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be a ring with unit and $ M $
2 KB (243 words) - 08:27, 6 June 2020
...oups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $ A $
be an associative, commutative ring with unit element. Witt vectors with components in $ A $
17 KB (2,502 words) - 17:25, 22 December 2019
[[Category:Ergodic theory]]
A continuous flow in [[Ergodic theory|ergodic theory]] is a family $\{T^t\}$ (where $t$ ranges over the real axis $\R$) of autom
3 KB (480 words) - 02:51, 15 February 2024
...{{Cite|Ma}}, {{Cite|Se}}, {{Cite|SaVa}}, {{Cite|GrSe}}) or as automorphism groups of large objects ({{Cite|HaMc}}; for example, the group of all bijections o
...logical space is completely determined by a certain perfect, locally free, and hence acyclic, group {{Cite|BeCa}}.
6 KB (873 words) - 08:18, 18 February 2022
...erous applications in algebraic geometry, class field theory and cobordism theory.
and field of residues $ k $ ,
17 KB (2,537 words) - 22:38, 15 December 2019
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and an Abelian group $ G $.
16 KB (2,386 words) - 16:47, 20 January 2024
...erous applications in algebraic geometry, class field theory and cobordism theory.
...\rightarrow G ( B )$. Here one can take as $4$ the completion of the local ring of $k$ at the unit element.
17 KB (2,520 words) - 13:36, 17 October 2019
...nd multiplication of them by elements of $K$ are defined in the usual way, and the commutator $[x,y]$ of two elements $x,y \in \mathfrak{g}$ is given by t
...rally identified with the set of all square matrices of order $n$ over $K$ and is denoted by $\mathfrak{gl}(n,K)$. Any linear Lie algebra is a subalgebra
3 KB (554 words) - 19:21, 24 February 2017
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...role in ring theory. Every ring with identity has maximal left (also right and two-sided) ideals. The quotient module $ M = R / I $
7 KB (1,160 words) - 08:00, 6 June 2020
...— associativity; this is the explanation of the term "semi-group" . Semi-groups are called ''monoids'' if they have, in addition, an identity element.
...ps of partial transformations), and the algebraic theory of automata (semi-groups of automata).
17 KB (2,435 words) - 09:18, 2 April 2023
its semi-simple and nilpotent components $s$ and $n$ should lie in ${\mathfrak g}$
case of a field $k$ of characteristic 0, a necessary and sufficient
3 KB (458 words) - 04:43, 4 January 2022
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...A}$ an [[Abelian category|Abelian category]] with exact infinite products, and $M : \mathcal{C} \rightarrow \mathcal{A}$ a covariant [[Functor|functor]].
9 KB (1,283 words) - 20:55, 8 February 2024
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is defined) and equivariant mappings is a sequence of contravariant functors $ H _ {G} ^
5 KB (772 words) - 11:12, 20 January 2021
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and is an acyclic mapping. This means that $ q _ {N} $
7 KB (1,010 words) - 16:47, 17 March 2023
...functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.).
...th the difference that the transition from geometry to algebra in homology theory is irreversible).
12 KB (1,885 words) - 23:48, 23 April 2017
...these theorems usually become invalid, and the Grothendieck group $K_0(R)$ and the Whitehead group $K_1(R)$ are, in a certain sense, a measure of their de
...jects may be studied with the aid of the homotopy theory of vector bundles and of topological
14 KB (2,405 words) - 22:14, 10 January 2015
...def\phi{\varphi}\phi:X\times X \to R$ that is linear in the first argument and satisfies the condition
Here $R$ is a [[unital ring|ring with a unit element]] and equipped with an involutory [[anti-automorphism]] $J$. In particular, $\phi
5 KB (831 words) - 17:13, 9 October 2016
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into the category of groups Gr. A homomorphism of a group object $ X $
6 KB (864 words) - 19:42, 5 June 2020
...of semi-linear transformations of $E$ preserving $f$ up to a scalar factor and an automorphism of $K$).
...be characterized as groups of those transformations of projective spaces (and also of certain varieties related to Grassmannians, see
12 KB (1,991 words) - 18:05, 29 November 2014
...ariski topology), is not a topological group with respect to this topology and therefore is not a compact group.
The following groups are two important classes of compact groups.
8 KB (1,117 words) - 20:03, 27 February 2021