By V. Srinivas

ISBN-10: 0817637028

ISBN-13: 9780817637026

ISBN-10: 0817647368

ISBN-13: 9780817647360

Algebraic K-Theory has develop into an more and more lively quarter of analysis. With its connections to algebra, algebraic geometry, topology, and quantity concept, it has implications for a large choice of researchers and graduate scholars in arithmetic. The ebook relies on lectures given on the author's domestic establishment, the Tata Institute in Bombay, and in different places. an in depth appendix on topology used to be supplied within the first version to make the therapy available to readers with a restricted history in topology. This new version additionally comprises an appendix on algebraic geometry that comprises the necessary definitions and effects had to comprehend the center of the publication; this makes the booklet obtainable to a much wider audience.

A significant a part of the publication is a close exposition of the guidelines of Quillen as contained in his vintage papers “Higher Algebraic K-Theory, I, II.” A extra ordinary evidence of the concept of Merkujev--Suslin is given during this variation; this makes the remedy of this subject self-contained. An functions is additionally given to modules of finite size and finite projective measurement over the neighborhood ring of a regular floor singularity. those effects lead the reader to a few attention-grabbing conclusions concerning the Chow staff of sorts.

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Taking the orientations into account one checks that r M is homotopic to rM, 9rM,,. 5. T h e K - T h e o r y of Rings and Schemes If R is a ring, let P(R) denote the category of finitely generated projective (left) R-modules. This is a full subcategory of the Abelian category of left R-modules, so that 7~(R) is an exact category where all exact sequences are split. We will prove the following result, comparing the plus and Q constructions, in Chapter 7. 1). There is a homotopy equivalence BGL(R) + , (I~BQ'P(R)) ~ which is natural up to homotopy.

Ai " 1-" Ai I,+,~ Ai+l -~-'- } An. 3. The Classifying Space of a Small Category 35 The classifying space of C is defined to be the geometric realization of NC and is denoted by BC (or B(C)); thus BC = INCl. Clearly, if F : C ; D is a functor between small categories C, :D then there is an induced map of simplicial sets NC ; N D , and hence an induced continuous map B F : BC ~ BD. A simple and useful example is given by the category {0 < 1}, consisting of two objects 0, 1 and a unique non-identity morphism 0 ; 1.

E~, 0 = EC~ and E~,q = 0 for q ~ 0. Now suppose F(R) is not acyclic; since ~ ( R ) + is simply connected and B G L ( R ) is connected, F(R) is path connected. Thus if q is the smallest integer such t h a t I-Iq(F(R), Z) r 0, then q > 0. Then Ep,q, 2 = Ep,q, r = 0 for all p, and all q' with 0 < q' < q. Since dr : E~,q * E~-r,q+r-l, we see t h a t E2o,q ~- Eq,+q1 E 0,q q+2 ~- E 0,q' ~176and E 0,q q+2 is the cokernel of dq+l 9E q+l,0 q+l E 0,q q+l 9 B u t E ~ , 0 = E n,0, ~176so that dq+l = 0 also.

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