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On the relation of Voevodsky’s algebraic cobordism to Quillen’s $$K$$-theory. (English) Zbl 1205.14023
The main result of this paper relates Voevodsky’s algebraic cobordism theory $$\text{MGL}^{*,*}$$ to Quillen’s $$K'$$-theory. These theories are analysed via their representing objects in the motivic stable homotopy category $$SH(S)$$, where $$S$$ is a Noetherian separated finite-dimensional scheme, as defined by V. Voevodsky [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, Vol. I, 579–604 (1998; Zbl 0907.19002)]. For $$S= k$$ a field, the algebraic cobordism $$\mathbb{P}^1$$-spectrum MGL of Voevodsky is considered as a commutative $$\mathbb{P}^1$$-ring spectrum. Setting $\text{MGL}^i= \bigoplus_{p- 2q=i} \text{MGL}^{p,q}$ one can regard the bi-graded theory $$\text{MGL}^{p,q}$$ as a graded theory. Then there is a unique ring morphism $$\phi: \text{MGL}^0(k)\to\mathbb{Z}$$ which sends the class $$[X]_{\text{MGL}}$$ of a smooth projective $$k$$-variety $$X$$ to its Euler characterestic $$\chi(X,{\mathcal O}_X)$$.
Let $${\mathcal S}mOp/k$$ be the category of pairs $$(X,U)$$ with $$X$$ a smooth variety over $$k$$ and $$U$$ open in $$X$$. Then the authors prove that there is a canonical grade preserving isomorphism of cohomology theories $\phi: \text{MGL}^*(X, X- Z)\otimes_{\text{MGL}^0(k)}\mathbb{Z}\simeq K_{-*}\;(X\text{ on }\mathbb{Z})= K_{-*}'(Z)$ on the category $${\mathcal S}mOp/k$$, with $$Z$$ closed in $$X$$. Here $$K_*$$ ($$X$$ on $$\mathbb{Z}$$) is Thomason-Trobaugh K-Theory and $$K_*'$$ is Quillen K-Theory. Both theories are oriented cohomology theories, according to the definition of I. Panin [K-Theory 30, No. 3, 265–314 (2003; Zbl 1047.19001)], and $$\phi$$ respects the orientations.

##### MSC:
 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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##### References:
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