有限控制与Novikov环迭代方法
Finite domination and Novikov rings. Iterative approach
摘要：
设C是Laurent多项式环L = R[x，1/x]上的自由模的有界链复形.则C是R-R-控制的，即在R上同伦等价于R-生成的投射R-模的有界链复形，当且仅当两个链复形C（（x））和C（（1/x））是非循环的，正如Ranicki所证明的.这里C（（x））是C与Novikov环R（（x））= R[[x]][1/x]（也称为x中的形式洛朗级数环）在L上的张量积;类似地，C（（1/x））是C与Novikov环R（（1/x））= R[[1/x]][x]在L上的张量积。在本文中，我们证明了这一标准的推广，使我们能够检测有限控制的下有界链链复形的投射模在劳伦特环在几个不定。
英文摘要：
Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x]. Then C is R-finitely dominated, ie, homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules, if and only if the two chain complexes C((x)) and C((1/x)) are acyclic, as has been proved by Ranicki. Here C((x)) is the tensor product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x]. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.
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