By V. Srinivas (auth.)
Algebraic K-Theory has develop into an more and more energetic sector of analysis. With its connections to algebra, algebraic geometry, topology, and quantity conception, it has implications for a wide selection 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 in different places. an in depth appendix on topology used to be supplied within the first version to make the therapy obtainable to readers with a restricted historical past in topology. the second one version additionally contains an appendix on algebraic geometry that comprises the necessary definitions and effects had to comprehend the center of the e-book; this makes the booklet obtainable to a much broader audience.
A critical a part of the ebook is a close exposition of the tips of Quillen as contained in his vintage papers "Higher Algebraic K-Theory, I, II." A extra straightforward evidence of the concept of Merkujev--Suslin is given during this variation; this makes the therapy of this subject self-contained. An program can be given to modules of finite size and finite projective size over the neighborhood ring of a standard floor singularity. those effects lead the reader to a few fascinating conclusions in regards to the Chow crew of varieties.
"It is a excitement to learn this mathematically appealing book..." ---WW.J. Julsbergen, arithmetic Abstracts
"The e-book does an admirable task of featuring the main points of Quillen's work..." ---Mathematical Reviews
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Additional resources for Algebraic K-Theory
This proves (b). The proof of (c) is by obstruction theory; we omit the details (however, see (A. 16)). 2). Let X = BS~, where S~o = U , > I So is the infinite permutation group. Let Aoo c So~ be the infinite alternating group [Jn>_l An. Then Aoo ,~ Soo = lh(BS~) is a perfect normal subgroup, so we can form the space B S +, with lrl(BS +) = Z/2Z. In fact, by Example the theorem of Barrat, Priddy and Quillen, one has a homotopy equivalence Z x B S + "~ fl~176176176176 where S o is the 0-sphere (discrete 2 point 22 Algebraic K-Theory space) (see S.
13). /] f, g 9 G the induced maps f +, g+ 9 B G preserving the respective base points. G L ( R ) are pseudo-conjugate, then B G L ( R ) + are homotopic as maps P r o o f . For any map f : G ; GL(R), let f ~ 1 denote the map given by x ~-. f ( x ) ~ l , for any x E G. Then f + is homotopic to ( f ~ l ) +, preserving the base point, because f ~ 1 = (u0)~ o f , where u0 E M is defined by 0(i) = 2 i - 1. Now suppose the given maps f, g satisfy g = (u~ o f ) a for ~ e G L ( R ) (where ( u ~ ~(u~ -1 for a l l x e G).
Further, the "switch map" S ~AS j - - ~ S j AS ~ has degree ( - 1 ) ~j (compute it on homology) so that if R is commutative, the homomorphism R | R ; R given by multiplication induces bilinear products , " Ki(R) | Kj(R) , Ki+j(R), i,j > 1 satisfying x , y = ( - 1 ) i J y , x for x e Ki(R), y E Kj(R). 3)). 30 Algebraic K-Theory Finally, we remark that Quillen has computed the K-groups of a finite field in the following paper: D. Quillen, On the Cohomology and K - T h e o r y of the General Linear Group over a Finite Field, Ann.