducible subschemes, is a local property. See also Global dimension.	dimension 0: Artinian schemes, 1: algebraic curves, 2: algebraic surfaces.
catenary	A scheme is catenary, if chains between two irreducible subschemes have all the same length.	(Virtually everything, e.g. varieties over a field)

In mathematics, a hyperconnected space (or irreducible space) is a topological space X that satisfies any of the following equivalent conditions: no two nonempty open sets are disjoint X cannot be written as the union of two proper closed sets every nonempty open set is dense in X the interior... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation XY = 0 is the union of the two lines X = 0 and Y = 0. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... This article does not cite its references or sources. ... Image File history File links No higher resolution available. ... In ring theory, a ring R is said to be reduced if it has no non-zero nilpotent elements. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, in the field of algebraic geometry, a normal scheme is a scheme X for which every stalk (local ring) OX,x of its structure sheaf OX is an integrally closed local ring; that is, each stalk is an integral domain such that its integral closure in its field... An ordered group G is called integrally closed iff for all elements a and b of G, an ≤ b for arbitrary high natural n implies a ≤ 1. ... In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension. ... Image File history File links No higher resolution available. ... In mathematics, a Cohen-Macaulay ring is a commutative noetherian local ring with Krull dimension equal to its depth. ... Image File history File links No higher resolution available. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In mathematics, a Noetherian topological space is a topological space in which closed subsets satisfy the descending chain condition. ... In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ... The global dimension of a ring (denoted gl dim()) is the supremum of the set of projective dimensions of all -modules. ... In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains p=p0 ⊂p1 . ...

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 of integers; so that any scheme S is over , and in a unique way. In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...

For the following definitions, we take as standard notation

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.

Notions related to the topological structure

A morphism of schemes is called open (closed) , if the underlying map of topological spaces is open (closed, respectively), i.e. if open subschemes of X are mapped to open subschemes of Y (and similarly for closed). For example, flat morphisms are open and proper maps are closed, see below. In topology, an open map is a function between two topological spaces which maps open sets to open sets. ...

A morphism is called dominant, if the image f(X) is dense. A morphism of affine schemes Spec A → Spec B is dense if and only if the corresponding map B → A is injective. In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...

A morphism is called quasi-compact, if for some (equivalently: every) open affine cover of Y by some Y_i = Spec B_i, the preimages f^-1(Y_i) is quasi-compact. “Compact” redirects here. ...

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 -algebras, i.e. closed immersions correspond locally to morphisms of rings , where I is the ideal of the closed subscheme Y. Equivalently, a morphism 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 is surjective. This is a glossary of scheme theory. ...

An immersion is an isomorphism of Y to an open subscheme of a closed subscheme of X.

Note, that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: SpecA / I may be homeomorphic to , without . 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.

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. In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

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 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. This article does not cite its references or sources. ... In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, that produces objects with the typical properties of projective spaces and projective varieties. ...

Separated and proper morphism

A separated morphism is a morphism f such that the fiber product of f with itself along f has its diagonal as a closed subscheme — in other words, the diagonal map is a closed immersion. In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ...

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 is separated.

Notice that for a topological space Y is Hausdorff iff the diagonal embedding Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

is closed. In algebraic geometry, the above formulation is used because a scheme is a Hausdorff space if and only if it is zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) , which is different from the product of topological spaces.

Any affine scheme Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence a closed immersion of schemes):

.

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

A morphism is proper if it is separated, universally closed (i.e. such that fiber products with it preserve closed immersions), and of finite type. Projective morphisms are 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. This is a glossary of scheme theory. ... In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ... This is a glossary of scheme theory. ...

Finite, quasi-finite, and finite type morphisms

A morphism is finite if Y may be covered by affine open sets Spec B such that each f ^{− 1}(Spec B) is affine -- say of the form Spec A -- and furthermore A is finitely generated as a B-module. See finite morphism. In mathematics, in algebraic geometry, a morphism of schemes is a finite morphism, if has an open cover by affine schemes such that for each , is an open affine subscheme , and the restriction of f to , which induces a map of rings makes a finitely generated module over . ...

The morphism f is locally of finite type if Y may be covered by affine open sets Spec B such that each inverse image f ^{− 1}(Spec B) is covered by affine open sets Spec A where each A is finitely generated as a B-algebra.

The morphism f is finite type if Y may be covered by affine open sets Spec B such that each inverse image f ^{− 1}(Spec B) is covered by finitely many affine open sets Spec A where each A is finitely generated as a B-algebra.

The morphism f has finite fibers if the fiber over each point is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers. A morphism of of schemes) is called quasi-finite if for every point the fibre (where is the residue field of and is the canonical morphism) has only a finite number of points. ...

Finite morphisms are quasi-finite, but not all morphisms having finite fibers are quasi-finite, and morphisms of finite type are usually not quasi-finite.

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. In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i. ... In ring theory, a homomorphism f from a ring R to a ring S is flat if S becomes a flat R-module when the action of R on S is given by f. ...

Unramified and étale morphisms

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

Let be the maximal ideal of , and let

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

is a finite, separable field extension. In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ... In mathematics, a field extension L/K is separable if it can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable minimal polynomial over K. The condition of...

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 , which coincides with the usual notion of étale map in differential geometry. In algebraic geometry, a field of mathematics, an étale morphism is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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. In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. ...