Algebraic k-theory by V. Srinivas

By V. Srinivas

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 booklet 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 version to make the remedy obtainable to readers with a restricted heritage in topology. This new version additionally comprises an appendix on algebraic geometry that includes the mandatory definitions and effects had to comprehend the center of the booklet; this makes the ebook available to a much wider audience.

A primary a part of the publication is an in depth exposition of the guidelines 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 remedy of this subject self-contained. An purposes is usually 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 attention-grabbing conclusions in regards to the Chow crew of types.

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A natural transformation of functors Aop , Set) we get a continuous map IFI , IGI . For any simplicial set F , a simplex ~ E F(_n) is called non-degenerate if it is not the degenerate simplex assigned to any ( n - 1)-simplex by one of the degeneracies. T h e n IF I is homeomorphic to a CW-complex, which has one n-cell corresponding to each non-degenerate n-simplex of F . If F, G are simplicial sets, let F x G denote the simplicial set whose n-simplices are F(_n) x G(_n), with obvious maps. , the product is formed in the category of compactly generated spaces).

By K u m m e r theory, we have an isomorphism k* | Z / m Z ~= H o m ( G a l ( k ' / k ) , pro), and by local class-field theory, we have an isomorphism Gal(k'/k) ~= k*/Nk,/k(k')* where Nk,/k is the norm (note that k l / k is a finite extension, since k* | Z / m Z is finite). Thus we have a pairing k* | k* ('); #m called the local ruth-power norm-residue symbol; one can check that it does indeed satisfy the conditions (i)-(iii) in the definition of a symbol. In fact, if k ~ C, then the m-torsion subgroup of B r ( k ) is known to be isomorphic to Z / m Z ~ pro; for a suitable choice of this isomorphism, one can identify the local norm residue symbol with the Galois Symbol (Ex.

The nerve of C, denoted NC (or N(C)) is defined to be the following simplicial set: an n-simplex of NC is a diagram A0 /~ ,A1 f2 ' A2 I~ ' " " ~ A~ with Ai E ObC, fi E MorC. ~_n in A, the corresponding map NC(n_) ~ N C ( m ) m a p s the above n-simplex to the msimplex Bo gx }B1 g~~,92 ~ - . g'%Bm where By = AI(j), and Bs ~- Bj+I is the composite map AI(j) } AI(j+I), where if f ( j ) = f ( j + 1), let AI(i) } Al(s+l) be the identity map. In particular the ith face of the above n-simplex is the ( n - 1)-simplex Ao I , AI } ....

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