$#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 $A \otimes_R A^\textrm{o} =
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...$S _ { T }$ is some [[Grothendieck topology|Grothendieck topology]] on the scheme $5$ [[#References|[2]]].
...$O _ { S }$ of $5$. In particular, this set is trivial if $5$ is an affine scheme. If $k$ is a global field (i.e. an algebraic number field or a field of alg
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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
12 KB (2,191 words) - 13:20, 24 November 2013
...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 $
7 KB (1,008 words) - 07:42, 20 March 2024
...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
11 KB (1,513 words) - 17:08, 7 February 2011
This page is a copy of the article [[Picard scheme]] in order to test [[User:Maximilian_Janisch/latexlist|automatic LaTeXifica
...} / S$ of schemes over the scheme $5$. The value of this functor on an $5$-scheme $S ^ { \prime }$ is the group
7 KB (1,013 words) - 13:44, 17 October 2019