The big de Rham-Witt complex

• 成果类型: Article
• 署名作者: Hesselholt, Lars
• 署名单位: Nagoya University; University of Copenhagen
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-015-0124-y
• 发表日期: 2015
• 页码: 135-207
• 关键词: k-theory; lambda-rings

摘要

This paper gives a new and direct construction of the multi-prime big de Rham-Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a lambda-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a lambda-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the lambda-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kahler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham-Witt complex possible. It is further shown that the big de Rham-Witt complex behaves well with respect to ,tale maps, and finally, the big de Rham-Witt complex of the ring of integers is explicitly evaluated.