On the homology of algebras of Whitney functions over subanalytic sets
On the homology of algebras of Whitney functions over subanalytic sets
复制标题
亚解析集上惠特尼函数代数的同调性
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
M. Pflaum
中科院分区：
文献类型：
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作者：
J. Brasselet;M. Pflaum
本文从非交换几何的角度研究了次解析集X <$Rn上Whitney函数代数E ∞（X）的几个同调理论.利用Teleman的局部化方法证明了E ∞（X）的一个Hochschild-Kostant-Rosenberg型定理，当X是Rn的一个正则子集，且其对角线是正则的.这包括子解析X的情况。我们还计算了E ∞（X）在正则集上的Hochschild上同调，并导出了循环和周期循环理论.证明了周期循环同调与de Rham上同调是一致的，从而推广了Feigin-Tsygan的一个结果.受Grothendieck的代数de Rham理论的启发，我们证明了对于次解析集，E ∞（X）的de Rham上同调与奇异上同调是一致的.为了证明这一结果，我们引入了双纯次解析三角剖分的概念，并证明了每个有界次解析集都允许这样的三角剖分。内容
In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ R n with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for E ∞ (X), when X is a regular subset of R n having regularly situated diagonals. This includes the case of subanalytic X. We also compute the Hochschild cohomology of E ∞ (X) for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories. It is shown that the periodic cyclic homology coincides with the de Rham cohomology, thus generalizing a result of Feigin-Tsygan. Motivated by the algebraic de Rham theory of Grothendieck we finally prove that for subanalytic sets the de Rham cohomology of E ∞ (X) coincides with the singular cohomology. For the proof of this result we introduce the notion of a bimeromorphic subanalytic triangulation and show that every bounded subanalytic set admits such a triangulation. Contents