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Filtrations on higher algebraic $$K$$-theory. (English) Zbl 0951.19003
Raskind, Wayne (ed.) et al., Algebraic $$K$$-theory. Proceedings of an AMS-IMS-SIAM summer research conference, Seattle, WA, USA, July 13-24, 1997. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 67, 89-148 (1999).
Let $$X$$ be a noetherian scheme and let $$K_0(X)$$ denote its Grothendieck group of locally free coherent $$\mathcal O_X$$-modules. On $$K_0(X)$$ one has the filtration by codimension of supports $$F^p_{\text{cod}}$$ and also the $$\gamma$$-filtration $$F^p_{\gamma}$$. It has been known since a long time that $$F^p_{\gamma}K_0(X)\subset F^p_{\text{cod}}K_0(X)$$. One also has these filtrations on the higher algebraic $$K$$-groups $$K_m(X)$$. Here a detailed account is given of the fact that one has, for $$X$$ of finite type over a field, the inclusions $$F^p_{\gamma}K_m(X)\subset F^{p-m} _{\text{cod}}K_m(X)$$. To prove this result a third filtration, the so called Brown filtration $$F^p$$ of $$K_m(X)$$, is introduced and both $$F^p_{\gamma}$$ and $$F^p_{\text{cod}}$$ are related to $$F^p$$. The paper consists of an introduction, seven sections and a bibliography.
The first section deals with the general setup of homotopy theory of simplicial sheaves on a Grothendieck site $$C$$. It is known that the category $$s{\mathcal T}$$ of sheaves of simplicial sets on $$C$$ is a closed simplicial model category. Its objects are called spaces. One may consider the subcategory $$s{\mathcal G}\subset s{\mathcal T}$$ of simplicial sheaves of groups and group homomorphisms. It is shown in detail that $$s{\mathcal G}$$ is a closed model category as well. So one has a good homotopy theory for $$s{\mathcal G}$$. One has the notion of Postnikov tower, Eilenberg-MacLane spaces $$K(\pi,n)$$, cohomology $$H^n(X,\pi)$$, etc. For two spaces $$X$$ and (pointed) $$Y$$ one has the Brown cohomological spectral sequence $$E_r^{p,q}$$, $$r\geq 2$$, given by $$E^{p,q}_2=H^p(X,\pi_{-q}Y)$$ for $$p+q\leq 0$$, and $$E^{p,q}_2=0$$ else. The $$E_{\infty}$$ is related to $$H^{p+q}(X,Y)$$.
The second section gives the comparison of two spectral sequences. Spectral sequences are presented by means of exact couples and it is shown how under a shift of filtration an isomorphism on $$E_2$$ terms is induced. Returning to geometry, let $$X$$ be a finite-dimensional noetherian topological space with a sheaf $$G$$ of simplicial groups on $$X$$. One may compute the homotopy type of the simplicial group $$\Gamma(X,G)= \text{Hom}(X,G)$$ by means of the Postnikov tower of $$G$$. One may also compute $$\Gamma(X,G)$$ via the filtration by coniveau (or codimension). The first point of view leads to the corresponding Brown spectral sequence $$E^{p,q}_r= E^{p,q}_r(X,G)$$ while the second gives the sequence $$E^{p,q}_{r,\text{cod}}=E^{p,q}_{r,\text{cod}}(X,G)$$. Under suitable conditions on $$G$$ and $$\Gamma$$ the main result of this section gives a morphism of spectral sequences $$E^{p,q}_r\rightarrow E^{p,q}_{r,\text{cod}}$$ compatible with the identity map on the abutments $$H^{-p-q}(X,G)$$. In particular, one has inclusions $$F^pH^m(X,G)\subset F^p_{\text{cod}}H^m(X,G)$$.
The next step is to implicate higher algebraic $$K$$-theory. $$K$$-theory for a locally ringed topos is defined. The notion of $$K$$-coherence for a space $$X$$ is introduced. Let $$T$$ be the category of sheaves on a category of schemes $${\mathcal C}$$. For $$X\in {\mathcal C}$$ let $$P(X)$$ denote the category of locally free and finitely generated $$\mathcal O_{T/X}$$-modules and let $$\Omega BQP$$ be the simplicial sheaf associated to that functor. Using a fibrant resolution of $$\Omega BQP$$ one has a natural map $$K_m(X)=\pi_m(\Omega BQP(X))\rightarrow H^{-m}(X,K)$$, $$K={\mathbb Z}\times{\mathbb Z}_{\infty}\text{BGL}$$, for all $$m\geq 0$$. For $$X$$ a noetherian scheme of finite Krull dimension, and $$T$$ either the big Zariski topos of all noetherian schemes of finite Krull dimension, or the big Zariski topos of all schemes of finite type over $$X$$ or the small Zariski topos of $$X$$, it is shown that $$K_m(X)\rightarrow H^{-m}(X,K)$$ is an isomorphism. For a field $$k$$, let $${\mathcal V}$$ denote the category of regular schemes of finite type over $$S=\text{Spec}(k)$$ equipped with the Zariski topology. Let $$s{\mathcal V}$$ be the category of simplicial objects in $$\mathcal V$$. For $$X \in s{\mathcal V}$$, let $$E$$ be a (finite rank) vector bundle on $$X$$ with associated projective bundle $$\pi:P(E)\rightarrow X$$. Then, for the filtration $$F^r$$ associated to the Brown spectral sequence, one has $$(\pi^*)^{-1}(F^rK_m( P(E)))=F^rK_m(X)$$.
In the fourth section it is shown that for a $$K$$-coherent space $$X$$ the $$H^0(X,K)$$ is a $$\lambda$$-algebra with involution and augmentation $$\varepsilon:H^0(X,K)\rightarrow H^0(X,{\mathbb Z})$$ induced by projection onto the first factor of $${\mathbb Z}\times{\mathbb Z}_{\infty}\text{BGL}$$. For $$m> 0$$, $$H^{-m}(X,K)$$ is an $$H^0(X,K)$$-$$\lambda$$-module with involution.
Section 5 deals with the Brown-Gersten spectral sequence and its relation to the Quillen spectral sequence. It leads to the main results of the paper: (i) For all $$m\geq 0$$ and $$p\geq 0$$ one has $$F^pK_m(X)\subset F ^p_{\text{cod}}(X)$$; (ii) When $$X$$ is of finite type over a field, then $$F^p_{\gamma}K_m(X)\subset F^{p-m}K_m(X)$$; (iii) When $$X$$ is regular of finite type over a field the Quillen spectral sequence coincides from $$E_2$$ on with the Brown-Gersten spectral sequence. In particular, $$F^p K_m(X)=F^p_{\text{cod}}K_m(X)$$.
Section 6 relates Chern classes and the $$\lambda$$-ring structure on higher $$K$$-groups to the effect that for a $$K$$-coherent space the total Chern class is a morphism of $$\lambda$$-rings with involution.
In the final section the Bloch-Lichtenbaum spectral sequence for a field $$F$$ (tensored with $${\mathbb Q}$$) is shown to degenerate from $$E_2$$ on, and to converge to the $$\gamma$$-filtration on $$K_{-p-q}(F)\otimes_{\mathbb Z}{\mathbb Q}$$.
For the entire collection see [Zbl 0931.00031].

##### MSC:
 19E08 $$K$$-theory of schemes 14C25 Algebraic cycles