Rigidity II: non-orientable case
In the proofs of the Suslin rigidity theorem and Gabber's rigidity theorem for algebraic \(K\)-groups [\textit{O. Gabber}, in: Algebraic \(K\)-theory, commutative algebra, and algebraic geometry, Proc. Jt. US-Italy Semin., Santa Margherita Ligure/Italy 1989, Contemp. Math. 126, 59--70 (1992; Zbl 0791.19002)] one needs the transfer structure on algebraic \(K\)-theory. Now consider a more general cohomology theory \(A\) on the category of smooth schemes over an algebraically closed field. For instance, any functor represented by a \(T\)-spectrum in the sense of Voevodsky is such a theory. It was shown by \textit{I. Panin} [``Push-forwards in oriented cohomology theories of algebraic varieties. II.'' Preprint, http://math.uiuc.edu/K-theory/0619] that a cohomology theory has transfer structure if and only if it is orientable. On the other hand, \textit{J. Hornborstel} [``\(\mathbb{A}^1\)-representability of hermitian \(K\)-theory and Witt groups.'' Preprint, http://www.math. uiuc.edu/K-theory/0578] has shown there are at least two interesting cohomology theories which are \(T\)-representable but not orientable, viz.\ higher Witt groups and hermitian \(K\)-theory. In order to prove rigidity for such cases, the author shows that it suffices to have a transfer structure for a small class of morphisms, the `trivial' ones. A trivial morphism from \(X\) to \(Y\) is built from a closed embedding \(\tau:X\hookrightarrow Y\times\mathbb A^n\) with trivial normal bundle, together with a trivialization \(\Theta\) of this normal bundle. The underlying morphism \(f:X\to Y\) is the composite of \(\tau\) with the projection on the factor \(Y\). The author associates some kind of Becker-Gottlieb transfer \(A(X)\to A(Y)\) with such data \((f,\Theta)\) and shows how to finish the proof of rigidity in this manner, for all cohomology theories represented by \(T\)-spectra.
scientific article; zbMATH DE number 2095715