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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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