Algebraic K-Theory by V. Srinivas (auth.) PDF

By V. Srinivas (auth.)

ISBN-10: 0817647368

ISBN-13: 9780817647360

ISBN-10: 0817647392

ISBN-13: 9780817647391

Algebraic K-Theory has develop into an more and more lively zone of study. With its connections to algebra, algebraic geometry, topology, and quantity idea, 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 variation to make the therapy available to readers with a restricted historical past in topology. the second one variation additionally comprises an appendix on algebraic geometry that comprises the mandatory definitions and effects had to comprehend the middle of the publication; this makes the publication obtainable to a much broader audience.

A critical a part of the e-book is a close exposition of the tips of Quillen as contained in his vintage papers "Higher Algebraic K-Theory, I, II." A extra simple facts of the theory of Merkujev--Suslin is given during this version; this makes the remedy of this subject self-contained. An software is additionally given to modules of finite size and finite projective size over the neighborhood ring of a typical floor singularity. those effects lead the reader to a few fascinating conclusions in regards to the Chow workforce of varieties.

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

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

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Let F be a finite field. Then K 2 ( F ) = 0. 14), and the fact t h a t the multiplicative group F* of non-zero elements of F is a cyclic group. Indeed, let u E F* be a generator. It suffices to prove t h a t {u, u} is trivial in K 2 ( F ) . If F has characteristic 2, then {u, u} = { u , - u } is trivial. So we may assume F has odd characteristic. If F has cardinality q, then u (q-1)/2 = - 1 , so {u, u (q+1)/2} = { u , - u } is trivial. , {u, u} has order at most 2. Now F* - {1 } is invariant under the bijection v H 1 - v.

O iM,,!. By definition t h e composite i! o q"M" corresponding to the d i a g r a m 0 q_M, M ' --, i M in C, represents an arrow u 90 ~ M in QC. On the other hand, t h e c o m p o s i t i o n law gives t h a t q: o i M,,! is represented by 0 0 • T T iM,I r B u t t h e r e is an isomorphism 0 x M" ~ p q )M" r" M > 1->M M - - ~ M t such t h a t OXM,,I~ M' c o m m u t e s , since 0 .. M~ i M q ~ M " - - * 0 is exact. T h u s u i! o q~4, a n d q! o i M,,! are represented by isomorphic d i a g r a m s , a n d so give equal a r r o w s in QC.

Let C be a small exact category, $ the category of (short) exact sequences in C. 4 of the form 0 ; M' ,M , M" ,0, where a morphism of complexes is a (commutative) diagram in ,4 M' ~ M ~ N 1 ~ M" ~ N" l N r ~ 0 ~ O. 1 Hence ~ is an exact category in a natural way, such that if E ' , E, E " E $, then 0 - ~ E ' , E ~E" , 0 is exact precisely when the corresponding diagram in C has exact rows and columns. 0 is the corresponding exact sequence in C. We can now state the first property of the functors Ki.

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