Introduction to Algebraic K theory

We shall start looking at some algebraic K theory.

There are several perspectives on this topic. The theory is motivated by ideas in several areas like homological algebra, the theory of vector bundles, homotopy theory, etc. There are some constructions like group completions (symmetric monoids), plus constructions ( rings), Q-constructions (exact categories), Waldhausen constructions (categories with cofibrations) which all coincide for the case of rings.

For rings, K-theory assigns a list of abelian groups and these assignments are functorial. These groups will be homotopy groups of certain spaces constructed out of objects related to R.
We shall look at definitions of lower K- groups and what information about the ring and related objects they encode.

basically is a way of getting a group structure from the weaker structure of monoid of finitely generated projective modules over  R. Projective modules are interesting objects in the category of R-modules for which lifting property is satisfied ( Whenever we have a map to the projective module we can lift it to any cover of it). Projective modules also arise in geometry: Sections of vector bundles over a space X give projective modules over C(X). More generally coherent sheaves of affine schemes give modules over commutative rings. Another way to think of projective module P is that the functor is exact. A more concrete way is that P is a direct summand of a free module.

For fields, we get finite dimensional vector spaces, hence the corresponding monoid is , is .
Similarly, over a PID we get . Projective modules are free over a PID because a projective module is a direct summand of a free module and submodules of free modules over a PID are free.
Ideals of Dedekind domain form non-free projective modules. In fact we shall later see that   , Cl(R) is ideal class group of R.

: Take the invertible matrices over R, GL(n,R) we can think all of these as a single object by thinking of GL(n,R) sitting naturally inside GL(n+1,R). This is a categorical direct limit if you like. This is an interesting object. We would like to take K_1(R) as the abelian part of GL i.e., quotient it with the commutator subgroup.

Higher K groups are harder to define. In this post, we shall look at the definitions. Later n we shall try to figure out the motivation behind the definitions.
Different mathematicians defined the higher K groups in different ways with more generalities.

One way to define is called the plus construction . Here refers to n-th homotopy group and refers to Classifying space of , plus sign refers to the plus construction made on . For a group G, is a space that is a universal object in the category of principal G bundles. Every principal G bundle can be seen as the pull of  inside the category.

Plus construction on a CW complex with a given normal subgroup of its fundamental group: Just take the generators loops of the subgroups and attach 2-cells along with them. This simplifies the fundamental group. We can make sure homology/cohomology doesn’t change by making changing some adjustments like adding 3-cells etc.  Now for BGL, we do the construction for the normal subgroup E generated by almost identity matrices with just one non zero off-diagonal entry.

We shall look at other constructions and generalizations in later posts.