Algebra extensions and nonsingularity
Algebra extensions and nonsingularity
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DOI：
10.1090/s0894-0347-1995-1303029-0
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发表时间：
1995-05
影响因子：
3.9
通讯作者：
J. Cuntz;D. Quillen
中科院分区：
文献类型：
--
作者：
J. Cuntz;D. Quillen
本文讨论了非对易代数的非奇异性概念，它自然地与循环同调有关。让我们考虑复数上的结合单位代数。我们称一个代数为拟自由代数，当它关于幂零扩张表现为自由代数时，任何同态A-+R/I，其中I是R中的幂零理想，都可以提升到同态A-+R。如果我们限制到有限生成的交换代数范畴，那么这个提升性质刻画了光滑代数，对应于非奇异仿射簇。这样，拟自由代数就表现为光滑代数的非交换类似。推广这个类比，我们甚至可以把拟自由代数看作流形的类似。本文的目的之一是通过证明拟自由代数为流形的某些方面的非对易形式提供了一个自然的环境，从而进一步发展了这个类比。举个例子，让我们考虑嵌入的模拟：扩张A=R/I，其中A和R是分别起到子流形和环境流形作用的拟自由代数。在流形情形下，I/I2是正规丛上的线性函数模，对称代数SA(III)是多项式函数的代数。现在，在从交换代数到非交换代数的过渡中，模的对称代数被双模的张量代数代替。
This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over the complex numbers. We call an algebra A quasi-free, when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism A -+ R/I, where I is a nilpotent ideal in R, can be lifted to a homomorphism A -+ R. If we restrict to the category of finitely generated commutative algebras, then this lifting property characterizes smooth algebras, the ones corresponding to nonsingular affine varieties. In this way quasi-free algebras appear as noncommutative analogues of smooth algebras. Stretching the analogy, we can even regard quasi-free algebras as analogues of manifolds. One of the aims of this paper is to develop the analogy further by showing that quasi-free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds. To give an example, let us consider the analogue of an embedding: an extension A = R/I, where A and R are quasi-free algebras playing the role of the submanifold and ambient manifold respectively. In the manifold situation, I/I2 is the module of linear functions on the nor2 mal bundle, and the symmetric algebra SA(III ) is the algebra of polynomial functions. Now in passing from commutative to noncommutative algebras, the symmetric algebra of a module is replaced by the tensor algebra of a bimodule.