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A universality theorem for Voevodsky’s algebraic cobordism spectrum. (English) Zbl 1162.14013

The paper contains the proof of a result, which can be viewed as an algebraic version of a theorem of Quillen showing that the formal group law associated to the complex cobordism spectrum \(MU\) is the universal one on the Lazard ring. As a consequence the set of orientations on a commutative ring spectrum \(E\) in the stable homotopy category is in 1-1 correspondence with the set of homomorphisms of ring spectra from \(MU\) to \(E\) in the stable homotopy category.
In the algebraic context there is a similar \(\mathbb{P}^1\)-ring spectrum \(MGL\) in the motivic stable homotopy category of a noetherian finite-dimensional scheme \(S\). But it is not known, for a general \(S\), if the formal group law associated to \(MGL\) is the universal one.
Nevertheless, in the case \(S\) is regular, the authors prove that a 1-1 correspondence exists, as in the topological case.
Theorem. Let \(S\) be a regular noetherian finite-dimensional scheme and let \(E\) be a commutative \(\mathbb{P}^1\)-ring spectrum over \(S\). Then the set of orientations on \(E\) is in bijection with the set of homomorphisms of \(\mathbb{P}^1\)-ring spectra from \(MGL\) to \(E\) in the motivic stable category over \(S\).

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
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