Algebraic K-theory

Algebraic K-theory is a part of mathematics that connects to many other areas, such as geometry, topology, ring theory, and number theory. In algebraic K-theory, mathematicians assign special groups called K-groups to different objects. These groups hold important information about those objects, but they can be very hard to figure out. One big challenge is finding the K-groups of the integers.

K-theory started in the late 1950s when Alexander Grothendieck studied intersection theory on algebraic varieties. At first, Grothendieck looked at K₀, the zeroth K-group. This still has many useful uses, like in the Grothendieck–Riemann–Roch theorem. Today, algebraic K-theory keeps growing and connects to topics such as motivic cohomology, Chow groups, quadratic reciprocity, and the study of number fields.

Early work focused on the simpler, "lower" K-groups. For example, for a field F, the group K₀(F) is related to the idea of vector space dimension. For a commutative ring R, K₀(R) connects to the Picard group. The group K₁(R) relates to the group of units, and for a field, it is exactly that group of units. Later, mathematicians like Daniel Quillen and Robert Thomason made big steps in understanding the more complex "higher" K-groups.

History

The history of algebraic K-theory started in the 1800s with work by Bernhard Riemann and Gustav Roch. They studied something called the Riemann–Roch theorem. This theorem looks at the properties of special spaces made from functions on a surface.

Later, in the middle of the 1900s, Friedrich Hirzebruch used this theorem for more complicated math ideas.

The name K-theory comes from work by Alexander Grothendieck in 1957. Grothendieck looked at groups of vector bundles, which are like families of vector spaces. He found a way to give these groups special numbers called invariants. These invariants are part of what we now call the Grothendieck group, written as ( K_0(X) ). His work helped connect K-theory to other parts of math, such as topology and number theory.

Lower K-groups

The lower K-groups are a key part of algebraic K-theory. This is a branch of mathematics that connects to geometry, topology, and number theory. These groups help us understand properties of rings and other algebraic structures.

K₀ is the simplest lower K-group. It looks at different ways to build structures called "projective modules" from a ring. For a commutative ring, there’s a special part of K₀ called the "reduced zeroth K-theory."

K₁ deals with the invertible elements of a ring and how they can be combined. It’s linked to the study of matrices and their properties. For fields (like the rational numbers), K₁ is easy to describe, but for more complex rings, it can be harder.

K₂ is another important group in K-theory. It was defined by John Milnor. For fields, there’s a clear way to understand K₂ using something called Steinberg symbols.

Milnor K-theory

Main article: Milnor K-theory

Milnor K-theory is a way to study special math groups called "higher" K-groups. It starts with a simple idea about the K-group of a field and builds more complex structures from it. These groups help connect algebra to geometry and number theory.

The theory uses tensor products and special rules to create new math objects called symbols. These symbols are important because they link K-theory to other areas of mathematics, like cohomology. Famous mathematicians have solved big questions about how these ideas fit together.

Higher K-theory

Higher algebraic K-theory is a part of mathematics that studies special groups linked to different math structures. These groups help mathematicians understand how algebra, geometry, and number theory are connected.

There are different ways to define these K-groups. One method, called the plus-construction, uses ideas from homotopy theory to build these groups. Another method, the Q-construction, works in more general cases and defines the K-groups directly. Both methods give the same results, showing that K-theory is consistent. These constructions help mathematicians discover important links in many areas of math.

Examples

Algebraic K-theory helps us understand different parts of math, like shapes and numbers. It can be hard to calculate, but mathematicians have found ways to do it.

One important calculation was done by Quillen for finite fields. These are special sets with a limited number of elements.

For a finite field with q elements, Quillen found specific results for K-groups:

• K₀ is always the group Z.
• For even indices greater than zero, the K-groups are zero.
• For odd indices, the K-groups have a specific structure related to q.

These discoveries help mathematicians learn more about how K-theory works with different kinds of numbers.

Applications and open questions

Algebraic K-groups help mathematicians study special values of L-functions and a theory called Iwasawa theory. They are also used to build something called higher regulators.

There are still many unsolved questions about these groups. One conjecture, called Parshin's conjecture, suggests that for certain types of numbers, these groups disappear except for a small part. Another important idea, known as Bass' conjecture, proposes that these groups are always built from a finite number of pieces.