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

By V. Srinivas (auth.)

Algebraic K-Theory has develop into an more and more energetic region of study. With its connections to algebra, algebraic geometry, topology, and quantity conception, it has implications for a large choice of researchers and graduate scholars in arithmetic. The booklet is predicated on lectures given on the author's domestic establishment, the Tata Institute in Bombay, and somewhere else. a close appendix on topology used to be supplied within the first version 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 includes the mandatory definitions and effects had to comprehend the middle of the ebook; this makes the e-book available to a much broader audience.

A principal a part of the publication is a close exposition of the tips of Quillen as contained in his vintage papers "Higher Algebraic K-Theory, I, II." A extra undemanding facts of the concept of Merkujev--Suslin is given during this version; this makes the remedy of this subject self-contained. An program is additionally given to modules of finite size and finite projective size over the neighborhood ring of an ordinary floor singularity. those effects lead the reader to a few attention-grabbing conclusions in regards to the Chow team of varieties.

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

"The e-book does an admirable task of offering the main points of Quillen's work..." ---Mathematical Reviews

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In fact, if k ~ C, then the m-torsion subgroup of B r ( k ) is known to be isomorphic to Z / m Z ~ pro; for a suitable choice of this isomorphism, one can identify the local norm residue symbol with the Galois Symbol (Ex. 16)). Now let K be a number field. For any place v of K which is either real or non-Archimedean, if re(v) is the number of roots of unity in the local field K~, let ( , )v denote the composition of K 2 ( K ) ----, K 2 ( K , ) with the local m(v)th power norm-residue symbol, taking values in the group /z.

For any associative ring R, we regard GL(R) as a topological group with the discrete topology, and let BGL(R) denote the 'classifying space' of GL(R). , BGL(R) is a connected space with ~I(BGL(R)) ~- GL(R), lr,(BGL(R)) = 0 for i > 2, and t h a t these properties characterize BGL(R) up to homotopy equivalence (since we are assuming that all spaces considered here have the homotopy type of a CW-complex). 10)). Since ~t (BGL(R)) ~ GL(R), ~rl(BGL(R)) has a perfect normal subgroup E(R). We will construct below a space BGL(R) + by applying the plus construction of Quillen to the pair (BGL(R), E(R)).

For any simplicial set F , a simplex ~ E F(_n) is called non-degenerate if it is not the degenerate simplex assigned to any ( n - 1)-simplex by one of the degeneracies. T h e n IF I is homeomorphic to a CW-complex, which has one n-cell corresponding to each non-degenerate n-simplex of F . If F, G are simplicial sets, let F x G denote the simplicial set whose n-simplices are F(_n) x G(_n), with obvious maps. , the product is formed in the category of compactly generated spaces). , if F has only a finite number of non-degenerate simplices, so that IF I is compact) then IF x G I is homeomorphic to IFI x IGI.

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