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.)
Full Text: