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Algebraic $$K$$-theory. (English) Zbl 0722.19001
Progress in Mathematics, 90. Boston, MA etc.: Birkhäuser. xv, 314 p. sFr. 88.00; DM 106.00 (1991).
This book is a welcome monograph on algebraic K-theory in the way it was founded by D. Quillen in the early 1970’s. It is based on lectures given by the author at the Tata Institute during 1986-1987 and consists of an Introduction, seven chapters on what may be called classical K-theory, an extensive account of the Merkurev-Suslin Theorem, a final chapter on localization for singular varieties and three appendices on topology, category theory and exact couples, respectively. After an introductory first chapter on Milnor’s K-theory of $$K_ 0$$, $$K_ 1$$ and $$K_ 2$$ of rings, the notion of symbols (norm residue symbol, Galois symbol, differential symbol) and Matsumoto’s theorem on $$K_ 2$$ of a (commutative) field, the reader is progressively led, via the plus- construction, Quillen’s first definition of $$K_ i(R)$$, R an associative ring, as the homotopy group $$\pi_ i(BGL(R)^+)$$, Loday’s theory of the H-space structure of $$BGL(R)^+$$ and his natural products $$K_ i(R)\otimes K_ j(R)\to K_{i+j}(R)$$, to the higher algebraic K-theory of (spectra of) rings and schemes. To this end the notion of the classifying space of a small category is extensively discussed. If $${\mathcal C}$$ is a small exact category (exact meaning additive and embedded as a full additive subcategory of an abelian category $${\mathcal A}$$ and “closed under extensions” in $${\mathcal A})$$ one can construct another small category Q$${\mathcal C}$$ with the same objects as $${\mathcal C}$$ and suitably defined morphisms, with classifying space BQ$${\mathcal C}$$, and then one defines, after Quillen, the higher algebraic K-groups $$K_ i({\mathcal C})$$ as the homotopy groups $$\pi_{i+1}(BQ{\mathcal C},\{0\})$$, $$i\geq 0$$, where 0 is a zero-object of $${\mathcal C}$$ so that $$\{$$ $$0\}$$ is a point of BQ$${\mathcal C}$$. Then, if $${\mathcal P}$$ is a full additive subcategory of the (small) exact category $${\mathcal M}$$, closed under extensions and taking subobjects in $${\mathcal M}$$ and such that any object of $${\mathcal M}$$ has a finite resolution by objects of $${\mathcal P}$$, one has the Resolution Theorem which says that $$K_ i({\mathcal P})\simeq K_ i({\mathcal M})$$ for all $$i\geq 0$$. Also, for a full abelian subcategory $${\mathcal B}$$ of an abelian category $${\mathcal A}$$, closed under taking subobjects, quotients and finite products and such that each object of $${\mathcal A}$$ admits a finite filtration with consecutive quotients in $${\mathcal B}$$, one has the Devissage Theorem: $$K_ i({\mathcal B})\simeq K_ i({\mathcal A})$$. As a third result one has the Localization Exact Sequence $...\to K_{i+1}({\mathcal C})\to K_ i({\mathcal B})\to K_ i({\mathcal A})\to K_ i({\mathcal C})\to...\to K_ 0({\mathcal A})\to K_ 0({\mathcal C})\to 0,$ where $${\mathcal B}$$ is a Serre subcategory of the abelian category $${\mathcal A}$$, and $${\mathcal C}$$ is the quotient category $${\mathcal A}/{\mathcal B}$$. Now, for an arbitrary scheme X, let $${\mathcal P}={\mathcal P}(X)$$ denote the category of locally free sheaves of finite rank on X. Then $${\mathcal P}$$ is an exact category and one defines $$K_ i(X)=K_ i({\mathcal P}(X))$$. For a noetherian scheme X, let $${\mathcal M}={\mathcal M}(X)$$ be the category of coherent sheaves on X. Then one defines $$G_ i(X)$$ (or $$K_ i'(X))$$ by $$G_ i(X)=K_ i({\mathcal M}(X))$$. For a regular noetherian scheme X one obtains, by the Resolution Theorem, $$K_ i(X)=G_ i(X)$$. For a (noetherian) ring R one defines $$K_ i(R)=K_ i(Spec(R))$$ $$(G_ i(R)=K_ i'(R)=G_ i(Spec(R)))$$. This definition agrees with the one given by the plus-construction. $$K_ i$$ is a contravariant functor from the category of schemes to abelian groups and $$G_ i$$ is a contravariant functor from the category of noetherian schemes and that morphisms to abelian groups. Tensoring with vector bundles induces an action of $$K_ 0(X)$$ on $$K_ i(X)$$ and $$G_ i(X)$$ when X is a noetherian scheme. For a proper morphism $$f:X\to Y$$ between noetherian schemes one has a direct image map $$f_*: G_ i(X)\to G_ i(Y)$$ under suitable finiteness conditions. Similarly for $$f_*: K_ i(X)\to K_ i(Y)$$. Also, there is a projection formula $$f_*(x)\cdot y=f_*(x\cdot f^*(y))$$, $$x\in K_ 0(X)$$, $$y\in K_ i(Y)$$ or $$G_ i(Y)$$, where $$\cdot$$ denotes the action of $$K_ 0(Y)$$ on $$K_ i(Y)$$ or $$G_ i(Y)$$, respectively.
Further basic topics discussed are the existence of a Mayer-Vietoris sequence for $$G_ i$$, the Homotopy Property which says that a flat map with affine fibers between noetherian schemes (e.g. a vector bundle) induces isomorphisms on the $$G_ i's$$ of the source and target schemes, the Projective Bundle Theorem, the Brown- Gersten-Quillen Spectral Sequence, Gersten’s Conjecture giving a long exact sequence relating $$G_ i(X)$$, X the spectrum of a regular local ring, to the $$K_{i-j}(k(x^{(j)}))$$, $$j=0,1,...,i$$, where the $$k(x^{(j)}))$$ are the residue fields of points of codimension j in X, its verification in some special cases, Bloch’s formula $$H^ p_{Zar}(X,{\mathcal K}_{pX})=CH^ p(X)$$, $$p\geq 0$$, where $${\mathcal K}_{p,X}$$ is the sheaf associated to the presheaf $$U\to K_ p(U)$$, and the $$K_ i$$ of Severi-Brauer schemes. Chapter 8 deals with the Merkurev- Suslin Theorem. It says that, for a field F of characteristic p and a positive integer n such that $$(p,n)=1$$, the Galois symbol $$R_{n,F}: K_ 2(F)\otimes_ ZZ/nZ\to H^ 2(F,\mu_ n^{\otimes 2})$$ is an isomorphism. This theorem is one of the milestones in K-theory and leads to interesting applications for the Chow groups of algebraic varieties, e.g., it is proved that the n-torsion subgroup of the Chow group $$CH^ 2$$ of a smooth, quasi-projective variety over an algebraically closed field of characteristic p, $$(p,n)=1$$, is finite. For a smooth affine surface it is zero, and for a smooth projective surface it coincides with the n-torsion subgroup of the Albanese variety (Roitman’s Theorem). The final chapter deals with a localization theorem of Quillen and a generalization due to Levine. As a corollary one obtains: Let X be a normal quasi-projective surface with only quotient singularities and let $$\pi$$ : $$Y\to X$$ be a resolution of singularities. Then $$\pi^*: CH^ 2(X)\to CH^ 2(Y)$$ is an isomorphism.
Summarizing, one can say that it is a pleasure to read this mathematically beautiful book but a critical remark may be made with respect to its typographic presentation. The table of contents lacks precision as a table of reference, there is no index, the text is poorly subdivided into sections, and the editor should have taken the trouble to have it rewritten in a more fashionable way (only the Introduction meets the usual standards) before publishing it in this expensive series.

##### MSC:
 19-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to $$K$$-theory 19-02 Research exposition (monographs, survey articles) pertaining to $$K$$-theory 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 19Dxx Higher algebraic $$K$$-theory 19Axx Grothendieck groups and $$K_0$$ 19Bxx Whitehead groups and $$K_1$$ 19Cxx Steinberg groups and $$K_2$$ 19D06 $$Q$$- and plus-constructions 19E08 $$K$$-theory of schemes 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry