Oriented cohomology theories of algebraic varieties
Let \(X\) be an algebraic variety over a field \(k\). Let \(A\) be a cohomology theory on the category of of pairs \((X,U)\) where \(X\) is a smooth variety and \(U\) is a Zariski open subset of \(X\), i.e., \(A\) is a homotopy invariant functor to the category of abelian groups satisfying excision and localization. The author studies the following three properties: (a) \(A\) being {orientable}, which means that given an algebraic variety \(X\), a closed subset \(Z\subset X\) and a vector bundle \(E/X\) we have a natural operator \(th^{E}_{Z}: A_{Z}(X)\to A_{Z}(E)\) which is a two-sided \(A(X)\)-module isomorphism satisfying: naturality on vector bundle isomorphisms, naturality on pullbacks, and naturality with respect to bundle sums. (b) Chern structures on \(A\), this consists of an assignment \(L\mapsto c(L)\) where \(L\) is a line bundle over a smooth variety \(X\), and \(c(L)\) is a universal central element in \(A(X)\). This assignment satisfies functoriality on line bundle isomorphisms, non degenerate, and \(c(\mathbf{1}_X)=0\), where \(\mathbf{1}_X\) is the trivial rank-one line bundle on \(X\). The element \(c(L)\) is called the Chern class of the line bundle \(L\). (c) Thom structures on \(A\), this consists of assigning to each line bundle \(L\) over \(X\) a universal central element \(th(L)\) in \(A_{X}(X)\) satisfying functoriality of line bundle isomorphisms, functoriality on pullback bundles, and non degeneracy: the cup product \(\cup \text{th} (\mathbf{1}):A(X)\to A_{X}(X\times \mathbb{A}^{1})\) is an isomorphism. The main theorem in this paper is that the existence of any of the above implies the existence of the others. The main examples are generalized cohomology theories, singular cohomology, Bloch's higher Chow groups, motivic cohomology, \(K\)-theory, and semi-topological complex and real \(K\)-theories.
scientific article; zbMATH DE number 2081549