By Bruce A. Magurn
This publication is either an creation to K-theory and a textual content in algebra. those roles are solely 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. nevertheless, K-theory is a common organizing precept for a standard subject matters of a moment direction in algebra, and those issues are offered conscientiously right here. The reader won't in basic terms examine algebraic K-theory, but in addition Dedekind domain names, category teams, semisimple earrings, personality thought, quadratic types, tensor items, localization, of completion, tensor algebras, symmetric algebras, external algebras, critical uncomplicated algebras, and Brauer teams. The presentation is self-contained, with all of the helpful historical past and proofs, and is split into brief sections with workouts to augment the guidelines and recommend extra traces of inquiry. the must haves are minimum: only a first semester of algebra (including Galois concept and modules over a vital excellent domain). No event with homological algebra, research, geometry, quantity thought, or topology is thought. the writer has successfuly used this article to educate algebra to first 12 months 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 12 months
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Extra resources for An Algebraic Introduction to K-Theory
31), ψ = k g and θ = [f h] , which are R-linear. By the exactness above and the hypotheses in (iii), matrix multiplication gives ψθ = 1 0 0 1 and θψ = 1 . 6) Deﬁnition. An R-linear short exact sequence f g 0 −−−→ L −−−→ M −−−→ N −−−→ 0 splits (or is split) if there is an R-linear map h : N → M with g ×ÐÑÖÒÕÓÔ h = iN . 7) Proposition. There is an R-linear isomorphism M ∼ = L ⊕ N if and only if there is a split R-linear short exact sequence: f g 0 −−−→ L −−−→ M −−−→ N −−−→ 0 . Proof. 5) (i) and (iii).
Suppose S is a set. The free monoid based on S is the set Mon(S) of all strings s1 s2 ... sm , where m ≥ 0 and each si ∈ S. Two strings are multiplied by concatenation: (s1 · · · sm )(s1 · · · sn ) = s1 · · · sm s1 · · · sn . The empty string (with m = 0) is denoted by 1, since it serves as the identity element of Mon(S). 19) Proposition. 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.
So, by injectivity of f and h, each element of M has an expression f (x) + h(y) for unique x ∈ L and y ∈ N . Deﬁne k : M → L by k(f (x) + h(y)) = x. A direct Projective Modules 46 check shows k is R-linear. For each x ∈ L, k(f (x)) = k(f (x) + h(0)) = x; so k Ð×ÑÖÒÕÓÔ f = iL . For each x ∈ L and y ∈ N , (f ×ÐÑÖÒÕÓÔ k)(f (x) + h(y)) = f (x) and (h Ð×ÑÖÒÕÓÔ g)(f (x) + h(y)) = h(0 + iN (y)) = h(y) . So (f Ð×ÑÖÒÕÓÔ k) + (h Ð×ÑÖÒÕÓÔ g) = iM . Under the hypotheses in (ii), f Ð×ÑÖÒÕÓÔ k is an idempotent in EndR (M ) with kernel • ker(k) and image f (k(M )) = f (L) = ker(g).
An Algebraic Introduction to K-Theory by Bruce A. Magurn