Homological and cohomological motives of algebraic varieties
Homological and cohomological motives of algebraic varieties
复制标题
代数簇的同调和上同调动机
DOI：
10.1007/s002220000091
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发表时间：
2000
影响因子：
3.1
通讯作者：
M. Hanamura
中科院分区：
文献类型：
--
作者：
M. Hanamura
The framework of mixed motives in [Ha] will be shown to be $ uited for the study of motives of quasi-projective, pessibly singulaur, varieties and their Chew groups. We define cohomological motives of quasi-prejective varieties; they ferm a cegtravariant imcter ftom the categery of q" asi--projective varieties eo the triallgulated categery ef inixed motives. There is atse the functer ef cempactly supported cohemolcgical metive $. These functers are eompatible with Wei} cohomelogy groups, resp. eompactly supperted cohomology groups. We use cubical hyperresolution (see [GNPP] for an elaborate exposition), a variant of hypercovering [De], both of which were used in the context of mixed Hodge theory. Hyperresolutiun has the technical advantage of being of finite length. One could formally say that this work is am adaptation of the method of hyperresolutions in the context of higher Chow groups and rnotives. For the hyperresolutions to exist, we assume resolutions of singularities exist for varieties over the ground field h. The main results of this paper are the following.(1)(See Theorems (2.3) and (2.9).) The homologica} cycle complex of S. B} och has descent property for hyperrese} ution $; thls is a consequeRce cf k $ locallzation pyoperty. Hence foi} ews ehe exlsteRee ofthe cgitSyavariant functor h. Åqcempactly supperted cekomeicgicai mgtive) frem the categery cf quasYprojective varieÅíies ever k alld proper maps to gke triallgu} ated category of mixed metives P (k). The dual of h. is the" Berel-Meore homoiogical metive" fuRctor.(2)(See Theorem I.) Let U be a quasi-projective variety and U. its hyperresolution. Let Xr (U.,•) be the cycle complex of codimension of r of U.. We may form the cohomological cgele complex Z'(U.)* as the deuble complex with terms X"(U.,•); one of the differentiaks is the differential of each Zr (U.,•), and the other differential comes from the pull-backs by the face maps of the hyperresolution.(More precisely, for the pull-backs to be defined one has to take quasi-isomorphic subcoinplexes of each Z"(U.,•).) The complex is independent of the hyperresolution in the derived category, and U e Z'(U.)" i $ contravariant} y fuRctorial.'The bomo} ogy of this cemplex, denoted CHC"(U, n),} s by definitioxx the higher C} iow cohomolegy gromp, er f;} etivic cehemelegy ef U.(3)(See Theorem II.) There exlsts a coRtravariallt functer h Åírem the category of qttasiprojeetive varieties over k (and ai} maps) te D (k). The cycle eomplex of h (U) is the cohomological cycle complex of U.(Each object of CL)(k) has its associated cycle complex.) h (U) is the cohomological motive of U, armd its dual h (U) V is the compactly supported homological motive of U.
混合动机和代数循环 II。
DOI：
--
发表时间：
2004
期刊：
Invent.Math. 158
影响因子：
--
作者：
A.C.Kable;A.Yukie;M.Hanamura
通讯作者：
M.Hanamura