Algebraic K-theory is a “universal” cohomology theory, which associates to any ring A, a family of invariants Kn(A), for all integers n, called the K-groups of A. These K-groups often encode knowledge of a geometric nature about the ring in question, so studying them can help us understand the ring better. The K-groups of rings were defined in the late 20th century by using homotopy theory to make constructions on their projective modules, and the definition has since been extended to define algebraic K-theory for schemes, and certain categories as well. In addition, calculations in algebraic K-theory have been used to solve problems in different areas such as Number theory and Geometric topology.
We will try to understand how some of the lower K-groups such as K0, and K1 were constructed, as they can usually be described more explicitly than higher K-groups. We will also try to see a rough overview of how higher K-groups of rings were defined, to then discuss some properties of algebraic K-theory.
