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Algebraic oriented cohomology theories. (English) Zbl 1051.14021

Vostokov, S. (ed.) et al., Algebraic number theory and algebraic geometry. Papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3267-0/pbk). Contemp. Math. 300, 171-193 (2002).
The author studies oriented cohomology theories defined by M. Levine and F. Morel [C. R. Acad. Sci., Paris, Sér. I, Math. 332, 723–728 (2001; Zbl 0991.19001)]. These are contravariant functors from the category \( \text{Sm}(F) \) of smooth, quasi-projective varieties to the category of graded commutative rings satisfying appropriate axioms. Levine and Morel have proved that in case of \(\text{char}(F)=0\) there exists a universal cohomology theory – the algebraic cobordism \({\Omega}^{*}.\)
For a given oriented cohomology theory \(A^{*}\) let \(A^{*}_{c}(X)\) denote the image of the universal morphism \({\Omega}^{*}(X)\rightarrow A^{*}(X).\) For arbitrary \(F\) to an oriented cohomology theory \(A^{*}(X)\) the author assigns a new theory \(\widetilde {A}^{*}(X)= A^{*}(X)\otimes {\mathbb Z}[ b]= A^{*}(X)[ b]\) where \({\mathbb Z}[ b]={\mathbb Z}[b_{1},b_{2},\dots ]\) is the polynomial ring in infinitely many variables. The author proves that \(\widetilde {H}^{*}_{c}(pt)\) and \(\widetilde {K}^{*}_{c}(pt)\) are both isomorphic to the Lazard ring. Here \(\widetilde {H}^{*}\) and \(\widetilde {K}^{*}\) denote respectively the Chow cohomology theory and the K-theory. For every projective variety \(X\in \text{Sm}{ (F)}\) the author defines the fundamental polynomial \(F^{H}_{X}\in {\mathbb Z}[ b]\) and proves that for every field \(F\) the fundamental polynomials of all projective \(X\) generate the Lazard ring considered as a subring of \({\mathbb Z}[ b].\) Using the invariant prime ideals in the Lazard ring the author assigns to every \(X\in \text{Sm} (F)\) and prime integer \(p\) the number \(n_{p}(X)\) and proves the following result:
Proposition. Let \(X\) and \(Y\) be projective smooth varieties such that \( \text{Mor}(Y,X)\neq \emptyset.\) Then \(\quad n_{p}(Y)\leq n_{p}(X)\) for every prime \(p.\)
For the entire collection see [Zbl 0997.00010].

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55N22 Bordism and cobordism theories and formal group laws in algebraic topology

Citations:

Zbl 0991.19001
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