Abstract:
Let k be a perfect field of characteristic p>0 and X be a separated scheme of finite type over k. In this talk, we will introduce a complex K_{n,X,log} via Grothendieck's coherent duality theory following Kato and build up a chain map from Bloch's cycle complex mod p^n to K_{n,X,log}. We show that this map is a quasi-isomorphism in the \'etale topology, and when k is algebraically closed, it is also a quasi-isomorphism in the Zariski topology. This result provides us with a perspective of higher Chow groups of zero cycles with \Z/p^n coefficients from coherent duality theory, and thus introduces the tools from coherent cohomology to the study of cycles. In particular, when k=\bar k, these higher chow groups are the Cartier invariant part of the hypercohomology of some coherent dualizing complex, so they can be reasonably regarded as an analog of log forms in the singular case. As corollaries, we deduce certain vanishing, étale descent properties as well as invariance under rational resolutions for higher Chow groups of 0-cycles with \Z/p^n-coefficients. We will also reproduce a finiteness result of Geisser.
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