Descent Properties of Homotopy K-Theory

Haesemeyer, Christian

doi:10.1215/s0012-7094-04-12534-5

Cited by 52 publications

(68 citation statements)

References 23 publications

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“…Given V with singular locus Z, choose a resolution of singularities, V ′ → V , and set Z ′ = Z × V V ′ . By Haesemeyer [20], KH(V ; Z/q) satisfies cdh descent. It follows that we have a fibration sequence…”

Section: Algebraic K-theory Of Varieties Over R Revisitedmentioning

confidence: 99%

The Witt group of real algebraic varieties

2016

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Abstract. The purpose of this paper is to compare the algebraic Witt group W (V ) of quadratic forms for an algebraic variety V over R with a new topological invariant, W R(V C ), based on symmetric forms on Real vector bundles (in the sense of Atiyah) on the space of complex points of V . This invariant lies between W (V ) and the group KO(V R ) of R-linear topological vector bundles on the space V R of real points of V .We show that the comparison maps W (V ) → W R(V C ) and W R(V C ) → KO(V R ) are isomorphisms modulo bounded 2-primary torsion. We give precise bounds for their exponent of the kernel and cokernel, depending upon the dimension of V. These results improve theorems of Knebusch, Mahé and Brumfiel.Along the way, we prove comparison theorem between algebraic and topological hermitian K-theory, and homotopy fixed point theorems for the latter. We also give a new proof (and a generalization) of a theorem of Brumfiel.

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Section: Algebraic K-theory Of Varieties Over R Revisitedmentioning

confidence: 99%

The Witt group of real algebraic varieties

2016

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“…If X is a hypersurface inside some smooth F -scheme U , we can factor its resolution of singularities locally into a sequence of blow-ups along regular sequences and finite abstract blow-ups; using induction on the dimension of X and the length of the resolution, we conclude that E(X) ∼ = H cdh (X, E). (See [16,Th. 6.1] for details of the proof in the case where E = KH.)…”

Section: Descent For the Cdh-topologymentioning

confidence: 99%

“…6.1] for details of the proof in the case where E = KH.) Next, if X is a local complete intersection, we use induction on the embedding codimension and Mayer-Vietoris for closed covers to prove that once again E(X) ∼ = H cdh (X, E) in this case (see [16,Corollary 6.2] for details). Finally, the general case follows from this because every integral F -scheme is locally a component of a complete intersection (see [16,Th.…”

Section: Descent For the Cdh-topologymentioning

confidence: 99%

Cyclic homology, cdh-cohomology and negative K-theory

et al. 2008

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We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.

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“…So, in order to apply the Guillén-Navarro descent theorem 2.1, we have to verify properties (F1) and (F2). The first one is immediate, while (F2) follows from Thomason's calculation in [T93,Théorème], of the algebraic K-theory of a blow up along a regularly immersed subscheme, as has been observed by many authors (see, for example, [H,Theorem 3.6], [GS,Theorem 5], or [CHSW,Remark 1.6…”

Section: Descent Algebraic K-theorymentioning

confidence: 68%

“…The homotopy limit spaces holimX n , n 0, in the sense of Bousfield-Kan, [BK, Chapter XI], define a fibrant spectrum holimX; see [T80, 5.6]. In fact, one can see that PreSp has a structure of a simplicial closed model category (see [S,Proposition 2.1.5]) so that we can apply the general theory of homotopy limits for these categories [H,Chapter 18].…”

Section: Homotopy Limitmentioning

confidence: 99%

Algebraic $K$-theory and cubical descent

Pascual

Pons

2009

Homology, Homotopy and Applications

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In this note we apply the Guillén-Navarro descent theorem to define a descent variant of the algebraic K-theory of varieties over a field of characteristic zero, KD(X), which coincides with K(X) for smooth varieties and to prove that there is a natural weight filtration on the groups KD * (X). After a result of Haesemeyer, we deduce that this theory is equivalent to the homotopy algebraic K-theory introduced by Weibel. IntroductionIn Théorème (2.1.5) of [GN02], F. Guillén and V. Navarro have proved a general result, which permits one to extend (in the presence of resolution of singularities) a contravariant functor compatible with smooth blow-ups on the category of smooth schemes to a functor on the category of all schemes in such a way that the extended functor is compatible with general blow-ups. In this paper we apply this result to algebraic K-theory. More specifically, we consider the algebraic K-theory functor, which to a smooth algebraic variety over a field of characteristic zero X, associates the spectrum of the cofibration category of perfect complexes, K(X). We apply Guillén-Navarro extension criterion to prove that this functor admits an (essentially unique) extension to all algebraic varieties, KD(X), which satisfies a descent property.Moreover, by using the extension theorem in analogy with Guillén and Navarro's paper [GN03], we are able to prove the existence of a natural filtration on the KDgroups associated to an algebraic variety. In fact, the KD-theory of an algebraic variety X is defined by cubical descent and therefore, if X • is a cubical hyperresolution of X (see [GNPP, I.(2.12)]), then there is a convergent spectral sequence, see Proposition 4.3,where we have written KD * (X) = π * (KD(X)). We prove that the associated filtration on KD * (X) is independent of the chosen hyperresolution X • of X.It is well known that algebraic K-theory of schemes does not satisfy descent. , satisfies descent for varieties over a field of characteristic zero. From the uniqueness of our extension KD and Haesemeyer's result, it follows that, for any variety X over a field of characteristic zero, the spectra KD(X) and KH(X) are weakly equivalent. We observe that Cortiñas, Haesemeyer and Weibel have analyzed in [CHW] the fiber of the morphism K −→ KH in terms of the negative cyclic homology functor.Following [GN02, Théorème (2.3.6)], we also find an extension of K to a functor with compact support, K c , which once again by uniqueness is weakly equivalent to the algebraic K-theory with compact support introduced by Gillet and Soulé in [GS]. By our techniques we recover the weight filtration introduced in [GS, Theorem 7] for algebraic K-theory with compact support.Some results of this paper have been obtained by other authors using the fibrant replacement functor for a closed model category structure on the category of presheaves on the category of schemes with a suitable topology; see the papers [CHSW, CHW] and [GS]. As remarked in [R2, Theorem 4.6], the two presentations are equivalent, but we think ...

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Descent Properties of Homotopy K-Theory

Cited by 52 publications

References 23 publications

The Witt group of real algebraic varieties

The Witt group of real algebraic varieties

Cyclic homology, cdh-cohomology and negative K-theory

Algebraic $K$-theory and cubical descent

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