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

By V. Srinivas (auth.)

Algebraic K-Theory has develop into an more and more lively zone of study. With its connections to algebra, algebraic geometry, topology, and quantity thought, 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 other places. an in depth appendix on topology was once supplied within the first variation to make the remedy obtainable to readers with a constrained heritage in topology. the second one variation additionally contains an appendix on algebraic geometry that includes the necessary definitions and effects had to comprehend the middle of the ebook; this makes the ebook available to a much broader audience.

A critical a part of the e-book is an in depth exposition of the information of Quillen as contained in his vintage papers "Higher Algebraic K-Theory, I, II." A extra trouble-free evidence of the concept of Merkujev--Suslin is given during this variation; this makes the remedy of this subject self-contained. An software can also be 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 attention-grabbing conclusions in regards to the Chow staff 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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Example text

Given rings R, R ~ and elements a e GL(R), a' e GL(R'), r enables us to convert a | a ' into an element of G L ( R | R~) 9 Thus we obtain a homomorphism GL(R) x GL(R') , G L ( R | R'), which clearly maps E ( R ) x E(R') into the commutator subgroup E ( R | R~), and hence induces a continuous map g: B G L ( R ) + x B G L ( R ' ) + ) B G L ( R | R') +. 9). Then by construction ~ is null homotopic when restricted to B G L ( R ) + VBGL(R') + - B G L ( R ) + • {X~o}U {x0} x BGL(R~) +. 7)) h. Ki(R) | Kj(R) ~ Ki+j(R | R'), from the homeomorphisms S ~ A SJ -~ S ~+j.

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}). There is a canonical admissible mono iM : 0 ~ M and an admissible epi qM : M --~ O, associated to any object M E C. Given any admissible mono i 9 M1 ~ M2, there is an arrow it : M1 ~ M2 in QC, corresponding to the (class of the) d i a g r a m 1 i M1 ~-- M1 ~-, M2.

Thus if q is the smallest integer such t h a t I-Iq(F(R), Z) r 0, then q > 0. Then Ep,q, 2 = Ep,q, r = 0 for all p, and all q' with 0 < q' < q. Since dr : E~,q * E~-r,q+r-l, we see t h a t E2o,q ~- Eq,+q1 E 0,q q+2 ~- E 0,q' ~176and E 0,q q+2 is the cokernel of dq+l 9E q+l,0 q+l E 0,q q+l 9 B u t E ~ , 0 = E n,0, ~176so that dq+l = 0 also. Hence E~,q = E 2O,q ~ =0as seen above. But on the other hand, E,O,q 2 = Ho(BGL(R) + Hq(F(R),Z) ~ 0 as we assumed Hq(F(R), Z) - Hq(F(R), Z) ~ 0. This contradiction proves t h a t F(R) is acyclic.

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