K-Theory also is a generalization of Linear Algebra.
K-Theory studies families of vector spaces, called vector bundles.
There is only one kind of family over a single point; i.e., a single vector space.
There is also only one kind of family over the reals; i.e.,
the product family, V x R
But there are two kinds of families over the circle, S1:
For more, see:
M. Walker with Eric Friedlander, Rational Isomorphisms between K-theories and cohomology theories,
Invent. Math., 154 (2003) 1-61.
Vector bundles give rise to rings and modules!
- Let E → X be a vector bundle over a space X
- Let R(X, k) be the set of maps to a field k
- Let H0(E/X) be the set of sections; i.e.,
the set of maps X → E such that
the composition X → E
→ X is the identity
- Then R(X, k) is a commutative ring and H0(E/X) is an
But what about Algebraic Geometry?