##
**Higher algebraic K-theory of schemes and of derived categories. Appendix A: Exact categories and the Gabriel-Quillen embedding. Appendix B: Modules versus quasi-coherent modules. Appendix C: Absolute noetherian approximation. Appendix D: Hypercohomology with supports. Appendix E: The Nisnevich topology. Appendix F: Invariance under change of universe.**
*(English)*
Zbl 0731.14001

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 247-435. Appendix A: 398-408; appendix B: 409-417; appendix C: 418-423; appendix D: 424-426; appendix E: 427-430; appendix F: p. 431 (1990).

[For the entire collection see Zbl 0717.00010.]

This paper develops a localization theorem for the K-theory of schemes. Previously D. Quillen had a satisfactory localization theory for \(K'\)-theory (or G-Theory, i.e. the K-Theory obtained by using coherent sheaves instead of vector bundles) of Noetherian schemes [in Higher Algebraic K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)]. However previous attempts to develop a localization sequence for K-theory only worked under restrictive hypotheses.

The authors work mostly with spectra. For a scheme X, K(X) is defined to be the K-theory spectrum of the complicial biWaldhausen category of perfect complexes of globally finite Tor-amplitude in the abelian category of \({\mathcal O}_ X\)-modules, and if Y is a closed subset of X, then K(X on Y) is similarly defined using those perfect complexes that are acyclic on X-Y (so that \(K(X)=K(X on X))\). The K-groups \(K_ i\) themselves are then the homotopy groups \(\pi_ i\) of these spectra (i\(\geq 0)\). The idea of using perfect complexes comes from Sémin. geómétrie algébrique 1966/67, SGA 6 [dirigé par P. Berthelot, A. Grothendieck and L. Illusie, “Théorie des intersections et théorème de Riemann-Roch”, Lect. Notes Math. 225 (1971; Zbl 0218.14001)], and Waldhausen’s K-Theory was developed by F. Waldhausen in Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006).

These topics are reviewed in the first two sections of the present paper. Roughly speaking a perfect complex is a complex that is locally quasi- isomorphic to a bounded complex of vector bundles, and quasi-isomorphisms are those that induce an isomorphism in cohomology. Perfect complexes are used in place of vector bundles because these extend (in a suitable sense) from an open subscheme to the whole scheme, whereas vector bundles do not. (This extension result is developed in section 5.) The spectra K(X) are contravariant in the scheme X, and if X has an ample family of line bundles (this includes all “reasonable” schemes, such as quasiprojective schemes over an affine scheme, and separated regular Noetherian schemes), then the new groups \(K_ i(X)\) agree with those of Quillen.

In section 6 a non-connective spectrum \(K^ B(X on Y)\) is defined, with a natural map K(X on Y)\(\to K^ B(X on Y)\) that induces isomorphisms on \(\pi_ n (n\geq 0)\). The homotopy groups of \(K^ B(X on Y)\) are denoted \(K_ n^ B(X on Y)\) (n\(\in {\mathbb{Z}}\), if \(n<0 \) \(K_ n^ B(X on Y)\) need not \(equal\quad 0).\) The reason for introducing \(K^ B\) is to correct for the failure of the restriction map \(K_ 0(X)\to K_ 0(U)\) to be surjective, by extending the localization sequence into negative K- groups. The main localization theorem then is the following theorem 7.4: Let X be a quasi-compact and quasi-separated scheme. Let j: \(U\to X\) be an open immersion with U quasi-compact. Set \(Y=X-U\). Let Z be a closed subscheme of X with X-Z quasi-compact. Then there are homotopy fibre sequences \(K^ B(X on Y)\to K^ B(X)\to K^ B(U);\) \(K^ B(X on Y\cap Z)\to K^ B(X on Z)\to K^ B(U on U\cap Z)\) with resulting sequences of homotopy groups \[ ...\quad \to K_ n^ B(X on Y)\to K_ n^ B(X)\to K_ n^ B(U)\to K^ B_{n-1}(X on Y)\to...\quad (n\in {\mathbb{Z}}) \]

\[ ...\quad \to K_ n^ B(X on Y\cap Z)\to K_ n^ B(X\text{ on } Z)\to K_ n^ B(U\text{ on } U\cap Z)\to K^ B_{in-1}(X on Y\cap Z)\to...\quad (n\in {\mathbb{Z}}). \] Along the way to prove the main theorem, improved versions of the projective space bundle theorem and Bass fundamental theorem are proved, and as consequences of theorem 7.4 improved versions of the Mayer-Viëtoris theorem, cohomological descent, and other important results are obtained in the last four sections of the paper.

This paper develops a localization theorem for the K-theory of schemes. Previously D. Quillen had a satisfactory localization theory for \(K'\)-theory (or G-Theory, i.e. the K-Theory obtained by using coherent sheaves instead of vector bundles) of Noetherian schemes [in Higher Algebraic K-Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)]. However previous attempts to develop a localization sequence for K-theory only worked under restrictive hypotheses.

The authors work mostly with spectra. For a scheme X, K(X) is defined to be the K-theory spectrum of the complicial biWaldhausen category of perfect complexes of globally finite Tor-amplitude in the abelian category of \({\mathcal O}_ X\)-modules, and if Y is a closed subset of X, then K(X on Y) is similarly defined using those perfect complexes that are acyclic on X-Y (so that \(K(X)=K(X on X))\). The K-groups \(K_ i\) themselves are then the homotopy groups \(\pi_ i\) of these spectra (i\(\geq 0)\). The idea of using perfect complexes comes from Sémin. geómétrie algébrique 1966/67, SGA 6 [dirigé par P. Berthelot, A. Grothendieck and L. Illusie, “Théorie des intersections et théorème de Riemann-Roch”, Lect. Notes Math. 225 (1971; Zbl 0218.14001)], and Waldhausen’s K-Theory was developed by F. Waldhausen in Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006).

These topics are reviewed in the first two sections of the present paper. Roughly speaking a perfect complex is a complex that is locally quasi- isomorphic to a bounded complex of vector bundles, and quasi-isomorphisms are those that induce an isomorphism in cohomology. Perfect complexes are used in place of vector bundles because these extend (in a suitable sense) from an open subscheme to the whole scheme, whereas vector bundles do not. (This extension result is developed in section 5.) The spectra K(X) are contravariant in the scheme X, and if X has an ample family of line bundles (this includes all “reasonable” schemes, such as quasiprojective schemes over an affine scheme, and separated regular Noetherian schemes), then the new groups \(K_ i(X)\) agree with those of Quillen.

In section 6 a non-connective spectrum \(K^ B(X on Y)\) is defined, with a natural map K(X on Y)\(\to K^ B(X on Y)\) that induces isomorphisms on \(\pi_ n (n\geq 0)\). The homotopy groups of \(K^ B(X on Y)\) are denoted \(K_ n^ B(X on Y)\) (n\(\in {\mathbb{Z}}\), if \(n<0 \) \(K_ n^ B(X on Y)\) need not \(equal\quad 0).\) The reason for introducing \(K^ B\) is to correct for the failure of the restriction map \(K_ 0(X)\to K_ 0(U)\) to be surjective, by extending the localization sequence into negative K- groups. The main localization theorem then is the following theorem 7.4: Let X be a quasi-compact and quasi-separated scheme. Let j: \(U\to X\) be an open immersion with U quasi-compact. Set \(Y=X-U\). Let Z be a closed subscheme of X with X-Z quasi-compact. Then there are homotopy fibre sequences \(K^ B(X on Y)\to K^ B(X)\to K^ B(U);\) \(K^ B(X on Y\cap Z)\to K^ B(X on Z)\to K^ B(U on U\cap Z)\) with resulting sequences of homotopy groups \[ ...\quad \to K_ n^ B(X on Y)\to K_ n^ B(X)\to K_ n^ B(U)\to K^ B_{n-1}(X on Y)\to...\quad (n\in {\mathbb{Z}}) \]

\[ ...\quad \to K_ n^ B(X on Y\cap Z)\to K_ n^ B(X\text{ on } Z)\to K_ n^ B(U\text{ on } U\cap Z)\to K^ B_{in-1}(X on Y\cap Z)\to...\quad (n\in {\mathbb{Z}}). \] Along the way to prove the main theorem, improved versions of the projective space bundle theorem and Bass fundamental theorem are proved, and as consequences of theorem 7.4 improved versions of the Mayer-Viëtoris theorem, cohomological descent, and other important results are obtained in the last four sections of the paper.

Reviewer: L.G.Roberts (Kingston / Ontario)

### MSC:

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

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