Cyclic homology and nonsingularity
Cyclic homology and nonsingularity
复制标题
循环同调性和非奇异性
DOI：
--
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
D. Quillen
中科院分区：
文献类型：
--
作者：
J. Cuntz;D. Quillen
从开拓性的工作康纳斯[上校]人们知道，周期循环的同源性可以被视为一个自然的延伸德拉姆上同调领域的非交换几何。本文的目的是给出Deligne [D]和哈茨霍恩[H]在代数几何中求解de Rham上同调的方法的非对易类比。在这种方法中，首先通过微分形式的de Rham复形获得非奇异代数簇的de Rham上同调。一个任意的品种，然后处理嵌入它在一个非奇异品种，并完成德拉姆复杂的后者沿着的子品种。在我们的非交换的版本代数簇被取代的结合酉代数在复数，和非奇异的品种成为代数是准自由的[CQ 1]。事实上，非奇异簇是由交换代数局部描述的，交换代数相对于交换代数的幂零扩张表现得像自由交换代数，而准自由代数是那些相对于幂零代数扩张表现得像自由代数的代数。像自由代数一样，拟自由代数R关于Hochschild上同调具有上同调维数：：; I，这意味着它的周期循环同调H PIJR，v E Z/2，由超复形计算
From the pioneering work of Connes [Col] one knows that periodic cyclic homology can be regarded as a natural extension of de Rham cohomology to the realm of noncommutative geometry. Our aim in this paper is to present the noncommutative analogue of the approach of Deligne [D] and Hartshorne [H] to de Rham cohomology in algebraic geometry. In this approach de Rham cohomology is first obtained for a nonsingular algebraic variety by means of the de Rham complex of differential forms. An arbitrary variety is then treated by embedding it in a nonsingular variety and completing the de Rham complex of the latter along the subvariety. In our noncommutative version algebraic varieties are replaced by associative unital algebras over the complex numbers, and nonsingular varieties become algebras which are quasi-free [CQ1]. Indeed, nonsingular varieties are described locally by commutative algebras which behave like free commutative algebras with respect to nilpotent extensions of commutative algebras, while quasi-free algebras are those algebras behaving like free algebras relative to nilpotent algebra extensions. Like a free algebra, a quasi-free algebra R has cohomological dimension::; I with respect to Hochschild cohomology, and this implies that its periodic cyclic homology H PIJR, v E Z/2 , is calculated by the supercomplex