Fundamental Groups in Arithmetic and Algebraic Geometry

Class field theory for singular varieties

speaker: Alexander Schmidt (Ruprecht-Karls-Universität Heidelberg)

abstract: The unramified class field theory of Kato and Saito establishes for smooth varieties $X$ over finite fields an isomorphism CH₀(X)^\wedge \cong \pi₁^ab(X), where \wedge denotes profinite completion. This was generalized to the tame fundamental group of quasiprojective varieties by Schmidt and Spiess. The role of the Chow group CH₀(X) is taken over by the integral Suslin homology in degree zero. The result fails already for normal, nonregular varieties.

In the talk we explain how to generalize tame class field theory to singular varieties. For connected schemes $X$ of finite type over finite fields we obtain an isomorphism H₁^{WS}(X,Z)^\wedge \cong \pi₁^{ab,\,tame}(X). Here H₁^{WS}(X, Z) is the Weil-Suslin homology defined by Geisser. (joint work with Th. Geisser)

timetable:
Mon 16 Dec, 12:00 - 13:00, Aula Dini
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