# Program of the baby seminar WS 2015/2016 on The K-Theory of Fields

AbstractNo Abstract Found

The aim of the seminar is understand the Higher K-Theory of fields. We will start from recalling the classical K-Theory and the construction for the higher K-theory as the homotopy groups of $$BGL(R)^{+}$$ given by Quillen.

## Plan of the Talks

### Introduction to K-theory

#### Talk 1: Higher Algebriac K-theory

In this talk the speaker should recall the definition of Higher K-Theory for rings. Explain the $$BGL^{+}$$ definition for rings. The speaker should recall the basic properties of the K-theory with out proofs.

• Porduct structure in the K-theory and the structure of $$H$$-space

• Localization theorem

• Projective Bundle Formula

• Fundamental exact sequence

• Genstern resolution for smooth-semilocal ring over $$k$$.

• The speaker should explain why it is important compute the K-theory of algebraically closed fields.

• Also it would be nice to give explicity computations of $$K_{0}$$, $$K_{1}$$, $$K_{2}$$ for fields.

### Borel’s theorem

#### Talk 2: Rank of $$K_{n}$$ over number fields

• Show how to reduce the problem to the computation of the real cohomology of $$SL(F)$$. Borel determined the ring $$H^{*}(SL_{m}(R);\mathbb{Q})$$ and its dual coalgebra. Relate the primitive elementes of to the indecomposables elementes and finally to $$H^{*}(SL(O_{F}),\mathbb{R})$$. A more precise survey is in Soulé’s paper Lecture one two papers of Borel. (Survey of Borel’s paper) (could be related later with $$K_{3}$$, Bloch groups and regulators)

Let $$A$$ be a finite-dimensional semisimple $$\mathbb{Q}-algebra$$. Then for every order $$R$$ in $$A$$ we have $$K_{n}(R)\otimes\mathbb{Q}\cong K_{n}(A)\otimes\mathbb{Q}$$.

\label{Borel-Ranks}

Let $$F$$ a number field and and let $$A$$ be a central simple $$F$$-algebra then rank $$K_{n}(A)\otimes\mathbb{Q}$$ is periodic with period four and equal to $$0$$, $$r_{1}+r_{2}$$, $$r_{2}$$

The pourpuse of this talk is assuming the computation of the real cohomology of $$SL(F)$$ prove Theorem \ref{Borel-Ranks}.

### K-theory for finite fields

Follow Mitchell, Notes on the K-theory of finite fields.

See Quillen, On the cohomology and K-theory of GL over a finite field for more details.

### K-theory of algebraically closed fields

The purpuse of this section is prove one of the Quillen-Lichtembaum conjectures

If $$F$$ is an algebraically closed field, then for $$n\geq 1$$, $$K_{n}(F)$$ is divisible and the torsion subgroup in $$K_{n}(F)=0$$ if $$n$$ is even and isomorphic to $$\coprod_{l\neq charF}\mathbb{Q}_{l}/Z_{l}(n)$$ if $$n$$ is odd.

#### Talk 5: K-theory with finite coefficients

In this talk the speaker should define the $$K$$ theory with finite coefficients, state the universal coefficient sequence to relate the mod $$l$$ $$K$$-groups to the usual $$K$$-groups. Explain the Example 2.5.2 (Bott Elements) in \cite{Kbook}

#### Suslin rigidity Theorem

In this talk we are going to discus the Rigidity Theorem and prove Theorem 1.1 y Thm 1.2 in Weibel’s book.

(Second Talk of this part) From the preliminar results describe the structure of $$K_{*}(F)$$ Corollary 1.3.1 and Proposition 1.4 The main references for this part are: Weibel’s book \cite{Kbook} VI.1 and Suslin’s paper

### Introduction to K-theory

#### Talk 1: Higher Algebriac K-theory

In this talk the speaker should recall the definition of Higher K-Theory for rings. Explain the $$BGL^{+}$$ definition for rings. The speaker should recall the basic properties of the K-theory with out proofs.

• Porduct in the K-theory and teh structure of $$H$$-space

• Localization theorem

• Projective Bundle Formula

• Fundamental exact sequence

• Genstern resolution for smooth-semilocal ring over $$k$$.

• Explain why it is important compute the K-theory of algebraically closed fields.

• Also it would be nice to give explicity computations of $$K_{0}$$, $$K_{1}$$, $$K_{2}$$ for fields.

#### Talk 2: K-theory for finite fields

Follow Mitchell, Notes on the K-theory of finite fields.

See Quillen, On the cohomology and K-theory of GL over a finite field for more details.

### Borel’s theorem

#### Talk 6: Rank of $$K_{n}$$ over number fields

• Show how to reduce the problem to the computation of the real cohomology of $$SL(F)$$. Borel determined the ring $$H^{*}(SL_{m}(R);\mathbb{Q})$$ and its dual coalgebra. Relate the primitive elementes of to the idecomposables elementes and finally to $$H^{*}(SL(O_{F}),\mathbb{R})$$. A more precise survey is in Soulé’s paper Lecture one two papers of Borel. (Survey of Borel’s paper) (could be related later with $$K_{3}$$, Bloch groups and regulators)

Let $$A$$ be a finite-dimensional semisimple $$\mathbb{Q}-algebra$$. Then for every order $$R$$ in $$A$$ we have $$K_{n}(R)\otimes\mathbb{Q}\cong K_{n}(A)\otimes\mathbb{Q}$$.

\label{Borel-Ranks}

Let $$F$$ a number field and and let $$A$$ be a central simple $$F$$-algebra then rank $$K_{n}(A)\otimes\mathbb{Q}$$ is periodic with period four and equal to $$0$$, $$r_{1}+r_{2}$$, $$r_{2}$$

The pourpuse of this talk is assuming the computation of the real cohomology of $$SL(F)$$ prove Theorem \ref{Borel-Ranks}.

#### Talk 11: K-theory of algebraically closed fields

The purpuse of this section is prove one of the Quillen-Lichtembaum conjectures

If $$F$$ is an algebraically closed field, then for $$n\geq 1$$, $$K_{n}(F)$$ is divisible and the torsion subgroup in $$K_{n}(F)=0$$ if $$n$$ is evan and isomorphic to $$\coprod_{l\neq charF}\mathbb{Q}_{l}/Z_{l}(n)$$ if $$n$$ is odd.

(First Talk of this part) In this talk we are going to discus the Rigidity Theorem and prove Theorem 1.1 y Thm 1.2 in Weibel’s book.

(Second Talk of this part) From the preliminar results describe the structure of $$K_{*}(F)$$ Corollary 1.3.1 and Proposition 1.4 The main references for this part are: Weibel’s book \cite{Kbook} VI.1 and Suslin’s paper

### References

1. Charles Weibel. The K-Book: An Introducccion to Algebraic K-Theory. American Mathematical Society, 2013.