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...finite type over a field (an [[algebraic variety]]) or over any Noetherian ring.

$#C+1 = 14 : ~/encyclopedia/old_files/data/R080/R.0800840 Regular scheme A [[Scheme|scheme]] $ ( X , {\mathcal O} _ {X} ) $

...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]]].

$#C+1 = 66 : ~/encyclopedia/old_files/data/N067/N.0607610 Normal scheme ...lity criterion holds [[#References|[1]]]: A [[Noetherian scheme|Noetherian scheme]] $ X $

is an affine [[Scheme|scheme]]. The scheme $ X $ scheme.

...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.

...$S$ 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]]].

$#C+1 = 30 : ~/encyclopedia/old_files/data/R080/R.0800820 Regular ring (in commutative algebra) A [[Noetherian ring|Noetherian ring]] $ A $

...ine algebraic set]]. An affine variety is a reduced [[Affine scheme|affine scheme]] $ X $ is the ring of polynomials over $ k $,

...means that the conductor determines a [[closed subscheme]] of the [[affine scheme]] $\mathrm{Spec}\,A$, consisting of the points that are not normal.

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

...cept of a non-singular [[Algebraic variety|algebraic variety]]. A [[Scheme|scheme]] $ X $ is called a smooth scheme (over $ k $)

$#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

is a [[Smooth scheme|smooth scheme]] (over the field $ k( y) $). A scheme $ X $

...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

$#C+1 = 33 : ~/encyclopedia/old_files/data/G044/G.0404620 Gorenstein ring ...e dimension (cf. [[Homological dimension|Homological dimension]]). A local ring $ A $

on an [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] $ X $ of the ring $ A $.

...41 : ~/encyclopedia/old_files/data/P075/P.0705360 Projective spectrum of a ring A [[Scheme|scheme]] $ X = \mathop{\rm Proj} ( R) $

$#C+1 = 74 : ~/encyclopedia/old_files/data/C022/C.0202970 Cohen\ANDMacaulay ring, ''Macaulay ring''

...{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.

The notion of a ''group scheme'' is a generalization of the concept of an schemes over a ground scheme $S$; a

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

...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)

$#C+1 = 71 : ~/encyclopedia/old_files/data/S086/S.0806590 Spectrum of a ring of a ring $ A $

is a [[Scheme|scheme]] and $ \Gamma $ is a [[Group scheme|group scheme]] over $ S $,

====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} =

scheme $ f ^ { - 1 } ( y) = X \otimes _ {Y} k ( y) $ scheme and $ Y _ {0} ^ {*} $

...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

[[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

...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:

...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} $

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 $

A ''scheme'' is a ...a scheme consists of a topological space $X$ (the underlying space of the scheme) and a

...ing; [[Analytic set|Analytic set]], and [[Spectrum of a ring|Spectrum of a ring]].

within the framework of the theory of [[scheme]]s. To define the Picard scheme for an arbitrary $ S $- scheme $ X $

...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

A transition from a commutative ring $ A $ to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ A [ S ^ {-1} ] $,

the local ring $ {\mathcal O} _ {X,x} $ is a flat module over the ring $ {\mathcal O} _ {Y,f( x) } $.

in a commutative associative ring $ R $'' Consider the quotient ring $ R/A $

[[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.

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 $

be a [[Noetherian ring|Noetherian ring]] and let $ I $ as the graded ring $ S = \oplus _ {k \geq 0 } I ^ {k} $(

...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 $

...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

...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.

be a [[Scheme|scheme]] and let $ p $ cf. [[Group scheme|Group scheme]]) such that the sequences

...braic variety|Complete algebraic variety]]; [[Projective scheme|Projective scheme]]. is a complete scheme over a field $ k $,

be a commutative ring with unity. To an element $ a \in A $ function) given by the ring homomorphism $ \mathbf Z [ T] \rightarrow A $,

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 $

...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.

The ''Chow ring'' of a non-singular quasi-projective algebraic variety ...Algebraic cycle|Algebraic cycle]]) on that variety. Multiplication in this ring is defined in terms of intersections of cycles (cf. [[Intersection theory|I

''to an algebraic variety or scheme $ X $ is the maximal ideal of the [[Local ring|local ring]] $ {\mathcal O} _ {X ,x } $

...e inclusion $\mathfrak{p}_x \subseteq \mathfrak{p}_y$ holds. When $A$ is a ring without zero divisors, the point $\{0\}$ is the generic one. The relation o

over a ring of Witt vectors $ W (k) $ ( is a left module over the ring $ D _{k} $ (

is an irreducible separable scheme of finite type over an algebraically closed field $ k $, is a Noetherian scheme, then they are coherent. If $ X $

The valuation ring $ R= \{ {a \in K } : {| a | \leq 1 } \} $ The ring $ \{ {f \in T _ {n} } : {\| f \| \leq 1 } \} $

...$, then $K(C)$ is the Witt–Grothendieck group of $k$ (cf. [[Witt ring|Witt ring]]). ...is the [[Picard group|Picard group]] $\mathrm{Pic}(A)$ of a ring (or of a scheme). It is the Grothendieck group associated to the commutative monoid of isom

be an associative, commutative ring with unit element. Witt vectors with components in $ A $ ...Witt vectors with the operations introduced above form a ring, called the ring of Witt vectors and denoted by $ W(A) $.

...[[Ring of polynomials|ring of polynomials]] is an associative-commutative ring without zero divisors (that is, a product of non-zero polynomials cannot be ...ivision with remainder can be carried out using the [[Horner scheme|Horner scheme]].

...sm of (left) modules with common ring of scalars (in this, the role of the ring is played by the domain of definition of the functors, and the functors the or a category of diagrams with scheme $ \mathfrak K $.

if the local ring $ {\mathcal O} _ {X,x} $ is naturally isomorphic to the integral closure of the ring $ {\mathcal O} _ {X,x} $

...generalization of the concept of a divisor of an element of a commutative ring. First introduced by E.E. Kummer ...ctorization, the elements of which are known as (integral) divisors of the ring $A$. The theory of divisors makes it possible to reduce a series of problem

is the category of modules over a Noetherian commutative ring $ \Lambda $, is a [[Scheme|scheme]], the converse statement holds for quasi-coherent $ {\mathcal O} _ {X ,

is an algebraic variety (or scheme) over a local field $ K $ is defined by a set of equations with coefficients from the ring $ A $

Then a connected affine formal scheme is a covariant functor $ H $ is said to be commutative. Every connected [[Group scheme|group scheme]] $ G $

...f rank $ n $. The set of all algebraic vector bundles of rank $ n $ on the scheme is in one-to-one correspondence with the cohomology set $ {H^{1}}(X,\operat ...thbf{P}(E) $, just like to a vector space one can associate a [[Projective scheme|projective space]].

be a [[Ring|ring]] (associative with one) and let $ A ^ {*} $ be a commutative ring and $ A $

is a locally [[Noetherian scheme|Noetherian scheme]], $ {\mathcal F} $ be a locally Noetherian scheme or a complex-analytic space, $ Z $

...thrm{Spec}(A)$, for $A$ a commutative [[ring with identity]]; cf. [[Affine scheme]].

...oetherian [[Local ring|local ring]] (cf. also [[Noetherian ring|Noetherian ring]]) with [[Maximal ideal|maximal ideal]] $\mathfrak{m}$ and $d = \operatorna ...f $A$ is a Buchsbaum ring, then $A _ { \mathfrak{p} }$ is a Cohen–Macaulay ring with $\operatorname { dim } A _ { \mathfrak { p } } = \operatorname { dim }

...xist for finding solutions of systems of such Diophantine equations in the ring of integers of any number field of finite degree over $ \mathbf Q $. be a system of polynomial equations having integral coefficients. The ring of all algebraic integers, $ {\widetilde{\mathbf Z} } $,

has the structure of an Abelian variety (cf. [[Picard scheme|Picard scheme]]). The operation of intersection of cycles makes it possible to define a m converting it into a commutative ring, called the Chow ring of the variety $ X $(

[[Ring|ring]]) over a ...\End{\textrm{End}}\End_L(\Ga)$ of the additive group scheme $\Ga/L$ is the ring of additive polynomials $f(x)\in L[x]$, i.e., of polynomials satisfying $f(

''simplicial scheme, abstract simplicial complex'' ...re is usually called an (abstract) simplicial complex; the term simplicial scheme would normally be understood to mean a simplicial object in the category of

is a [[Chow ring|Chow ring]], or $ H ^ {. } X $ is the ring associated to the Grothendieck ring $ K ^ {0} ( X) $ (see [[#References|[2]]], [[#References|[7]]]). Let $

...ension]]) one, and "adding a point" $\infty$ to the corresponding [[Scheme|scheme]] $\operatorname {Spec}( \mathbf{Z})$ makes it look like a complete curve. In higher dimension, given a regular projective flat scheme $X$ over $\mathbf{Z}$, one considers pairs $( Z , g )$ consisting of an alg

be a scheme over a field be a scheme and let

is an associative algebra over a commutative ring $ K $ is the ring of integers, $ G $

Any Boolean algebra is a [[Boolean ring|Boolean ring]] with a unit element with respect to the operations of "addition" ( $ + any Boolean ring with a unit element can be considered as a Boolean algebra.

...n $fg=0$ on $L$ implies that $f$ or $g$ is zero; in that case the quotient ring $K_L = K/I_L$ does not have divisors of zero (cf. ...o divisor]]) and is called the ring of polynomials on $L$ (here $K$ is the ring of all polynomials).

of all unitary left modules over an arbitrary associative ring $ R $ with diagram scheme $ \mathfrak D $

...functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.). ...in of application of homological algebra is the category of modules over a ring. Most of the results known for modules may be applied to abelian categories

...ring of polynomials in a finite number of variables over a field or over a ring of integers any ideal is generated by a finite number of elements (has a fi be a ring of polynomials over $ k $,

be the [[field of fractions]] of a regular discretely-normed ring $ O $ then its minimal [[Néron model|Néron model]] is a smooth group scheme $ X \rightarrow \mathop{\rm Spec} O $,

...ery such is axiomatizable by a single sentence and an "axiom of infinity" -scheme. ...matrix ring over an algebraically closed field; every superstable division ring is an algebraically closed field. G.L. Cherlin (middle of the 1970's) conje

...s (this principle becomes a theorem if "space" is understood to be "affine scheme" or "compact topological space" , and "algebra" is understood to mean "C*-a over a commutative ring $ k $

of a scheme $ V $ The cohomology ring of a smooth complex projective hypersurface can be expressed completely in

...between them. A configuration can also be defined by means of an abstract scheme, for example, in the table (Fig. b) the incidences are shown (denoted by cr gon such that its vertices and sides are pairwise incident. The abstract scheme of a configuration $ ( 3 _ {2} ) $

...distinguish between sets and classes. The most commonly used is the axiom scheme of Gödel–Bernays–von Neumann. of (right) modules over a fixed ring $ R $,

...ns for some function $f ( x , t )$ and coefficients $u_m ( x , t )$ from a ring ...an elementary Darboux transformation on a [[Differential ring|differential ring]] [[#References|[a10]]], [[#References|[a11]]]. A particular case of this n

...in Hodge theory, yields information about the structure of the cohomology ring $ H ^ {*} M $ but rather a commutative ring $ F $(

...pen subset $U\subseteq X$ an Abelian group $F(U)$ (a ring, a module over a ring, etc.) and to every pair of open sets $V\subseteq U$ a homomorphism $F_V^U: $\G_c(X,\cF)$) be the group (ring, module, etc.) of all sections of

be an [[Algebraic variety|algebraic variety]] or a [[Scheme|scheme]] of finite type over a field $ k $. is said to be singular if the corresponding local ring $ {\mathcal O} _{X,x} $

...terminal. The same model was used for the performance evaluation of token ring local area network in the 1980s. It can also be applied to patrolling machi

...oup]]). Considering such groups as group schemes (cf. [[Group scheme|Group scheme]]), that is, as a family of groups $G ( K )$, where $K$ varies over some cl where $e _ { \mu }$ denotes a formal basis element of the integral group ring ${\bf Z} \Lambda ( n )$ corresponding to $\mu$. The formal character $\chi_

be the maximal ideal of the local ring $ {\mathcal O} _ {x} $ ($ x \in X $). ...s see [[#References|[9]]]; for the general case, see ). In particular, any scheme of finite type over a complete normed field $ k $

...eaf $\widetilde \omega_X$ is invertible if and only if $X$ is a Gorenstein scheme (cf. [[Gorenstein ring|Gorenstein ring]]).