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

By V. Srinivas (auth.)

Algebraic K-Theory has develop into an more and more lively zone of analysis. 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 somewhere else. an in depth appendix on topology was once supplied within the first version to make the remedy obtainable to readers with a restricted heritage in topology. the second one version additionally contains an appendix on algebraic geometry that includes the necessary definitions and effects had to comprehend the center of the booklet; this makes the e-book available to a much broader audience.

A crucial 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 hassle-free facts of the theory 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 a regular 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 attractive book..." ---WW.J. Julsbergen, arithmetic Abstracts

"The publication does an admirable task of proposing the main points of Quillen's work..." ---Mathematical Reviews

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Let C, C~ be small categories, C x C~ their product, so that Ob(C x C')= (ObC) x (ObC'), and Homcxc,((A, B), (C, D)) = Homc(A, C) x Homc,(B, D). Then clearly N ( C x C') is the product simplicial set N C x NC ~. Hence B(C xC ~) ~. BC x BC ~ is a continuous bijection, which is a homeomorphism if the product on the right is given the compactly generated topology; in particular it is a homeomorphism if either BC or BC ~ is locally compact. Thus B(C x {0 < 1}) ~ BC x I . h functor H : C x {0 < 1} ; :/9 is just a pair of functors F, G : C ; :D (given by F ( A ) = H ( ( A , O ) ) , G(A) = H ( ( A , 1))) together with a natural transformation 77: F , G (given by r/(A) = H((1A, 0 , 1))).

41)). 9). Let u0, v0 E M be given by Uo(i) = 2i - 1, vo(i) = 2i, for all i E N. Then (U0)o(a) = a 9 1, and (V0)o(a) = 1 ~ a for any c~ E GL(R). By construction, the map + : B G L ( R ) + x B G L ( R ) + B G L ( R ) +, when restricted to B G L ( R ) + x {,}, is homotopic to (u0) +, and when restricted to {,} x B G L ( R ) +, is homotopic to (v0) +. Hence the base point * E B G L ( R ) + is a 2-sided identity, up to homotopy, for the operation +. By definition, this makes ( B G L ( R ) +, + ) a H-space.

3)). Let X be a topological space, and let S ( X ) denote the total singular complex (Ex. 1)). Then there is a continuous surjective map f : IS(X)I } X. , S(x0)(n) C S ( X ) ( n ) for all n) such that IS(x0)[ is a point whose image under f is x0. We have: f " (JS(X)l, IS(x0)l) , (X, xo) induces isomorphisms on homotopy groups. Hence if X is a CW-complex, it is a homotopy equivalence Theorem. 6)). 4). Let (S n) be the simplicial set described in Ex. 2). Then the geometric realization I(S r')l is naturally homeomorphic to the n-sphere S ~.

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