[Paper Review] Duality via cycle complexes

Thomas Geisser|ArXiv.org|Aug 18, 2006

Homotopy and Cohomology in Algebraic Topology19 references4 citations

TL;DR

This paper establishes duality theorems for torsion étale sheaves on separated schemes of finite type over perfect fields, finite fields, local fields of mixed characteristic, and rings of integers in number fields by using Bloch’s complex of relative zero-cycles as a dualizing complex. The key result is a quasi-isomorphism $ R ext{Hom}_X( ext{F}, bZ^c_X) o R ext{Hom}_Y(Rf_!F, bZ^c_Y) $, which unifies $p$-adic and $ar{bQ}_l$-adic duality and generalizes Rojtman’s theorem to normal, projective schemes.

ABSTRACT

We show that Bloch's complex of relative zero-cycles can be used as a dualizing complex over perfect fields and number rings. This leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields, finite fields, local fields of mixed characteristic, and rings of integers in number rings, generalizing results which so far have only been known for smooth schemes or in low dimensions, and unify the p-adic and l-adic theory. As an application, we generalize Rojtman's theorem to normal, projective schemes.

Motivation & Objective

• To extend duality theorems for torsion étale sheaves beyond smooth schemes and low dimensions.
• To unify $p$-adic and $ar{bQ}_l$-adic duality theories using a single dualizing complex.
• To generalize Rojtman’s theorem on the Albanese map for zero-cycles to normal, projective schemes.
• To establish duality over rings of integers in number fields and local fields of mixed characteristic.
• To provide a homological framework for the Albanese variety in terms of Chow groups of zero-cycles.

Proposed method

• Uses Bloch’s complex of relative zero-cycles $ bZ^c_X $ as a dualizing complex over perfect fields and Dedekind rings of characteristic 0.
• Applies the adjunction $ R ext{Hom}_X( ext{F}, bZ^c_X) o R ext{Hom}_Y(Rf_!F, bZ^c_Y) $ to derive duality via quasi-isomorphism.
• Employs the purity property of $ bZ^c_X $ for closed embeddings and the homotopy invariance $ p_*bZ^c_{X imesbA^r}(n) o bZ^c_X(n-r)[2r] $.
• Relies on the Beilinson-Lichtenbaum conjecture to relate Zariski and étale cohomology of $ bZ^c_X/m(n) $.
• Uses the spectral sequence $ E^{1}_{s,t} = igoplus_{x o X^{(s)}} H^{s-t}(k(x), bZ(s-n)) o H_{s+t}(X, bZ^c(n)) $ to compute cohomology.
• Applies Artin-Verdier duality and compactly supported cohomology to derive perfect pairings over number rings and local fields.

Experimental results

Research questions

• RQ1 Can Bloch’s complex of zero-cycles serve as a dualizing complex for duality theorems over non-smooth schemes?
• RQ2 How can $p$-adic and $ar{bQ}_l$-adic duality be unified in a single framework for torsion sheaves?
• RQ3 Does Rojtman’s theorem on the Albanese map extend to normal, projective schemes over algebraically closed fields?
• RQ4 What is the role of $ bZ^c_X $ in duality for schemes over rings of integers in number fields?
• RQ5 How does the dualizing complex $ bZ^c_X $ behave under base change and compactly supported cohomology in mixed characteristic?

Key findings

• For a separated scheme $ f:X o Y $ of finite type over a perfect field $ k $, there is a quasi-isomorphism $ R ext{Hom}_X( ext{F}, bZ^c_X) o R ext{Hom}_Y(Rf_!F, bZ^c_Y) $, establishing duality.
• Over an algebraically closed field, perfect pairings $ ext{Ext}^{1-i}_X( ext{F}, bZ^c_X) imes H^i_c(X_{ ext{et}}, ext{F}) o bQ/bZ $ hold for constructible $ ext{F} $, with $ CH_0(X,i,bZ/m) o H^i_c(X_{ ext{et}}, bZ/m)^* $ an isomorphism.
• The abelianized fundamental group satisfies $ ar{bpi}_1^{ab}(X)^0 o CH_0(ar{X},1)^{ ext{hat}}_G $, with a short exact sequence $ 0 o CH_0(ar{X},1)^{ ext{hat}}_G o ar{bpi}_1^{ab}(X)^0 o CH_0(ar{X})^G o 0 $.
• Over finite fields, perfect pairings $ ext{Ext}^{2-i}_X( ext{F}, bZ^c_X) imes H^i_c(X_{ ext{et}}, ext{F}) o bQ/bZ $ are established, generalizing Deninger, Spieß, and Moser.
• Over a Dedekind ring of characteristic 0 with perfect residue fields, assuming the Beilinson-Lichtenbaum conjecture, the same duality holds: $ R ext{Hom}_X( ext{F}, bZ^c_X) o R ext{Hom}_S(Rf_!F, bZ^c_S) $.
• Over rings of integers in number fields, perfect pairings $ ext{Ext}^{2-i}_X( ext{F}, bZ^c_X) imes H^i_c(X_{ ext{et}}, ext{F}) o bQ/bZ $ are obtained via Artin-Verdier duality, generalizing Milne and Spieß.