[Paper Review] An explicit approach to residues on and dualizing sheaves of arithmetic surfaces

Matthew Morrow|arXiv (Cornell University)|Nov 3, 2009

Algebraic Geometry and Number Theory47 references21 citations

TL;DR

This paper develops an explicit theory of residues for arithmetic surfaces using two-dimensional local fields, establishes a reciprocity law for differential forms around points, and constructs the relative dualizing sheaf via residue maps. The key contribution is an explicit formula for the dualizing sheaf in terms of integral residues, generalizing classical results to arithmetic surfaces over Dedekind domains with finite residue fields.

ABSTRACT

We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface. These are generalisations of known results for surfaces over a perfect field. In an appendix, explicit local ramification theory is used to recover the fact that in the case of a local complete intersection the dualizing and canonical sheaves coincide.

Motivation & Objective

• To develop a theory of residues for arithmetic surfaces using two-dimensional local fields.
• To establish a reciprocity law for differential forms around points on normal, complete local rings of dimension two.
• To explicitly construct the relative dualizing sheaf of an arithmetic surface using local residue maps.
• To generalize the classical relation between canonical and dualizing sheaves to the arithmetic setting, particularly for local complete intersections.
• To provide an explicit adèlic approach to Grothendieck duality in arithmetic geometry.

Proposed method

• Define a relative residue map on continuous differential forms over two-dimensional local fields of mixed characteristic.
• Prove functoriality of the residue map under finite extensions of local fields.
• Establish reciprocity for differential forms on normal, complete, two-dimensional local rings with finite residue fields.
• Use local residue maps at closed points and curves to define a global section condition for the dualizing sheaf.
• Apply explicit local ramification theory to relate the different ideal to the Jacobian determinant in complete intersection rings.
• Prove that for local complete intersections, the dualizing and canonical sheaves coincide via the Jacobian determinant.

Experimental results

Research questions

• RQ1 How can residues be explicitly defined for differential forms on arithmetic surfaces in mixed characteristic?
• RQ2 What is the precise form of the reciprocity law for differential forms on two-dimensional, normal, local rings over a complete discrete valuation ring?
• RQ3 How can the relative dualizing sheaf of an arithmetic surface be constructed explicitly using residue maps?
• RQ4 Under what conditions do the canonical and dualizing sheaves coincide in the arithmetic setting?
• RQ5 What is the relationship between the different ideal and the Jacobian determinant in local complete intersection rings over mixed characteristic local rings?

Key findings

• The reciprocity law holds: for any differential form ω on the function field of a normal, complete, two-dimensional local ring over a local field K, the sum of residues over all height one primes vanishes.
• The dualizing sheaf ω_π of a flat, projective arithmetic surface π:X→Spec(𝒪_K) is explicitly given by the set of differential forms ω for which Res_{x,y}(fω) lies in the completion of K at π(x), for all x∈y⊂U and f∈𝒪_{X,y}.
• For local complete intersection arithmetic surfaces, the canonical and dualizing sheaves coincide, generalizing the classical algebraic geometry result.
• The codifferent of a finite extension of normal rings is given by the inverse of the Jacobian determinant, i.e., ℂ(A/B) = J^{-1}A.
• In the case of a complete intersection over a complete discrete valuation ring, the dualizing module is isomorphic to the inverse of the Jacobian ideal.
• The residue map is functorial: Res_F ∘ Tr_{F'/F} = Res_{F'} for finite extensions F'/F of two-dimensional local fields.