In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K ).

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory.

A k-division algebra is a division ring that is also a k-algebra. For example, H is an R-division algebra, and a Q-division algebra, although one usually views it only as an R-

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K -groups. These are groups in the sense of abstract algebra.

Let R be an associative ring spectrum. We set K(R) = K(Modperf. R ). We will refer to K(R) as the algebraic K-theory space of R. Remark 4. It may appear that De nition 3 is a very special case of the construction described in Lecture. 16.

carry relevant information about the ring. To name a few simple examples, K 0(R) = Z Cl(R) computes the class group of a ring when Ris a Dedekind domain2, and K 1(F) = F for any eld. Therefore, it was to be expected that higher invariants would contain other valuable information that would help further our understanding of these structures.

1. R is a valuation ring of K. 2. R is local, and is maximal among local subrings of K under the partial ordering of domination. 3. There exists an algebraically closed field L and a homomorphism θ:R → L (not necessarily injective) with respect to which R is maximal: if R ⊆ R0 ⊆ K and θ:R0 → L prolongs θ, then R = R0.

DEFINITIONS OF HIGHER K-THEORY The higher algebraic K-groups of a ring Rare defined to be the homotopy groups Kn(R) = πnK(R) of a certain topological space K(R), which we shall construct in this chapter. Of course, the space K(R) is rigged so that if n= 0,1,2 then πnK(R) agrees with the groups Kn(R) constructed in chapters II and III.

A ring $A$ equipped with a ring homomorphism $k \to Z(A)$ is called a $k$-algebra. More explicitly, this means that $A$ has a structure of vector space over $k$ and also a ring structure such that: (1) The operations " $+$ " coming from the vector space structure and the ring structure, respectively, are the same.

If Y ⊆An Y ⊆ A n is an affine algebraic set, we define the affine coordinate ring A(Y) A (Y) of Y Y to be A/I(Y) A / I (Y)... ... A(Y) A (Y) is a finitely generated k k -algebra. Conversely, any finitely generated k k -algebra B B which is a domain is the affine coordinate ring of some affine variety.