The tame site of a scheme
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
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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
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