An Algebraic Introduction to K-Theory - download pdf or read online

By Bruce A. Magurn

ISBN-10: 0521106583

ISBN-13: 9780521106580

ISBN-10: 0521800781

ISBN-13: 9780521800785

This ebook is either an advent to K-theory and a textual content in algebra. those roles are fullyyt suitable. at the one hand, not anything greater than the fundamental algebra of teams, earrings, and modules is required to give an explanation for the clasical algebraic K-theory. nonetheless, K-theory is a traditional organizing precept for a standard subject matters of a moment path in algebra, and those themes are awarded rigorously right here. The reader won't simply study algebraic K-theory, but in addition Dedekind domain names, classification teams, semisimple jewelry, personality conception, quadratic kinds, tensor items, localization, final touch, tensor algebras, symmetric algebras, external algebras, crucial uncomplicated algebras, and Brauer teams. The presentation is self-contained, with all of the precious history and proofs, and is split into brief sections with workouts to enhance the guidelines and recommend additional strains of inquiry. the must haves are minimum: only a first semester of algebra (including Galois conception and modules over a significant excellent domain). No adventure with homological algebra, research, geometry, quantity thought, or topology is believed. the writer has successfuly used this article to educate algebra to first yr graduate scholars. chosen issues can be utilized to build a number of one-semester classes; insurance of the total textual content calls for a whole yr

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Each function ψ : S → N , from a set S into a monoid N , has one and only one extension to a monoid homomorphism ψ : Mon(S) → N . Proof. Such an extension must satisfy ψ(s1 · · · sm ) = ψ(s1 ) · · · ψ(sm ) = ψ(s1 ) · · · ψ(sm ) , and this formula defines a monoid homomorphism ψ extending ψ. 20) Definition. If S is any set, the free ring based on S is the monoid ring Z[Mon(S)] of the free monoid based on S. 21) Corollary. Each function θ : S → R, from a set S into a ring R, has one and only one extension to a ring homomorphism θ : Z[Mon(S)] → R.

31), the R-linear map (fij ) : M → N is left multiplication of a column of columns by a t × s matrix A = (Aij ) whose entries Aij are matrices. If also L = L1 ⊕ · · · ⊕ Lr , where Li = Rd(i)×1 , and (gij ) : L → M is left multiplication by a matrix B = (Bij ), where Bij ∈ Rc(i)×d(j) , then (fij ) ×ÐÑÖÒÕÓÔ (gij ) : L → N is left multiplication by the product (Aij )(Bij ) = (Cij ) , where Cij = Aik Bkj k is a matrix sum of matrix products. ” In more detail, a matrix A = (Aij ) ⎡ A11 . = ⎣ .. At1 c(1)col’s ··· ··· ⎤ A1s } b(1) rows ..

10) Corollary. Every R-module M is R-linearly isomorphic to a quotient of a free R-module. For each positive integer n, every R-module generated by a set of n elements is R-linearly isomorphic to a quotient of Rn . Proof. If R is trivial, M = {0}, which is already free. If R is nontrivial, and M = S for nonempty S, the inclusion S → M extends to a surjective Rlinear map f : FR (S) → M with some kernel K, and this induces an R-linear isomorphism f : FR (S)/K ∼ = M . If M = v1 , · · · , vn , there is a surjective R-linear map f : Rn → M with f (ei ) = vi for each i .

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An Algebraic Introduction to K-Theory by Bruce A. Magurn


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