...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) $
3 KB (468 words) - 09:09, 6 January 2024
$#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.
6 KB (884 words) - 09:29, 13 December 2016