By Bruce A. Magurn

This publication is either an advent to K-theory and a textual content in algebra. those roles are totally appropriate. at the one hand, not anything greater than the fundamental algebra of teams, jewelry, and modules is required to give an explanation for the clasical algebraic K-theory. nevertheless, K-theory is a ordinary organizing precept for a standard subject matters of a moment direction in algebra, and those issues are awarded rigorously right here. The reader won't purely study algebraic K-theory, but additionally Dedekind domain names, type teams, semisimple jewelry, personality concept, quadratic varieties, tensor items, localization, crowning glory, tensor algebras, symmetric algebras, external algebras, critical easy algebras, and Brauer teams. The presentation is self-contained, with the entire precious heritage and proofs, and is split into brief sections with workouts to enhance the tips and recommend extra traces of inquiry. the necessities are minimum: only a first semester of algebra (including Galois concept and modules over a imperative perfect domain). No event with homological algebra, research, geometry, quantity conception, or topology is believed. the writer has successfuly used this article to coach algebra to first yr graduate scholars. chosen subject matters can be utilized to build various one-semester classes; assurance of the full textual content calls for an entire yr

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**Extra resources for An Algebraic Introduction to K-Theory**

**Example text**

If R is a ring; b, c, and d are rows of positive integers; and A, A ∈ Rb×c and B ∈ Rc×d are partitioned matrices, then erase(A + A ) = erase(A) + erase(A ) , and erase(AB) = erase(A) erase(B) . Proof. The ﬁrst equation is immediate, since block addition A + A is entrywise addition (Aij ) + (Aij ) = (Aij + Aij ) and Aij + Aij is entrywise addition over R. In the block product (Aij )(Bij ) = (Cij ), the i , j -entry of Cij is the sum over k of the i , j -entries of each Aik Bkj ; so it is the sum over k of the i -row of Aik times the j -column of Bkj .

7), any two split short exact sequences 0 −−−→ L −−−→ M1 −−−→ N −−−→ 0 0 −−−→ L −−−→ must have M1 ∼ =L⊕N ∼ = M2 . M2 −−−→ N −−−→ 0 Consider again internal direct sums. 9) Deﬁnition. If L is a submodule of M , a complement of L in M is any • submodule N of M with M = L ⊕ N . Call L a direct summand of M if L has a complement in M . 10) Example. In the real vector space R2 , a line V through the origin is a direct summand of R2 , and any other line through the origin is a complement of V . 2), L is a direct summand of M if and only if L = e(M ) for an idempo• tent e ∈ EndR (M ).

En is impossible because (r1 , . . , rn ) = (0, . . , 0) implies ri = 0 for each i. (iv) Suppose R is a nontrivial ring and S is a nonempty set. A function f : S → R is said to have ﬁnite support if f (s) = 0R for all but ﬁnitely many s ∈ S. Let ⊕S R denote the set of all functions f : S → R with ﬁnite support. Under pointwise sum and scalar multiplication, ⊕S R is an R-module. If s ∈ S, let s : S → R denote the characteristic function of s, deﬁned by 1R 0R s(t) = if t = s , if t = s . Then the set S = {s : s ∈ S} is a basis of ⊕S R : For if f ∈ ⊕S R, then f = f (s)s ; s∈S and an R-linear combination f (s)s (= f ) s∈S is the zero map ( = the zero element of ⊕S R) if and only if every f (s) = 0R .