Power Towers: Purely Inseparable Galois Theory and Foliations in Positive Characteristic
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911196112748544 |
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| author | Grabowski, Przemysław |
| author_facet | Grabowski, Przemysław |
| contents | We build a purely inseparable Galois theory using non-derived commutative algebra. Our theory works on fields and on normal varieties. It says that a purely inseparable morphism corresponds to a finite (saturated) subalgebra of differential operators. Our approach unifies most of the literature about purely inseparable morphisms and shows new research directions.
Indeed, our theory extends to a theory that covers all (saturated) subalgebras of differential operators. The extended theory gives a correspondence between the subalgebras and a new notion of \textbf{power towers}. A power tower is an object analogous to a foliation from differential geometry. In particular, it admits its own versions of many key results about foliations, such as a ``fibrations inject into foliations'' and a ``Frobenius theorem''. The latter has a nontrivial twist: the local structure is trivial at every point, but it may differ between points!
In general, our analogy between power towers and foliations explains why purely inseparable morphisms are foliation-like, because these morphisms are power towers in an explicit way.
Finally, we use our theory to produce a formula for pullbacks of canonical divisors for arbitrary purely inseparable morphisms. We use it to conclude some ``not-L{ü}roth theorems'': if the characteristic is high enough, then any variety purely inseparably covered by an $n$-dimensional projective space has Iitaka dimension equal to $-\infty$ or $n$, i.e., it is not zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Power Towers: Purely Inseparable Galois Theory and Foliations in Positive Characteristic Grabowski, Przemysław Algebraic Geometry Commutative Algebra 12F15, 12F20, 13B05, 13N10, 13N15, 16S32 We build a purely inseparable Galois theory using non-derived commutative algebra. Our theory works on fields and on normal varieties. It says that a purely inseparable morphism corresponds to a finite (saturated) subalgebra of differential operators. Our approach unifies most of the literature about purely inseparable morphisms and shows new research directions. Indeed, our theory extends to a theory that covers all (saturated) subalgebras of differential operators. The extended theory gives a correspondence between the subalgebras and a new notion of \textbf{power towers}. A power tower is an object analogous to a foliation from differential geometry. In particular, it admits its own versions of many key results about foliations, such as a ``fibrations inject into foliations'' and a ``Frobenius theorem''. The latter has a nontrivial twist: the local structure is trivial at every point, but it may differ between points! In general, our analogy between power towers and foliations explains why purely inseparable morphisms are foliation-like, because these morphisms are power towers in an explicit way. Finally, we use our theory to produce a formula for pullbacks of canonical divisors for arbitrary purely inseparable morphisms. We use it to conclude some ``not-L{ü}roth theorems'': if the characteristic is high enough, then any variety purely inseparably covered by an $n$-dimensional projective space has Iitaka dimension equal to $-\infty$ or $n$, i.e., it is not zero. |
| title | Power Towers: Purely Inseparable Galois Theory and Foliations in Positive Characteristic |
| topic | Algebraic Geometry Commutative Algebra 12F15, 12F20, 13B05, 13N10, 13N15, 16S32 |
| url | https://arxiv.org/abs/2510.06058 |