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Mixed motives. (English) Zbl 0902.14003
Mathematical Surveys and Monographs. 57. Providence, RI: American Mathematical Society (AMS). x, 515 p. (1998).
The search for a good category of mixed motives has been going on for more than a decade now, and although several attempts and (partial) results exist, the theory of mixed motives is still in its infancy. One would like to have, for an arbitrary scheme \(S\), an abelian tensor category \(\mathcal{MM}_S\) with a duality involution, functors \(f^*\), \(f_*\), \(f^!\) and \(f_!\) for a map of schemes \(f:T\rightarrow S\) as in the theory of sheaves, satisfying the standard relations of functoriality, adjointness, and duality. For a smooth \(S\)-scheme \(X\) one should have the associated motive \(M(X)\) in \(\mathcal{MM}_S\). For two \(S\)-schemes one should have a Künneth isomorphism for the product of the corresponding motives. There should be Tate objects \({\mathcal Z}(q)\), \(q\in\mathbb{Z}\), such that the ‘motivic’ cohomology groups \(H^p_{\mu}(X,{\mathcal Z}(q)):=\text{Ext}^p_{\mathcal{MM}_S}({\mathcal Z}(0),M(X)\otimes{\mathcal Z}(q))\) form a Bloch-Ogus cohomology theory. This cohomology theory should be universal. One should have a formalism of Chern classes relating the algebraic \(K\)-theory of a scheme to its motivic cohomology. The category \(\mathcal{MM}_S\otimes{\mathbb Q}\) should be Tannakian with Betti and étale realizations giving a fiber functor, and finally there should be a natural weight filtration on the objects of \(\mathcal{MM}_S\otimes{\mathbb Q}\) such that each graded object is semi-simple.
The book under review is the first book devoted entirely to the subject of mixed motives, and as the author writes, rather than attempting the construction of \(\mathcal{MM}_S\), he considers a more modest problem: The construction of a triangulated tensor category which has the expected properties of the bounded derived category of \(\mathcal{MM}_S\). This results in the construction of a triangulated tensor category \(\mathcal{DM}(S)\) for each reduced scheme \(S\), generated by objects \({\mathcal Z}_X(q)\), \(q\in{\mathbb Z}\), for smooth quasi-projective \(S\)-schemes \(X\). The construction of \(\mathcal{DM}(S)\) proceeds in several steps, where the incorporation of a good cycle formalism is one of the main obstacles. First, one constructs a motivic DG (differential graded) category \(\mathcal{A}=\mathcal{A}_{\text{mot}}(\mathcal{V})\) for a strictly full subcategory \(\mathcal{V}\) of the category of smooth quasi-projective \(S\)-schemes which are essentially of finite type over \(S\), i.e. which are localizations of schemes of finite type over \(S\), where \(S\) is assumed to be reduced. Then one takes the triangulated homotopy category associated to the differential graded category of bounded complexes in \(\mathcal{A}\), localizes this last one to impose the relations (homotopy invariance, Mayer-Viëtoris, etc.) of a Bloch-Ogus cohomology theory, finally one takes the pseudo-abelian hull (Karoubi envelope) to get the triangulated tensor category \(\mathcal{DM}(S)\). As a matter of fact, one has a contravariant (pseudo-)functor \(\mathcal{DM}\) from the category of (noetherian) schemes to the category of triangulated categories. The construction of \(\mathcal{DM}\) can be generalized to (co)simplicial schemes, and more generally, to diagrams of schemes. Motivic cohomology is defined by: \[ H^p(X,{\mathcal Z}(q)):=\text{Hom}_{\mathcal{DM}(\mathcal{V})}({\mathcal Z}_S(0),{\mathcal Z}_X(q)[p]), \] where \({\mathcal Z}_S(0)\) is the \(1\)-object of the tensor category \(\mathcal{DM}(\mathcal{V})\). One can define the Chow group CH\((\Gamma)\) for an object \(\Gamma\) of \(\mathcal{DM}(\mathcal{V})\) and make a cycle class map \(\text{cl}(\Gamma):\text{CH}(\Gamma)\rightarrow\text{Hom}_{\mathcal{DM}(\mathcal{V})}(1,\Gamma)\). Under suitable conditions on the base scheme \(S\) and on the category \(\mathcal{V}\), \(\text{cl}(\Gamma)\) is shown to be an isomorphism. Also, Bloch’s construction of his higher Chow groups for schemes over a field, can be extended to the case of an arbitrary base scheme to get groups \(\text{CH}^p(X,q)\) and cycle class maps \(\text{cl}_X^{q,p}:\text{CH}^q(X,2q-p)\rightarrow H^p(X,{\mathcal Z}(q))\). Again, for suitable \(S\) and \(\mathcal{V}\), the \(\text{cl}_X^{q,p}\) can be shown to be isomorphisms. For \(X\) in \(\mathcal{V}\), there are natural isomorphisms \(K_{2q-p}(X)^{(q)}\rightarrow H^p(X,{\mathbb Q}(q))\), where \(K_n(X)^{(q)}\) is the weight \(q\) eigenspace of \(K_n(X)\otimes{\mathbb Q}\) for the Adams operators. The relationship between \(K\)-theory and motivic cohomology is given by Chern classes and the Riemann-Roch theorem. The constructions are extensively discussed, also for diagrams. The notion of Grothendieck’s lambda ring is recalled, and Chern classes and the Chern character for higher \(K\)-theory are constructed. A sketch of Bloch’s argument which shows that the Chern character induces an isomorphism \(K_p(X)\otimes{\mathbb Q}{\buildrel\sim\over\rightarrow}\bigoplus_qH^{2q-p}(X,{\mathbb Q}(q))\) for \(S=\text{Spec}(k)\), \(k\) a field, or \(S\) smooth and of dimension at most one over a field, is given.
The importance of the notion of duality in tensor categories has been known since Grothendieck’s ideas on motives and Tannakian categories. Of course, when dealing with cohomology theories, the notion of duality is at the heart of the matter. Here it is shown that for \(X\) in \(\mathcal{V}\), \(X\) smooth and projective over \(S\), one may define the dual \({\mathcal Z}_X(a)[b]^D\) of the motive \({\mathcal Z}_X(a)[b]\) by \({\mathcal Z}_X(a)[b]^D={\mathcal Z}_X(d-a)[2d-b]\), where \(d\) is the dimension of \(X\). This notion of duality induces an exact duality on \(\mathcal{DM}(\mathcal{V})^{\text{pr}}\), the smallest strictly full triangulated subcatecory of \(\mathcal{DM}(\mathcal{V})\) containing the objects \({\mathcal Z}_X(a)\) with \(X\) in \(\mathcal{V}\) smooth and projective over \(S\) and closed under taking summands. In particular, if \(S=\text{Spec}(k)\) and one has resolution of singularities, one obtains a rigid triangulated tensor category \(\mathcal{DM}(k)\). For \(S=\text{Spec}(k)\) the action of correspondences as homomorphisms in \(\mathcal{DM}(S)\) implies that the category of Chow motives over a field \(k\) admits a fully faithful embedding into the triangulated tensor category \(\mathcal{DM}(k)\). The homological motive of a smooth \(S\)-scheme \(X\) is defined as its dual: \({\mathcal Z}_X^h:={\mathcal Z}_X^D\). The motivic homology of \(X\) is given by \(H_p(X,{\mathcal Z}(q)):=\text{Hom}_{\mathcal{DM}(S)}(1,{\mathcal Z}^h_X(-q)[-p])\). The Borel-Moore motive of \(X\) (assumed equidimensional of dimension \(d\)) is defined by \({\mathcal Z}_X^{\text{B.M.}}:={\mathcal Z}_X(d)[2d]\), and the Borel-Moore homology of \(X\) is given by \(H_p^{\text{B.M.}}(X,{\mathcal Z}(q)):=\text{Hom}_{\mathcal{DM}(S)}(1,{\mathcal Z}_X^{\text{B.M.}}(-q)[-p])\). For \(X\) such that the motive \({\mathcal Z}_X\) is in \(\mathcal{DM}(\mathcal{V})^{\text{pr}}\), one defines the motive of \(X\) with compact support over \(S\) by \({\mathcal Z}^{c/S}_X:=({\mathcal Z}_Z^{\text{B.M.}})^D\). The compactly supported motivic cohomology of \(X\) is \(H^p_{c/S}(X,{\mathcal Z}(q)):=\text{Hom}_{\mathcal{DM}(S)}(1,{\mathcal Z}_X^{c/S}(q)[p])\). One can extend the notion of Borel-Moore motive and motive with compact support to non-smooth schemes \(X\). The main result becomes a Riemann-Roch theorem for singular schemes.
The triangulated Tate motivic category \(\mathcal{DTM}(S)\) is defined as the strictly full triangulated subcategory of \(\mathcal{DM}(S)\) generated by objects of the form \({\mathcal Z}_S(q)\), \(q\in{\mathbb{Z}}\) (actually this can be given in a more general setting). If the Beilinson-Soulé vanishing conjecture for \(K_p(F)^{(q)}\), \(F\) a field, holds, then \(\mathcal{DTM}(\text{Spec}(F))_{\mathbb Q}\) has a canonical \(t\)-structure with heart \(\mathcal{MTM}(F)_{\mathbb Q}\), the (Tannakian) category of mixed Tate motives. The category \(\mathcal{MTM}(F)_{\mathbb Q}\) plays an important role in the Beilinson-Deligne theory of polylogarithms.
Motives have realizations in cohomology theories. For a Grothendieck site \((\mathfrak{G},\mathfrak{T})\) and a tensor category \(\mathcal{B}\) one writes \(\text{Sh}^{\mathcal{B}}_{\mathfrak{G},\mathfrak{T}}\) for the category of sheaves on \(\mathfrak{G}\) with values in \(\mathcal{B}\). Its derived category (of bounded below complexes) is denoted \(\mathbb{D}^+ (\text{Sh}^{\mathcal{B}}_{\mathfrak{G}, \mathfrak{T}})\). Similarly, for an object \(X\) of \(\mathfrak{G}\) one writes \(\mathbb{D}^+ (\text{Sh}^{\mathcal{B}}_{\mathfrak{G}, \mathfrak{T}}(X))\). A construction of a realization functor \(\mathfrak{R}_{\mathcal F}: \mathcal{DM}(\mathcal{V})\rightarrow \mathbb{D}^+(\text{Sh}^{\mathcal{B}}_{\mathfrak{G},\mathfrak{T}}(*))\) for a geometric cohomology theory \(\mathcal{F}\) is given. Concrete examples of such realization functors are the Betti realization, the étale realization and the Hodge realization. These are discussed in detail. Also, the realizations in Saito’s mixed Hodge modules and in Beilinson’s and Jannsen’s category of mixed absolute Hodge complexes are constructed.
The last chapter deals with motivic constructions and comparisons. Milnor’s \(K\)-theory is interpreted in terms of the category \(\mathcal{DM}\), and a proof of the Steinberg relation, based solely on formal properties of \(\mathcal{DM}\), is given. Also, Beilinson’s construction of the rational motivic polylogarithm is discussed in terms of \(\mathcal{DM}\). Finally, for a field \(k\) admitting resolution of singularities, one may construct an exact functor \(\Psi:\mathcal{DM}(k)\rightarrow DM_{gm}(k)\), where \(DM_{gm}(k)\) is Voevodsky’s category of geometric motives over \(k\). Then \(\Psi\) is shown to be an equivalence of triangulated categories.
The book is subdivided in two parts, part I covering the topics mentioned above. All topics all treated in detail, but reading the text is not always easy, though fortunately every chapter begins with an introduction that briefly resumes its contents. Sometimes notation is cumbersome, especially in the first chapter which is the most discouraging because of many concepts and categories that appear. Also, referencing from the beginning so many times to the second part of the book (cf. infra) is not quite stimulating. Fortunately, once one has mastered the first chapter, things become much more transparent. Part I ends with two interesting appendices, one on Suslin’s and Voevodsky’s theory of equidimensional cycles, and one on algebraic \(K\)-theory.
The second part, part II, is called “Categorical algebra”. The reader should be urged to read this part, at least superficially, to get acquainted with the author’s style and notations, before embarking on part I. It is divided into four chapters:
(i) Symmetric monoidal structures,
(ii) DG categories and triangulated categories,
(iii) Simplicial and cosimplicial constructions, and
(iv) Canonical models for cohomology.
These chapters are fairly self-contained and extremely useful to understand part I. The book closes with an extensive bibliography of 130 references, a subject index, and an index of notation. All in all, everyone interested in mixed motives and willing to take a serious look at the topic, should try his/her hand on this impressive work.

14A20 Generalizations (algebraic spaces, stacks)
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
19-02 Research exposition (monographs, survey articles) pertaining to \(K\)-theory
14C25 Algebraic cycles
14C15 (Equivariant) Chow groups and rings; motives
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C40 Riemann-Roch theorems
19D45 Higher symbols, Milnor \(K\)-theory
19E08 \(K\)-theory of schemes
19E20 Relations of \(K\)-theory with cohomology theories
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14F99 (Co)homology theory in algebraic geometry