Algebraic Jf-theory and Trace Invariants
Algebraic Jf-theory and Trace Invariants
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代数 Jf 理论和迹不变量
DOI：
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发表时间：
2003
期刊：
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通讯作者：
L. Hesselholt
中科院分区：
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作者：
L. Hesselholt
Bokstedt-Hsiang-Madsen的分圆迹是Bokstedt在京都大会上演讲的主题，它是从奎伦的代数A”-理论到康纳斯循环同调的拓扑精化的亲阿贝尔群K-f（A）^.TR;（A;p）的映射。在过去的十年里，我们对目标及其与A”理论的关系的理解有了显著的进步。这一点和未来可能的发展是我演讲的主题。分圆迹取被称为Frobenius的算子F固定的子集中的值。已知诱导映射K*（A，Z/PV）-^ TR;（A;P，Z/PV）F=1是同构的，例如，如果A是正则局部FP-代数，或如果A是具有可分闭剩余域的混合特征（0，p）的Henselian离散赋值环.对于更广泛的一类环，用分圆迹来评价A”理论是可能的，但是精确的联系要稍微复杂一些。亲阿贝尔群TR*（A;p）通常非常大。但是它们配备了许多运算符，并且组合的代数结构相当严格。这种结构有一个普遍的例子--德·拉姆-维特复形--它首先由布洛赫-德利涅-伊卢西结合格罗滕迪克的结晶上同调来考虑。一般来说，标准映射W.QqA^TR-q（A-p）是一个同构，如果q < 1，并且更高的群也可以用德·拉姆-维特群来表示。例如，如果A是正则FP-代数，或者A是局部数域上的整数环上的光滑代数，则这是真的。在后一种情况下的计算验证了焦点数域的LichtenbaumQuillen猜想，或者更一般地，几何类型的亨塞尔离散赋值域。
The cyclotomic trace of Bokstedt-Hsiang-Madsen, the subject of Bokstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K-fiA) ^.TR;(A;p) from Quillen's algebraic A"-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relation to A"-theory has been significantly advanced. This and possible future development is the topic of my lecture. The cyclotomic trace takes values in the subset fixed by an operator F called the Frobenius. It is known that the induced map K*(A,Z/pv) -^ TR;(A;P,Z/PV)F=1 is an isomorphism, for instance, if A is a regular local Fp-algebra, or if A is a henselian discrete valuation ring of mixed characteristic (0,p) with a separably closed residue field. It is possible to evaluate A"-theory by means of the cyclotomic trace for a wider class of rings, but the precise connection becomes slightly more complicated to spell out. The pro-abelian groups TR*(A;p) are typically very large. But they come equipped with a number of operators, and the combined algebraic structure is quite rigid. There is a universal example of this structure — the de Rham-Witt complex — which was first considered by Bloch-Deligne -Illusie in connection with Grothendieck's crystalline cohomology. In general, the canonical map W.QqA^TR-q(A-p) is an isomorphism, if q < 1, and the higher groups, too, can often be expressed in terms of the de Rham-Witt groups. This is true, for example, if A is a regular Fp-algebra, or if A is a smooth algebra over the ring of integers in a local number field. The calculation in the latter case verifies the LichtenbaumQuillen conjecture for focal number fields, or more generally, for henselian discrete valuation fields of geometric type.