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ADDENDUM: REAL TOPOLOGICAL HOCHSCHILD HOMOLOGY OF SCHEMES

Published online by Cambridge University Press:  08 May 2024

Jens Hornbostel*
Affiliation:
Bergische Universität Wuppertal, Fakultät für Mathematik und Naturwissenschaften, Fachgruppe Mathematik und Informatik, Gaußstrasse 20, 42119 Wuppertal, Germany
Doosung Park
Affiliation:
Bergische Universität Wuppertal, Fakultät für Mathematik und Naturwissenschaften, Fachgruppe Mathematik und Informatik, Gaußstrasse 20, 42119 Wuppertal, Germany ([email protected])
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Abstract

Type
Addendum
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

For commutative rings A in which 2 is invertible, as well as for $A=\mathbb {Z}$ , [Reference Hornbostel and Park3, Proposition 2.3.5] is true. However, for some rings A in which 2 is not invertible, the result is not correct as stated. The reason is the description of the ideal T in the proof of loc. cit. is not correct in this case, and this might imply that the map $\alpha $ in loc. cit. is not an isomorphism. In more detail, let A be a commutative ring. Due to [Reference Dotto, Moi, Patchkoria and Reeh1, Corollary 5.2], $\underline {\pi }_0\mathrm {THR}(A)$ is isomorphic to the Mackey functor

where $\mathrm {res}(x\otimes y)=xy$ for $x,y\in A$ , $\mathrm {tran}(a)=2a\otimes 1$ for $a\in A$ , and $T_A$ is the subgroup generated by $x\otimes a^2 y-a^2 x\otimes y$ and $x\otimes 2ay-2ax\otimes y$ for $a,x,y\in A$ . Let $2A$ denote the ideal $(2)$ in A. We have the monomorphism $2A\to (A\otimes A)/T_A$ given by $2a\mapsto 2a\otimes 1$ for $2a\in 2A$ . Its cokernel is isomorphic to $(A/2\otimes A/2)/T_{A/2}$ , where $A/2:=A/2A$ . Observe that $T_{A/2}$ is the subgroup generated by $x\otimes a^2 y-a^2 x\otimes y$ for $a,x,y\in A/2$ . Hence, we have the short exact sequence

$$\begin{align*}0 \to 2A \to (A\otimes A)/T_A \to A/2\otimes_{\varphi,A/2,\varphi} A/2 \to 0, \end{align*}$$

where $\varphi \colon A/2\to A/2$ denotes the Frobenius (i.e., the squaring map). In particular, [Reference Hornbostel and Park3, Proposition 2.3.5] holds if and only if $\varphi $ is surjective.

The only statement in [Reference Hornbostel and Park3] where this proposition is used is the following one in the proof of [Reference Hornbostel and Park3, Proposition 3.2.2], for which we now provide an alternative proof.

Proposition 1. Let $A\to B$ be an étale homomorphism of commutative rings. Then the induced morphism of Mackey functors

$$\begin{align*}\underline{\pi}_0\mathrm{THR}(\iota A)\square_{\iota A}\iota B \to \underline{\pi}_0\mathrm{THR}(\iota B) \end{align*}$$

is an isomorphism.

Proof. By [Reference Hornbostel and Park2, Lemma 5.1], the Mackey functor $\underline {\pi }_0\mathrm {THR}(\iota A)\square _{\iota A}\iota B$ is isomorphic to

Using the above computations, we have the following commutative diagram where the vertical maps are induced by multiplication:

We only need to show that $\beta $ is an isomorphism. Since B is flat over A, the rows are short exact sequences, and $\alpha $ is an isomorphism. The induced square of commutative rings

is coCartesian by [4, Tag 0EBS] since $A/2\to B/2$ is étale. It follows that $\gamma $ is an isomorphism. Hence, $\beta $ is an isomorphism by the five lemma.

References

Dotto, E., Moi, K., Patchkoria, I. and Reeh, S. P., Real topological Hochschild homology, J. Eur. Math. Soc. (JEMS) 23 (2021), 63152.10.4171/jems/1007CrossRefGoogle Scholar
Hornbostel, J. and Park, D., Real topological Hochschild homology of perfectoid rings, Preprint, 2023, arXiv:2310.11183.10.1017/S1474748023000178CrossRefGoogle Scholar
Hornbostel, J. and Park, D., Real topological Hochschild homology of schemes, JIMJ (2023) 158. doi:10.1017/S1474748023000178.Google Scholar
Stacks Project Authors , The Stacks Project. https://stacks.math.columbia.edu, 2022.Google Scholar