3) In the category of Abelian groups, the functor $ \mathop{\rm Hom} ( A , Y ) $
...roups is the left adjoint of the functor of taking the torsion part of any Abelian group.
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...assigns the sum in the usual sense to any convergent series (an [[Abelian theorem]]). The series
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there is associated a sequence of Abelian groups $ H ^ { n } ( G, A) $,
is an Abelian group and $ G $
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are Abelian for all $ 0 \leq i < n $ );
is a one-dimensional (Abelian) Lie algebra for $ 0 \leq i < m $ .
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Given an arbitrary Abelian group $ \pi $
one can define a simplicial set (in fact, a simplicial Abelian group) $ E ( \pi , n) $.
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...his 1963 notes [[#References|[a7]]]. This facilitated a proof of a duality theorem of A. Grothendieck (cf. [[#References|[a5]]]). Let $ {\mathcal C} $
...seful to keep the example below in mind: the category of complexes over an Abelian category (and algebraic mapping cones, the corresponding long exact sequenc
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...orem of local [[Class field theory|class field theory]] there is for every Abelian extension of local fields $ K / k $
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...among all algebraic number fields is illustrated by the [[Kronecker–Weber theorem]], which states that a finite extension $ K/ \mathbf Q $
is Abelian if and only if $ K \subset K _ {n} $
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In the special case of the group algebra of a locally compact Abelian group (with convolution taken as multiplication in the algebra, cf. also [[
...[[#References|[a2]]], Sect. 11.13). A well-known such theorem is Wiener's theorem (cf. also [[#References|[a1]]], Chapt. XI, Sect. 2): If $ f ( t) = \sum _
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...ty|Rational variety]], [[Unirational variety|Unirational variety]]). Since Abelian varieties can never be rational, the main interest is in rationality theore
...ective). Fairly complete results are obtained for tori which split over an Abelian extension of the ground field (see [[#References|[5]]]). The first example
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...y studies the cohomology of a [[Galois group|Galois group]]. Let $M$ be an Abelian group, let $G(K/k)$ be the
defined for a non-Abelian group $M$. Namely,
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...of) Wedderburn's theorem (cf. [[Wedderburn–Artin theorem|Wedderburn–Artin theorem]]), that every finite-dimensional$C^*$-algebra is isomorphic to the direct
is the countable Abelian group of formal differences of equivalence classes of projections in matrix
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...References|[a11]]]). Eklof and Trlifaj attribute the inspiration for their theorem to a construction of [[#References|[a9]]]. D. Quillen [[#References|[a10]]]
...gn="top">[a11]</td> <td valign="top"> L. Salce, "Cotorsion theories for Abelian groups" , ''Symp. Math.'' , '''23''' , Amer. Math. Soc. (1979) pp. 11–3
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is an Abelian [[Semi-group|semi-group]] and $ E $
...great importance in dynamical systems theory, owing to the Smale–Birkhoff theorem: A discrete-time [[Dynamical system|dynamical system]] containing a [[Homoc
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of Abelian varieties $ T _ {\mathbf C} ^ {N} $(
of Abelian surfaces $ T _ {\mathbf C} ^ {2} $
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is exact; for an Abelian group $ G $,
is free (Stalling's theorem, see [[Homological dimension|Homological dimension]]). If $ G $
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...arbitrary dimension) of the classical [[Riemann–Roch theorem|Riemann–Roch theorem]] (see [[#References|[2]]]). After higher algebraic $ K $-
...ield theory]] in higher dimensions describes the Galois group of a maximal Abelian extension of rational function fields of arithmetic schemes of dimension $
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...w} K_{n - 2} \stackrel{\partial}{\longrightarrow} \cdots $ is a complex of abelian groups, and a continuous mapping of spaces induces homomorphisms of their r
...([[#References|[4]]]), the latter was extended so as to apply to arbitrary abelian categories with enough injective objects, and became applicable to arithmet
12 KB (1,885 words) - 23:48, 23 April 2017
...proof (cf. [[Schneider method|Schneider method]]) is based on the addition theorem for the exponential function $e ^ { z _ { 1 } + z _ { 2 } } = e ^ { z _ { 1
...elliptic integrals of the first or second kind [[#References|[a9]]] and of Abelian integrals [[#References|[a10]]], including the transcendence of the values
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consists of functions on a locally compact Abelian group and $ {\mathcal A} $
...includes the classical problems of harmonic synthesis on a locally compact Abelian group (see [[Harmonic analysis, abstract|Harmonic analysis, abstract]]), wh
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