in an [[Abelian category|Abelian category]] $ C $
1) The category of Abelian groups has enough injective objects. These objects are the complete (divisi
4 KB (643 words) - 22:12, 5 June 2020
The classical Torelli theorem relates to the case of curves (see [[#References|[1]]], [[#References|[2]]]
be a basis of the Abelian differentials (cf. [[Abelian differential|Abelian differential]]) and let the $ ( g \times 2g) $-
6 KB (967 words) - 08:26, 6 June 2020
A theorem proved by J.-P. Serre in 1965 about the cohomology of pro-$p$-groups which
...dexing set $I$, where $\textbf{Z}/p$ is cyclic of order $p$). Then Serre's theorem asserts that there exist non-trivial $\mod p$ cohomology classes $v_1,...,v
6 KB (868 words) - 22:16, 5 February 2021
...iaofmath.org/legacyimages/f/f120/f120130/f12013073.png" /> with elementary
Abelian quotient groups <img align="absmiddle" border="0" src="https://www.encyclop
...opediaofmath.org/legacyimages/f/f120/f120130/f12013092.png" />, then, by a
theorem of Burnside, <img align="absmiddle" border="0" src="https://www.encyclopedi
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...larized algebraic variety|Polarized algebraic variety]]; [[Abelian variety|Abelian variety]]), which is not always true for $ T _ {G} ^ {n} ( X) $.
as well as a duality between the Abelian varieties $ T _ {W} ^ {n} ( X) $
6 KB (953 words) - 12:29, 29 December 2021
...of Orlicz). The result subsequently came to be known as the Orlicz–Pettis theorem (see [[#References|[a3]]] for a historical discussion).
...measure and integration theory, there have been attempts to generalize the theorem in several directions. For example, A. Grothendieck remarked that the resul
5 KB (714 words) - 15:30, 1 July 2020
from an (Abelian) [[Semi-group|semi-group]] $ H $
to subsets of an (Abelian) semi-group $ G $
2 KB (318 words) - 16:09, 1 April 2020
be an [[Abelian group|Abelian group]] and let $ A \subset G $.
...degree of the minimal polynomial of the Grasmann derivative, the following theorem is true [[#References|[a3]]]: Let $ p $
4 KB (577 words) - 10:26, 10 December 2023
...nsion $K/k$ is Kummer (for a given $n$) if and only if $K/k$ is a normal [[Abelian extension]] and the Galois group $\mathrm{Gal}(K/k)$ is annihilated by $n$.
...$\mathrm{Gal}(K/k_0)$.) By the above proposition, many problems concerning Abelian extensions of exponent $n$ of a field $k$ can be reduced to the theory of K
5 KB (938 words) - 20:00, 18 September 2017
...ance of loops in the theory of quasi-groups is determined by the following theorem: Any quasi-group is isotopic (see [[Isotopy|Isotopy]]) to a loop. Therefore
Albert's theorem). In particular, isotopic groups are isomorphic. Some other classes of loop
8 KB (1,291 words) - 06:59, 30 March 2024
An Abelian variety is
implies severe restrictions on an Abelian variety. Thus, an Abelian
8 KB (1,216 words) - 20:39, 5 March 2012
$#C+1 = 17 : ~/encyclopedia/old_files/data/C023/C.0203650 Comparison theorem (algebraic geometry)
A theorem on the relations between homotopy invariants of schemes of finite type over
2 KB (271 words) - 13:16, 6 April 2023
Khinchin's theorem on the factorization of distributions: Any probability distribution $P$ adm
...tions on the line, in which factorization theorems analogous to Khinchin's theorem are valid.
2 KB (326 words) - 16:26, 9 April 2016
...is denoted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that the Abelian group $\mathrm{NS}(X)$ is finitely generated.
...heory of the base (see, for example, [[#References|[1]]]), a proof of this theorem using topological and transcendental tools. The first abstract proof (valid
4 KB (687 words) - 05:47, 15 April 2023
...additive category]] with set of objects $\mathrm{Ob}(C)$ and let $G$ be an Abelian group. A mapping $\phi: \mathrm{Ob}(C) \to G$ is said to be additive if for
...f coherent and locally free sheaves on schemes in proving the Riemann–Roch theorem. See [[K-functor|$K$-functor]] in algebraic geometry. The group $K(C)$ is u
4 KB (701 words) - 06:11, 26 March 2023
''EGZ theorem''
...\{1,\ldots,2m-1\}$ of cardinality $m$ such that $\sum_{i\in I}a_i=0$. This theorem was first shown in [[#References|[a5]]].
10 KB (1,573 words) - 17:25, 28 January 2020
consisting of the principal ideals. The divisor class group is Abelian and is usually denoted by $ C ( A) $.
Nagata's theorem). If $ B $
5 KB (820 words) - 19:36, 5 June 2020
$#C+1 = 187 : ~/encyclopedia/old_files/data/O110/O.1100050 O\AApNan\ANDScott theorem
A reduction theorem for the class of finite primitive permutation groups, distributing them in
11 KB (1,611 words) - 08:03, 6 June 2020
...s a description of all Abelian extensions (finite Galois extensions having Abelian Galois groups) of a field $ K $
In local class field theory, each finite Abelian extension $ L/K $
17 KB (2,620 words) - 07:48, 13 February 2024
...2 : ~/encyclopedia/old_files/data/K055/K.0505920 Krull\ANDRemak\ANDSchmidt theorem
...or a ring. The lattice-theoretical version of the result is known as Ore's theorem (see [[Modular lattice|Modular lattice]]). For a group $ G $
5 KB (786 words) - 22:15, 5 June 2020