By Barry Dayton, Charles Weibel (auth.), J. F. Jardine, V. P. Snaith (eds.)

ISBN-10: 9400923996

ISBN-13: 9789400923997

ISBN-10: 9401075808

ISBN-13: 9789401075800

A NATO complicated learn Institute entitled "Algebraic K-theory: Connections with Geometry and Topology" used to be held on the Chateau Lake Louise, Lake Louise, Alberta, Canada from December 7 to December eleven of 1987. This assembly used to be together supported via NATO and the usual Sciences and Engineering learn Council of Canada, and was once backed partly by means of the Canadian Mathematical Society. This e-book is the quantity of lawsuits for that assembly. Algebraic K-theory is largely the research of homotopy invariants coming up from jewelry and their linked matrix teams. extra importantly maybe, the topic has turn into primary to the examine of the connection among Topology, Algebraic Geometry and quantity thought. It attracts on all of those fields as an issue in its personal correct, however it serves in addition to a good translator for the applying of thoughts from one box in one other. The papers during this quantity are consultant of the present country of the topic. they're, for the main half, examine papers that are basically of curiosity to researchers within the box and to these desiring to be such. there's a part on difficulties during this quantity which might be of specific curiosity to scholars; it incorporates a dialogue of the issues from Gersten's recognized checklist of 1973, in addition to a brief checklist of recent problems.

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A+). A= R mAl m ... A+}. be a commutative graded ring. are naturally continuous W(R)-submodules In particular. 4 Corollary: of 23 R contains the rational numbers, is naturally an R-module. 3 does not apply, but arguments of [Vorst] do). APPENDIX. PATCHING VECfOR BUNDLES It is well known that a rank n vector bundle P on a scheme be obtained by patching free 0U-modules on some open cover X via matrices guy € GLn(U n V} on each U n V. this data, the determinant line bundle 0u together via the units SL (U n n V), det(guv}, the line bundle Anp Anp If P ~ X may = {U} of is given by is formed by patching the so if each is trivial.

0), that A A c l (L,II II) = c l (L, VII II)' But if 'Y(~l)(L,1I II)) s is a holomorphic section of L. = 'Y(div(s) , log IIs11 2 ) = (odiv(s),dc log 2 IIsll ) 1 = (od'IV ( S ) ,

Xoo ~ Yoo compatible with the push forward maps on cycles and on forms. and compatible with pull-backs and products via the usual projection formula. 5. A variation on the ideas above is to consider smooth projective varieties over the complex numbers. e. 2, X. with conjugate complex structures be the antiholomorphic identification map). oH*(X) for X smooth and projective over We then ~. with the same formal properties as for arithmetic varieties, except that the product on oH*(X) ~. is defined using coefficients in There is an obvious restriction map, if Z rather than X is an arithmetic variety Finally we remark that arithmetic varieties over over ~ are both special cases of the more general notion of an arithmetic variety over an arithmetic ring; see details.

### Algebraic K-Theory: Connections with Geometry and Topology by Barry Dayton, Charles Weibel (auth.), J. F. Jardine, V. P. Snaith (eds.)

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