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].

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].

Reviewer: W.W.J.Hulsbergen (Haarlem)