Clifford模与二次型的不变量
Clifford modules and invariants of quadratic forms
摘要：
设A是一个交换环，A中有1/2。本文定义了具有非退化二次型的非线性生成投射A-模V的新的特征类。这些类属于A.当A是紧空间X上的连续函数环时，它们在某种意义上推广了拓扑K理论中经典的“同类相食”Bott类.为了定义这些类，我们用A-模和C（V）-模之间的Morita等价来代替拓扑Thom同构，其中C（V）表示V的Clifford代数，假设A的分次Brauer群中的C（V）类是平凡的。然后，我们基本上使用的想法回到阿蒂亚，博特和夏皮罗连同另一种定义的亚当斯行动由于阿蒂亚。当C（V）在分次Brauer群中不是平凡的时，特征类取其值的代数模拟是扭曲K-理论。最后，我们还利用J写的一封信。P. Serre的方法，以便把这些类解释为定义在环A的Witt群W（A）上。引理3.5总结了这封信的一个方面，其中证明了在我们的情况下，Bott类在A的K-理论中有一个标准的平方根。
英文摘要：
Let A be a commutative ring with 1/2 in A. In this paper, we define new characteristic classes for finitely generated projective A-modules V provided with a non degenerate quadratic form. These classes belong to the usual K-theory of A. They generalize in some sense the classical "cannibalistic" Bott classes in topological K-theory, when A is the ring of continuous functions on a compact space X. To define these classes, we replace the topological Thom isomorphism by a Morita equivalence between A-modules and C(V)-modules, where C(V) denotes the Clifford algebra of V, assuming that the class of C(V) in the graded Brauer group of A is trivial. We then essentially use ideas going back to Atiyah, Bott and Shapiro together with an alternative definition of the Adams operations due to Atiyah. When C(V) is not trivial in the graded Brauer group, the characteristic classes take their values in an algebraic analog of twisted K-theory. Finally, we also make use of a letter written by J.-P. Serre to the author, in order to interpret these classes as defined on the Witt group W(A) of the ring A. One aspect of this letter is summarized in our Lemma 3.5 where it is shown that in our situation the Bott class has a canonical square root in the K-theory of A.
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