...finite type over a field (an [[algebraic variety]]) or over any Noetherian ring.
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$#C+1 = 14 : ~/encyclopedia/old_files/data/R080/R.0800840 Regular scheme
A [[Scheme|scheme]] $ ( X , {\mathcal O} _ {X} ) $
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...al ring at any point does not contain non-zero nilpotent elements. For any scheme $\left({ X,\mathcal{O}_X }\right)$ there is a largest closed reduced subsch
...nilpotent elements of the ring $\mathcal{O}_{X,x}$. A [[Group scheme|group scheme]] over a field of characteristic 0 is reduced [[#References|[3]]].
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$#C+1 = 66 : ~/encyclopedia/old_files/data/N067/N.0607610 Normal scheme
...lity criterion holds [[#References|[1]]]: A [[Noetherian scheme|Noetherian scheme]] $ X $
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is an affine [[Scheme|scheme]]. The scheme $ X $
scheme.
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...acobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring.
...rum $\mathrm{Spec}(A)$; this definition leads to the concept of a Jacobson scheme.
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...$S$ is the spectrum of a field $k$ (cf. [[Spectrum of a ring|Spectrum of a ring]]) and $X=P_k^n$ is a projective space over $k$, then the set of rational $
...X/S)$ over all $P\in\mathbf Q(z)$. For any connected ground scheme $S$ the scheme $\operatorname{Hilb}^P(X/S)$ is also connected [[#References|[2]]].
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$#C+1 = 30 : ~/encyclopedia/old_files/data/R080/R.0800820 Regular ring (in commutative algebra)
A [[Noetherian ring|Noetherian ring]] $ A $
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...ine algebraic set]]. An affine variety is a reduced [[Affine scheme|affine scheme]] $ X $
is the ring of polynomials over $ k $,
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...means that the conductor determines a [[closed subscheme]] of the [[affine scheme]] $\mathrm{Spec}\,A$, consisting of the points that are not normal.
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A commutative group scheme is a group scheme $G$ over a basis scheme $S$, the value of which on any
...e is an Abelian group. Examples of commutative group schemes are [[Abelian scheme|Abelian schemes]] and [[Algebraic torus|algebraic tori]]. A generalization
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...cept of a non-singular [[Algebraic variety|algebraic variety]]. A [[Scheme|scheme]] $ X $
is called a smooth scheme (over $ k $)
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$#C+1 = 47 : ~/encyclopedia/old_files/data/E036/E.0306760 Excellent ring
...es not inherent in arbitrary Noetherian rings. The concept of an excellent ring makes it possible to take the most important properties of geometric rings
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is a [[Smooth scheme|smooth scheme]] (over the field $ k( y) $).
A scheme $ X $
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...es $ x_{0},\ldots,x_{n} $ on $ \mathbf{P}_{\mathbb{k}}^{n} $, a projective scheme is given by a system of homogeneous algebraic equations:
...(compact in the case $ \mathbb{k} = \mathbb{C} $). Conversely, a complete scheme is projective if there is an [[Ample sheaf|ample]], [[Invertible sheaf|inve
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$#C+1 = 33 : ~/encyclopedia/old_files/data/G044/G.0404620 Gorenstein ring
...e dimension (cf. [[Homological dimension|Homological dimension]]). A local ring $ A $
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on an [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] $ X $
of the ring $ A $.
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...41 : ~/encyclopedia/old_files/data/P075/P.0705360 Projective spectrum of a ring
A [[Scheme|scheme]] $ X = \mathop{\rm Proj} ( R) $
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$#C+1 = 74 : ~/encyclopedia/old_files/data/C022/C.0202970 Cohen\ANDMacaulay ring,
''Macaulay ring''
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...{f} $ is the [[Localization in a commutative algebra|localization]] of the ring $ A $ with respect to the multiplicative system $ \{ f^{n} \}_{n \in \Bbb{N
...chemes. A scheme is a ringed space that is locally isomorphic to an affine scheme.
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The notion of a ''group scheme'' is a generalization of the concept of an
schemes over a ground scheme $S$; a
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A generalization of the concepts of a [[Scheme|scheme]] and an [[Algebraic variety|algebraic variety]]. This generalization is th
...roposition reveals the geometric sense of an algebraic space as a quotient scheme according to the étale equivalence relation. Morphisms of an algebraic spa
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...losely related to the [[Divisor class group|divisor class group]] for this ring.
...iety]] $ X $ has a natural algebraic structure (see [[Picard scheme|Picard scheme]]). The reduced connected component of the zero of $ \operatorname{Pic}(X)
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$#C+1 = 71 : ~/encyclopedia/old_files/data/S086/S.0806590 Spectrum of a ring
of a ring $ A $
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is a [[Scheme|scheme]] and $ \Gamma $
is a [[Group scheme|group scheme]] over $ S $,
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====Separable algebra over a ring====
An algebra $A$ over a commutative ring $R$ is separable if $A$ is projective as a left $A \otimes_R A^\textrm{o} =
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scheme $ f ^ { - 1 } ( y) = X \otimes _ {Y} k ( y) $
scheme and $ Y _ {0} ^ {*} $
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...efines the Zariski topology of the [[affine scheme]] ${\rm Spec}\; A$ of a ring $A$ (sometimes called the spectral topology) — the closed sets
{{Cite|Se}}. An affine scheme endowed with the Zariski
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[[Ring of polynomials|ring of polynomials]] $k[T]=k[T_1,\dots,T_n]$. The set ${\mathfrak A}_X$ of all
an affine algebraic set with the unit ideal. The quotient ring $k[X]=k[T]/{\mathfrak A}_X$ is
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...016.png" /> (cf. [[Hopf order|Hopf order]]), then the [[Group
scheme|group
scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
...wledge of Tate–Oort Galois extensions can also be used to characterize the
ring of integers of certain degree-<img align="absmiddle" border="0" src="https:
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...nected [[Affine scheme|affine scheme]], be the spectrum of the commutative ring $ A $(
cf. [[Spectrum of a ring|Spectrum of a ring]]), let $ {\mathcal F} $
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A [[Group scheme|group scheme]] associated to an [[Abelian variety|Abelian variety]] and having a certain
is a local Henselian [[discrete valuation ring]] with residue field $ k $
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A ''scheme'' is a
...a scheme consists of a topological space $X$ (the underlying space of the scheme) and a
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...ing; [[Analytic set|Analytic set]], and [[Spectrum of a ring|Spectrum of a ring]].
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within the framework of the theory of [[scheme]]s. To define the Picard scheme for an arbitrary $ S $-
scheme $ X $
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...are the concept of an induced fibration in topology, and extension of the ring of scalars in the theory of modules.
...ncyclopediaofmath.org/legacyimages/b/b015/b015310/b01531022.png" /> is the scheme
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A transition from a commutative ring $ A $
to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ A [ S ^ {-1} ] $,
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the local ring $ {\mathcal O} _ {X,x} $
is a flat module over the ring $ {\mathcal O} _ {Y,f( x) } $.
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in a commutative associative ring $ R $''
Consider the quotient ring $ R/A $
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[[Local ring|Local ring]]). In defining a morphism $(f,f^\sharp)$ between local ringed spaces $(X,\
[[Scheme|Scheme]]) constitute a full subcategory of the category of local ringed spaces.
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A mapping of modules over a commutative ring which is an analogue of the concept of a differential operator. Let $ R $
be a commutative ring, let $ S $
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be a [[Noetherian ring|Noetherian ring]] and let $ I $
as the graded ring $ S = \oplus _ {k \geq 0 } I ^ {k} $(
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...aximum of the injective (or projective) dimensions of the modules over the ring and is the analogue of the Lebesgue dimension of $ X $.
...homological dimension of a topological space for an algebraic variety or a scheme with a selected cohomology theory. Let $ X $
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...d let $\mathfrak{M}$ be its maximal ideal. Then the spectrum of the graded ring
...e, if the tangent cone is reduced, normal or regular, then so is the local ring $\mathcal{O}_{X,x}$. The dimension and multiplicity of $X$ at $x$ are the s
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...h $ X $ such as the [[Picard group|Picard group]]; the [[Chow ring|Chow ring]]; the [[K-functor| $ K $ -functor]], and the cohomology group are a tool
...or a smooth hypersurface, then the connected component of the unit of this scheme is a variety.
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be a [[Scheme|scheme]] and let $ p $
cf. [[Group scheme|Group scheme]]) such that the sequences
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...braic variety|Complete algebraic variety]]; [[Projective scheme|Projective scheme]].
is a complete scheme over a field $ k $,
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be a commutative ring with unity. To an element $ a \in A $
function) given by the ring homomorphism $ \mathbf Z [ T] \rightarrow A $,
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be a prime number; an $ \ell $-adic sheaf on a scheme $ X $
...e of the category of locally constant constructible sheaves on a connected scheme $ X $
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...ideals (cf. [[Divisorial ideal|Divisorial ideal]]) of a [[Krull ring|Krull ring]] $ A $
into irreducible factors. Thus, a factorial ring has trivial divisor class group.
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