TY - JOUR
T1 - Corrigendum to “On stably pointed varieties and generically stable groups in ACVF” [Ann. Pure Appl. Log. 170(2) (2019) 180–217] (Annals of Pure and Applied Logic (2019) 170(2) (180–217), (S0168007218301076), (10.1016/j.apal.2018.09.003))
AU - Halevi, Yatir
N1 - Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2022/1/1
Y1 - 2022/1/1
N2 - In the paper [2], a functor was constructed from the category of pairs [Formula presented], where V is an algebraic variety over a non-trivially valued algebraically closed field and p is a Zariski dense generically stable type concentrated on V. For an affine variety V, it associates to each pair [Formula presented] as above the scheme [Formula presented] over [Formula presented], where [Formula presented]. The functor was first defined for affine varieties and then extended to general varieties by gluing. The gluing process does not work as stated there due to the following issues. We were not able to remedy them. I We have not shown that the functor preserves open immersions (and do not know if this is even true in general).II Even if gluing is possible in some situation, this might depend on the affine cover that was chosen.As a result, the following main results from the paper must be crossed out: (1) Proposition 4.2.4 (the existence of the functor from pairs [Formula presented] where V is any variety over K).(2) Theorem 5.3.2 (correspondence between generically stable groups and group schemes over [Formula presented]).(3) Proposition 5.4.7 (correspondence between generically stable algebraic groups and universally closed group schemes).The rest of the results are, as a whole, correct. In particular the other main results of the paper are still correct: (1) Theorem 3.2.4 (surjectivity of the map [Formula presented]).(2) Theorem 5.2.2 (maximum modulus principle for group schemes over [Formula presented]).The suitable modifications, as well as other minor corrections that arised since the paper was published, are listed below. A corrected version is available on arXiv, see [1]. (1) In the following places the “irreducible” assumption should be changed to “integral”: (a) Proposition 3.1.4 (and Claim 1 in the proof)(b) The remark after Proposition 3.1.5 should assume that [Formula presented] is integral.(c) Proposition 4.1.4(d) Lemma 4.1.6(e) Proposition 4.1.7(f) Lemma 4.3.4(g) Proposition 4.3.5(h) Proposition 5.1.4(i) Theorem 5.2.2(2) The statement of Lemma 3.2.2 should be (the proof is more or less the same): Let [Formula presented] be an irreducible affine scheme which is flat over [Formula presented] and let b be a K-generic point of [Formula presented]. If [Formula presented] is dominant then (a) [Formula presented];(b) and if we further assume that [Formula presented] is integral and [Formula presented] has a [Formula presented]-point then for any [Formula presented], where L is an algebraically closed field for which [Formula presented], there exists a valuation ring [Formula presented] such that [Formula presented] and [Formula presented].(3) The proof of Proposition 3.2.3 should be: Since every open subscheme of [Formula presented] is Zariski dense and the given [Formula presented]-point must land in some affine open subscheme of [Formula presented], we may reduce to the case where [Formula presented] is affine. Assume that [Formula presented] is affine. We will show that any basic open subscheme of [Formula presented] has an [Formula presented]-point. Let f be a regular function on [Formula presented] and assume it is defined over [Formula presented] for some [Formula presented]. Since [Formula presented] has an [Formula presented] point so does the reduced induced subscheme [Formula presented] and consequently [Formula presented] has a [Formula presented]-point. By Lemma 3.2.2 there exists [Formula presented] and [Formula presented] which is a [Formula presented]-generic of [Formula presented]. Since [Formula presented] is irreducible (again, by Proposition 3.1.5) and thus has unique generic type, [Formula presented] as required. If [Formula presented] is of finite type over [Formula presented], there thus exists [Formula presented] with [Formula presented].(4) The first paragraph of the proof of Theorem 3.2.4 should be: Since [Formula presented] has an [Formula presented] point, [Formula presented] has a [Formula presented]-point. Every [Formula presented]-point (resp. k-point) factors through [Formula presented] (resp. [Formula presented]) so we may assume that [Formula presented] and [Formula presented] are reduced. By Proposition 3.1.4, [Formula presented] is faithfully flat. We first assume that [Formula presented] is affine, say [Formula presented], where [Formula presented] may be infinite but small. The last line of the proof should read: since [Formula presented] is also irreducible and flat over [Formula presented], by Lemma 3.2.2, we may reduce to the affine case.(5) Since the functor is only defined for affine varieties, Subsection 4.2 should deal with affine varieties only. In particular (a) The objects in the category in Definition 4.2.2 are pairs [Formula presented] where V is affine.(b) Proposition 4.2.4 is not true.(c) The proof of Proposition 4.2.20 is incorrect. This does not affect the rest of the paper since every affine scheme is separated.(6) Subsection 4.3 should only deal with affine varieties. The functor is only defined for affine varieties (e.g. in the introduction the subsection, in Lemma 4.3.7, Proposition 4.3.8, etc.).(7) In Lemma 4.3.7, the type should concentrate on every [Formula presented], whenever it is non-empty.(8) Theorem 5.3.2 and Proposition 5.4.7 should be deleted, since the functor is not defined for general varieties.
AB - In the paper [2], a functor was constructed from the category of pairs [Formula presented], where V is an algebraic variety over a non-trivially valued algebraically closed field and p is a Zariski dense generically stable type concentrated on V. For an affine variety V, it associates to each pair [Formula presented] as above the scheme [Formula presented] over [Formula presented], where [Formula presented]. The functor was first defined for affine varieties and then extended to general varieties by gluing. The gluing process does not work as stated there due to the following issues. We were not able to remedy them. I We have not shown that the functor preserves open immersions (and do not know if this is even true in general).II Even if gluing is possible in some situation, this might depend on the affine cover that was chosen.As a result, the following main results from the paper must be crossed out: (1) Proposition 4.2.4 (the existence of the functor from pairs [Formula presented] where V is any variety over K).(2) Theorem 5.3.2 (correspondence between generically stable groups and group schemes over [Formula presented]).(3) Proposition 5.4.7 (correspondence between generically stable algebraic groups and universally closed group schemes).The rest of the results are, as a whole, correct. In particular the other main results of the paper are still correct: (1) Theorem 3.2.4 (surjectivity of the map [Formula presented]).(2) Theorem 5.2.2 (maximum modulus principle for group schemes over [Formula presented]).The suitable modifications, as well as other minor corrections that arised since the paper was published, are listed below. A corrected version is available on arXiv, see [1]. (1) In the following places the “irreducible” assumption should be changed to “integral”: (a) Proposition 3.1.4 (and Claim 1 in the proof)(b) The remark after Proposition 3.1.5 should assume that [Formula presented] is integral.(c) Proposition 4.1.4(d) Lemma 4.1.6(e) Proposition 4.1.7(f) Lemma 4.3.4(g) Proposition 4.3.5(h) Proposition 5.1.4(i) Theorem 5.2.2(2) The statement of Lemma 3.2.2 should be (the proof is more or less the same): Let [Formula presented] be an irreducible affine scheme which is flat over [Formula presented] and let b be a K-generic point of [Formula presented]. If [Formula presented] is dominant then (a) [Formula presented];(b) and if we further assume that [Formula presented] is integral and [Formula presented] has a [Formula presented]-point then for any [Formula presented], where L is an algebraically closed field for which [Formula presented], there exists a valuation ring [Formula presented] such that [Formula presented] and [Formula presented].(3) The proof of Proposition 3.2.3 should be: Since every open subscheme of [Formula presented] is Zariski dense and the given [Formula presented]-point must land in some affine open subscheme of [Formula presented], we may reduce to the case where [Formula presented] is affine. Assume that [Formula presented] is affine. We will show that any basic open subscheme of [Formula presented] has an [Formula presented]-point. Let f be a regular function on [Formula presented] and assume it is defined over [Formula presented] for some [Formula presented]. Since [Formula presented] has an [Formula presented] point so does the reduced induced subscheme [Formula presented] and consequently [Formula presented] has a [Formula presented]-point. By Lemma 3.2.2 there exists [Formula presented] and [Formula presented] which is a [Formula presented]-generic of [Formula presented]. Since [Formula presented] is irreducible (again, by Proposition 3.1.5) and thus has unique generic type, [Formula presented] as required. If [Formula presented] is of finite type over [Formula presented], there thus exists [Formula presented] with [Formula presented].(4) The first paragraph of the proof of Theorem 3.2.4 should be: Since [Formula presented] has an [Formula presented] point, [Formula presented] has a [Formula presented]-point. Every [Formula presented]-point (resp. k-point) factors through [Formula presented] (resp. [Formula presented]) so we may assume that [Formula presented] and [Formula presented] are reduced. By Proposition 3.1.4, [Formula presented] is faithfully flat. We first assume that [Formula presented] is affine, say [Formula presented], where [Formula presented] may be infinite but small. The last line of the proof should read: since [Formula presented] is also irreducible and flat over [Formula presented], by Lemma 3.2.2, we may reduce to the affine case.(5) Since the functor is only defined for affine varieties, Subsection 4.2 should deal with affine varieties only. In particular (a) The objects in the category in Definition 4.2.2 are pairs [Formula presented] where V is affine.(b) Proposition 4.2.4 is not true.(c) The proof of Proposition 4.2.20 is incorrect. This does not affect the rest of the paper since every affine scheme is separated.(6) Subsection 4.3 should only deal with affine varieties. The functor is only defined for affine varieties (e.g. in the introduction the subsection, in Lemma 4.3.7, Proposition 4.3.8, etc.).(7) In Lemma 4.3.7, the type should concentrate on every [Formula presented], whenever it is non-empty.(8) Theorem 5.3.2 and Proposition 5.4.7 should be deleted, since the functor is not defined for general varieties.
UR - http://www.scopus.com/inward/record.url?scp=85116208253&partnerID=8YFLogxK
U2 - 10.1016/j.apal.2021.103045
DO - 10.1016/j.apal.2021.103045
M3 - Comment/debate
AN - SCOPUS:85116208253
SN - 0168-0072
VL - 173
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1
M1 - 103045
ER -