Oriented cohomology, Borel-Moore homology, and algebraic cobordism

Levine, Marc

doi:10.1307/mmj/1220879423

Cited by 11 publications

(17 citation statements)

References 9 publications

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“…Possibly this could be supplied by the method used by Cai in [2, §6.3] to define pull back maps in his theory CK Cai ; for regular embeddings. The results of this section can be interpreted as saying that, in case k admits resolution of singularities, the oriented duality theory .CK 0 ; ;CK ; / on Sch k defined using the results of [14] is isomorphic to .C G ; ;CK ; / (ignoring the cap product structure).…”

Section: G-theory and Connective G-theorymentioning

confidence: 96%

“…We recall the construction of the homotopy coniveau tower from [13] and relate the homotopy coniveau tower for K-theory to connective K-theory in §2. In §3, we recall from [14] the notion of an oriented duality theory as well as how an oriented motivic ring spectrum gives rise to an oriented duality theory, and we apply these constructions to K-theory and connective K-theory in §4. We also extend the computation of connective K-theory from §2 to give an explicit description of the "geometric part" of connective Ktheory in this section.…”

Section: Introductionmentioning

confidence: 99%

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Connective AlgebraicK-theory

Dai¹,

Levine²

2014

J K-Theor

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We examine the theory of connective algebraic K-theory, CK, defined by taking the 1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend CK to a bi-graded oriented duality theory .CK 0 ; ;CK ; / when the base scheme is the spectrum of a field k of characteristic zero. The homology theory CK 0 ; may be viewed as connective algebraic G-theory. We identify CK 0 2n;n .X / for X a finite type k-scheme with the image of K 0 .M .n/ .X // in K 0 .M .nC1/ .X //, where M .n/ .X / is the abelian category of coherent sheaves on X with support in dimension at most n; this agrees with the (2n,n) part of the theory of connective algebraic K-theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies CK 0 2 ; with the universal oriented Borel-Moore homology theory CK WD ˝L ZOEˇ having formal group law u C v ˇuv with coefficient ring ZOEˇ. As an application, we show that every pure dimension d finite type k-scheme has a well-defined fundamental class OEX CK in CK d .X /, and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.

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Section: G-theory and Connective G-theorymentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Connective AlgebraicK-theory

Dai¹,

Levine²

2014

J K-Theor

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“…For We recall the following result from [10]: Via the universal property of the Lazard ring, the relation (4.1) is equivalent to the identity…”

Section: Algebraic Cobordism and Oriented Duality Theoriesmentioning

confidence: 99%

Connective algebraic K-theory

Dai¹,

Levine²

2012

Preprint

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We examine the theory of connective algebraic K-theory, CK, defined by taking the -1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend CK to a bi-graded oriented duality theory (CK ′ * , * , CK * , * ) in case the base scheme is the spectrum of a field k of characteristic zero. The homology theory CK ′ * , * may be viewed as connective algebraic G-theory. We identify CK ′ 2n,n (X) for X a finite type k-scheme with the image of, where M (n) (X) is the abelian category of coherent sheaves on X with support in dimension at most n; this agrees with the (2n,n) part of the theory of connective algebraic K-theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies CK ′ 2 * , * with the universal oriented Borel-Morel homology theory Ω CK *As an application, we show that every pure dimension d finite type k scheme has a well-defined fundamental class [X] CK in Ω CK d (X), and this fundamental class is functorial with respect to pull-back for lci morphisms. Furthermore, the fundamental class [X] CK maps to the usual fundamental classes [X] Chow , reap. [X] K under the natural mapsgiven by inverting β, resp. moding out by β.

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“…In fact, we prove more. In [2], we have shown how one can extend MGL * , * to a bi-graded oriented duality theory (MGL * , * , MGL * , * ) on quasi-projective k-schemes, Sch k , and how ϑ MGL extends to a natural transformation ϑ MGL : Ω * → MGL 2 * , * .…”

Section: Mu 2 * (X(c))mentioning

confidence: 99%