$#C+1 = 244 : ~/encyclopedia/old_files/data/C023/C.0203130 Cohomology of groups
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...$A$ is called a left polygon over a monoid $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that
and
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[[Chow ring|Chow ring]] $A(G_{n,m})$, and for $k=\C$ — a basis for the homology group $H_*(G_{n,m},\Z)$.
...operties. Hilbert's 15th problem concerns a foundation for the enumeration theory developed by Schubert (see
3 KB (391 words) - 16:26, 9 December 2023
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...eld (cf. also [[Algebraic number|Algebraic number]]; [[Field|Field]]) with ring of integers $ R $
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the structure of a module over the integer group ring $ \mathbf Z (J ^ {a} ) $
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...a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way us
Let $\mathfrak{g}$ be a [[Lie algebra]] and $x \in \mathfrak{g}$ an element of $\mathfrak{g}$ for which $\mathrm{ad}(x)
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...a generalized manifold) with coefficients in a locally constant system of groups (modules) $ {\mathcal G} $,
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...operations, that is, addition, multiplication and transitions to negative and inverse elements (the latter is defined only on the set of non-zero element
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A category $\mathfrak C$ in which for any two objects $X$ and $Y$ an Abelian group structure is defined on the set of morphisms $\Hom_{\m
...of a category]]) as well as the product $X\times Y$ of any two objects $X$ and $Y$.
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The theory that studies arithmetic properties of linear algebraic groups (cf.
...he principal objects of study in the arithmetic theory of linear algebraic groups are arithmetic subgroups of an algebraic group $G$ (see
11 KB (1,671 words) - 18:19, 24 May 2019
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A [[Morphism|morphism]] $\mu : M \rightarrow P$ of groups together with an action of the [[Group|group]] $P$ on the group $M$ satisfy
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...)$ is a derivation of $V$ (cf. also [[Derivation in a ring|Derivation in a ring]]). Thus one denotes $\{ L ( x , y ) \} _ { \text{span} }$ by $\operatornam
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and any non-zero vector $ v \in V $
...the theorem on finite generation of the algebra of invariants of reductive groups of automorphisms of an algebra of finite type over a field (see [[Hilbert t
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An invariant of cohomology type associated with schemes in [[Algebraic K-theory|algebraic $ K $-
theory]]. More precisely, in algebraic $ K $-
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...fference set|Difference set]] (Vol. 3). For an extensive discussion of the theory of Abelian difference sets, see also [[#References|[a1]]], Chap. VI.
9 KB (1,331 words) - 19:36, 13 February 2024
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...tre $Z _ { G }$ of $G$ (cf. also [[Centre of a group|Centre of a group]]), and coincides with $Z _ { G }$ if $G$ is connected. The image $\text{Ad}( G )$
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groups ${\rm Gr}$. The category $S-{\rm Gr}$ of group schemes over $S$ is defined
multiplicative) group of the ring of sections of the structure sheaf
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and negatively graded if $ A _ {n} = 0 $
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...her with a unary [[relation symbol]] for being an element of the valuation ring, or a binary relation symbol for valuation divisibility $v(x)\leq v(y)$ (cf
...t in the proof that Hilbert's 10th problem has a positive solution for the ring of all algebraic integers (see [[Algebraic Diophantine equations|Algebraic
12 KB (1,846 words) - 09:28, 26 November 2016
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The groups
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