By V. Srinivas (auth.)
Algebraic K-Theory has turn into an more and more energetic sector of analysis. With its connections to algebra, algebraic geometry, topology, and quantity concept, it has implications for a wide selection of researchers and graduate scholars in arithmetic. The ebook 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 variation to make the therapy obtainable to readers with a constrained historical past in topology. the second one version additionally contains an appendix on algebraic geometry that comprises the mandatory definitions and effects had to comprehend the center of the e-book; this makes the e-book available to a much broader audience.
A principal a part of the e-book is a close exposition of the guidelines 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 is usually given to modules of finite size and finite projective measurement over the neighborhood ring of a standard floor singularity. those effects lead the reader to a couple attention-grabbing conclusions concerning the Chow team of varieties.
"It is a excitement to learn this mathematically appealing book..." ---WW.J. Julsbergen, arithmetic Abstracts
"The booklet does an admirable activity of proposing the main points of Quillen's work..." ---Mathematical Reviews
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Extra resources for Algebraic K-Theory
Equivalently, L is a (~l(X,x)/g)-module. ()~+,)~) 9 But this complex is acyclic, so Hn(X+,X;L) = 0 Vn. ,(X,i*L) ~ Hn(X+,L) Vn, where i : X -4 X +. 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.
Since G has an initial object (any object is an initial object) B G is contractible. Thus G ~ 7rl(BG), and 7ri(BG) = 0 for i # 0. One also sees that B G ~. G_is a principal G-bundle with contractible total space, which is the usual defining property of the classifying space, which characterizes it up to homotopy equivalence. 4. Exact Categories and Quillen's Q-Construction For our purposes, an exact category C is an additive category C e m b e d d e d as a full (additive) subcategory of an Abelian category ,4, such t h a t if 0 -, M ~ --, M ~M" ~ 0 is an exact sequence in j [ with M ' , M " E C, t h e n M is isomorphic to an object of C.
3) G e o m e t r i c R e a l i z a t i o n . To each simplicial set F 9A ~ , Set, we can associate a topological space IF[ called the geometric realization of F. This is defined to be the quotient space where for each n >_ 0, F(_n) is regarded as a discrete space. The equivalence relation ~ is defined as follows: given f 9m__ , n in A, let f 9 Am , An be the continuous map defined in Ex. 1) above. ~ (F(f)~, y) for all y E Am, where (~, ](y)) E F ( n ) x An, and (F(f)~, y) E F(m__)x Am. Let ~ be the equivalence relation so generated, and let IFI be the quotient space, with the quotient topology (an open subset is a set whose inverse image in I_In F(n_) x An is open).
Algebraic K-Theory by V. Srinivas (auth.)