A general construction of partial Grothendieck transformations
A general construction of partial Grothendieck transformations
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部分格洛腾迪克变换的一般构造
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发表时间：
2002
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通讯作者：
Joerg Schuermann
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作者：
Joerg Schuermann
富尔顿和麦克弗森引入了与黎曼-洛克定理相关的双变理论的概念，特别是在奇异空间的背景下。这是一个强有力的形式主义，它是一对逆变和协变理论的同时推广。双变理论的自然变换称为格罗滕迪克变换，它们推广了一对普通的自然变换。但在许多情况下，这样的双变理论或相应的格罗滕迪克变换只是“部分已知”的：奇异空间的特征类（例如Stiefel-Whitney或Chern类），上同调运算（例如Chow群的奇异亚当斯黎曼-罗克和斯廷罗德运算）或等变理论（例如莱夫谢茨黎曼-罗克）。我们在本文中介绍了一个更简单的概念，部分（弱）双变理论和部分Grothendieck变换，适用于所有这些例子。我们的主要定理表明，一个自然变换的协变理论，它与外部产品交换，自动扩展到这样一个部分Grothendieck变换的适当的部分（弱）双变理论！在上面的几何情形中，我们必须只考虑态射，它的目标是光滑流形，或者更一般地说，一个合适的“同调流形”（在一般的双变语言中，这与合适的强定向的存在有关）。我们说明我们的主要定理的例子，将其与相应的已知的黎曼-罗克定理。
Fulton and MacPherson introduced the notion of bivariant theories related to Riemann-Roch-theorems, especially in the context of singular spaces. This is powerful formalism, which is a simultaneous generalization of a pair of contravariant and covariant theories. Natural transformations of bivariant theories are called Grothendieck transformations, and these generalize a pair of ordinary natural transformations. But there are many situations, where such a bivariant theory or a corresponding Grothendieck transformation is only ”partially known”: characteristic classes of singular spaces (e.g. Stiefel-Whitney or Chern classes), cohomology operations (e.g. singular Adams Riemann-Roch and Steenrod operations for Chow groups) or equivariant theories (e.g. Lefschetz RiemannRoch). We introduce in this paper a simpler notion of partial (weak) bivariant theories and partial Grothendieck transformations, which applies to all these examples. Our main theorem shows, that a natural transformation of covariant theories, which commutes with exterior products, automatically extends uniquely to such a partial Grothendieck transformations of suitable partial (weak) bivariant theories ! In the above geometric situations one has for example to consider only morphisms, whose target is a smooth manifold, or more generally, a suitable ”homology manifold” (and in the general bivariant language this is related to the existence of suitable strong orientations). We illustrate our main theorem for the examples above, relating it to corresponding known Riemann-Roch theorems.