By V. Srinivas (auth.)
Algebraic K-Theory has turn into an more and more energetic quarter of analysis. With its connections to algebra, algebraic geometry, topology, and quantity idea, 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 in other places. an in depth appendix on topology was once supplied within the first version to make the therapy available to readers with a restricted heritage 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 publication; this makes the ebook obtainable to a much wider audience.
A significant a part of the booklet is an in depth exposition of the information of Quillen as contained in his vintage papers "Higher Algebraic K-Theory, I, II." A extra straight forward facts 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 typical floor singularity. those effects lead the reader to a few attention-grabbing conclusions in regards to the Chow crew of varieties.
"It is a excitement to learn this mathematically attractive book..." ---WW.J. Julsbergen, arithmetic Abstracts
"The ebook does an admirable task of providing the main points of Quillen's work..." ---Mathematical Reviews
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Additional resources for Algebraic K-Theory
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. Let A(n) = n o m ~ (-,_n_n) be the simplicial set naturally associated to n_ E Ob A; then one sees easily that IA(n)l = An. 8)). Finally, we note t h a t the homology of IFI can be computed as follows. Let Cn(F) denote the free Abelian group on F(n_), and let 0~' 9 34 Algebraic K-Theory Cn(F) ~ Cn-I(F) be the map induced by F(O[').
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 ~. Indeed, the only 2 non-degenerate simplices of (S n) are eo e (S n)(O) and fn e (S n)(n), so I(S")I is a C W complex with exactly 2 cells. If n > 0, all faces of f . are degenerate, so that I(S")I is obtained by attaching an n-cell to the 0-cell le0] with a constant attaching map. This is the standard description of S n as a CW-complex. 5) T h e C l a s s i f y i n g S p a c e of a C a t e g o r y .
In fact, there exist spaces with self-maps not homotopic to the identity, but which induce the identity on homology and homotopy (where we assume the map fixes a base point, to make sense of the statement about homotopy groups). M. Lemaire has constructed such an example. D e f i n i t i o n . Let G be a group. Two homomorphisms f, g" G ". G L ( R ) are pseudo-conjugate if there exists u E M such t h a t either u. o f and g, or u. o g and f, are conjugate by an element of GL(R). 13). /] f, g 9 G the induced maps f +, g+ 9 B G preserving the respective base points.