Weight filtrations in algebraic \(K\)-theory.

*(English)*Zbl 0803.19001
Jannsen, Uwe (ed.) et al., Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 55, Pt. 1, 207-237 (1994).

The paper gives a survey of algebraic \(K\)-theory, from its very foundations up to the recent conjectural formalism of motivic cohomology in relation to the Beilinson conjectures. In the introduction the author says: ‘We give a basic description of algebraic \(K\)-theory and explain how Quillen’s idea that the Atiyah-Hirzebruch spectral sequence of topology may have an algebraic analogue guides the search for motivic cohomology.’ This program is carried out in the subsequent twenty-one sections giving a wealth of material on related matters as well. The starting principle is to realize an interesting group (such as the Grothendieck group of a ring) as a (low dimensional) homotopy group of some space constructed in some combinatorial way out of the algebraic inputs and then study the homotopy type of this space. Notions as simplicial sets, geometric realization, nerve of a category, classifying spaces of groups, simplicial abelian groups and Eilenberg-MacLane spaces are briefly recalled to set the stage. A small section mentions the original definitions of \(K_ 0\), \(K_ 1\) and \(K_ 2\) of a ring. Then a more detailed account is given of Quillen’s plus-construction and his \(Q\)-construction for exact categories with its ensuing definition of the higher \(K\)-groups as homotopy groups (of the associated loop spaces). Also, Waldhausen’s and Gillet and Grayson’s definitions of the higher \(K\)-groups are briefly discussed. Quillen’s four fundamental theorems, i.e., the additivity theorem, the Jordan-Hölder theorem (implying the relation of algebraic cycles to algebraic \(K\)-theory), the resolution theorem and the localization theorem are nicely explained. Then comes an overview of known results such as the \(K\)-groups of finite fields (Quillen), of algebraically closed fields (Suslin), \(K_ 2 (\mathbb{Q})\) (Tate), and Quillen and Borel’s results on the ranks of the \(K\)-groups of the ring of integers of a number field, as well as the first four explicitly known \(K\)-groups of \(\mathbb{Z}\).

From now on the rings \(R\) on which the \(K\)-functors operate are assumed to be commutative. This implies that one can define a product on \(\bigoplus_{n=0}^ \infty K_ n(R)\), thus making it a skew- commutative graded ring. One defines the Milnor \(K\)-group \(K^ M_ n(F)\) of a field as the degree \(n\) part of the quotient of the tensor algebra of \(F^ \times\) by the ideal generated by the so-called Steinberg relations. Then \(K_ i^ M(F)= K_ i(F)\) for \(i=0,1,2\), but not for \(i>2\). The Adams operations \(\psi^ k\) on \(K_ n(R)\) are defined in a heuristic way as \(k\)th-power maps on the elements of \(R\) entering in the construction of an element in the \(K\)-groups. Their properties are resumed. If now the projective \(R\)-module \(P\) can be written as \(P\simeq L_ 1\oplus L_ 2\oplus \cdots\oplus L_ n\) with \(L_ i\) projective of rank 1, one defines \(x_ i= [L_ i]-1\) and \(x= x_ 1+ \cdots+ x_ n= [P] -n\) and then \(\gamma^ k (x)\) by the formula \(\sum_{k=0}^ n \gamma^ k (x) t^ k= \prod_{i=1}^ n (1+ x_ i t)\), and one verifies that \(\psi^ k (\gamma^ r (x))= k^ r \gamma^ r (x)+ g(x_ 1, \dots, x_ n)\), where \( g(x_ 1,\dots, x_ n)\) is some symmetric polynomial in \(x_ 1,\dots, x_ n\) of degree \(\geq n+1\). One defines a filtration \(\{F^ k_ \gamma K_ 0(R)\}_{k \geq 1}\) on \(K_ 0 (R)\) by taking \(F^ k_ \gamma K_ 0(R)\) to be the subgroup of \(K_ 0(R)\) generated by all products of the form \(\gamma^{k_ 1} (y_ 1) \cdots \gamma^{k_ m} (y_ m)\) with the \(y_ i\in K_ 0(R)\) of rank \(0\) and such that \(\sum_{i=1}^ m k_ i \geq k\). At the same time one shows that \(K_ 0(R)_ \mathbb{Q}= K_ 0(R) \otimes \mathbb{Q}\) splits as a direct sum \(K_ 0 (R)_ \mathbb{Q}= \bigoplus_{i=0}^ \infty K_ 0(R)^{(i)}_ \mathbb{Q}\), where \(K_ 0(R)^{(i)}_ \mathbb{Q}\) is the eigenspace for \(\psi^ k\) with eigenvalue \(k^ i\). Then there is an isomorphism \(K_ 0(R)^{(i)}_ \mathbb{Q} {\overset \sim \rightarrow} F^ i_ \gamma K_ 0 (R)_{\mathbb{Q}}/ F^{i+1}_ \gamma K_ 0 (R)_ \mathbb{Q}\). For a non-singular quasi-projective variety \(X\) defined over a field one has \(K_ 0 (X)^{(i)}_ \mathbb{Q}\cong \text{CH}^ i(X)_ \mathbb{Q}\), the \(i\)th Chow space of \(X\). For the higher \(K\)-groups one also defines the weight \(i\) part of \(K_ n(R)\), \(K_ n(R)^{(i)}_ \mathbb{Q}\), as the eigenspace of \(\psi^ k\) with eigenvalue \(k^ i\). One can also define a gamma-filtration on \(K_ n(R)\) and show that \(K_ n(R)_ \mathbb{Q}\) splits as the direct sum of the \(K_ n (R)^{(i)}_ \mathbb{Q}\). For a field \(F\) the image of \(K_ n^ M (F)\) in \(K_ n(F)\) is of weight \(n\). For fields one has some results on the weights, but in general not much is known.

For a finite cell complex \(X\) one has the Atiyah-Hirzebruch theorem relating topological \(K\)-theory to cohomology. It says that there is a spectral sequence with \(E_ 2\)-term \(E_ 2^{pq}= H^ p (X, \mathbb{Z} (- q/2))\) abutting to \(K_{-p-q}(X)\). Furthermore, one can show a corollary of the Atiyah-Hirzebruch result that says that \(K_ n (X^{\text{top}})^{(i)}_ \mathbb{Q} \cong H^{2i-n} (X, \mathbb{Q}) (i))\), where \(K_ n (X^{\text{top}})= K_ n (\mathbb{C} (X^{\text{top}}))\) denote the topological \(K\)-groups and where \(\mathbb{Q} (i)\) denotes \(\mathbb{Q}\) together with the action of the Adams operations given by \(\psi^ k (x) =k^ i x\). By using the Postnikov tower of \(BU\), Dwyer and Friedlander were able to construct an Atiyah-Hirzebruch spectral sequence for a scheme \(X\). For an algebraic variety \(X\) Beilinson, possibly guided by the Atiyah- Hirzebruch result, defined motivic cohomology groups \(H^{2i-n}_{\mathcal M} (X, \mathbb{Q} (i)):= K_ n (X)^{(i)}_{\mathbb{Q}}\), and stated some intriguing conjectures about them. In modern terms these motivic cohomology groups should be higher Yoneda extensions of certain objects (Tate motives) in an abelian category (yet to be constructed) of mixed motives. An interesting question is whether one can define integral motivic cohomology groups \(H^ m_{\mathcal M} (X, \mathbb{Z} (i))\). Such a construction would follow from a weight filtration of the algebraic \(K\)- space \(K(X)\) by taking as cohomology groups the homotopy groups of the graded pieces thus obtained. This would lead to the construction of a so- called motivic complex sought for by Beilinson and Lichtenbaum to solve some deep conjectures in arithmetic algebraic geometry related to \(L\)- functions. Several attempts to construct motivic complexes are mentioned. Combining known and conjectural properties of the motivic cohomology groups one is led to hope for a prospective Atiyah-Hirzebruch spectral sequence for an algebraic variety \(X\) of the form \(E_ 2^{pq}= H^{p- q}_{\mathcal M} (X, \mathbb{Z} (-q))\to K_{-p-q} (X)\). This can be graphically represented in a handy chart where one can read off many known and conjectural properties of \(K\)-groups, Milnor \(K\)-groups, Chow groups, motivic cohomology, \(\dots\), etc. One may also ask for étale cohomology and étale \(K\)-theory. Here one has a result of Dwyer and Friedlander giving a fourth quadrant spectral sequence \(E_ 2^{pq}= H^ p_{\acute{e}t} (X, \mathbb{Z} (-q/2)) \to K^{\acute{e}t}_{-p-q} (X; \mathbb{Z}_ \ell)\) for a scheme of finite \(\mathbb{Z}/ \ell\)-cohomological dimension, converging in positive degrees. This can be applied to rephrase the Quillen-Lichtenbaum conjecture.

Related work of S. Mitchell on a completely explicit description of \(K_ n (\mathbb{Z})\), \(n\geq 2\), is briefly discussed, as well as results of Thomason. The final section recalls the relation of \(K\)-theory and/or motivic cohomology to other cohomology theories, such as Deligne- Beilinson cohomology, via Chern class maps and Chern character maps. This is at the heart of Beilinson’s conjectures.

For the entire collection see [Zbl 0788.00053].

From now on the rings \(R\) on which the \(K\)-functors operate are assumed to be commutative. This implies that one can define a product on \(\bigoplus_{n=0}^ \infty K_ n(R)\), thus making it a skew- commutative graded ring. One defines the Milnor \(K\)-group \(K^ M_ n(F)\) of a field as the degree \(n\) part of the quotient of the tensor algebra of \(F^ \times\) by the ideal generated by the so-called Steinberg relations. Then \(K_ i^ M(F)= K_ i(F)\) for \(i=0,1,2\), but not for \(i>2\). The Adams operations \(\psi^ k\) on \(K_ n(R)\) are defined in a heuristic way as \(k\)th-power maps on the elements of \(R\) entering in the construction of an element in the \(K\)-groups. Their properties are resumed. If now the projective \(R\)-module \(P\) can be written as \(P\simeq L_ 1\oplus L_ 2\oplus \cdots\oplus L_ n\) with \(L_ i\) projective of rank 1, one defines \(x_ i= [L_ i]-1\) and \(x= x_ 1+ \cdots+ x_ n= [P] -n\) and then \(\gamma^ k (x)\) by the formula \(\sum_{k=0}^ n \gamma^ k (x) t^ k= \prod_{i=1}^ n (1+ x_ i t)\), and one verifies that \(\psi^ k (\gamma^ r (x))= k^ r \gamma^ r (x)+ g(x_ 1, \dots, x_ n)\), where \( g(x_ 1,\dots, x_ n)\) is some symmetric polynomial in \(x_ 1,\dots, x_ n\) of degree \(\geq n+1\). One defines a filtration \(\{F^ k_ \gamma K_ 0(R)\}_{k \geq 1}\) on \(K_ 0 (R)\) by taking \(F^ k_ \gamma K_ 0(R)\) to be the subgroup of \(K_ 0(R)\) generated by all products of the form \(\gamma^{k_ 1} (y_ 1) \cdots \gamma^{k_ m} (y_ m)\) with the \(y_ i\in K_ 0(R)\) of rank \(0\) and such that \(\sum_{i=1}^ m k_ i \geq k\). At the same time one shows that \(K_ 0(R)_ \mathbb{Q}= K_ 0(R) \otimes \mathbb{Q}\) splits as a direct sum \(K_ 0 (R)_ \mathbb{Q}= \bigoplus_{i=0}^ \infty K_ 0(R)^{(i)}_ \mathbb{Q}\), where \(K_ 0(R)^{(i)}_ \mathbb{Q}\) is the eigenspace for \(\psi^ k\) with eigenvalue \(k^ i\). Then there is an isomorphism \(K_ 0(R)^{(i)}_ \mathbb{Q} {\overset \sim \rightarrow} F^ i_ \gamma K_ 0 (R)_{\mathbb{Q}}/ F^{i+1}_ \gamma K_ 0 (R)_ \mathbb{Q}\). For a non-singular quasi-projective variety \(X\) defined over a field one has \(K_ 0 (X)^{(i)}_ \mathbb{Q}\cong \text{CH}^ i(X)_ \mathbb{Q}\), the \(i\)th Chow space of \(X\). For the higher \(K\)-groups one also defines the weight \(i\) part of \(K_ n(R)\), \(K_ n(R)^{(i)}_ \mathbb{Q}\), as the eigenspace of \(\psi^ k\) with eigenvalue \(k^ i\). One can also define a gamma-filtration on \(K_ n(R)\) and show that \(K_ n(R)_ \mathbb{Q}\) splits as the direct sum of the \(K_ n (R)^{(i)}_ \mathbb{Q}\). For a field \(F\) the image of \(K_ n^ M (F)\) in \(K_ n(F)\) is of weight \(n\). For fields one has some results on the weights, but in general not much is known.

For a finite cell complex \(X\) one has the Atiyah-Hirzebruch theorem relating topological \(K\)-theory to cohomology. It says that there is a spectral sequence with \(E_ 2\)-term \(E_ 2^{pq}= H^ p (X, \mathbb{Z} (- q/2))\) abutting to \(K_{-p-q}(X)\). Furthermore, one can show a corollary of the Atiyah-Hirzebruch result that says that \(K_ n (X^{\text{top}})^{(i)}_ \mathbb{Q} \cong H^{2i-n} (X, \mathbb{Q}) (i))\), where \(K_ n (X^{\text{top}})= K_ n (\mathbb{C} (X^{\text{top}}))\) denote the topological \(K\)-groups and where \(\mathbb{Q} (i)\) denotes \(\mathbb{Q}\) together with the action of the Adams operations given by \(\psi^ k (x) =k^ i x\). By using the Postnikov tower of \(BU\), Dwyer and Friedlander were able to construct an Atiyah-Hirzebruch spectral sequence for a scheme \(X\). For an algebraic variety \(X\) Beilinson, possibly guided by the Atiyah- Hirzebruch result, defined motivic cohomology groups \(H^{2i-n}_{\mathcal M} (X, \mathbb{Q} (i)):= K_ n (X)^{(i)}_{\mathbb{Q}}\), and stated some intriguing conjectures about them. In modern terms these motivic cohomology groups should be higher Yoneda extensions of certain objects (Tate motives) in an abelian category (yet to be constructed) of mixed motives. An interesting question is whether one can define integral motivic cohomology groups \(H^ m_{\mathcal M} (X, \mathbb{Z} (i))\). Such a construction would follow from a weight filtration of the algebraic \(K\)- space \(K(X)\) by taking as cohomology groups the homotopy groups of the graded pieces thus obtained. This would lead to the construction of a so- called motivic complex sought for by Beilinson and Lichtenbaum to solve some deep conjectures in arithmetic algebraic geometry related to \(L\)- functions. Several attempts to construct motivic complexes are mentioned. Combining known and conjectural properties of the motivic cohomology groups one is led to hope for a prospective Atiyah-Hirzebruch spectral sequence for an algebraic variety \(X\) of the form \(E_ 2^{pq}= H^{p- q}_{\mathcal M} (X, \mathbb{Z} (-q))\to K_{-p-q} (X)\). This can be graphically represented in a handy chart where one can read off many known and conjectural properties of \(K\)-groups, Milnor \(K\)-groups, Chow groups, motivic cohomology, \(\dots\), etc. One may also ask for étale cohomology and étale \(K\)-theory. Here one has a result of Dwyer and Friedlander giving a fourth quadrant spectral sequence \(E_ 2^{pq}= H^ p_{\acute{e}t} (X, \mathbb{Z} (-q/2)) \to K^{\acute{e}t}_{-p-q} (X; \mathbb{Z}_ \ell)\) for a scheme of finite \(\mathbb{Z}/ \ell\)-cohomological dimension, converging in positive degrees. This can be applied to rephrase the Quillen-Lichtenbaum conjecture.

Related work of S. Mitchell on a completely explicit description of \(K_ n (\mathbb{Z})\), \(n\geq 2\), is briefly discussed, as well as results of Thomason. The final section recalls the relation of \(K\)-theory and/or motivic cohomology to other cohomology theories, such as Deligne- Beilinson cohomology, via Chern class maps and Chern character maps. This is at the heart of Beilinson’s conjectures.

For the entire collection see [Zbl 0788.00053].

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

##### MSC:

19-02 | Research exposition (monographs, survey articles) pertaining to \(K\)-theory |

19E20 | Relations of \(K\)-theory with cohomology theories |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

19E08 | \(K\)-theory of schemes |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |