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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bloch-Kato conjectures ： ウィキペディア英語版 Norm residue isomorphism theorem In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime $\backslash ell$ and any natural number $n$. John Milnor〔Milnor(1970)〕 speculated that this theorem might be true for $\backslash ell=2$ and all $n$, and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato 〔Bloch, Spencer and Kato, Kazuya, "p-adic étale cohomology.", Inst. Hautes Études Sci. Publ. Math. No. 63 (1986), p.118, http://www.numdam.org/item?id=PMIHES_1986__63__107_0〕 and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of ''L''-functions.〔Bloch, Spencer and Kato, Kazuya, "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, 333–400, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990.〕 The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost. ==Statement== For any integer ℓ invertible in a field ''k'' there is a map $\backslash partial\; :\; k^$ *\rightarrow H^1(k, \mu_\ell) where $\backslash mu\_\backslash ell$ denotes the Galois module of ℓ-th roots of unity in some separable closure of ''k''. It induces an isomorphism $k^\backslash times/(k^\backslash times)^\backslash ell\; \backslash cong\; H^1(k,\; \backslash mu\_\backslash ell)$. The first hint that this is related to ''K''-theory is that $k^\backslash times$ is the group ''K''1(''k''). Taking the tensor products and applying the multiplicativity of étale cohomology yields an extension of the map $\backslash partial$ to maps: :$\backslash partial^n\; :\; k^\backslash times\; \backslash otimes\; \backslash cdots\; \backslash otimes\; k^\backslash times\; \backslash rightarrow\; H^n\_(k,\; \backslash mu\_\backslash ell^).$ These maps have the property that, for every element ''a'' in $k\; \backslash setminus\; \backslash$, $\backslash partial^n(\backslash ldots,a,\backslash ldots,1-a,\backslash ldots)$ vanishes. This is the defining relation of Milnor ''K''-theory. Specifically, Milnor ''K''-theory is defined to be the graded parts of the ring: :$K^M\_$ *(k) = T(k^\times)/(\), where $T(k^\backslash times)$ is the tensor algebra of the multiplicative group ''k''× and the quotient is by the two-sided ideal generated by all elements of the form $a\; \backslash otimes\; (1\; -\; a)$. Therefore the map $\backslash partial^n$ factors through a map: :$\backslash partial^n\; \backslash colon\; K^M\_n(k)\; \backslash to\; H^n\_(k,\; \backslash mu\_\backslash ell^).$ This map is called the Galois symbol or norm residue map.〔Srinivas (1996) p.146〕〔Gille & Szamuely (2006) p.108〕〔Efrat (2006) p.221〕 Because étale cohomology with mod-ℓ coefficients is an ℓ-torsion group, this map additionally factors through $K^M\_n(k)\; /\; \backslash ell$. The norm residue isomorphism theorem (or Bloch–Kato conjecture) states that for a field ''k'' and an integer ℓ that is invertible in ''k'', the norm residue map :$\backslash partial^n\; :\; K\_n^M(k)/\backslash ell\; \backslash to\; H^n\_(k,\; \backslash mu\_\backslash ell^)$ from Milnor K-theory mod-ℓ to étale cohomology is an isomorphism. The case is the Milnor conjecture, and the case is the Merkurjev–Suslin theorem.〔〔Srinivas (1996) pp.145-193〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Norm residue isomorphism theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.