The Witt filtration of Lubin-Tate deformation rings
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| Format: | Preprint |
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2026
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| _version_ | 1866912964430266368 |
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| author | Rezk, Charles |
| author_facet | Rezk, Charles |
| contents | This note is a meditation on a generalization $\mathbb{W}_E$ of the classical p-typical Witt vectors $\mathbb{W}_p$, which arises (geometrically) from isogenies of deformations of formal groups, or (topologically) from the theory of power operations on Morava $E$-theory. For formal groups of height $1$ we have $\mathbb{W}_E=\mathbb{W}_p$, but the $\mathbb{W}_E$ are richer when height is $\geq 2$. We show that $\mathbb{W}_p$ splits naturally from $\mathbb{W}_E$.
A key property of $\mathbb{W}_E$ is the isomorphism $π_0E\approx \mathbb{W}_E(π_0E/\mathfrak{m})$, the ``cofreeness of the Morava $E$-theory'' proved by Burklund, Schlank, and Yuan. This isomorphism determines a natural ``Witt filtration'' on $π_0 E$. We describe how this Witt filtration interpolates between the $p$-adic filtration and a geometric filtration on $π_0E/(p)$. We use this to give a new proof of cofreeness. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_12490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Witt filtration of Lubin-Tate deformation rings Rezk, Charles Algebraic Topology 55S12 (Primary), 13K05 (Secondary) This note is a meditation on a generalization $\mathbb{W}_E$ of the classical p-typical Witt vectors $\mathbb{W}_p$, which arises (geometrically) from isogenies of deformations of formal groups, or (topologically) from the theory of power operations on Morava $E$-theory. For formal groups of height $1$ we have $\mathbb{W}_E=\mathbb{W}_p$, but the $\mathbb{W}_E$ are richer when height is $\geq 2$. We show that $\mathbb{W}_p$ splits naturally from $\mathbb{W}_E$. A key property of $\mathbb{W}_E$ is the isomorphism $π_0E\approx \mathbb{W}_E(π_0E/\mathfrak{m})$, the ``cofreeness of the Morava $E$-theory'' proved by Burklund, Schlank, and Yuan. This isomorphism determines a natural ``Witt filtration'' on $π_0 E$. We describe how this Witt filtration interpolates between the $p$-adic filtration and a geometric filtration on $π_0E/(p)$. We use this to give a new proof of cofreeness. |
| title | The Witt filtration of Lubin-Tate deformation rings |
| topic | Algebraic Topology 55S12 (Primary), 13K05 (Secondary) |
| url | https://arxiv.org/abs/2603.12490 |