By V. Srinivas (auth.)

Algebraic K-Theory has turn into an more and more lively quarter of study. With its connections to algebra, algebraic geometry, topology, and quantity thought, it has implications for a wide selection of researchers and graduate scholars in arithmetic. The publication is predicated on lectures given on the author's domestic establishment, the Tata Institute in Bombay, and somewhere else. an in depth appendix on topology was once supplied within the first version to make the therapy obtainable to readers with a constrained history in topology. the second one variation additionally contains an appendix on algebraic geometry that includes the mandatory definitions and effects had to comprehend the center of the e-book; this makes the publication obtainable to a much broader audience.

A primary a part of the booklet is a close exposition of the tips of Quillen as contained in his vintage papers "Higher Algebraic K-Theory, I, II." A extra straight forward facts of the concept of Merkujev--Suslin is given during this variation; this makes the remedy of this subject self-contained. An software is additionally given to modules of finite size and finite projective measurement over the neighborhood ring of a typical floor singularity. those effects lead the reader to a couple attention-grabbing conclusions in regards to the Chow workforce of varieties.

"It is a excitement to learn this mathematically appealing book..." **--- WW.J. Julsbergen, arithmetic Abstracts**

"The ebook does an admirable activity of featuring the main points of Quillen's work..." *---Mathematical Reviews*

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A/B is a generalization of the construction of the localization of a ring, and of modules over the ring, with respect to a central multiplicative set. 9) (Localization). A/B. C be the natural exact functors. Then BQB ~ BQA BQ8 ~ BQC BQp 4. 27) of B Qp)). ----, K (B) , -, , Ko(C) ,... a) o. 0); we discuss this below. This may give t h e reader a little practice in thinking about the Q-construction. Example ( 4 . 1 0 ) . As noted above, the isomorphism O" Ko(C) ~-, Irl(BQC,{O}) is explicitly given by associating to each M E C a certain loop rM based at 0, such t h a t r = [rM] e lrl(BQC, {0}).

We also have WorB (A, F p - I N ) = 0. Assume that r is an isomorphism, and that Tor B (A, FvN ) = 0; we will deduce that Cv is an isomorphism, and that Tor~(A, Fv_IN ) = O. 5. The K-Theory of Rings and Schemes From the sequence 0 have an exact sequence , Fp_ 1N , FpN Tor~(A, FpN) --4 Tor~(A, FpN/Fp_IN) ---. A | ~ A| l[ 49 ~ FpN/Fp_ 1N Fp_IN ', A | , 0 we F~,N--, (FpN/Fp_,N) ---, O. ), A @B Fp_ 1N , A | B Fp N is an isomorphism in degrees _< p - 1, and A| Fp_ 1N vanishes in all higher degrees. Hence the map i is injective, and so T o r f (A, FpN/Fp_ I N) = O.

Tn) where ti -- ~ Sj , l(j)=~ where we define tj = 0 if {j I f(J) = i} is empty. ~Top. 2) ("The n-sphere"). } {era} U {all degenerate m-simplices associated to f , } , if 0 _< m < n ifm=n if m > n. T h e faces of em all equal era-1 (for m >_ 1), and the degenerate simplices of em all equal em+l (for m >_ 0). T h e faces of fn all equal en-1, if n > 0 (if 3. The Classifying Space of a Small Category 33 n - 0, there are no face maps to be defined). We leave it to the reader to check t h a t these conditions do uniquely define a simplicial set; the reason for the name ("The n-sphere") is explained below (Ex.