MIXED MOTIVES AND ALGEBRAIC CYCLES, I
MIXED MOTIVES AND ALGEBRAIC CYCLES, I
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混合动机和代数循环，I
DOI：
10.4310/mrl.1995.v2.n6.a12
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
M. Hanamura
中科院分区：
文献类型：
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作者：
M. Hanamura;M. Hanamura
动机的概念可以追溯到 A. Grothendieck（参见 [Kl] [Ma]）。当他研究有限域 k 上的簇 X 的 zeta 函数的韦伊猜想时，他需要研究不同 (cid:4) 的 Frobenius 映射在 (cid:4) -adic etale 上同调 H i ( X ⊗ k ¯ k, Q (cid:4) ) 上的作用；为了证明该猜想，必须证明特征多项式是独立于 (cid:4) 的 Q 多项式，并且其特征值具有预期的绝对值。他对代数环类提出了某些猜想，称为标准猜想，并观察到它们暗示着韦尔猜想。标准猜想不仅涉及 (cid:4) -adic 上同调，还涉及任何其他 Weil 上同调（例如 Betti 或 de Rham 上同调），其后果影响深远。它们表明（纯格罗腾迪克）动机理论的存在：存在一个格罗腾迪克动机 M ( k ) （其中 k 是任意基域）的半简单阿贝尔范畴，满足以下性质：
The idea of motives goes back to A. Grothendieck (cf. [Kl] [Ma]). As he worked on the Weil conjecture on the zeta function of a variety X over a finite field k , he was in need of studying the actions of the Frobenius maps on the (cid:4) -adic etale cohomologies H i ( X ⊗ k ¯ k, Q (cid:4) ) for different (cid:4) ; to prove the conjecture one must show that the characteristic polynomial is a Q -polynomial independent of (cid:4) , and its eigenvalues have expected absolute values. He postulated certain conjectures on algebraic cycle classes, called the standard conjectures, and observed that they imply the Weil conjecture. The standard conjectures are concerned not only with (cid:4) -adic cohomology, but with any other Weil cohomology (such as Betti or de Rham cohomology) as well, and the consequences are further-reaching. They indicate the existence of the theory of (pure Grothendieck) motives : there is a semi-simple abelian category of Grothendieck motives M ( k ) (where k is any ground field) satisfying the properties: