### Glossary of scheme theory

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 and glossary of classical algebraic geometry and glossary of commutative algebra and glossary of stack theory

## A-E

- affine
- 1. Affine space is roughly a vector space where one has forgotten which point is the origin
- 2. An affine variety is a variety in affine space
- 3. 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*-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms._{X} - arithmetic genus
- The arithmetic genus of a variety is a variation of the Euler characteristic of the trivial line bundle; see Hodge number.
- catenary
- A scheme is catenary, if all chains between two irreducible closed subschemes have the same length. Examples include virtually everything, e.g. varieties over a field, and it is hard to construct examples that are not catenary.
- closed
- 1. _Z) is a scheme called the
**closed subscheme defined by the quasi-coherent sheaf of ideals**.*J*^{[1]}The reason the definition of closed subschemes relies on such a construction is that, unlike open subsets, a closed subset of a scheme does not have a unique structure as a subscheme. - Cohen–Macaulay
- A scheme is called Cohen-Macaulay
if all local rings are Cohen-Macaulay.
For example, regular schemes, and
*Spec k*[*x,y*]/(*xy*) are Cohen–Macaulay, but is not. - connected
- The scheme is
*connected*as a topological space. Since the connected components refine the irreducible components any irreducible scheme is connected but not vice versa. An affine scheme*Spec(R)*is connected iff the ring*R*possesses no idempotents other than 0 and 1; such a ring is also called a**connected ring**. Examples of connected schemes include affine space, projective space, and an example of a scheme that is not connected is*Spec*(*k*[*x*]×*k*[*x*]) - dimension
- The dimension, by definition the maximal length of a chain of irreducible closed subschemes, is a global property. It can be seen locally if a scheme is irreducible. It depends only on the topology, not on the structure sheaf. See also Global dimension. Examples: equidimensional schemes in dimension 0: Artinian schemes, 1: algebraic curves, 2: algebraic surfaces.
- dominant
- A morphism is called
*dominant*, if the image*f*(*Y*) is dense. A morphism of affine schemes*Spec A*→*Spec B*is dense if and only if the kernel of the corresponding map*B*→*A*is contained in the nilradical of*B*. - étale
- 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 an algebraically closed field, étale morphisms are precisely those inducing an isomorphism of tangent spaces $df:\; T\_\{x\}\; X\; \backslash rightarrow\; T\_\{f(x)\}\; Y$, 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.

## F-J

- final
- 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 $\backslash mathbb\{Z\}$ of integers; so that any scheme $S$ is*over*$\backslash textrm\{Spec\}\; (\backslash mathbb\{Z\})$, and in a unique way. - finite
- The morphism $f$ is
**finite**if $X$ may be covered by affine open sets $\backslash text\{Spec\; \}B$ such that each $f^\{-1\}(\backslash text\{Spec\; \}B)$ is affine — say of the form $\backslash text\{Spec\; \}A$ — and furthermore $A$ is finitely generated as a $B$-module. See finite morphism. The morphism $f$ is**locally of finite type**if $X$ may be covered by affine open sets $\backslash text\{Spec\; \}B$ such that each inverse image $f^\{-1\}(\backslash text\{Spec\; \}B)$ is covered by affine open sets $\backslash text\{Spec\; \}A$ where each $A$ is finitely generated as a $B$-algebra. The morphism $f$ is**finite type**if $X$ may be covered by affine open sets $\backslash text\{Spec\; \}B$ such that each inverse image $f^\{-1\}(\backslash text\{Spec\; \}B)$ is covered by finitely many affine open sets $\backslash text\{Spec\; \}A$ where each $A$ is finitely generated as a $B$-algebra. The morphism $f$ has**finite fibers**if the fiber over each point $x\; \backslash in\; X$ is a finite set. A morphism is**quasi-finite**if it is of finite type and has finite fibers. 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. If*y*is a point of*Y*, then the morphism*f*is**of finite presentation at**(or*y***finitely presented at**) if there is an open affine subset*y**U*of*f(y)*and an open affine neighbourhood*V*of*y*such that*f*(*V*) ⊆*U*and $\backslash mathcal\{O\}\_Y(V)$ is a finitely presented algebra over $\backslash mathcal\{O\}\_X(U)$. The morphism*f*is**locally of finite presentation**if it is finitely presented at all points of*Y*. If*X*is locally Noetherian, then*f*is locally of finite presentation if, and only if, it is locally of finite type.^{[2]}The morphism*f*is**of finite presentation**(or) if it is locally of finite presentation, quasi-compact, and quasi-separated. If*Y*is finitely presented over*X**X*is locally Noetherian, then*f*is of finite presentation if, and only if, it is of finite type.^{[3]} - flat
- 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 points of $X$, the geometric meaning of flatness could roughly be described by saying that the fibers $f^\{-1\}(x)$ do not vary too wildly.
- image
- If
*f*:*Y*→*X*is any morphism of schemes, the**scheme-theoretic image**of*f*is the unique*closed*subscheme*i*:*Z*→*X*which satisfies the following universal property:*f*factors through*i*,- if
*j*:*Z*′ →*X*is any closed subscheme of*X*such that*f*factors through*j*, then*i*also factors through*j*.^{[4]}^{[5]}

*f*,*f*(*Y*). For example, the underlying space of*Z*always contains (but is not necessarily equal to) the Zariski closure of*f*(*Y*) in*X*, so if*Y*is any open (and not closed) subscheme of*X*and*f*is the inclusion map, then*Z*is different from*f*(*Y*). When*Y*is reduced, then*Z*is the Zariski closure of*f*(*Y*) endowed with the structure of reduced closed subscheme. But in general, unless*f*is quasi-compact, the construction of*Z*is not local on*X*. - immersion
**Immersions***f*:*Y*→*X*are maps that factor through isomorphisms with subschemes. Specifically, an**open immersion**factors through an isomorphism with an open subscheme and a**closed immersion**factors through an isomorphism with a closed subscheme.^{[6]}Equivalently,*f*is a closed immersion if, and only if, it induces a homeomorphism from the underlying topological space of*Y*to a closed subset of the underlying topological space of*X*, and if the morphism $f^\backslash sharp:\; \backslash mathcal\{O\}\_X\; \backslash to\; f\_*\; \backslash mathcal\{O\}\_Y$ is surjective.^{[7]}A composition of immersions is again an immersion.^{[8]}Some authors, such as Hartshorne in his book*Algebraic Geometry*and Q. Liu in his book*Algebraic Geometry and Arithmetic Curves*, define immersions as the composite of an open immersion followed by a closed immersion. These immersions are immersions in the sense above, but the converse is false. Furthermore, under this definition, the composite of two immersions is not necessarily an immersion. However, the two definitions are equivalent when*f*is quasi-compact.^{[9]}Note that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: $\backslash operatorname\{Spec\}\; A/I$ and $\backslash operatorname\{Spec\}\; A/J$ may be homeomorphic but not isomorphic. This happens, for example, if*I*is the radical of*J*but*J*is not a radical ideal. When specifying a closed subset of a scheme without mentioning the scheme structure, usually the so-called*reduced*scheme structure is meant, that is, the scheme structure corresponding to the unique radical ideal consisting of all functions vanishing on that closed subset.- integral
- A scheme that is both reduced and irreducible is called
*integral*. For locally Noetherian schemes, to be integral is equivalent to being 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.) For example, the scheme*Spec k*[*t*]/*f*,*f*irreducible polynomial is integral, while*Spec A*×*B*. (*A*,*B*≠ 0) is not. - irreducible
- 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*. Using the correspondence of prime ideals and points in an affine scheme, this means*X*is irreducible iff*X*is connected and the rings A_{i}all have exactly one minimal prime ideal. (Rings possessing exactly one minimal prime ideal are therefore also called irreducible.) 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 and projective space are irreducible, while*Spec**k*[*x,y*]/(*xy*) = is not.

## K-P

- local
- Most important properties of schemes are
*local in nature*, i.e. a scheme*X*has a certain property*P*if and only if for any cover of*X*by open subschemes*X*, i.e._{i}*X*=$\backslash cup$*X*, every_{i}*X*has the property_{i}*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. Consider a scheme*X*and a cover by affine open subschemes*Spec A*. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings_{i}*A*. A property_{i}*P*is local in the above sense, iff the corresponding property of rings is stable under localization. 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. 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*= $\backslash cup$*Spec A*be a covering of a scheme by open affine subschemes. For definiteness, let_{i}*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. Connected, irreducible, reduced, integral, normal, regular, Cohen-Macaulay, locally noetherian, dimension, catenary, - locally of finite type
- The morphism $f$ is
**locally of finite type**if $X$ may be covered by affine open sets $\backslash text\{Spec\; \}B$ such that each inverse image $f^\{-1\}(\backslash text\{Spec\; \}B)$ is covered by affine open sets $\backslash text\{Spec\; \}A$ where each $A$ is finitely generated as a $B$-algebra. - locally Noetherian
- The
*A*are Noetherian rings. If in addition a finite number of such affine spectra covers_{i}*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. For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but $GL\_\backslash infty\; =\; \backslash cup\; GL\_n$ is not. - normal
- An integral scheme is called
*normal*, if the*A*are integrally closed domains. For example, all regular schemes are normal, while singular curves are not._{i} - open
- 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*Y*are mapped to open subschemes of*X*(and similarly for closed). For example, finitely presented flat morphisms are open and proper maps are closed. - An
**open subscheme**of a scheme*X*is an open subset*U*with structure sheaf $\backslash mathcal\{O\}\_X.\_U$.^{[7]} - point
- A scheme $S$ is a locally ringed space, so
*a fortiori*a topological space, but the meanings 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) $\backslash textrm\{Spec\}(K)$, where $K$ is a field, is a morphism from $\backslash textrm\{Spec\}\; (\backslash overline\{K\})$ to $S$ where $\backslash overline\{K\}$ is an algebraic closure of $K$.

*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 $S^\{\backslash prime\}\; \backslash to\; S$ is thought of as- $S^\{\backslash prime\}\; \backslash times\_\{S\}\; \backslash textrm\{Spec\}(\backslash overline\{K\})$.

- projective
- Projective morphisms are defined similarly to affine morphisms: $f$ is called
**projective**if it factors as a closed immersion followed by the projection of a projective space $\backslash mathbb\{P\}^\{n\}\_X:=\; \backslash mathbb\{P\}^n\; \backslash times\_\{\backslash mathrm\{Spec\}\backslash mathbb\; Z\}\; X$ to $X$.^{[10]}Note that this definition is more restrictive than that of EGA, II.5.5.2. The latter defines $f$ to be projective if it is given by the global**Proj**of a quasi-coherent graded*O*-Algebra $\backslash mathcal\; S$ such that $\backslash mathcal\; S\_1$ is finitely generated and generates the algebra $\backslash mathcal\; S$. Both definitions coincide when $X$ is affine or more generally if it is quasi-compact, separated and admits an ample sheaf,_{X}^{[11]}e.g. if $X$ is an open subscheme of a projective space $\backslash mathbb\; P^n\_A$ over a ring $A$. - proper
- A morphism is
**proper**if it is separated,*universally closed*(i.e. such that fiber products with it are closed maps), 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.

## Q-Z

- quasi-compact
- A morphism
*f*:*X*→*Y*is called*quasi-compact*, if for some (equivalently: every) open affine cover of*Y*by some*U*=_{i}*Spec B*, the preimages_{i}*f*^{−1}(*U*) are quasi-compact._{i} - quasi-finite
- The morphism $f$ has
**finite fibers**if the fiber over each point $x\; \backslash in\; X$ is a finite set. A morphism is**quasi-finite**if it is of finite type and has finite fibers. - quasi-separated
- A morphism
*f*:*X*→*Y*is called**quasi-separated**or () if the diagonal morphism*X*is quasi-separated over*Y**X*→*X*×_{Y}*X*is quasi-compact. A scheme*X*is called**quasi-separated**if*X*is quasi-separated over Spec(**Z**).^{[12]} - reduced
- The
*A*are reduced rings. Equivalently, none of its rings of sections $\backslash mathcal\; O\_X(U)$ (_{i}*U*any open subset of*X*) has any nonzero nilpotent element. Allowing non-reduced schemes is one of the major generalizations from varieties to schemes. Any variety is reduced (by definition) while*Spec k*[*x*]/(*x*^{2}) is not. - regular
- The
*A*are regular. For example, smooth varieties over a field are regular, while_{i}*Spec k*[*x,y*]/(*x*^{2}+*x*^{3}-*y*^{2})= is not. - separated
- 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*. 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 $X\; \backslash rightarrow\; \backslash textrm\{Spec\}\; (\backslash mathbb\{Z\})$ is separated. Notice that a topological space*Y*is Hausdorff iff the diagonal embedding- $Y\; \backslash stackrel\{\backslash Delta\}\{\backslash longrightarrow\}\; Y\; \backslash times\; Y$

*affine*scheme*Spec A*is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes):*$A\; \backslash otimes\_\{\backslash mathbb\; Z\}\; A\; \backslash rightarrow\; A,\; a\; \backslash otimes\; a\text{'}\; \backslash mapsto\; a\; \backslash cdot\; a\text{'}$*.

The higher-dimensional analog of étale morphisms are *smooth morphisms*. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness:

- 1) for any
*y*∈*Y*, there are open affine neighborhoods*V*and*U*of*y*,*x*=*f*(*y*), respectively, such that the restriction of*f*to*V*factors as an étale morphism followed by the projection of affine*n*-space over*U*. - 2)
*f*is flat, locally of finite presentation, and for every geometric point $\backslash bar\{y\}$ of*Y*(a morphism from the spectrum of an algebraically closed field $k(\backslash bar\{y\})$ to*Y*), the geometric fiber $X\_\{\backslash bar\{y$ :=X\times_Y \mathrm{Spec} (k(\bar{y})) is a smooth

*n*-dimensional variety over $k(\backslash bar\{y\})$ in the sense of classical algebraic geometry.

}}

**subscheme**, without qualifier, of

*X*is a closed subscheme of an open subscheme of

*X*.

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

Let $\backslash mathfrak\{m\}$ be the maximal ideal of $\backslash mathcal\{O\}\_\{X,f(y)\}$, and let

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

be the ideal generated by the image of $\backslash mathfrak\{m\}$ in $\backslash mathcal\{O\}\_\{Y,y\}$. The morphism $f$ is **unramified** if it is locally of finite presentation and if for all $y$ in $Y$, $\backslash mathfrak\{n\}$ is the maximal ideal of $\backslash mathcal\{O\}\_\{Y,y\}$ and the induced map

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

is a finite, separable field extension. This is the geometric version (and generalization) of an unramified field extension in algebraic number theory.