Errata to “Discriminants in the Grothendieck ring”
Errata to “Discriminants in the Grothendieck ring”
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“格洛腾迪克环中的判别式”勘误表
DOI：
10.1215/00127094-2020-0001
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发表时间：
2020
影响因子：
2.5
通讯作者：
M. Wood
中科院分区：
文献类型：
--
作者：
R. Vakil;M. Wood
1.1 节中 M 的定义应该是 K0(VarK) 通过 [X] − [Y ] 形式的关系得到的商，只要 X → Y 是 K 上簇的根满射态射，并且本文中的所有进一步的陈述都应该使用这个更正的定义。格洛腾迪克环的商通常用于动机整合的应用（参见[Mus11，第 7.2 节] 和 [CNS18，第 4.4 节]）。当 K 的特征为 0 时，这些附加关系在 K0(VarK) 中就已经微不足道了（例如参见 [Mus11, Prop 7.25]）。有限域上点计数的动机度量仍然是 M 的新定义的因素。这一修正是必要的，以便论文中的证明，特别是定理 1.13 和第 5 节中的证明是正确的。这些论点声称 [X] 和 [Y ] 的 M 相等，其中态射 X → Y 在任何代数闭域上的点上都是双射的。这样的论证在上面 M 的更正定义中是有效的（[Mus11，备注 A.22]），但未知在 K0(VarK) 中是否有效。我们感谢 Margaret Bilu 和 Sean Howe 指出了这个错误并进行了必要的纠正。有关此问题的进一步讨论，请参阅 [BH19]。
The definition ofM in Section 1.1 should be the quotient of K0(VarK) by relations of the form [X] − [Y ] whenever X → Y is a radicial surjective morphism of varieties over K, and all further statements in the paper should use this corrected definition. This quotient of the Grothendieck ring is often taken for applications to motivic integration (see [Mus11, Section 7.2] and [CNS18, Section 4.4]). When K has characteristic 0, these additional relations were already trivial in K0(VarK) (e.g. see [Mus11, Prop 7.25]). The motivic measure of point counting over a finite field still factors through this new definition ofM. This correction is necessary so that the proofs in the paper, in particular those of Theorem 1.13 and in Section 5, are correct. The arguments claim equality inM of [X] and [Y ] where we have a morphism X → Y that is bijective on points over any algebraically closed field. Such an argument is valid in the corrected definition ofM above ([Mus11, Remark A.22]), but is not known to be valid in K0(VarK). We thank Margaret Bilu and Sean Howe for pointing out this mistake and the necessary correction. See [BH19] for further discussion of this issue.