The valuation ring $ R= \{ {a \in K } : {| a | \leq 1 } \} $
The ring $ \{ {f \in T _ {n} } : {\| f \| \leq 1 } \} $
8 KB (1,161 words) - 08:25, 6 June 2020
...$, 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
4 KB (701 words) - 06:11, 26 March 2023
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) $.
17 KB (2,502 words) - 17:25, 22 December 2019
A module $N$ over a ring of Witt vectors $V ( k )$ (cf. [[Witt vector|Witt vector]]), where $k$ is a
...definition, $N$ is a left module over the ring $D _ { k }$ (the Dieudonné ring) generated by $V ( k )$ and two variables $H ^ { \prime }$ and $V$ connecte
4 KB (566 words) - 13:36, 17 October 2019
...[[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]].
9 KB (1,497 words) - 10:44, 27 June 2015
...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 $.
4 KB (489 words) - 16:56, 23 November 2023
if the local ring $ {\mathcal O} _ {X,x} $
is naturally isomorphic to the integral closure of the ring $ {\mathcal O} _ {X,x} $
8 KB (1,143 words) - 16:48, 19 February 2022
...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 (2,805 words) - 02:18, 6 January 2022
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 (793 words) - 16:10, 1 April 2020
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
...\rightarrow G ( B )$. Here one can take as $4$ the completion of the local ring of $k$ at the unit element.
If $4$ is the ring $k [ [ X _ { 1 } , \ldots , X _ { 2 } ] ]$ of formal power series in $12$ v
17 KB (2,520 words) - 13:36, 17 October 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 (2,169 words) - 08:12, 14 December 2016
be a [[Ring|ring]] (associative with one) and let $ A ^ {*} $
be a commutative ring and $ A $
9 KB (1,440 words) - 16:43, 4 June 2020
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
|[[Noetherian ring]]
|[[Normal scheme]]
5 KB (641 words) - 11:27, 6 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 (1,874 words) - 06:08, 17 April 2024