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

    Abstract

    No 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.

    Main Theorems

    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

    Quillen’s work in Adam’s conjecture

    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.

    Talk 4:

    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 (Weibel 2013)

    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 (Weibel 2013) VI.1 and Suslin’s paper

    \(K_{3}\) of fields and the relation with the Bloch group

    Talk 12: \(K_{3}\) of fields and the relation with the Bloch group












    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

    Quillen’s work in Adam’s conjecture

    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.

    Talk 5:

    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 (Weibel 2013) VI.1 and Suslin’s paper

    \(K_{3}\) of fields and the relation with the Bloch group

    Talk 12: \(K_{3}\) of fields and the relation with the Bloch group