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.

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.

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

##### 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 |