The tame site of a scheme
- PDF / 648,294 Bytes
- 65 Pages / 439.37 x 666.142 pts Page_size
- 57 Downloads / 168 Views
The tame site of a scheme Katharina Hübner1 · Alexander Schmidt2
Received: 20 April 2020 / Accepted: 30 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 Definition of the tame site . . . . . . . . . . . . . . . . . 3 Topological invariance and excision . . . . . . . . . . . . 4 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 5 The tame fundamental group . . . . . . . . . . . . . . . . 6 Joins of hensel rings . . . . . . . . . . . . . . . . . . . . ˇ 7 Comparison with Cech cohomology . . . . . . . . . . . . 8 Comparison with étale cohomology . . . . . . . . . . . . 9 Finiteness in dimension 1 . . . . . . . . . . . . . . . . . 10 Review of Huber pairs . . . . . . . . . . . . . . . . . . . 11 Joins of henselian Huber pairs . . . . . . . . . . . . . . . 12 Riemann–Zariski morphisms . . . . . . . . . . . . . . . ˇ 13 Cech cohomology of discretely ringed adic spaces . . . . 14 Comparison between algebraic and adic tame cohomology 15 Purity and homotopy invariance . . . . . . . . . . . . . . 16 Connection to Suslin homology . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
B Alexander Schmidt
[email protected] Katharina Hübner [email protected]
1
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Giv’at Ram, Jerusalem, Israel
2
Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
123
K. Hübner, A. Schmidt
1 Introduction The étale fundamental group of a scheme plays a similar role in algebraic geometry as the topological fundamental group in algebraic topology. For a scheme X of characteristic p > 0 however, the p-part of π1et (X ) is not well-behaved, e.g., it is not (A1 -)homotopy invariant. Therefore the tame fundamental group has been studied in positive and mixed characteristic (Grothendieck–Murre [1], Kerz–Schmidt [2]). Unfortunately, lacking an associated tame cohomology theory, sometimes ad hoc arguments have to be used in applications. It would be helpful to have a tame Grothendieck topology whose associated fundamental group is the tame fundamental group. Such a topology would provide a tame cohomology theory and, in addition, higher tame homotopy groups. The latter seem to be even more desirable because the higher étale homotopy groups of affine varieties vanish in pos
Data Loading...
