[Paper Review] Graded quotients of ramification groups of local fields with imperfect residue fields

Takeshi Saito|arXiv (Cornell University)|Apr 8, 2020

Algebraic Geometry and Number Theory16 references4 citations

TL;DR

This paper proves that the graded quotients of ramification groups in henselian discrete valuation fields with imperfect residue fields of characteristic $p > 0$ are $\mathbf{F}_p$-vector spaces. Using two proofs—reduction to perfect residue fields and geometric construction via cotangent complexes—it establishes an injection from the character group of each graded quotient into a twisted cotangent space, generalizing Kato's refined Swan conductor and enabling future construction of characteristic cycles in mixed characteristic.

ABSTRACT

We prove that the graded quotients of the filtration by ramification groups of any henselian discrete valuation field of residue characteristic $p>0$ are $F_p$-vector spaces. We define an injection of the character group of each graded quotient to a twisted cotangent space defined using a cotangent complex.

Motivation & Objective

• To establish that graded quotients of ramification groups in local fields with imperfect residue fields are $\mathbf{F}_p$-vector spaces.
• To generalize the refined Swan conductor to non-perfect residue fields via a geometric construction using the cotangent complex.
• To provide a functorial characterization of the upper numbering ramification filtration using reduction to the classical perfect residue field case.
• To lay the foundation for constructing the characteristic cycle of a constructible sheaf in mixed characteristic via the injection (4.20).

Proposed method

• Reduction to the classical case by constructing a tangentially dominant extension $K'$ with perfect residue field, leveraging functoriality of the cotangent complex.
• Geometric construction of a Galois covering of a vector space $\Theta^{(r)}$ over $\bar{F}$ with Galois group $\mathrm{Gr}^rG$, using a smooth group scheme structure on the covering.
• Application of a general criterion in Section 2.2 to show that the covering is a morphism of group schemes, enabling the use of $\mathrm{Tor}_1$-computations in the cotangent complex.
• Definition of the characteristic form (4.20) as an injection $\mathrm{Hom}(\mathrm{Gr}^rG, \mathbf{F}_p) \to \mathrm{Hom}_{\bar{F}}(\mathfrak{m}_{\bar{K}}^r / \mathfrak{m}_{\bar{K}}^{r+}, H_1(L_{\bar{F}/S}))$, derived from the cotangent complex $L_{\bar{F}/S}$.
• Use of the exact sequence (0.1) involving $H_1(L_{\bar{F}/S})$, $\mathfrak{m}_K / \mathfrak{m}_K^2 \otimes_F \bar{F}$, and $\Omega^1_F \otimes_F \bar{F}$ to relate the tangent space to the cotangent complex.
• Functoriality of the injection (4.20) under field extensions, with commutative diagrams (4.19) and (4.23) showing compatibility under ramification index scaling.

Experimental results

Research questions

• RQ1 Are the graded quotients $\mathrm{Gr}^rG = G^r / G^{r+}$ of the upper numbering ramification filtration on the Galois group of a henselian discrete valuation field with imperfect residue field of characteristic $p > 0$ necessarily $\mathbf{F}_p$-vector spaces?
• RQ2 Can the refined Swan conductor be generalized to non-perfect residue fields via a geometric construction involving the cotangent complex?
• RQ3 Does the injection (4.20) from characters of $\mathrm{Gr}^rG$ to a twisted cotangent space provide a canonical, functorial invariant of the ramification filtration?
• RQ4 What is the role of tangentially dominant extensions in reducing the general case to the classical perfect residue field case?
• RQ5 How does the wild inertia group act on $\mathrm{Gr}^rG$, and what constraints does this impose on the ramification index and the denominator of $r$?

Key findings

• The graded quotients $\mathrm{Gr}^rG = G^r / G^{r+}$ are proven to be $\mathbf{F}_p$-vector spaces for all $r > 1$, extending the classical result from perfect residue fields.
• An injection (4.20) is constructed: $\mathrm{Hom}(\mathrm{Gr}^rG, \mathbf{F}_p) \to \mathrm{Hom}_{\bar{F}}(\mathfrak{m}_{\bar{K}}^r / \mathfrak{m}_{\bar{K}}^{r+}, H_1(L_{\bar{F}/{\mathcal{O}}_K}))$, generalizing Kato's non-logarithmic refined Swan conductor.
• The injection (4.20) is compatible with base change under field extensions, as shown by the commutative diagram (4.19), ensuring functoriality.
• The wild inertia group $P = G^{1+}$ acts trivially on $\mathrm{Gr}^rG$, and if $\mathrm{Gr}^rG \neq 1$, then the prime-to-$p$ part of the denominator of $r$ divides the ramification index $e_{L/K}$.
• For extensions $K' / K$ with $e_{K'/K} \neq 1$, the composition of (4.20) with the morphism $H_1(L_{\bar{F}/{\mathcal{O}}_K}) \to \Omega^1_{S/S_0} \otimes_{\mathcal{O}_S} \bar{F}$ yields an injection (4.22), which is also functorial.
• The construction of the tangent space $\Theta_{K,\bar{F}} = \mathrm{Spec}\, S(H_1(L_{\bar{F}/S}))$ via the symmetric algebra on $H_1(L_{\bar{F}/S})$ provides a geometric model for the ramification data, with $H_1(L_{\bar{F}/S})$ fitting into the exact sequence (0.1).