$#C+1 = 90 : ~/encyclopedia/old_files/data/A010/A.0100200 Abelian category
...me of the characteristic properties of the category of all Abelian groups. Abelian categories were introduced as the basis for an abstract construction of hom
10 KB (1,515 words) - 18:19, 31 March 2020
...blem attributed, to J.H.C. Whitehead, which asks for a characterization of Abelian groups $ A $
is free (see [[Free Abelian group|Free Abelian group]]). This condition has been proved to be necessary if $ A $
4 KB (665 words) - 08:29, 6 June 2020
$#C+1 = 42 : ~/encyclopedia/old_files/data/F041/F.0401790 Frobenius theorem
...ors of zero; it was proved by G. Frobenius [[#References|[1]]]. Frobenius' theorem asserts that:
5 KB (790 words) - 19:40, 5 June 2020
$#C+1 = 145 : ~/encyclopedia/old_files/data/A010/A.0100210 Abelian differential
...g on the nature of their singular points, one distinguishes three kinds of Abelian differentials: I, II and III, with proper inclusions $ I \subset II \sub
11 KB (1,603 words) - 16:08, 1 April 2020
Then the category of sheaves of Abelian groups on $ X _ {et} $
is an Abelian category with a sufficient collection of injective objects. The functor $
5 KB (746 words) - 11:54, 8 April 2023
of an [[Abelian category|Abelian category]] $ \mathfrak A $
is Abelian.
3 KB (469 words) - 16:39, 17 March 2023
extensions with an Abelian Galois group (Abelian extensions) is a part
The fundamental result on Galois groups is the following theorem,
3 KB (494 words) - 21:56, 5 March 2012
...are isomorphic over a finite extension of $K$. One of the marvels of this theorem is the fact that the construction of the period $q$ starting from $E$, and
...], and it was used in the theory of compactifications of moduli schemes of Abelian varieties.
4 KB (680 words) - 21:50, 21 December 2014
$#C+1 = 99 : ~/encyclopedia/old_files/data/A010/A.0100220 Abelian function
is called an Abelian function if there exist $ 2p $
11 KB (1,602 words) - 16:08, 1 April 2020
coincides with the set of one-dimensional non-Abelian cohomology $ H ^{1} ( S _{T} ,\ \Gamma ) $.
group is trivial (Lang's theorem). This theorem also holds if $ k $
5 KB (854 words) - 10:51, 20 December 2019
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