[Paper Review] Constructible 1-motives and exactness

Simon Pepin Lehalleur|arXiv (Cornell University)|Dec 4, 2017

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper establishes the t-exactness of pullbacks, Betti, and étale realization functors for the constructible motivic t-structure on cohomological 1-motives over a noetherian, finite-dimensional, excellent scheme S. Under a mild resolution of singularities hypothesis, it proves that the t-structure restricts to compact 1-motives and that every constructible 1-motive admits a stratification of S into regular subschemes where its restriction becomes a Deligne 1-motive.

ABSTRACT

We prove that arbitrary pullbacks, as well as Betti and étale realisation functors, are t-exact for the constructible motivic t-structure on the category of cohomological 1-motives over a base scheme.

Motivation & Objective

• To prove that the constructible motivic t-structure on cohomological 1-motives is preserved under arbitrary pullbacks.
• To establish the t-exactness of Betti and étale realization functors for this t-structure.
• To show that the t-structure restricts to the full subcategory of compact 1-motives.
• To demonstrate that every constructible 1-motive admits a stratification of the base scheme into regular subschemes such that its restriction to each stratum is a Deligne 1-motive.
• To provide a new proof of the t-exactness of pullbacks using a different method than Vaish’s gluing approach, with potential for future unification of methods.

Proposed method

• Uses the motivic t-structure on cohomological 1-motives over a base scheme S, constructed in prior work under a resolution of singularities hypothesis.
• Applies induction on the dimension of S, leveraging the fact that for a closed immersion i:Z→S with Z of lower dimension, the pullback functor i* is t-exact by induction hypothesis.
• Employs distinguished triangles from the localization sequence for open and closed immersions to analyze the cohomological truncation of motives.
• Applies absolute purity for smooth motives to compute the !-pullback of a 1-motive along a regular immersion, using the formula k_i^!M ≃ k_i^*M(−c_i)[−2c_i] for a regular immersion of codimension c_i.
• Uses the fact that ω¹ preserves compact objects under the resolution hypothesis, and applies the weight truncation functor ω¹ to restrict to 1-motives.
• Relies on the structure of Deligne 1-motives (lattices and semi-abelian schemes) and their behavior under base change to establish the stratification result.

Experimental results

Research questions

• RQ1 Are arbitrary pullbacks t-exact for the constructible motivic t-structure on cohomological 1-motives?
• RQ2 Are Betti and étale realization functors t-exact when the target categories are equipped with their standard t-structures?
• RQ3 Does the motivic t-structure on 1-motives restrict to the subcategory of compact 1-motives?
• RQ4 Can every constructible 1-motive be stratified over S into regular subschemes such that its restriction to each stratum is a Deligne 1-motive?
• RQ5 Can the t-exactness results be proven via a method distinct from Vaish’s gluing of t-structures using weight truncations?

Key findings

• The motivic t-structure on the category of cohomological 1-motives over S restricts to the full subcategory of compact 1-motives, confirming that the heart of the restricted t-structure is well-defined.
• Arbitrary pullbacks along morphisms of schemes are t-exact for the constructible motivic t-structure on 1-motives, provided S allows resolution of singularities by alterations.
• The Betti and étale realization functors are t-exact when the target categories are equipped with their standard t-structures.
• For any constructible 1-motive M over S, there exists a stratification of S into regular subschemes such that the restriction of M to each stratum is a Deligne 1-motive.
• The !-pullback of a smooth constructible 1-motive along a closed immersion i:Z→S is computed via ω¹i!M ≃ ω¹(k₀* k₀* i*M(−1))[−2], where k₀ is the inclusion of the codimension-one part of Z.
• The morphism H₀(ω¹∂f_*ℚ[+1]) → ℚ induced by the boundary of a proper curve is an isomorphism, which implies t-exactness in the key step of the proof.