...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
16 KB (2,808 words) - 16:01, 27 July 2024
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 ,
4 KB (643 words) - 22:12, 5 June 2020
is an algebraic variety (or scheme) over a local field $ K $
is defined by a set of equations with coefficients from the ring $ A $
5 KB (794 words) - 18:12, 18 July 2024
Then a connected affine formal scheme is a covariant functor $ H $
is said to be commutative. Every connected [[Group scheme|group scheme]] $ G $
17 KB (2,537 words) - 22:38, 15 December 2019
...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]].
14 KB (2,170 words) - 05:55, 19 July 2024
if the local ring $ {\mathcal O} _ {X,x} $
is naturally isomorphic to the integral closure of the ring $ {\mathcal O} _ {X,x} $
8 KB (1,138 words) - 07:29, 21 July 2024
be a [[Ring|ring]] (associative with one) and let $ A ^ {*} $
be a commutative ring and $ A $
9 KB (1,429 words) - 18:05, 1 July 2024
is a locally [[Noetherian scheme|Noetherian scheme]], $ {\mathcal F} $
be a locally Noetherian scheme or a complex-analytic space, $ Z $
10 KB (1,477 words) - 22:17, 5 June 2020
...thrm{Spec}(A)$, for $A$ a commutative [[ring with identity]]; cf. [[Affine scheme]].
3 KB (527 words) - 07:33, 24 November 2023
...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 }
27 KB (4,003 words) - 17:43, 1 July 2020
...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} } $,
12 KB (1,775 words) - 11:58, 4 April 2020
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 $(
12 KB (1,877 words) - 17:38, 16 July 2024
[[Ring|ring]]) over a
...\End{\textrm{End}}\End_L(\Ga)$ of the additive group scheme $\Ga/L$ is the ring of additive polynomials $f(x)\in L[x]$, i.e., of polynomials satisfying $f(
19 KB (3,204 words) - 20:11, 14 April 2012
''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
11 KB (1,773 words) - 09:56, 13 February 2024
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 $
10 KB (1,385 words) - 03:10, 2 March 2022
...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
8 KB (1,219 words) - 21:00, 13 July 2020
be a scheme over a field
be a scheme and let
41 KB (5,916 words) - 11:24, 26 March 2023
is an associative algebra over a commutative ring $ K $
is the ring of integers, $ G $
19 KB (2,870 words) - 09:48, 26 March 2023
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.
14 KB (2,077 words) - 21:17, 17 January 2021
...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).
11 KB (1,916 words) - 00:44, 12 August 2019