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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 (mathematics)|sheaf]] and [[scheme (mathematics)|scheme]]. The concern here is to list the fundamental technical definitions and properties of scheme theory.
| | All of the trophies from all within the members in your family get added up and furthermore divided by 2 identify your clans overall awards. Playing many different kinds of games assists your gaming time more fun. and your league also determines any battle win bonus. 5 star rating and is defined as known to be somewhat [http://www.Twitpic.com/tag/addictive addictive] as players frequently devote several hours enjoying the game. She focuses on beauty salon business startup and client fascination.<br><br> |
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| See also [[list of algebraic geometry topics]] and [[glossary of classical algebraic geometry]] and [[glossary of commutative algebra]] and [[glossary of stack theory]]
| | Once as a parent you are always concerned with movie recreation content, control what online mods are put globe sport. These down loadable mods are usually developed by players, perhaps not that gaming businesses, therefore there is no ranking system. Using thought was a reasonably un-risky game can revert a lot worse by means of any of these mods.<br><br>Last component There are a associated with Apple fans who play in the above game all internationally. This generation has hardly been the JRPG's best; in fact it's been doing unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Decide to put with all 4 Mission: Impossible movies). Although it is a special day's grand gifts and gestures, one Valentines Day can now blend into another increasingly easily. clash of clans is one among the quickest rising video games as of late.<br><br>Take note of how money your teen would be shelling out for playing games. These kinds most typically associated with products aren't cheap as well as , then there is very much the option of placing your order for much more add-ons during the game itself. In case you loved this article and you want to receive details regarding [http://circuspartypanama.com Clash Of Clans Unlimited Gems Apk] kindly visit our own website. Establish month-to-month and yearly plans available restrictions on the figure of money that is likely to be spent on applications. Also, have conversations for the youngsters about having a budget.<br><br>The company's important to agenda you are apple is consistently locate from association war problems because association wars are usually fought inside a improved breadth absolutely -- this war zone. About the war region, your adapt and advance warfare bases instead of acknowledged villages; therefore, your villages resources, trophies, and absorber are never in risk.<br><br>This particular information, we're accessible in order to alpha dog substituting respects. Application Clash of Clans Cheats' data, let's say for archetype you appetite 1hr (3, 600 seconds) on bulk 20 gems, and then 1 day (90, 700 seconds) to help group 260 gems. A number of appropriately stipulate a task for this kind about band segment.<br><br>You don''t necessarily really need one of the improved troops to win victories. A mass volume of barbarians, your first-level troop, will totally destroy an assailant village, and strangely it''s quite enjoyable to take a the virtual carnage. |
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| {{Compact ToC|short1|sym=yes|
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| seealso=yes|refs=yes}}
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| ==A-E==
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| {{gloss}}
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| {{term|affine}}
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| {{defn|no=1|[[Affine space]] is roughly a vector space where one has forgotten which point is the origin}}
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| {{defn|no=2|An [[affine variety]] is a variety in affine space}}
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| {{defn|no=3|1=
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| 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|global '''Spec''']] construction for sheaves of ''O<sub>X</sub>''-Algebras, defined by analogy with the [[spectrum of a ring]]. Important affine morphisms are [[vector bundle]]s, and [[finite morphism]]s.
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| }}
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| {{term|1=arithmetic genus}}
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| {{defn|1=The [[arithmetic genus]] of a variety is a variation of the Euler characteristic of the trivial line bundle; see [[Hodge number]].}}
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| <span id="B">
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| <span id="C">
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| {{term|1=catenary}}
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| {{defn|1=
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| A scheme is [[Catenary (ring theory)|catenary]], if all chains between two irreducible closed subschemes have the same length.
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| Examples include virtually everything, e.g. varieties over a field, and it is hard to construct examples that are not catenary.
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| }}
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| {{term|1=closed}}
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| {{defn|no=1|1=
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| '''Closed subschemes''' of a scheme ''X'' are defined to be those occurring in the following construction. Let ''J'' be a [[Quasi-coherent sheaf|quasi-coherent]] sheaf of <math>\mathcal{O}_X</math>-[[sheaf of ideals|ideals]]. The [[support of a sheaf|support]] of the [[quotient sheaf]] <math>\mathcal{O}_X/J</math> is a closed subset ''Z'' of ''X'' and <math>(Z,(\mathcal{O}_X/J)|_Z)</math> is a scheme called the '''closed subscheme defined by the [[Quasi-coherent sheaf|quasi-coherent]] [[sheaf of ideals]] ''J'''''.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=4.1.2 and 4.1.3}}</ref> 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.
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| }}
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| {{term|Cohen–Macaulay}}
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| {{defn|1=
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| A scheme is called Cohen-Macaulay
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| if all local rings are [[Cohen-Macaulay ring|Cohen-Macaulay]].
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| For example, regular schemes, and ''Spec k''[''x,y'']/(''xy'') are Cohen–Macaulay, but
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| [[Image:Non cohen macaulay scheme thumb.png|50px]] is not.
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| }}
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| {{term|1=connected}}
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| {{defn|1=
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| The scheme is ''[[Connected space|connected]]'' as a topological space. Since the [[connected component (topology)|connected components]] refine the [[irreducible component]]s any irreducible scheme is connected but not vice versa. An [[affine scheme]] ''Spec(R)'' is connected [[iff]] the ring ''R'' possesses no [[idempotent]]s other than 0 and 1; such a ring is also called a '''connected ring'''.
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| Examples of connected schemes include [[affine space]], [[projective space]], and an example of a scheme that is not connected is
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| ''Spec''(''k''[''x'']×''k''[''x''])
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| }}
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| <span id="D">
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| {{term|1=dimension}}
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| {{defn|1=
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| The [[Krull dimension|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]].
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| Examples: [[equidimensional scheme]]s in dimension 0: [[Artinian ring|Artinian]] schemes, 1: [[algebraic curves]], 2: [[algebraic surfaces]].
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| }}
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| {{term|1=dominant}}
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| {{defn|1=
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| A morphism is called ''dominant'', if the image ''f''(''Y'') is [[dense set|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''.
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| }}
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| <span id="E">
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| {{term|1=étale}}
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| {{defn|1=
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| A morphism <math> f </math> is [[étale morphism|étale]] if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties <math> X </math> and <math> Y </math> over an algebraically closed [[field (mathematics)|field]], étale morphisms are precisely those inducing an isomorphism of tangent spaces <math> df: T_{x} X \rightarrow T_{f(x)} Y</math>, which coincides with the usual notion of étale map in differential geometry.
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| 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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| }}
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| {{glossend}}
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| ==F-J==
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| {{gloss}}
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| {{term|1=final}}
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| {{defn|1=
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| 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 <math> \mathbb{Z} </math> of integers; so that any scheme <math> S </math> is ''over'' <math> \textrm{Spec} (\mathbb{Z}) </math>, and in a unique way.
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| }}
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| {{term|1=finite}}
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| {{defn|1=
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| The morphism <math> f </math> is '''finite''' if <math>X</math> may be covered by affine open sets <math> \text{Spec }B </math> such that each <math> f^{-1}(\text{Spec }B) </math> is affine — say of the form <math> \text{Spec }A </math> — and furthermore <math> A </math> is finitely generated as a <math> B </math>-module. See [[finite morphism]].
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| The morphism <math> f </math> is '''locally of finite type''' if <math> X </math> may be covered by affine open sets <math> \text{Spec }B </math> such that each inverse image <math>f^{-1}(\text{Spec }B)</math> is covered by affine open sets <math>\text{Spec }A</math> where each <math> A </math> is finitely generated as a <math>B</math>-algebra.
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| The morphism <math> f </math> is '''finite type''' if <math> X </math> may be covered by affine open sets <math> \text{Spec }B </math> such that each inverse image <math>f^{-1}(\text{Spec }B)</math> is covered by finitely many affine open sets <math>\text{Spec }A</math> where each <math> A </math> is finitely generated as a <math>B</math>-algebra.
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| The morphism <math> f </math> has '''finite fibers''' if the fiber over each point <math> x \in X </math> is a finite set. A morphism is '''[[quasi-finite morphism|quasi-finite]]''' if it is of finite type and has finite fibers.
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| 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.
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| If ''y'' is a point of ''Y'', then the morphism ''f'' is '''of finite presentation at ''y''''' (or '''finitely presented at ''y''''') if there is an open affine subset ''U'' of ''f(y)'' and an open affine neighbourhood ''V'' of ''y'' such that ''f''(''V'') ⊆ ''U'' and <math>\mathcal{O}_Y(V)</math> is a [[finitely presented algebra]] over <math>\mathcal{O}_X(U)</math>. 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.<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=§1.4}}</ref>
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| The morphism ''f'' is '''of finite presentation''' (or '''''Y'' is finitely presented over ''X''''') if it is locally of finite presentation, quasi-compact, and quasi-separated. If ''X'' is locally Noetherian, then ''f'' is of finite presentation if, and only if, it is of finite type.<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=§1.6}}</ref>
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| }}
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| {{term|1=flat}}
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| {{defn|1=
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| A morphism <math> f </math> is [[flat morphism (ring theory)|flat]] if it gives rise to a [[Flat map (ring theory)|flat map]] on stalks. When viewing a morphism as a family of schemes parametrized by the points of <math> X </math>, the geometric meaning of flatness could roughly be described by saying that the fibers <math>f^{-1}(x)</math> do not vary too wildly.
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| }}
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| <span id="G">
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| <span id="H">
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| <span id="I">
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| {{term|1=image}}
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| {{defn|1=
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| If {{nowrap|''f'' : ''Y'' → ''X''}} is any morphism of schemes, the '''scheme-theoretic image''' of ''f'' is the unique ''closed'' subscheme {{nowrap|''i'' : ''Z'' → ''X''}} which satisfies the following [[universal property]]:
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| #''f'' factors through ''i'',
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| #if {{nowrap|''j'' : ''Z''′ → ''X''}} is any closed subscheme of ''X'' such that ''f'' factors through ''j'', then ''i'' also factors through ''j''.<ref>{{harvnb|Hartshorne|1977|loc=Exercise II.3.11(d)}}</ref><ref>[http://www.math.columbia.edu/algebraic_geometry/stacks-git/morphisms.pdf The Stacks Project], Chapter 21, §4.</ref>
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| This notion is distinct for that of the usual set-theoretic image of ''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''.
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| }}
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| {{term|1=immersion}}
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| {{defn|1=
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| '''Immersions''' {{nowrap|''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.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=4.2.1}}</ref> 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 <math>f^\sharp: \mathcal{O}_X \to f_* \mathcal{O}_Y</math> is surjective.<ref name="HartshorneII3"/> A composition of immersions is again an immersion.<ref>{{harvnb|Grothendieck|Dieudonné|1960|loc=4.2.5}}</ref>
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| 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.<ref>Q. Liu, ''Algebraic Geometry and Arithmetic Curves, exercise 2.3</ref>
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| Note that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: <math>\operatorname{Spec} A/I</math> and <math>\operatorname{Spec} A/J</math> 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.
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| }}
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| {{term|1=integral}}
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| {{defn|1=
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| {{Anchor|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 domain]]s. (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.)
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| For example, the scheme ''Spec k''[''t'']/''f'', ''f'' [[irreducible polynomial]] is integral, while ''Spec A''×''B''. (''A'', ''B'' ≠ 0) is not.
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| }}
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| {{term|1=irreducible}}
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| {{defn|1=
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| A scheme ''X'' is said to be ''[[hyperconnected space|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<sub>i</sub> all have exactly one minimal [[prime ideal]]. (Rings possessing exactly one minimal prime ideal are therefore also called [[Irreducible ring|irreducible]].) Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its [[irreducible component]]s. [[Affine space]] and [[projective space]] are irreducible, while
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| ''Spec'' ''k''[''x,y'']/(''xy'') = [[Image:Reducible scheme.png|50px]] is not.
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| }}
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| <span id="J">
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| {{glossend}}
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| ==K-P==
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| {{gloss}}
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| <span id="L">
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| {{term|1=local}}
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| {{defn|1=
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| 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<sub>i</sub>'', i.e. ''X''=<math>\cup</math> ''X<sub>i</sub>'', every ''X<sub>i</sub>'' 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]].
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| Consider a scheme ''X'' and a cover by affine open subschemes ''Spec A<sub>i</sub>''. Using the dictionary between [[commutative ring|(commutative) rings]] and [[affine scheme]]s local properties are thus properties of the rings ''A<sub>i</sub>''. A property ''P'' is local in the above sense, iff the corresponding property of rings is stable under [[Localization of a ring|localization]].
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| For example, we can speak of ''[[noetherian scheme|locally Noetherian]]'' schemes, namely those which are covered by the spectra of [[Noetherian ring]]s. 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 ring|reduced]] (i.e., has no non-zero [[nilpotent]] elements), then so are its localizations.
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| 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.
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| The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let ''X'' = <math>\cup</math> ''Spec A<sub>i</sub>'' be a covering of a scheme by open affine subschemes. For definiteness, let ''k'' denote a [[field (mathematics)|field]] in the following. Most of the examples also work with the integers '''Z''' as a base, though, or even more general bases.
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| Connected,
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| irreducible,
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| reduced,
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| integral,
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| normal,
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| regular,
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| Cohen-Macaulay,
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| locally noetherian,
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| dimension,
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| catenary,
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| }}
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| {{term|1=locally of finite type}}
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| {{defn|1=
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| The morphism <math> f </math> is '''locally of finite type''' if <math> X </math> may be covered by affine open sets <math> \text{Spec }B </math> such that each inverse image <math>f^{-1}(\text{Spec }B)</math> is covered by affine open sets <math>\text{Spec }A</math> where each <math> A </math> is finitely generated as a <math>B</math>-algebra.
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| }}
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| {{term|1=locally Noetherian}}
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| {{defn|1=
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| The ''A<sub>i</sub>'' are [[Noetherian ring|Noetherian]] rings. If in addition a finite number of such 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. | |
| For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but
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| <math>GL_\infty = \cup GL_n</math> is not.
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| }}
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| <span id="M">
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| <span id="N">
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| {{term|1=normal}}
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| {{defn|1=
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| An integral scheme is called ''[[normal scheme|normal]]'', if the ''A<sub>i</sub>'' are [[integrally closed]] domains.
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| For example, all regular schemes are normal, while singular curves are not.
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| }}
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| <span id="O">
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| {{term|1=open}}
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| {{defn|1=
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| A morphism of schemes is called ''open'' (''closed''), if the underlying map of topological spaces is [[open map|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.
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| }}
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| {{defn|2=
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| An '''open subscheme''' of a scheme ''X'' is an open subset ''U'' with structure sheaf <math>\mathcal{O}_X|_U</math>.<ref name="HartshorneII3">{{harvnb|Hartshorne|1977|loc=§II.3}}</ref>
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| }}
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| <span id="P">
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| {{term|1=point}}
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| {{defn|1=
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| A scheme <math> S </math> is a [[locally ringed space]], so ''a fortiori'' a [[topological space]], but the meanings of ''point of <math> S </math>'' are threefold:
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| #a point <math> P </math> of the underlying topological space;
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| #a <math> T </math>-valued point of <math>S</math> is a morphism from <math> T </math> to <math> S </math>, for any scheme <math> T </math>;
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| #a ''geometric point'', where <math> S </math> is defined over (is equipped with a morphism to) <math> \textrm{Spec}(K) </math>, where <math> K </math> is a [[field (mathematics)|field]], is a morphism from <math> \textrm{Spec} (\overline{K}) </math> to <math> S </math> where <math> \overline{K} </math> is an [[algebraic closure]] of <math>K</math>.
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| Geometric points are what in the most classical cases, for example [[algebraic varieties]] that are [[complex manifold]]s, would be the ordinary-sense points. The points <math> P </math> of the underlying space include analogues of the [[generic point]]s (in the sense of [[Zariski]], not that of [[André Weil]]), which specialise to ordinary-sense points. The <math> T </math>-valued points are thought of, via [[Yoneda's lemma]], as a way of identifying <math> S </math> with the [[representable functor]] <math> h_{S} </math> 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 <math> T </math>-valued points were a massive further step.
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| As part of the predominating [[Grothendieck's relative point of view|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 product]]s of two morphisms. For example, a '''geometric fiber''' of a morphism <math> S^{\prime} \to S </math> is thought of as
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| :<math> S^{\prime} \times_{S} \textrm{Spec}(\overline{K}) </math>.
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| This makes the extension from [[affine scheme]]s, where it is just the [[tensor product of R-algebras]], to all schemes of the fiber product operation a significant (if technically anodyne) result.
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| }}
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| {{term|1=projective}}
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| {{defn|1=
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| Projective morphisms are defined similarly to affine morphisms: <math> f </math> is called '''[[projective variety|projective]]''' if it factors as a closed immersion followed by the projection of a [[projective space]] <math> \mathbb{P}^{n}_X := \mathbb{P}^n \times_{\mathrm{Spec}\mathbb Z} X</math> to <math> X </math>.<ref>{{harvnb|Hartshorne|1977|loc=II.4}}</ref> Note that this definition is more restrictive than that of [[Éléments de géométrie algébrique|EGA]], II.5.5.2. The latter defines <math> f </math> to be projective if it is given by the [[global Proj|global '''Proj''']] of a quasi-coherent graded ''O<sub>X</sub>''-Algebra <math>\mathcal S</math> such that <math>\mathcal S_1</math> is finitely generated and generates the algebra <math>\mathcal S</math>. Both definitions coincide when <math>X</math> is affine or more generally if it is quasi-compact, separated and admits an ample sheaf,<ref>[[Éléments de géométrie algébrique|EGA]], II.5.5.4(ii).</ref> e.g. if <math>X</math> is an open subscheme of a projective space <math>\mathbb P^n_A</math> over a ring <math>A</math>.
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| }}
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| {{term|1=proper}}
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| {{defn|1=
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| A morphism is '''[[proper morphism|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.
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| }}
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| {{glossend}}
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| ==Q-Z==
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| {{gloss}}
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| {{term|1=quasi-compact}}
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| {{defn|1=
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| A morphism {{nowrap|''f'' : ''X'' → ''Y''}} is called ''[[quasi-compact morphism|quasi-compact]]'', if for some (equivalently: every) open affine cover of ''Y'' by some ''U<sub>i</sub>'' = ''Spec B<sub>i</sub>'', the preimages ''f''<sup>−1</sup>(''U<sub>i</sub>'') are [[quasi-compact]].
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| }}
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| {{term|1=quasi-finite}}
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| {{defn|1=
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| The morphism <math> f </math> has '''finite fibers''' if the fiber over each point <math> x \in X </math> is a finite set. A morphism is '''[[quasi-finite morphism|quasi-finite]]''' if it is of finite type and has finite fibers.
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| }}
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| {{term||1=quasi-separated}}
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| {{defn|1=
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| A morphism {{nowrap|''f'' : ''X'' → ''Y''}} is called '''[[quasi-separated]]''' or ('''''X'' is quasi-separated over ''Y''''') if the diagonal morphism {{nowrap|''X'' → ''X'' ×<sub>''Y''</sub>''X''}} is quasi-compact. A scheme ''X'' is called '''quasi-separated''' if ''X'' is quasi-separated over Spec('''Z''').<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=1.2.1}}</ref>
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| }}
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| <span id="R">
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| {{term|1=reduced}}
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| {{defn|1=
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| The ''A<sub>i</sub>'' are [[reduced ring]]s. Equivalently, none of its rings of sections <math>\mathcal O_X(U)</math> (''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.
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| Any [[algebraic variety|variety]] is reduced (by definition) while
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| ''Spec k''[''x'']/(''x''<sup>2</sup>) is not.
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| }}
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| {{term|1=regular}}
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| {{defn|1=
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| The ''A<sub>i</sub>'' are [[Regular ring|regular]].
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| For example, [[smooth scheme|smooth varieties]] over a field are regular, while
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| ''Spec k''[''x,y'']/(''x''<sup>2</sup>+''x''<sup>3</sup>-''y''<sup>2</sup>)=[[Image:Non regular scheme thumb.png|50px]] is not.
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| }}
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| <span id="S">
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| {{term|1=separated}}
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| {{defn|1=
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| A separated morphism is a morphism <math> f </math> such that the [[fiber product]] of <math> f </math> with itself along <math> f </math> has its [[diagonal]] as a closed subscheme — in other words, the diagonal map is a ''closed immersion''. | |
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| As a consequence, a scheme <math> X </math> is '''separated''' when the diagonal of <math> X </math> within the ''scheme product'' of <math> X </math> 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 <math>X \rightarrow \textrm{Spec} (\mathbb{Z})</math> is separated.
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| Notice that a [[topological space]] ''Y'' is Hausdorff iff the diagonal embedding
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| :<math>Y \stackrel{\Delta}{\longrightarrow} Y \times Y</math>
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| is closed. In algebraic geometry, the above formulation is used because a scheme which is a Hausdorff space is necessarily empty or 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) <math>X \times_{\textrm{Spec} (\mathbb{Z})} X</math>, which is different from the product of topological spaces.
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| Any ''affine'' scheme ''Spec A'' is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes):
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| :''<math>A \otimes_{\mathbb Z} A \rightarrow A, a \otimes a' \mapsto a \cdot a'</math>''.
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| }}
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| {{term|1=smooth}}
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| {{defn|1=
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| {{main|smooth morphism}}
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| The higher-dimensional analog of étale morphisms are ''smooth morphisms''. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness:
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| :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 space|affine ''n''-space]] over ''U''.
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| :2) ''f'' is flat, locally of finite presentation, and for every geometric point <math>\bar{y}</math> of ''Y'' (a morphism from the spectrum of an algebraically closed field <math>k(\bar{y})</math> to ''Y''), the geometric fiber <math>X_{\bar{y}}:=X\times_Y \mathrm{Spec} (k(\bar{y}))</math> is a smooth ''n''-dimensional variety over <math>k(\bar{y})</math> in the sense of classical algebraic geometry.
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| }}
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| {{term|1=subscheme}}
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| {{defn|1=
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| A '''subscheme''', without qualifier, of ''X'' is a closed subscheme of an open subscheme of ''X''.
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| }}
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| <span id="T">
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| <span id="U">
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| {{term|1=universally}}
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| {{defn|1=
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| A morphism has some property universally if all base changes of the morphism have this property. Examples include [[universally catenary]], [[universally injective]].
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| }}
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| {{term|1=unramified}}
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| {{defn|1=
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| For a point <math> y </math> in <math> Y </math>, consider the corresponding morphism of local rings
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| :<math>f^\# \colon \mathcal{O}_{X, f(y)} \to \mathcal{O}_{Y, y}.</math>.
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| Let <math> \mathfrak{m} </math> be the maximal ideal of <math> \mathcal{O}_{X,f(y)} </math>, and let
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| : <math>\mathfrak{n} = f^\#(\mathfrak{m}) \mathcal{O}_{Y,y}</math>
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| be the ideal generated by the image of <math> \mathfrak{m} </math> in <math>\mathcal{O}_{Y,y} </math>. The morphism <math> f </math> is '''unramified''' (resp. '''G-unramified''') if it is locally of finite type (resp. locally of finite presentation) and if for all <math> y </math> in <math> Y </math>, <math> \mathfrak{n} </math> is the maximal ideal of <math> \mathcal{O}_{Y,y} </math> and the induced map
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| :<math>\mathcal{O}_{X,f(y)}/\mathfrak{m} \to \mathcal{O}_{Y,y}/\mathfrak{n} </math>
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| is a [[finite field extension|finite]] [[separable field extension]].<ref>The notion G-unramified is what is called "unramified" in EGA, but we follow Raynaud's definition of "unramified", so that [[closed immersion]]s are unramified. See [http://stacks.math.columbia.edu/tag/02G4 Tag 02G4 in the Stacks Project] for more details.</ref> This is the geometric version (and generalization) of an [[ramification|unramified field extension]] in [[algebraic number theory]].
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| }}
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| {{glossend}}
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| <span id="V">
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| <span id="W">
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| <span id="X">
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| <span id="Y">
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| <span id="Z">
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{EGA | book=I}}
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| *{{EGA | book=II}}
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| *{{EGA | book=III-1}}
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| *{{EGA | book=III-2}}
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| *{{EGA | book=IV-1}}
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| *{{EGA | book=IV-2}}
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| *{{EGA | book=IV-3}}
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| *{{EGA | book=IV-4}}
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| *{{Hartshorne AG}}
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| {{DEFAULTSORT:Glossary Of Scheme Theory}}
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| [[Category:Glossaries of mathematics|Scheme theory]]
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| [[Category:Scheme theory| ]]
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