Tilting and untilting for ideals in perfectoid rings
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| Format: | Preprint |
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2023
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| _version_ | 1866917965317799936 |
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| author | Dine, Dimitri Ishizuka, Ryo |
| author_facet | Dine, Dimitri Ishizuka, Ryo |
| contents | For an (integral) perfectoid ring $R$ of characteristic $0$ with tilt $R^{\flat}$, we introduce and study a tilting map $(-)^{\flat}$ from the set of $p$-adically closed ideals of $R$ to the set of ideals of $R^{\flat}$ and an untilting map $(-)^{\sharp}$ from the set of radical ideals of $R^{\flat}$ to the set of ideals of $R$. The untilting map $(-)^{\sharp}$ is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic $p$ introduced by the first author. We prove that these two maps, $(-)^{\flat}$ and $(-)^{\sharp}$, define an inclusion-preserving bijection between the set of ideals $J$ of $R$ such that the quotient $R/J$ is perfectoid and the set of $p^{\flat}$-adically closed radical ideals of $R^{\flat}$, where $p^{\flat}\in R^{\flat}$ corresponds to a compatible system of $p$-power roots of a unit multiple of $p$ in $R$. Furthermore, we prove that the maps send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of the spectrum of $R$ consisting of prime ideals $\mathfrak{p}$ of $R$ such that $R/\mathfrak{p}$ is perfectoid and the subspace of the spectrum of $R^{\flat}$ consisting of $p^{\flat}$-adically closed prime ideals of $R^{\flat}$. In particular, we obtain a generalization and a new proof of the main result of the first author's previous research which concerned prime ideals in perfectoid Tate rings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_09600 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Tilting and untilting for ideals in perfectoid rings Dine, Dimitri Ishizuka, Ryo Algebraic Geometry Commutative Algebra Number Theory For an (integral) perfectoid ring $R$ of characteristic $0$ with tilt $R^{\flat}$, we introduce and study a tilting map $(-)^{\flat}$ from the set of $p$-adically closed ideals of $R$ to the set of ideals of $R^{\flat}$ and an untilting map $(-)^{\sharp}$ from the set of radical ideals of $R^{\flat}$ to the set of ideals of $R$. The untilting map $(-)^{\sharp}$ is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic $p$ introduced by the first author. We prove that these two maps, $(-)^{\flat}$ and $(-)^{\sharp}$, define an inclusion-preserving bijection between the set of ideals $J$ of $R$ such that the quotient $R/J$ is perfectoid and the set of $p^{\flat}$-adically closed radical ideals of $R^{\flat}$, where $p^{\flat}\in R^{\flat}$ corresponds to a compatible system of $p$-power roots of a unit multiple of $p$ in $R$. Furthermore, we prove that the maps send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of the spectrum of $R$ consisting of prime ideals $\mathfrak{p}$ of $R$ such that $R/\mathfrak{p}$ is perfectoid and the subspace of the spectrum of $R^{\flat}$ consisting of $p^{\flat}$-adically closed prime ideals of $R^{\flat}$. In particular, we obtain a generalization and a new proof of the main result of the first author's previous research which concerned prime ideals in perfectoid Tate rings. |
| title | Tilting and untilting for ideals in perfectoid rings |
| topic | Algebraic Geometry Commutative Algebra Number Theory |
| url | https://arxiv.org/abs/2308.09600 |