K-Theory also is a generalization of Linear Algebra.

K-Theory studies families of vector spaces, called vector bundles.

Examples:

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 RR.

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.


Note:

Vector bundles give rise to rings and modules!


Question:

But what about Algebraic Geometry?