F −→ H0(V, F)
on the category of quasi-coherent k-regulous sheaves on V is exact.
Note that every non-empty k-regulous open subset U of Knis an affine k-regulous variety. Indeed, if U = D ( f ) for a k-regulous function f on Kn(Corollary11.11), then U is isomorphic to the affine k-regulous subvariety
V := Z (y f (x) − 1) ⊂ Kxn× Ky.
Remark 13.9 Consider a smooth algebraic subvariety X of the affine space KAn. We may look at the set V := X(K ) of its K -points both as an algebraic variety X(K ) and as a k-regulous subvariety V of Kn. Every function f : V → K that is k-regulous on V in the second sense remains, of course, k-regulous on X(K ) in the sense of the definition from the beginning of Sect.11.
Open problem. The problem whether the converse implication is true for k> 0 is unsolved as yet.
For k = 0 the answer is in the affirmative and follows immediately from Theo-rem10.2. Indeed, every continuous hereditarily rational function f on X(K ) extends to a continuous rational function on Kn, whence f is regulous on V . This theorem was proven for real and p-adic varieties in [27].
Finally, we wish to give a criterion for a continuous function to be regulous. It relies on Theorem10.2on extending continuous hereditarily rational functions on algebraic K -varieties.
Proposition 13.10 Let V be an affine regulous subvariety of Knand f : V → K a function continuous in the K -topology. Then a necessary and sufficient condition for
f to be a regulous function is the following:
(*) For every Zariski closed subset Z of Kn, there exist a Zariski dense open subset U of the Zariski closure of V∩ Z in Knand a regular function g on U such that
f(x) = g(x) for all x ∈ V ∩ Z ∩ U.
Proof By Corollary11.5, the necessary condition is clear, because f is the restriction to V of a regulous function on Kn(Corollary13.3to Cartan’s theorem B).
In order to prove the sufficient condition, we proceed with induction with respect to the dimension d of the set V which is a closed (in the K -topology) constructible subset of Kn. The case d= 0 is trivial. Assuming the assertion to hold for dimensions
less than d, we shall prove it for d. So suppose V is of dimension d. By Corollary11.2, the Zariski closure W of V in Knis of dimension d and we have
W0:= {x ∈ W : W is smooth of dimension d at x} ⊂ V.
Obviously, W∗:= W\W0is a Zariski closed subset of Knof dimension less than d.
Therefore, Y := V ∩ W∗is a regulous closed subset of V of dimension less than d and
W = W0∪ W∗⊂ V ∪ W∗⊂ W.
Since the restriction f|Y satisfies condition (*), it is a regulous function on Y by the induction hypothesis. It is thus the restriction to Y of a regulous function F on Kn (Corollary13.3to Cartan’s theorem B).
Further, the function f and the restriction F|W∗can be glued to a function
g: W = V ∪ W∗→ K, g(x) =
! f(x) : x ∈ V F(x) : x ∈ W∗
which satisfies condition (*) as well. Now, it follows from Theorem10.2that g extends to a regulous function G on Kn. Since f is the restriction to V of the function G which
is regulous, so is f , as required.
Acknowledgements The author wishes to express his gratitude to János Kollár and Wojciech Kucharz for several stimulating discussions on the topics of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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